Bifurcation analysis of chemical reactors and reacting flows

Bifurcation analysis of chemical reactors and reacting flowsVemuri Balakotaiah, Sandra M. S. Dommeti, and Nikunj Gupta Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 9, 13 (1999); doi: 10.1063/1.166377 View online: http://dx.doi.org/10.1063/1.166377 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/9/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Bifurcation characteristics of fractional reaction-diffusion systems AIP Conf. Proc. 1493, 290 (2012); 10.1063/1.4765503 Defect mediated turbulence in a locally quasiperiodic chemical medium J. Chem. Phys. 133, 044909 (2010); 10.1063/1.3464493 Kinetic theory and hydrodynamics of dense, reacting fluids far from equilibrium J. Chem. Phys. 120, 6325 (2004); 10.1063/1.1648012 On slow manifolds of chemically reactive systems J. Chem. Phys. 117, 1482 (2002); 10.1063/1.1485959 Gravity effects on triple flames: Flame structure and flow instability Phys. Fluids 11, 3449 (1999); 10.1063/1.870203

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Bifurcation analysis of chemical reactors and reacting flowsVemuri Balakotaiah,a) Sandra M. S. Dommeti, and Nikunj GuptaDepartment of Chemical Engineering, University of Houston, Houston, Texas 77204-4792

~Received 30 October 1998; accepted for publication 30 December 1998!

In this work we review the local bifurcation techniques for analyzing and classifying the steady-stateand dynamic behavior of chemical reactor models described by partial differential equations~PDEs!. First, we summarize the formulas for determining the derivatives of the branching equationand the coefficients in the amplitude equations for the most common singularities. We also illustratethe procedure for the numerical computation of these coefficients. Next, the application of theselocal results to various reactor models described by PDEs is discussed. Specifically, we review therecent literature on the bifurcation features of convection-reaction and convection-diffusion-reactionmodels in one and more spatial dimensions, with emphasis on the features introduced due tocoupling between the flow, heat and mass diffusion and chemical reaction. Finally, we illustrate theuse of dynamical systems concepts in developing low dimensional~effective orpseudohomogeneous! models of reactors and reacting flows, and discuss some problems of currentinterest. © 1999 American Institute of Physics.@S1054-1500~99!01801-7#

The mathematical models describing the steady-state anddynamic behavior of chemical reactors and reacting flowsare nonlinear partial differential equations containing alarge number of parameters and exhibiting various bifur-cations and spatio-temporal patterns. This work summa-rizes a systematic approach for analyzing such models byproviding the defining conditions for various singularitiesand procedures for their numerical computation. Theusefulness of the local bifurcation techniques and the fea-tures of various reactor models are illustrated by a dozenexamples.

I. INTRODUCTION

Mathematical models describing the steady-state and dy-namic behavior of chemical reactors are obtained by writingdown the continuity, momentum, species, and energy bal-ances, and combining these with the various constitutiveequations for the transport and reaction rate processes. De-pending on the level of detail included, these models canvary from a pair of ordinary differential equations to severalpartial differential equations in three spatial coordinates andtime. In addition, the model equations describing chemicalreactors are highly nonlinear due to the strong coupling be-tween the transport processes and the chemical kinetics andthe nonlinear dependence of the kinetic and transport coeffi-cients on the state variables~temperature or concentrations!.The model equations also contain a large number of param-eters, some of which can only be estimated with some un-certainty. It is well known that these nonlinear models ex-hibit multiple steady states, periodic states, and variousspatio-temporal patterns.

Since the pioneering work of Aris and Amundson~1958!on the dynamic behavior of the continuous flow stirred tank

reactor~CSTR! model, considerable progress has been madein the understanding of chemical reactors using local bifur-cation techniques. Uppalet al. ~1974! and Poore~1973! in-troduced the Hopf bifurcation formulas and the concept ofhigher codimension singularities to construct phase diagramsin the parameter space that classify the different types ofdynamic behavior. Zeldovich and Zysin~1941!, Golubitskyand Keyfitz ~1980! and Balakotaiah and Luss~1982, 1983,1984! applied the singularity theory to analyze the steady-state behavior of lumped reactor models and construct phasediagrams in the parameter space that classify the differenttypes of steady-state behavior. At present, there is an exten-sive literature dealing with the steady-state and dynamic bi-furcation behavior of lumped reactor models described byordinary differential equations. We refer to the review ar-ticles by Hudson and Ro¨ssler ~1986!, Razon and Schmitz~1987!, Schneider and Munster~1991!, Balakotaiah and Luss~1986!, Planeauxet al. ~1986! and the books by Field andBurger ~1985!, Marek and Schreiber~1991!, Gray and Scott~1990!, and Scott~1991! for a summary of the results onlumped reactor models.

The main purpose of this article is to review the variouslocal bifurcation techniques that are useful in examining andclassifying the steady-state and dynamic behavior of chemi-cal reactor models described by partial differential equationsin one or more spatial dimensions. Specifically, we focus onreacting systems in which the coupling between the convec-tion ~or fluid flow!, heat and mass transfer, and chemicalreaction leads to various instabilities and spatio-temporalpatterns. In addition, we focus mainly on reactor models inwhich either convection is dominant or is strongly coupledwith the diffusion or conduction processes. We shall excludepurely diffusion-reaction type models since there is alreadyan extensive literature on this subject.

In the next section, we summarize some local bifurcationresults that are useful for analyzing the features of reactormodels described by PDEs. In Sec. III, we review some pro-

a!Author to whom correspondence should be addressed; electronic mail:[email protected]

CHAOS VOLUME 9, NUMBER 1 MARCH 1999

131054-1500/99/9(1)/13/23/$15.00 © 1999 American Institute of Physics


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cedures based on the local results to compute singular pointsof convection-reaction and diffusion-convection-reactionmodels in one and more spatial dimensions. In Sec. IV, wepresent the bifurcation features of various convection-reaction and diffusion-convection-reaction models in onespatial dimension. In Sec. V, we review the steady-state anddynamical features of heterogeneous catalytic reactor modelsand combined homogeneous-heterogeneous reaction models.Section VI reviews various spatio-temporal patterns thatarise due to the interaction between fluid flow, heat and massdiffusion, and physical property variation. In Sec. VII, wediscuss some recent work on the application of dynamicalsystems concepts to develop low dimensional models of re-actors and reacting flows. In the last section, we discusssome problems of current interest.

II. LOCAL BIFURCATION RESULTS FOR PDEMODELS

In this section we review some local bifurcation resultsthat can be used to analyze the behavior of chemical reactormodels described by partial differential equations. We con-sider a nonlinear mathematical model of the form

Cdu

dt5F~u,p,l!, ~2.1a!

B~u,p,l!50, on ]V, boundary condition~s!,~2.1b!

u5uo @t50, initial condition, ~2.1c!

where C is a capacitance matrix~which is not necessarilyinvertible!, u is a vector of state variables,F is a nonlinearoperator,l is a distinguished~bifurcation! parameter, andpis a vector of parameters. We assume that the nonlinear op-eratorF has Fre´chet derivatives of all orders with respect tou andl. We further assume that the system has some basecase steady-state solution, i.e.,

F~uo ,p,lo!50. ~2.2!

For simplicity of notation, we assume thatuo50, lo50and suppress the parameters vectorp. The first step in thelocal analysis involves linearizing the original system aroundthe base case solution to obtain the linearized system for theperturbed vectorz,

Cdz

dt5Lz; L5DuF~0,l!, ~2.3a!

BL~0,l!z50, on ]V ~ linearized homogeneous BCs!.

~2.3b!Using the separation of variables techniquez5y exp(mt),one obtains the following spatial eigenvalue problem:

Ly5mCy. ~2.4!

The corresponding adjoint eigenvalue problem can be shownto be,

L* v5mC* v. ~2.5!

Here,L is the linearized operator,L* , the adjoint operator,and m, the eigenvalues of the system. IfL has a discretespectrum and all the eigenvalues have a negative real part,the base case solution (0,0) is stable and new solutions donot bifurcate from the base state. If, however, one or more ofthe eigenvalues cross the imaginary axis into the right halfplane, the base case solution is unstable and new solutionsappear in the system. This can happen~generically! in one oftwo different ways:~i! a real eigenvalue (m50) crosses theimaginary axis leading to a simple bifurcation. From thesesimple bifurcation points, new stationary solutions appear inthe system.~ii ! a pair of eigenvalues become purely imagi-nary (m56 iv) and cross the imaginary axis leading toHopf bifurcation. From these Hopf points, new oscillatorysolutions appear in the system. When additional parameters~or spatial symmetries! are present in the problem, either thecoefficients of some nonlinear terms could vanish or moreeigenvalues can cross the imaginary axis, leading to highercodimension singular points and correspondingly more com-plex dynamics near these points. We summarize here variousformulas for analyzing the local bifurcations.

A. Single state variable steady-state bifurcationformulas

The simplest case corresponds to whenL has a simplezero eigenvalue@dim(Ker(L ))51#. Let

Lyo50 ~eigenvector!, ~2.6a!

L* vo50 ~adjoint eigenvector!. ~2.6b!

Then, all the local steady-state solutions of Eq.~2.1! are ofthe form

u5uo1xyo1h.o.t., ~2.7!

where the scalar state variablex is given by the branchingequation

g~x,l!5^vo ,F~xyo1W~xyo ,l!,l!&50. ~2.8!

Here, the vector of slave variablesW is defined by the im-plicit equation

EF~xyo1W~xyo ,l!,l!50, ~2.9!

whereE is the projection operator~onto range ofL ) definedby

Ey5y2^y,vo&yo . ~2.10!

~The notation^•,•& refers to inner product in the functionspace of interest. It is also assumed thatyo and vo are nor-malized so that yo ,vo&51.) The derivatives of the branch-ing equation are given by@Golubitsky and Schaeffer~1985!,Balakotaiahet al. ~1985!#

g~0,0!5gx~0,0!50, ~2.11!

gxx~0,0!5^vo ,Duu2 F~0,0!.~yo ,yo!&, ~2.12!

gxxx~0,0!5^vo ,Duuu3 F~0,0!.~yo ,yo ,yo!

13Duu2 F~0,0!.~yo ,z1!&, ~2.13!

14 Chaos, Vol. 9, No. 1, 1999 Balakotaiah, Dommeti, and Gupta


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gxxxx~0,0!5^vo ,Duuuu4 F~0,0!.~yo ,yo ,yo ,yo!16Duuu

3 F~0,0!.~yo ,yo ,z1!13Duu2 F„0,0!.~z1 ,z1!14Duu

2 F~0,0!.~yo ,z2!&,~2.14!

gxxxxx~0,0!5^vo ,Duuuuu5 F~0,0!.~yo ,yo ,yo ,yo ,yo!110Duuuu

4 F~0,0!.~yo ,yo ,yo ,z1!

115Duuu3 F~0,0!.~yo ,z1 ,z1!110Duuu

3 F~0,0!.~yo ,yo ,z2!110Duu2 F~0,0!.~z1 ,z2!15Duu

2 F~0,0!.~yo ,z3!&.

~2.15!

The derivatives ofg w.r .t.l are given by

gl~0,0!5^vo ,DlF~0,0!&, ~2.16!

gxl~0,0!5^vo ,Duu2 F~0,0!.~z4 ,yo!1Dul

2 F~0,0!.yo&,

~2.17!

gll~0,0!5^vo ,Duu2 F~0,0!.~z4 ,z4!

12Dul2 F~0,0!.z41Dll

2 F~0,0!&. ~2.18!

Here the vectors,zi , are given by the solution of the linearequations

Lz152EDuu2 F~0,0!.~yo ,yo!, ~2.19a!

Lz252EDuuu3 F~0,0!.~yo ,yo ,yo!

23EDuu2 F~0,0!.~yo ,z1!, ~2.19b!

Lz352EDuuuu4 F~0,0!.~yo ,yo ,yo ,yo!

26EDuuu3 F~0,0!.~yo ,yo ,z1!

23EDuu2 F~0,0!.~z1 ,z1!24EDuu

2 F~0,0!.~yo ,z2!,

~2.19c!

Lz452EDlF~0,0!, ~2.19d!

^zi ,yo&50; i 51,2,3,4. ~2.19e!

The bifurcation diagram (x versusl) of the branchingequation changes qualitatively only when the parameterscross one of the three codimension one singularities~Golu-bitsky and Schaeffer, 1985!. The first is the hysteresis varietydefined by

g5gx5gxx50. ~2.20!

The second is the isola~or bifurcation! variety defined by

g5gx5gl50. ~2.21!

The third is the double limit variety defined by

g~xi ,l!5gx~xi ,l!50; i 51,2; x1Þx2 . ~2.22!

In many physical problems, the nonlinear operatorF(u,l) commutes with a group of symmetries, i.e.,

F~gu,l!5gF~u,l!,

where g is any element of a groupG. For example, themodel equations may be invariant when a spatial coordinatej is replaced by2j. In this case,F(u,l) is said to commutewith the group of reflections~or Z2 symmetry!. Similarly,periodic boundary conditions in a spatial coordinate,j, leadto O„2… symmetry. In this case, the model equations areinvariant to rotations and reflections on the unit circle.

When the model equations possessZ2 symmetry it canbe shown that the branching equation should be odd inx,i.e.,

g~x,l!5xh~x2,l!. ~2.23!

Thus, all the even derivatives of branching equationw.r.t. xand derivativesw.r.t. l vanish. The simplest bifurcation inthis case would be a pitchfork bifurcation with local behaviorgiven by

g~x,l!5Ax31Bxl1 . . . 50, ~2.24!

where

A5 16 ^Duuu

3 F~0,0!.~yo ,yo ,yo!,vo&

1 12 ^Duu

2 F~0,0!.~yo ,z1!,vo&, ~2.25a!

B5^Dul2 F~0,0!.yo ,vo&. ~2.25b!

In this case,B50 defines the isola variety andA50 definesthe hysteresis variety~sub/supercritical transition!.

B. Center manifold reduction of PDEs and amplitudeequations for local bifurcations

WhenL has a discrete spectrum of eigenvalues, the cen-ter manifold reduction may be used to obtain a low dimen-sional model of the original system that describes the large-time dynamics of the system. The equations describing thislow dimensional model are called amplitude equations. Wegive here the form of the amplitude equations for a generalsystem of PDEs of the form given by Eq.~2.1! for the fourmost common local bifurcations, namely,~i! simple zero ei-genvalue,~ii ! double zero eigenvalue~Takens-Bogdanov sin-gularity!, ~iii ! double zero eigenvalue with geometric multi-plicity of 2, and~iv! simple Hopf bifurcation. The details ofthe derivation of these amplitude equations may be found inthe references~Carr, 1981; Kuramoto, 1984; Subramanianand Balakotaiah, 1995; and Wiggins, 1990!.

1. Simple zero eigenvalue

The dynamic behavior near a simple zero eigenvalue isgiven by

da

dt5Al1Ba21Cal1Da31h.o.t. ~2.26!

Here,

A5^DlF~0,0!,vo&, ~2.27a!

B5 12 ^Duu

2 F~0,0!.~yo ,yo!,vo&, ~2.27b!

15Chaos, Vol. 9, No. 1, 1999 Balakotaiah, Dommeti, and Gupta


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C5^Dul2 F~0,0!.yo ,vo&1^Duu

2 F~0,0!.~yo ,wo!,vo&,

~2.27c!

D5 16 ^Duuu

3 F~0,0!.~yo ,yo ,yo!,vo&

1^Duu2 F~0,0!.~yo ,w1!,vo&, ~2.27d!

where

Lwo52EDlF~0,0!, ~2.27e!

Lw152 12 EDuu

2 F~0,0!.~yo ,yo!, ~2.27f!

and the projection,E, may be expressed as

Ey5y2^y,vo&Cyo . ~2.27g!

It is assumed that the vectorsyo andvo are normalized suchthat

^Cyo ,vo&51. ~2.28!

When the model Eq.~2.1! has eitherZ2 or O„2… [email protected]., bothF(u,l) andC commute with the group#, the coef-ficientsA andB in Eq. ~2.26! vanish and the amplitude equa-tion is simplified to the form

da

dt5Cal1Da31h.o.t. ~2.29!

Equation~2.29! is often called the Landau equation.

2. Double zero eigenvalue (Takens-Bogdanovsingularity)

WhenL has a zero eigenvalue of multiplicity 2 with oneeigenvector and one generalized eigenvector, it may beshown that the local behavior is given by the amplitudeequations

dai

dt5Ail1Bia21Cia1l1Dia2l1Eia1

21Fia1a2

1Gia221h.o.t. ~ i 51,2!, ~2.30!

where

Ai5^DlF~0,0!,vi&, ~2.31a!

B150, B25^y1 ,v2&, ~2.31b!

Ci5^Dul2 F~0,0!.y1 ,vi&1^Duu

2 F~0,0!.~y1 ,wo!,vi&,

~2.31c!

Di5^Dul2 F~0,0!.y2 ,vi&1^Duu

2 F~0,0!.~y2 ,wo!,vi&,

~2.31d!

Ei512 ^Duu

2 F~0,0!.~y1 ,y1!,vi&, ~2.31e!

Fi5^Duu2 F~0,0!.~y1 ,y2!,vi&, ~2.31f!

Gi512 ^Duu

2 F~0,0!.~y2 ,y2!,vi&. ~2.31g!

The different vectors are expressed as follows:

Ly150, Ly25y1 , ~2.32a!

L* v150, L* v25v1 , ~2.32b!

Lwo52EDlF~0,0!. ~2.32c!

The projection,E, is expressed as

Ey5y2^y,v1&Cy12^y,v2&Cy2 . ~2.32d!

The vectors are normalized such that

^Cyi ,vj&5di j . ~2.33!

Again, when the model equations have symmetries, some ofthe coefficients vanish. In addition, Eq.~2.30! can be put in asimpler normal form by a near identity local change of co-ordinates.

3. Double zero eigenvalue

When L has two zero eigenvalues with two eigenvec-tors, the dynamic behavior is given by the following twoamplitude equations:

dai

dt5Ail1Bia1l1Cia2l1Dia1

21Eia1a21Fia22

1Gia131Hia2

31I ia12a21Jia1a2

21h.o.t.

~ i 51,2!. ~2.34!

Here,

Ai5^DlF~0,0!,vi&, ~2.35a!

Bi5^Dul2 F~0,0!.y1 ,vi&1^Duu

2 F~0,0!.~y1 ,wo!,vi&,

~2.35b!

Ci5^Dul2 F~0,0!.y2 ,vi&1^Duu

2 F~0,0!.~y2 ,wo!,vi&,

~2.35c!

Di512 ^Duu

2 F~0,0!.~y1 ,y1!,vi&, ~2.35d!

Ei5^Duu2 F~0,0!.~y1 ,y2!,vi&, ~2.35e!

Fi512 ^Duu

2 F~0,0!.~y2 ,y2!,vi&, ~2.35f!

Gi516 ^Duuu

3 F~0,0!.~y1 ,y1 ,y1!,vi&

1^Duu2 F~0,0!.~y1 ,w1!,vi&, ~2.35g!

Hi516 ^Duuu

3 F~0,0!.~y2 ,y2 ,y2!,vi&

1^Duu2 F~0,0!.~y2 ,w3!,vi&, ~2.35h!

I i512 ^Duuu

3 F~0,0!.~y1 ,y1 ,y2!,vi&

1^Duu2 F~0,0!.~y1 ,w2!,vi&

1^Duu2 F~0,0!.~y2 ,w1!,vi&, ~2.35i!

Ji512 ^Duuu

3 F~0,0!.~y1 ,y2 ,y2!,vi&

1^Duu2 F~0,0!.~y1 ,w3!,vi&

1^Duu2 F~0,0!.~y2 ,w2!,vi&. ~2.35j!

The different vectors are expressed as follows:

Lwo52EDlF~0,0!, ~2.36a!

Lw152 12 EDuu

2 F~0,0!.~y1 ,y1!, ~2.36b!

Lw252EDuu2 F~0,0!.~y1 ,y2!, ~2.36c!



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Lw352 12 EDuu

2 F~0,0!.~y2 ,y2!. ~2.36d!

The projection,E, is defined by

Ey5y2^y,v1&Cy12^y,v2&Cy2 . ~2.36e!

Again, when the governing Eqs.~2.1! possessZ2 orO„2… symmetry, the above amplitude equations simplify,since many of the coefficients vanish. For example, for thereaction driven natural convection problem discussed in Sec.VI ~which hasZ2 symmetry!, the amplitude equations formode 1 and 2 interaction can be reduced to the form~Dan-gelmayr and Armbruster, 1986!,

da1

dt5B1a1l1E1a1a2 ,

~2.37a!da2

dt5C2a2l1D2a1

21H2a23,

while for all remaining mode interactions (m and m11,mÞ1) the truncated normal form is given by

da1

dt5B1a1l1G1a1

31J1a1a22,

~2.37b!da2

dt5C2a2l1H2a2

31I 2a12a2 .

4. Simple Hopf bifurcation

Hopf bifurcation ~leading to periodic solutions! occurswhen a pair of eigenvalues ofL cross the imaginary axistransversely while all other eigenvalues remain in the lefthalf plane. The local bifurcation behavior is then given bythe following complex amplitude equation:

da

dt5Aal1Ba2a1h.o.t. ~2.38!

Here,

A5^Dul2 F~0,0!.yo ,vo&1^Duu

2 F~0,0!.~yo ,wo!,vo&,

~2.39a!

B5 12 ^Duuu

3 F~0,0!.~yo ,yo ,yo!,vo&

1^Duu2 F~0,0!.~yo ,w2!,vo&

1^Duu2 F~0,0!.~ yo ,w1!,vo&. ~2.39b!

The vectors appearing in Eq.~2.39! are determined by thefollowing linear equations:

Lyo5 ivCyo , ~2.40a!

L* vo52 ivC* vo , ~2.40b!

Lwo52DlF~0,0!, ~2.40c!

~L22ivC!w152 12 Duu

2 F~0,0!.~yo ,yo!, ~2.40d!

Lw252Duu2 F~0,0!.~yo ,yo!, ~2.40e!

^Cyo ,vo&51; ^Cyo ,vo&50, ~2.40f!

^Cyo ,vo&50; ^Cyo ,vo&51. ~2.40g!

When a trivial solution exists for alll, the second term onthe right-hand side of Eq.~2.39a! vanishes. The local behav-ior of the periodic branch is determined by the complex con-stantsA andB. By converting Eq.~2.38! to two real ampli-tude equations, it is easily seen that the condition Re(B)50 defines the sub/supercritical transition, while the condi-tion Re(A)50 determines the appearance or disappearanceof two Hopf points.

We note that the computation of the numerical values ofthe various coefficients appearing in the amplitude equationsinvolves the determination of Fre´chet derivatives of the non-linear operator in Eq.~2.1!, solution of the eigenvalue prob-lems defined by Eqs.~2.4! and~2.5!, solution of linear PDEs~for vectors such aswo ,w1, etc.!, and evaluation of someinner products.

The Frechet derivatives can be determined using the for-mulas

DuF~uo ,lo!.y5]

]s~F~uo1sy,lo!! @s50 , ~2.41a!

DlF~uo ,lo!5]

]s~F~uo ,lo1s!! @s50 , ~2.41b!

Duu2 F~uo ,lo!.~v,w!

5]

]s1

]

]s2

~F~uo1s1v1s2w,lo!! @s15s250 ,

~2.41c!

Dul2 F~uo ,lo!.y

5]

]s1

]

]s2

~F~uo1s1y,lo1s2!! @s15s250 ,

~2.41d!

Duuu3 F~uo ,lo!.~y,v,w!

5]

]s1

]

]s2

]

]s3

~F~uo1s1y1s2v1s3w,lo!!

@s15s25s350 . ~2.41e!

Similar expressions can be used for determining higher orderFrechet derivatives. As an example, we consider the nonlin-ear operator,

F~u,l!5S ¹2c1Ra

a2

]u

]x

2]u

]x

]c

]z1

]c

]x

]u

]z1¹2u1Br~c,u!

2Le]c

]x

]c

]z1Le

]c

]x

]c

]z1¹2c2r ~c,u!

D ,

~2.42!

where



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

u5S c~x,z,t!

u~x,z,t!

c~x,z,t!D , ¹25

1

a2

]2

]x21

]2

]z2,

r ~c,u!5f2c expS gu

g1u D .

We take the Rayleigh number, Ra, as the bifurcation variable(l5Ra!, and for the base one-dimensional~conduction!state,

uo5S 0uo~z!

co~z!D .

Let

y5S y1~x,z,t!

y2~x,z,t!

y3~x,z,t!D , v5S v1~x,z,t!

v2~x,z,t!

v3~x,z,t!D ,

and

w5S w1~x,z,t!

w2~x,z,t!

w3~x,z,t!D .

Using the above formulas, we get

DuF~uo ,l!.y5Ly5S ¹2 Ra

a2

]

]x0

]uo

]z

]

]x¹21Bs1 Bs2

Le]co

]z

]

]x2s1 ¹22s2

D S y1

y2

y3

D 5S ¹2y11Ra

a2

]y2

]x

]uo

]z

]y1

]x1¹2y21Bs1y21Bs2y3

Le]co

]z

]y1

]x2s1y21¹2y32s2y3

D ,

DlF~u,l!5S 1

a2

]u

]x

00

D , DlF~uo ,l!5S 000D ,

Duu2 F~uo ,l!.~y,w!5S 0

2]w2

]x

]y1

]z2

]y2

]x

]w1

]z1

]w1

]x

]y2

]z1

]w2

]z

]y1

]x1BR2

2LeS ]w3

]x

]y1

]z1

]y3

]x

]w1

]zD 1LeS ]w1

]x

]y3

]z1

]w3

]z

]y1

]xD 2R2

D ,

Duuu3 F~uo ,l!.~y,v,w!5S 0

B21

D R3 ,

where

s15]r

]u; s25

]r

]c,

R25]2r

]c2w3y312

]2r

]c]u~w2y31w3y2!1

]2r

]u2w2y2 ,

R35]3r

]c3w3v3y31

]3r

]c2]u~w2v3y31w3v3y21w3v2y3!

1]3r

]c]u2~w2v2y31w3v2y21w2v3y2!

1]3r

]u3w2v2y2 ,

and the derivatives ofr are evaluated at the base point@uo(z),co(z)#.

III. COMPUTATION OF SINGULARITIES ANDCOEFFICIENTS IN AMPLITUDE EQUATIONS

In this section, we review briefly how the various steady-state and dynamic singularities and the coefficients in theamplitude equations can be computed for PDE models. Wefirst consider convection-reaction and diffusion-convection-reaction problems~parabolic or hyperbolic PDEs in one spa-tial coordinate and time!. To illustrate the computation of thesingularities, we consider a pair of parabolic equations of thetype,

]ui

]t5Di

]2ui

]j21 f iS j,

]um

]j,um ,l D ; 0,j,1,t.0,

~3.1a!

with unmixed boundary conditions,



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

aiui~0,t !1biui8~0,t !5 j i , ~3.1b!

ciui~1,t !1diui8~1,t !5ki , ~ i 51,2; m51,2!. ~3.1c!

~The procedure can easily be generalized to any number ofequations and more general boundary conditions.! At steadystate, Eqs.~3.1a! reduce to

Di

d2ui

dj21 f iS j,

dum

dj,um ,l D 50, ~ i 51,2; m51,2!,

~3.2!

with boundary conditions as given by Eqs.~3.1b! and~3.1c!.To compute the steady-state singularities, we can use the

shooting method, in which the boundary value problem isconverted into an initial value problem by guessing the val-ues of variables at one end of the boundary~sayj50). Thedifferential equations are then integrated using these guesses,and adjusted so that the boundary conditions at the other end~say j51), a set of algebraic equations, dependent on theguessed values, are satisfied. Thus, the solution of the bound-ary value problem is reduced to finding zeros of a set ofalgebraic equations. Depending on the sensitivity of integra-tion in a particular direction, values are either guessed atj51 or j50 ~also, multiple shooting can be used!. For illus-trative purposes, we assume that valuesuio(0)5wi ( i51,2) are guessed and the initial value problem~IVP! to besolved reduces to

Di

d2yio

dj21 f iS j,

dymo

dj,ymo ,l D 50, ~3.3a!

yio~0!5wi , ~3.3b!

yio8 ~0!5j i2aiyio~0!

bi

, ~ i 51,2; m51,2!. ~3.3c!

The boundary conditions at the other end@Eq. ~3.1c!# arethen written as a set of algebraic equations dependent onwi ,

Fi~w1 ,w2 ,l!5ciyio~1!1diyio8 ~1!2ki50, ~ i 51,2!.

~3.4!@The second subscript zero ony denotes steady-state profileand yi5ui when Eqs.~3.4! are satisfied.# The Liapunov-Schmidt reduction is now applied to Eqs.~3.4! and the localbifurcation formulas given in Sec. II can be used. For ex-ample, to determine the hysteresis variety, the followingequations have to be satisfied:

F~w1 ,w2 ,l!50, ~3.5a!

det DF~w1 ,w2 ,lo!5U ]F1

]w1

]F1

]w2

]F2

]w1

]F2

]w2

U@lo

50, ~3.5b!

^vo ,D2F~w1 ,w2 ,lo!.~yo,yo!&50, ~3.5c!

wherevo is the adjoint eigenvector andyo is the eigenvectorcorresponding to zero eigenvalue of the linearized matrix,A5DF(w1 ,w2 ,lo). For the system described by Eq.~3.4!,the conditions for evaluating hysteresis locus reduces tosolving Eq.~3.5a! along with

Uc1q111d1q118 c1q211d1q218

c2q121d2q128 c2q221d2q228U

@j51

50 ~3.6!

and

^~v11,v12!,D2F~w1 ,w2 ,lo!.~yo,yo!&@j5150, ~3.7!

where,

D2F~w1 ,w2 ,lo!.~yo,yo!5S ~c1q311d1q318 !u112 12~c1q511d1q518 !u11u121~c1q411d1q418 !u12

2

~c2q321d2q328 !u112 12~c2q521d2q528 !u11u121~c2q421d2q428 !u12

2 D . ~3.8!

Here, q1i5]yio /]w1 ; q2i5]yio /]w2 ; q3i5]2yio /]w12 ;

q4i5]2yio /]w22; andq5i5]2yio /]w1]w2 are the sensitivity

functions. It can be seen that in order to solve for a hysteresispoint we need to evaluate the ten sensitivity functions atj51. Differentiating Eq.~3.3! w.r.t. w1 andw2 , we obtain thefollowing boundary value problems~BVPs! describing thesensitivity functionsqi j :

Di

d2q1i

dj21

d

dw1

f iS j,dymo

dj,ymo ,l D 50, ~3.9a!

Di

d2q2i

dj21

d

dw2

f iS j,dymo

dj,ymo ,l D 50, ~3.9b!

Di

d2q3i

dj21

d2

dw12

f iS j,dymo

dj,ymo ,l D 50, ~3.9c!

Di

d2q4i

dj21

d2

dw22

f iS j,dymo

dj,ymo ,l D 50, ~3.9d!

Di

d2q5i

dj21

d2

dw1dw2

f iS j,dymo

dj,ymo ,l D 50

~ i 51,2; m51,2!, ~3.9e!

with the following boundary conditions:

aiqji ~0!1biqji8 ~0!50, ~3.10a!

ciqji ~1!1diqji8 ~1!50, ~ i 51,2;j 51, . . . ,5!. ~3.10b!

The above equations are converted to an IVP and inte-grated to get the values of the sensitivity functions requiredin the evaluation of the hysteresis point. Evaluation of thiscodimension one singularity involves solving four equations@Eq. ~3.5!# in four unknowns. Two of the unknowns are the



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

values of the guessed variablesw1 andw2, the two remain-ing guesses can be the bifurcation parameter,l, and oneother parameter chosen from the vectorp* .

The isola and boundary limit varieties can be determinedin a similar way. We refer to the article by Subramanian andBalakotaiah~1996! for details and numerical examples.

To illustrate the computation of the dynamic singulari-ties, we consider again Eq.~3.1!. Linearizing around a basestate, we obtain the system

]Yi

]t5DiYi91 (

m51

2 S ] f i

]ym8D Ym8 1 (

m51

2 S ] f i

]ymD Ym ;

0,j,1 ~ i 51,2!, ~3.11!

subject to the homogeneous boundary conditions,

aiYi~0,t !1biYi8~0,t !50, ~3.12a!

ciYi~1,t !1diYi8~1,t !50. ~3.12b!

HereYi is the perturbation function defined by

Yi~j,t !5yi~j,t !2yio~j!. ~3.12c!

The system of linear partial differential equations de-fined by Eq.~3.11! may be solved by the separation of vari-ables technique. Substituting,Yi(j,t)5zi(j)exp(mt), givesthe eigenvalue problem for the spatial perturbation functionszi(j),

Dizi91 (m51

2 S ] f i

]ym8D zm8 1 (

m51

2 S ] f i

]ymD zm5mzi , ~3.13!

with the boundary conditions,

aizi~0!1bizi8~0!50, ~3.14a!

cizi~1!1dizi8~1!50, ~ i 51,2!. ~3.14b!

At a Takens-Bogdanov double zero singularity, the follow-ing conditions need to be satisfied:

Ly150, ~3.15!

Ly25y1 , ~3.16!

while all other eigenvalues ofL are in the left half plane.Here,L is the linear operator defined by

L5Dd2

dj21Q1

d

dj1Q2 , ~3.17!

where the matricesD,Q1 , andQ2 are defined by

D5S D1 0

0 D2D , ~3.18a!

Q15S ] f 1

]y18

] f 1

]y28

] f 2

]y18

] f 2

]y28

D@~y1o ,y2o!

, ~3.18b!

and

Q25S ] f 1

]y1

] f 1

]y2

] f 2

]y1

] f 2

]y2

D@~y1o ,y2o!

. ~3.18c!

Denotingy15(z21

z11), y25(z22

z12) and settingm50 in Eq.~3.13!,

we obtain

Dizi19 1 (m51

2 S ] f i

]ym8D zm18 1 (

m51

2 S ] f i

]ymD zm150, ~3.19a!

Dizi29 1 (m51

2 S ] f i

]ym8D zm28 1 (

m51

2 S ] f i

]ymD zm25zi1

~ i 51,2!.

~3.19b!

The boundary conditions are obtained from Eq.~3.14! byreplacingzi with zi1 andzi2 . @The second subscriptm on zim

refers to the eigenvector (m51) and the generalized eigen-vector (m52) corresponding to the zero eigenvalue.# Toevaluate a double zero point, valueszim(0) (i 51,2;m51,2) are guessed atj50. It should be noted that since thelinearized equations have two zero eigenvalues, the linearityof the problem allows us to fixz1m (m51,2) arbitrarily, thusfreeing two guesses. Hence, two valuesz2m (m51,2) andtwo valuesyi1 ( i 51,2), describing the steady state, alongwith two parameters are chosen. The differential equationscorresponding to the perturbed vector and the steady state areintegrated and the guesses are adjusted until the followingsix conditions are satisfied atj51:

ciyi1~1!1diyi18 ~1!2ki50, ~3.20a!

cizim~1!1dizim8 ~1!50, ~ i 51,2;m51,2!. ~3.20b!

Other dynamic singularities can be computed in a similarmanner~Subramanian and Balakotaiah, 1996!.

It should be pointed out that in the special case in whichthe base case solution is a constant~independent of the spa-tial coordinate! or is a linear function of the spatial coordi-nates, the linear eigenvalue problem can be solved exactly~since the linear operator has constant coefficients!, and it ispossible to calculate the various singularities analytically.This is the case for many classical examples such as theRayleigh-Benard problem, the Lapwood problem, and thedoubly diffusive ~thermohaline! convection problems. Formost reacting systems such analytical determination of thecoefficients in the amplitude equation is not possible and onehas to resort to numerical computations.

The procedure outlined above may also be used to com-pute the singularities of PDE models in two or three spatialdimensions provided the base case solution is dependentonly on a single spatial coordinate. In this case, the linear-ized equations have constant coefficients in the other spatialdirections, and hence the corresponding eigenfunctions aredefined by eigenvalue problems with constant coefficients.Thus, all the local bifurcation calculations reduce to solvingboundary value problems in one spatial coordinate and the



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

shooting method outlined above can be used again. A largeclass of bifurcation problems describing reactors and react-ing flows fall into this category.

When the shooting technique is not convenient or cannotbe used~e.g., bifurcation problems in two or three spatialdimensions!, the standard discretization methods such as fi-nite difference, finite element, orthogonal collocation orspectral methods may be used to reduce Eqs.~2.1! into afinite dimensional system,

CdU

dt5F~U,l!, UPRN, ~3.21!

which can be analyzed using the formulas given in Sec. II.We refer to the articles by Jensen and Ray~1982!, Stroh andBalakotaiah~1991!, Kubicek and Marek~1983!, and Doedelet al. ~1997! for bifurcation analysis of reactor problems us-ing this approach.

IV. CONVECTION-REACTION AND DIFFUSION-CONVECTION-REACTION MODELS IN 1D

In this section, we review recent results on the bifurca-tion features of the convection-reaction and diffusion-convection-reaction models in one dimension. For eachmodel, we present the model equations and briefly reviewthe known bifurcation features. The details of the analysis ornumerical computations will be omitted.

A. Recycle reactor model

The simplest convection-reaction model that shows com-plex bifurcation features is that of the recycle reactor modeldescribing the conversion~x! and temperature (u) in a tubu-lar recycle reactor. The model is given by a pair of hyper-bolic PDEs,

]x

]t52

~R11!

Da

]x

]j1~12x!expS gu

g1u D ,

~4.1a!

Le]u

]t52

~R11!

Da

]u

]j1B~12x!expS gu

g1u D2a~u2uc!; 0,j,1,t.0,

with boundary conditions

x~0,t!5R

R11x~1,t!,

~4.1b!

u~0,t!5R

R11u~1,t!.

Here, it is of interest to classify the steady-state and dynamicbehavior when the residence time, or equivalently, theDamkohler number (Da), is taken as the bifurcation vari-able. Subramanian and Balakotaiah~1996! applied local bi-furcation theory techniques described in Secs. II and III togive a complete classification of this model. Figures 1 and 2show a sample of their results. Here, the different possiblebifurcation diagrams of the exit temperatureu(1) versusDaare shown~Fig. 2! as the parameters are varied. The bound-aries~in the phase diagram shown in Fig. 1! that separate the

different parameter regions corresponding to qualitativelydifferent types of bifurcation diagrams were determined us-ing the formulas and numerical techniques described in Secs.II and III. These diagrams contain up to four limit points andfour Hopf points. Though the number of bifurcation dia-grams is large, complex dynamics such as quasiperiodic orchaotic solutions was not found. Also, the existence of suchsolutions is neither confirmed~by computing higher codi-mension singularities! nor disproved.

B. Autothermal reactor models

The classical autothermal reactor model is a convection-reaction model that accounts for heat exchange and reactionin concentric tubes. It is described by the equations

]y1

]t1

]y1

]j52St~y12y2!,

Le]y2

]t2

]y2

]j5bDa~12x!expS gy2

11y2D

1St~y12y2!2ay2 ,~4.2a!

]x

]t2

]x

]j5Da~12x!expS gy2

11y2D , 0,j,1,t.0,


FIG. 1. Phase diagram for the classification of the steady-state and dynamicbehavior of the recycle reactor model.~The bottom diagram is a schematicblow-up of the top diagram nearB510.2.)



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

y1~0,t!5yin ,

x~1,t!50, ~4.2b!

y1~1,t!5y2~1,t!.

Here,y1 and y2 are the temperatures in the preheating andthe reaction sections, respectively, whilex is the conversion.

Lovo and Balakotaiah~1992! analyzed the bifurcationfeatures of the above model. It was shown that for the adia-batic case (a50), at most three solutions exist and a com-plete classification of the steady-state and dynamic behavioris possible. However, when there is heat loss (aÞ0), therecan be an arbitrarily large number of solutions and the num-ber of possible bifurcation diagrams is also arbitrarily large.The dynamic behavior of this model has not been analyzed.

C. Adiabatic tubular reactor model

The classical adiabatic tubular reactor model, with axialdispersion of heat and mass, is given by the equations

]x

]t5

1

fm2

]2x

]j22

1

Da

]x

]j1~12x!expS gu

g1u D ,

~4.3a!

Le]u

]t5

1

fh2

]2u

]j22

1

Da

]u

]j1B~12x!expS gu

g1u D ;

0,j,1,t.0,

with boundary conditions,

1

fm2

]x

]j2

x

Da50,

1

fh2

]u

]j2

u

Da50, @j50,

~4.3b!]x

]j5

]u

]j50, @j51.

Here, x and u are the conversion and temperature, respec-tively. The parametersfm

2 andfh2 are the mass and thermal

Thiele moduli. The above model was analyzed by Hlavacekand Hoffman~1970! and more recently by Subramanian andBalakotaiah~1996!. It was found that the maximum numberof solutions is three and the phase diagram contains six re-gions corresponding to six qualitatively different bifurcationdiagrams ofu(1) versusDa. These bifurcation diagramscontain at most two Hopf points, and oscillatory behaviormay exist in the feasible region (Le.1) only if fm

2 ,fh2 .

Again, the existence of more complex behavior is neitherconfirmed nor disproved.

D. Cooled tubular reactor model

The mathematical model of a wall cooled tubular reactorwith axial dispersion of heat and mass is given by the fol-lowing pair of parabolic PDEs for the conversion~x! andtemperature (u):

]x

]t5

1

fm2

]2x

]j22

1

Da

]x

]j1~12x!expS gu

g1u D ;

0,j,1,t.0,~4.4!

Le]u

]t5

1

fh2

]2u

]j22

1

Da

]u

]j1B~12x!expS gu

g1u D2a~u2uc!.

The boundary conditions are the same as those given by Eq.~4.3b!. Here,a is the dimensionless wall heat transfer coef-ficient anduc is the coolant~or wall! temperature.

It should be pointed out that there is an extensive litera-ture on the steady-state and the dynamic behavior of theabove model. The article by Morbidelliet al. ~1987! summa-rized the work prior to that date. More recently, Alexander~1990! and Strohet al. ~1990! have shown that Eq.~4.4! canhave an arbitrarily large number of steady-state solutions.Subramanian and Balakotaiah~1996! analyzed the specialcase offm

2 5` and constructed some phase diagrams. It wasfound that even for this special case, the number of differentpossible bifurcation diagrams is rather large and these dia-grams contained up to four Hopf points. For the general caseof finite fm

2 andfh2 , it is also known that the above model is

a two state variable problem~i.e., there can be two zeroeigenvalues with two eigenvectors!, and hence even thesteady-state bifurcation diagrams have to be presented in athree-dimensional space. A complete classification of thesteady-state behavior or the confirmation of the existence ofmore complex dynamics of the cooled tubular reactor modelare still lacking.

FIG. 2. Different possible bifurcation diagrams of the recycle reactor model.The dots denote Hopf bifurcation points.



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E. Reverse-flow reactor model

A reverse-flow reactor~RFR! is a packed-bed reactor inwhich the flow direction is periodically reversed to trap a hotzone within the reactor. Matros and Bunimovich~1996! re-viewed various practical applications of the RFR. We reviewhere briefly some interesting bifurcation features of the RFR.

Khinast and Luss~1997! analyzed the bifurcation behav-ior of a RFR using the singularity theory techniques pre-sented in Sec. II. They used a one-dimensional pseudohomo-geneous model for their analysis. The dimensionless energyand species balances for a single exothermic first-order reac-tion in the RFR are given by

1

z

]u

]t2

1

Peh

]2u

]j21 f

]u

]j2DaBR~u!~12x!50,

0,j,1,~4.5a!

1

Le

]x

]t2

1

Pem

]2x

]j21 f

]x

]j2DaR~u!~12x!50,

whereu is the dimensionless temperature,x is the conver-sion, andz is the dimensionless cycle time.

Here, one full cycle of operation implies one flow rever-sal, i.e., for half the period the flow is in one direction, char-acterized byf 51, and for the remaining half, it is in thereverse direction withf 521. The spatial boundary condi-tions are given by

]z

]j5Pe z @j50,

]z

]j50 @j51, 0,t,1, ~4.5b!

and the temporal symmetry boundary condition is

z~t50,j!5z~t51,12j!, ~4.5c!

where

z5S uxD , Pe5S Peh 0

0 PemD ,

R~u!5exp$gu/g1u%

11Dapm exp$gu/g1u%.

When only the periodic states are considered, the modelequations may be formulated as a boundary value problem inj andt. The different codimension one singularities such ashysteresis, double-limit, and isola varieties were calculatedby solving the model equations simultaneously with a set ofdefining conditions for each singularity. The stability of theperiodic solution was also calculated using Floquet theory.For the case of a single reaction they used Damko¨hler num-ber as the bifurcation variable (umax versusDa) and showedthe existence of five different types of steady-state bifurca-tion diagrams for an adiabatic RFR.

Khinast et al. ~1998! further extended the analysis to acooled RFR and showed that it can have complex dynamicfeatures. They illustrated the existence of unsymmetric,m-periodic and quasiperiodic states.

F. Differential „cross … flow model

The differential flow instability~DIFI! is based on theinterplay of activator-inhibitor kinetics@Rovinsky and Men-zinger ~1992, 1993!#. The activator is an autocatalytic spe-cies and the inhibitor is a component that counteracts thistendency. In the context of the exothermic reaction in thepacked-bed reactor, the reaction heat represents the activator,while the reactive matter is the inhibitor. When inhibitionprevails over activation, the system may attain a uniformsteady state. The stability of this state is maintained dynami-cally: when the activator deviates from the stationary state,the inhibitor responds promptly and counteracts this devia-tion. However, the activator may be disengaged from theinhibitor through their differential flow. This releases the in-herent tendency of the activator to grow, and to depart fromits steady state. The evolution of activator induces the corre-sponding changes in the inhibitor component, and as a con-sequence of the differential flow the whole system attains anew, spatio-temporally patterned state~Yakhnin et al.,1996!.

Yakhninet al. ~1994! demonstrated that differential bulkflow can lead to spatio-temporal patterns even in very simplesystems. They used a tubular cross-flow reactor model fortheir analysis. The dimensionless model equations for thecase of a single exothermic first-order reaction could be writ-ten as

]x

]t5

1

fm2

]2x

]j22

1

Da

]x

]j1~12x!expS gu

g1u D2b rx;

0,j,1,t.0,~4.6a!

Le]u

]t5

1

fh2

]2u

]j22

1

Da

]u

]j1B~12x!expS gu

g1u D2b ru

2a~u2uc!.

Here, b r is the cross-flow parameter and other parametershave the same meaning as in the previous models. It shouldbe noted that the above model reduces to the cooled tubularreactor model for the special case of no cross-flow, i.e.,b r

50.Since the above tubular cross-flow reactor model has

spatially uniform steady states, Yakhninet al. ~1994! usedthe following periodic boundary conditions:

x~0,t!5x~1,t!, u~0,t!5u~1,t!. ~4.6b!

They used spatially uniform steady state as the initial state ofthe system and slightly modified it by applying sinusoidalperturbation. They found that the perturbations grow andlead to a spatio-temporally patterned state at some selectedvalues of parameters, which can be calculated using localbifurcation techniques. It should be pointed out that the pe-riodic boundary conditions used by these authors are notrealistic and hence this model does not describe any of theexisting chemical reactors.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

V. BIFURCATION FEATURES OF HETEROGENEOUS/HOMOGENEOUS-HETEROGENEOUS REACTORMODELS

When the interphase mass and heat transfer resistancesbetween the solid and fluid phases are significant, the con-centration and temperature in the two phases would be quitedifferent and a pseudohomogeneous description of a catalyticreactor is not quite appropriate. In this case, heterogeneousor two-phase reactor models need to be used. Heterogeneousreactor models use separate conservation equations for thesolid and fluid phases.

The steady-state and dynamic behavior of heterogeneousmodels could be quite different, due to the appearance oflimit and Hopf points at the boundaries of the feasible region~for example, zero residence time or temperature!. In addi-tion, the highest order singularity~steady-state or dynamic!may be on the boundary of the feasible parameter space andthe qualitative nature of a bifurcation diagram may changewhen parameters cross loci such as boundary limit set orboundary Hopf sets. A complete mathematical description ofthe theory for such systems does not exist~especially for thedynamical behavior!. Partial descriptions may be found inthe papers by Balakotaiah and Luss~1988!, Subramanian andBalakotaiah~1997b!, and Dommetiet al. ~1999!. We reviewhere some recent results on the features of simple heteroge-neous reactor models.

A. One-dimensional convection-reaction model

The simplest heterogeneous reactor model with spatialgradients, namely, the one-dimensional plug flow catalyticreactor model, is given by

e]xf

]t52

1

Da

]xf

]j1

xs2xf

Dapm

, ~5.1!

~12e!]xs

]t52

xs2xf

Dapm

1~12xs!expS us

11 ~us/g!D ,

~5.2!

L]u f

]t52

1

Da

]u f

]j1

us2u f

Daph

, ~5.3!

and

s~12e!]us

]t52

us2u f

Daph

1B~12xs!expS us

11 ~us /g!D ,

~5.4!


xf50, u f50, @j50, ~5.5!

and some initial conditions. Here,xf , xs , u f , andus repre-sent the dimensionless conversions and temperatures in thefluid and solid phases, respectively. The dimensionless vari-ables and parameter that appear in the above equations are asfollows: Dapm is the particle mass Damko¨hler number whichis the ratio of interphase mass transfer time to reaction time,Daph is the particle heat Damko¨hler number which is theratio of interphase heat transfer time to reaction time, Lep is

the particle Lewis number which is the ratio ofDapm toDaph , i.e., the ratio of interphase mass transfer time to in-terphase heat transfer time,Da is the Damko¨hler numberwhich is the ratio of convection time to reaction time,B isthe nondimensional heat of reaction,g is the dimensionlessactivation energy, andL is the dimensionless heat capaci-tance parameter.

The steady-state version of this model that has been ana-lyzed by Dommetiet al. ~1998! is a differential-algebraicsystem in terms of the solid and fluid temperatures,

du f

dDa5

Lep

Dapm

~us2u f !, u f50@Da50, ~5.6a!

us2u f5Dapm~B2u f !exp~us/~11 us /g!!

Lep@11Dapmexp~us/~11 us /g!!#. ~5.6b!

The corresponding conversion profiles are obtained by usingthe relations

xf5u f

B, ~5.7!

xs5xf1Dapm exp~us/~11 us /g!!

@11Dapm exp~us/~11 us /g!!#. ~5.8!

Dommeti et al. ~1999! have done a complete steady-state classification of the two-phase plug flow reactor modeldescribed by Eq.~5.6! for the limiting case ofg→` usingsingularity theory methods described in Sec. II. They showthat the phase diagram of the steady-state model in(B,Dapm ,Lep) space is divided into as many as ten regions,each with a different bifurcation diagram~of particle tem-perature versus Damko¨hler number! shown in Fig. 3. Thephase diagram could be obtained by plotting three differentloci viz., hysteresis variety, isola locus, and boundary limitset. When the hysteresis locus is crossed, an ignition and anextinction point appear or disappear. When the isola locus iscrossed, isolated branches appear or disappear, and when the

FIG. 3. Different possible bifurcation diagrams of the heterogeneous adia-batic convection-reaction model.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

boundary limit set is crossed a limit point appears or disap-pears atDa50, corresponding to zero residence time. There-fore, particle could ignite or extinguish at zero residencetime. The isola locus does not exist for Lep>1. They alsoshow that when Lep,1, the temperature of the solid phasecould exceed the adiabatic temperature rise~B! and could beas large asB/Lep , and is always observed at the reactorinlet. The authors show that the steady-state, two-phasemodel has three classification diagrams in (B,Dapm) spacecorresponding to 0,Lep,0.5, 0.5,Lep,1, and Lep>1,and ten different bifurcation diagrams. A representativephase diagram for Lep50.1 is shown in Fig. 4, and in thiscase the boundary limit, the hysteresis, and the isola locidivide the parameter space into seven regions that have bi-furcation diagrams of type~i!, ~ii !, ~iii !, ~iv!, ~v!, ~vi!, and~vii ! of Fig. 3.

B. Stagnation point flow „axisymmetric boundarylayer … model with homogeneous-heterogeneousreaction

Songet al. ~1991! and Olsenet al. ~1994! have studiedthe stagnation point flow model with homogeneous-heterogeneous combustion of propane. The physical pictureof the stagnation surface on which the heterogeneous reac-tion takes place is shown in Fig. 5~a!. The steady-state two-dimensional model describing mass, momentum, energy, andspecies balances is transformed to a one-dimensional modelusing Levy-Lees coordinate transformation. The transformedequations for dimensionless stream function (f ), temperature(u), and reactant mass fraction~m! in terms of the trans-formed distance coordinate (h) are,

d3f

dh31 f

d2f

dh21

1

2F u

ue

2S d f

dh D 2G50, ~5.9a!

d2u

dh21 f Pr

du

dh1b Pr r 50, ~5.9b!

d2m

dh21 f Sc

dm

dh2Sc r 50. ~5.9c!

The boundary conditions at the surface (h50) are given by

f 5d f

dh50,

dm

dh5r s ,

du

dh52br s2Aus

ue

@PW2kr~us42ue

4!2hL~us2ue!#,

and those at the inlet (h5`) are given by

d f

dh51, m5me , u5ue .

Here,r and r s are the dimensionless homogeneous and sur-face reaction rates, respectively, and are of the form

r 5f expS g2g

u Dg1~m,me ,ae!,

r s5fs expS gs2gs

usD g2~m,me ,ae!.

The model has 14 parametersviz., the Prandtl number~Pr!, the Schmidt number~Sc!, the dimensionless heat ofreaction (b), the dimensionless activation energies of thehomogeneous and surface reactions (g andgs , respectively!,the Thiele moduli of the homogeneous and surface reac-

FIG. 4. Phase diagram of the heterogeneous adiabatic convection-reactionmodel in (B,Dapm) space for Lep50.1.

FIG. 5. Schematic diagram of the stagnation point flow~top! and the bifur-cation set in the surface temperature-fuel concentration plane for thehomogeneous-heterogeneous reaction~Olsenet al., 1994!.



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tions (f andfs , respectively!, the dimensionless heat trans-fer coefficient (hL), the dimensionless power input to thesurface (PW), the radiation heat loss coefficient of the cata-lytic surface (kr), dimensionless feed temperature (ue), inletvelocity, mass fraction of the limiting reactant in the feed(me), and the stoichiometric ratio of the reactants in the feed(ae).

Olsenet al. ~1994! have computed some bifurcation dia-grams of surface temperature versus power input to the sur-face, at different inlet velocities and reactant compositionsusing the shooting method. The limit points of the bifurca-tion diagrams predict heterogeneous ignition/extinctionpoints and a homogeneous ignition point. The bifurcationdiagrams of surface temperature versus power input at agiven velocity and reactant composition could be a singlevalued curve corresponding to homogeneous ignition in theabsence of heterogeneous ignition or an S-shaped curvemarking the ignition of both heterogeneous and homoge-neous ignition. They show that the heterogeneous ignitioncould occur at zero power input. The bifurcation diagrams ofvelocity versus surface temperature and fuel compositionversus surface temperature were shown to have isolatedbranches at zero power input. They have computed severalbifurcation sets in the surface temperature-inlet velocity, sur-face temperature-fuel composition, and fuel/oxidant massfraction in feed-inlet velocity planes, and a bifurcation set inthe surface temperature-reactant composition plane is asshown in Fig. 5~b!. Their results indicate that the heteroge-neous ignition temperature depends weakly on velocity whileheterogeneous extinction, homogeneous ignition, and auto-thermal temperature depend strongly on velocity. The com-position range for which the homogeneous ignition occurswas found to decrease sharply with increasing velocities. Atlow velocities, homogeneous ignition was found to occur inthe absence of heterogeneous ignition. Increasing velocityhad the effect of decreasing the composition at which hetero-geneous ignition occurs and increasing the range of autother-mal operation.

C. Spatio-temporal patterns in catalytic reactors

Shvartsman and Sheintuch~1995! investigated thespatio-temporal patterns generated by a one-dimensional het-erogeneous model of a fixed-bed reactor. The model used issimilar to Eqs.~5.1!–~5.5! with some modifications. Theyinclude the conduction term in the solid phase energy bal-ance@Eq. ~5.4!# and ignore the accumulation terms in Eqs.~5.1!–~5.3!. Also, they add an additional equation for cata-lyst activity variation and assume that the rate of change ofactivity is a linear function of the solid temperature and theactivity. Their numerical simulations showed the existenceof almost homogeneous oscillations, periodic pulses, excit-able waves, and oscillatory fronts.

VI. BIFURCATIONS AND CONVECTIVE PATTERNSIN REACTING FLOWS

In this section, we review some two- and three-dimensional models of reactors in which the interaction be-

tween the fluid flow, transport processes, and reaction leadsto bifurcations and spatial patterns, which we term as con-vective patterns.

The first example we consider is the transverse instabil-ity reported by Balakotaiahet al. ~1999! for forced convec-tion problem in packed-bed reactors. The second exampledeals with convection patterns in reaction driven natural con-vection that was studied by Subramanian and Balakotaiah~1994, 1995, and 1997a!. The third example deals with con-vective patterns in down-flow reactors in which the interac-tion between externally imposed forced convection and reac-tion driven natural convection leads to flow maldistributions~Stroh and Balakotaiah, 1991!.

A. Pattern formation in forced convection problem

Balakotaiahet al. ~1999! studied transverse pattern for-mation in an adiabatic cylindrical packed-bed reactor withuniform one-dimensional flow field (u), in which a reactionof the typeA1nB→P, with Langmuir-Hinshelwood kinet-ics occurs. The pseudohomogeneous mathematical model de-scribing the concentrations of the species and temperature is

]CA

]t1u

]CA

]z85DeAz

]2CA

]z821DeArF 1

r 8

]

]r 8S r 8

]CA

]r 8D

11

r 82

]2CA

]w2 G2R8~CA ,CB ,T!, ~6.1a!

]CB

]t1u

]CB

]z85DeBz

]2CB

]z821DeBrF 1

r 8

]

]r 8S r 8

]CB

]r 8D

11

r 82

]2CB

]w2 G2nR8~CA ,CB ,T!, ~6.1b!

rmCpm

]T

]t1r fCp fu

]T

]z8

5lez

]2T

]z821lerF 1

r 8

]

]r 8S r 8

]T

]r 8D 1

1

r 82

]2T

]w2G1~2DHR!R8~CA ,CB ,T!, ~6.1c!


Deiz

]Ci

]z85u~Ci2Ci0!,

lez

]T

]z85r fCp fu~T2T0!, @z850, ~6.2a!

]Ci

]z850,

]T

]z850, @z85L, ~6.2b!



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]Ci

]r 850,

]T

]r 850, @r 850,R, ~6.2c!

Ci~z8,r 8,w,t !5Ci~z8,r 8,w12p,t !, ~ i 5A,B!,~6.2d!

T~z8,r 8,w,t !5T~z8,r 8,w12p,t !, ~6.2e!

whereCA andCB are the concentration of speciesA andB,respectively, andT is the temperature. The dependence ofphysical properties on temperature is neglected. The authorshave assumed a reaction rate expression of the form

R8~CA ,CB ,T!5kR0KA0KB0CACB exp@~ER2EdA2EdB !/RG~1/T0 2 1/T!#

$11KA0CA exp@EdA /RG~1/T 2 1/T0!#1KB0CB exp@EdB /RG~1/T 2 1/T0!#%2, ~6.3!

wherekR0 , KA0 , andKB0 are the reaction rate constant andadsorption equilibrium constants of speciesA andB, respec-tively, at feed temperature (T0). The model equations werewritten in nondimensional form as

s]cA

]t1

]cA

]z5

1

PeAz

]2cA

]z21

1

PeArF1

r

]

]r S r]cA

]rD

11

r 2

]2cA

]w2 G2DaR~cA ,cB ,y!, ~6.4a!

s]cB

]t1

]cB

]z5

1

PeBz

]2cB

]z21

1

PeBrF1

r

]

]r S r]cB

]rD

11

r 2

]2cB

]w2 G2Da

mR~cA ,cB ,y!, ~6.4b!

]y

]t1

]y

]z5

1

Pez

]2y

]z21

1

PerF1

r

]

]r S r]y

]r D11

r 2

]2y

]w2G1bDaR~cA ,cB ,y!, ~6.4c!

the dimensionless reaction rate is given by

R5) i 5A

B Kici exp~2 g i y/11y!exp~gRy/11y!

@11( i 5AB Kici exp~2 g i y/11y!#2

, ~6.5!

and the boundary conditions become

1

Peiz

]ci

]z2ci1150,

1

Pehz

]y

]z2y50, @z50,

~6.6a!

]ci

]z50,

]y

]z50, @z51, ~6.6b!

]ci

]r50,

]y

]r50, @r 50,1, ~6.6c!

ci~z,r ,w,t!5ci~z,r ,w12p,t!, ~ i 5A,B!, ~6.6d!

y~z,r ,w,t!5y~z,r ,w12p,t!. ~6.6e!

The dimensionless variables and parameters that appear inEqs.~6.4!–~6.6! are defined as

z5z8

L, r 5

r 8

R, t5

ut

L, ci5

Ci

Ci0

, y5T2T0

T0

,

Da5L

u

kR0

CA0

, m5CB0

nCA0

, s5r fCp f

rmCpm

, g i5Edi

RT0

,

b5~2DHR!CA0

r fCp fT0

, Ki5Ki0Ci0 ,

Peiz5uL

Deiz

5~Peiz!pS L

dpD ,

Peir 5uR2

DeirL5~Peir !pS D

2dpD 2S dp

LD ,

Pez5r fCp fuL

lez

5~Pez!pS L

dpD ,

Per5r fCp fuR2

lerL5~Per !pS D

2dpD 2S dp

LD ,

~Peik!p5udp

Deik

,

~Pek!p5r fCp fudp

lek

, ~ i 5A,B!, ~k5z,r !,

whereD is the reactor diameter,dp is the particle diameter,g i is the nondimensional activation energy,b is the dimen-sionless heat of reaction,Da is the Damko¨hler number,Peik

is the mass Peclet number of speciesi (5A,B) in the k(5r ,z) direction, Pek is the heat Peclet number in thekdirection, (Peik)p and (Pek)p are the corresponding particlePeclet numbers. The Peclet numbersPeik ,Pek are expressedas functions of the corresponding particle Peclet numbersand the number of particles in the axial direction (L/dp) andthe radial direction (D/dp).

The authors studied the stability of one-dimensional~i.e.,only z-dependent! steady-state solutions to transverse pertur-bations using linear stability theory. The one-dimensionalsteady-state base solution is obtained by solving the modelequations@Eqs. ~6.4!–~6.6!# after dropping the time deriva-tive terms and the transverse spatial gradient terms. For some



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

range of parameter values, the bifurcation diagrams of theone-dimensional model are S-shaped. They found transverseinstability ~which leads to nonuniformities in ther and wdirections! on the ignited branch of the S-shaped curve whenthe reactor to particle diameter exceeds about 4. Figure 6shows a typical neutral stability curve that defines the onsetof transverse patterns for various azimuthal and radial modes~defined by the wave numberkmn). For the parameter valuesconsidered, the authors have observed only stationary~timeindependent! nonuniformities. It is interesting to note that thefirst transverse pattern that emerges is three dimensional(m51), and transverse nonuniformities could occur instrongly convective systems even when the physical propertydependence on temperature is ignored. These transverse pat-terns are similar to the Turing patterns in diffusion-reactionsystems and occur due to different thermal and mass diffu-sivities in the transverse direction.

B. Convective patterns in reaction driven naturalconvection

Subramanian and Balakotaiah~1994, 1995, and 1997a!have studied the bifurcations and convective patterns gener-ated due to the heat released by an exothermic reactionoccurring in a porous medium. Assuming the validityof Darcy’s law and Boussinesq approximation, thepseudohomogeneous model for the stream functionc, thedimensionless temperatureu, and the dimensionless concen-tration c for a rectangular box~2D!, is given by

Cdu

dt5F~u,l!, ~6.7a!

where,

u5S c~x,z,t!

u~x,z,t!

c~x,z,t!D , C5S 0 0 0

0 1 0

0 0 LesD

0,x,1, 0,z,1, t.0, ~6.7b!

and F(u,l) is as defined in Eq.~2.42!, with the followingboundary conditions:

c50,]u

]x50,

]c

]x50, at x50,1, ~6.7c!

c50,]u

]z50,

]c

]z50, at z50, ~6.7d!

]c

]z50, u50, c51, at z51. ~6.7e!

The dimensionless variables and parameters that appear inthe above equations are defined below,

z5z8

H, x5

x8

L, t5

u* t

sH, u5gS T2T0

T0D ,

c5cA f

cA f0

, a5L

H, Le5

lm

Deff

, g5E

RT0

,

~6.8!

B5~2DH !cA f0

r fCp fT0

3Deff

lm

3E

RT0

,

s5er fCp f

rmCpm

, f25k~T0!H2

Deff

, Ra5r f

2Cp fgHbT0k

gmkeff

.

Here,u* is the reference velocity, Ra is the Rayleigh num-ber, and defines the ratio of characteristic time for heat con-duction to that of natural convection,a is the aspect ratio,f2

is the Thiele modulus, and is the ratio of characteristic timefor diffusion to that of reaction, Le is the Lewis number, andis the ratio of thermal to mass diffusivities.

The above model has purely conductive solutions@forwhich c(x,z)50 and u and c depend only onz] for allvalues of the Rayleigh number. When the reaction is exo-thermic (B.0), these can be either one or three conductionstates, and these could be oscillatory, especially when thereacting fluid is a liquid.

Subramanian and Balakotaiah~1997a! analyzed the sta-bility of the one-dimensional conduction states and showedthat they could be unstable and lead to convective flows.Both pure and mixed mode convective patterns arise fromthe conduction state and the interaction between differentmodes can lead to secondary bifurcations. All the local bi-furcations can be analyzed using the amplitude equations andformulas given in Sec. II~note that the governing equationshere possessZ2 symmetry!. We review here two sample re-sults. Figure 7 shows the local bifurcation diagram of theconduction and convection solutions close to a horizontalmode 1-mode 2 bicritical point. Here, the primary bifurca-tions lead to stationary convective pattern, but the interactionbetween the two stationary patterns leads to a secondary bi-furcation giving rise to oscillatory convection. Figure 8

FIG. 6. Neutral stability curves for transverse patterns in a packed-bedreactor~Balakotaiahet al., 1999!.



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shows a classification diagram and the different possible bi-furcation diagrams of the pure mode 1 convective solutions.The parameter plane~Le, Ra! is divided into five regions,each with a different bifurcation diagram. In region~i! onlyconduction solution exists and is stable~no convective solu-tions!. In region~ii !, the convective mode 1 branch bifurcatessupercritically at two points and is connected. In region~v!,the convective branch is isolated and is not connected to theconduction branch. It is interesting to note that in region~iv!,the convective solution bifurcates subcritically and hence in-troduces an ignition point in the system.

C. Convective patterns in down-flow reactors

Stroh and Balakotaiah~1991! studied the interaction be-tween forced and natural convection leading to flow maldis-tributions in an adiabatic down-flow packed-bed reactor,with a constant heat source~corresponding to zero activationenergy!. The flow maldistributions arise due to the instabili-ties ~bifurcations! caused by the dependence of the physicalproperties on the temperature. The physical system consistsof a packed column through which a fluid moves downward.Heat is generated in the system~due to reaction! and the fluidis heated up as it moves through the column. The one-dimensional solution can become unstable and lead to 2 or3D flows. The simple model analyzed by them assumes the

validity of Darcy’s law and Boussinesq approximation in themomentum balance. The model is described by the continu-ity, momentum, and energy balances,

¹•u50, ~6.9a!

¹p52m

ku1r0@12b~T2T0!#gez , ~6.9b!

rmCpm

]T

]t1r fCp fu•¹T5leff¹

2T1S, ~6.9c!

with the boundary conditions

us50,]T

]s50, @s50,D, ~6.10a!

T5T0 , uL5u0 , @L50, ~6.10b!

p5p1 ,]T

]L50, @L5L1 , ~6.10c!

and defining the following nondimensional parameters andvariables:

x5s

L1

, z5L

L1

, v5u

u0

, t5tu0

L

r fCp f

rmCpm

,

FIG. 7. Local bifurcations close to amode 1-mode 2 bicritical point for thereaction driven natural convectionproblem.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

y5r0Cp fu0~T2T0!

SL1

, Pe5r fCp fu0L

leff

,

L5SL1gkb

m0u02Cp f

, a5L1

D,

)5k~p2r0gzL1!

m0u0L1

, )05k~p12r0gL1!

m0u0L1

,

the model Eq.~6.9! may be written in the following nondi-mensional form~for the two-dimensional case!:

]vx

]x1

]vz

]z50,

])

]x52vx ,

])

]z52vz2Ly,

]y

]t1vx

]y

]x1vz

]y

]z5

1

PeS ]2y

]x21

]2y

]z2D 11,

and the boundary conditions become

y50, vz51, @z50,

]y

]z50, )5)0 , @z51,

vx50,]y

]x50, @x50,a.

The viscosity variation with temperature could be includedby defining

m5m0@12b* ~T2T0!#,

L5SL1gkb

m0u02Cp f

S gbr02m0b* u0

kD ,

wherem0 is the viscosity at the inlet. The three parameters ofthe model are the Darcy buoyancy number (L), the Pecletnumber~Pe!, and the aspect ratio (a).

One solution of the above model~for all L anda) is theone-dimensional solution for whichvx50, vz51 and thepressure and temperature vary only in the flow direction ()andy depend only onz). Stroh and Balakotaiah~1991! ana-lyzed the stability of this solution and showed that it be-comes unstable at a critical value ofL ~which depends onaandPe!. They used the amplitude equations given in Sec. IIto show that the uniform flow becomes unstable through asubcritical pitchfork bifurcation. The local bifurcation resultswere followed by numerical computations of the solutionbranches. Figure 9 shows a typical bifurcation diagram aswell as the different types of nonuniform flows that couldexist in the system. The numerical results show that asL isincreased, there could be a jump from the uniform solution toa periodic or a chaotic solution~as shown in Fig. 9!.

VII. LOW DIMENSIONAL MODELS OF REACTORS ANDREACTING SYSTEMS USING CENTER ANDINVARIANT MANIFOLD TECHNIQUES

Mathematical models of homogeneous and catalytic re-actors that include spatial and time dependence of velocity,species concentration, and temperature for the fluid~andsolid! phase~s! are often too complex. These detailed modelsmay be in the form of several partial differential equations intwo or three spatial dimensions and time. Under certain con-ditions defined in terms of the spatial/temporal time scalespresent in the model, the detailed model may be simplified tolow dimensional model for an averaged quantity. For ex-ample, the heterogeneous two-phase reactor model describedby two separate species and energy balances for the fluid andsolid phases, may be simplified under certain conditions to

FIG. 8. Classification diagram for the mode 1 convective solutions anddifferent types of bifurcation diagrams for the reaction driven convectionproblem.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

one that gives average concentration and temperature of thetwo phases. This low dimensional model is termed aspseudohomogeneous, or effective model, in the literature.Another example is the classical Taylor-Aris dispersionproblem for laminar flow in a tube, in which the mathemati-cal model for the three-dimensional spatially dependent sol-ute concentration is reduced to the cross-sectional averageconcentration that varies along the reactor length.

Dynamical systems concepts, specifically, center and in-variant manifold theorems, may be used to derive these lowdimensional models systematically. The mathematical as-pects of the center and invariant manifolds are discussed byCarr ~1981!, Wiggins ~1990!, and Mercer and Roberts~1990!. In this section, we illustrate the usefulness of thesetechniques with two examples related to reactor problemsthat were studied in the recent literature.

A. Low dimensional models for laminar flow in a tubewith a nonlinear bulk reaction „Taylor-Arisdispersion …

Balakotaiah and Chang~1995! applied center and invari-ant manifold theorems to derive an effective equation to in-finite order for the classical Taylor-Aris problem of disper-sion of nonreactive solute in fully developed laminar channelflow. They have also derived the effective transport equationfor the case of nonlinear bulk or volumetric reaction. The

transport equation~without axial molecular diffusion! for thelocal concentration of the reacting species in nondimensionalform is given by

]C

]t1p f~x,y!

]C

]z5¹

*2 C2f2r ~C!;

0,z,1, t.0, ~x,y!PV, ~7.1a!

¹* C•n50 on ]V, ~7.1b!

whereV is the channel cross section with]V as its bound-ary,x andy are the dimensionless transverse coordinates,z isthe dimensionless axial coordinate,C is the dimensionlessconcentration, r (C) is the dimensionless reaction rate,f (x,y) is the dimensionless fully developed velocity profilein the channel,¹

*2 is the Laplacian operator inV, andn is

the unit outward normal to]V. The time scales present inthis simple problem of fluid flow with transverse diffusionand homogeneous bulk reaction are convective time scale,tC5L/u, transverse diffusion time scale,tD5a2/Dm , andreaction time scale,tR5CR /R(CR). Here, u is the averagevelocity, Dm is the molecular diffusivity,a is the transverselength scale~tube radius!, L is the length scale in the flowdirection~length of the tube!, andR(CR) is the dimensionalreaction rate at some reference concentrationCR . The di-mensionless parameterp in Eq. ~7.1a! is the ratio of charac-teristic time for transverse diffusion to convection,p5tD /tC , andf2 is the square of the Thiele modulus, whichis the ratio of transverse diffusion time to reaction time. Thedimensionless variables and parameters that appear in Eq.~7.1! are defined as

x5x8

a, y5

y8

a, z5

z8

L, f ~x,y!5 f 8~ax,ay!,

t5t8

tD

, p5a2u

DmL5

tD

tC

, Da5tC

tR

5R~CR!L

CRu,

f25R~CR!a2

CRDm

5pDa5tD

tR

.

When the transverse diffusion time is much smallercompared to convection and reaction time scales, i.e.,p!1andf2;O(p), the center manifold theorem can be used toeliminate transverse spatial dependency of concentration andobtain an effective equation for cross-sectional average con-centration,

Cm~z,t !51

VEV

C~x,y,z,t !dV.

The authors show that the effective equation to orderp ~orPe21, where Pe548/p) is the standard one-dimensionaldiffusion-convection-reaction model,

]Cm

]t1

]Cm

]z5

1

Pe

]2Cm

]z2 2Dar~Cm!1O~Pe22!, ~7.2!

with the effective dispersion constant the same as the Taylor-Aris dispersion constant without reaction. They also derivedan effective transport equation for the case fast reactionf2

FIG. 9. Different possible flow patterns and bifurcation diagram for thepacked-bed with down-flow reactor.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

;O(1) andp!1 ~i.e., the transverse diffusion time and re-action time are of the same order and are small compared toconvection time! using the invariant manifold theorem.

Balakotaiah and Chang~1995! also applied these tech-niques to derive effective equations for Taylor-Aris disper-sion ~or laminar flow in a tube! with surface reaction, andshowed that the effective velocity, diffusivity, and rate con-stant~cross-section or volume averaged! depend on the sur-face reaction rate constant.

B. Low dimensional or pseudohomogeneous modelsof catalytic reactors

Balakotaiah and Dommeti~1999! applied the centermanifold theorem to reduce two-phase packed-bed reactormodels to pseudohomogeneous models. They have studiedthe classical problem of heat transfer in packed-beds withand without nonlinear heat source, mass transfer in packed-beds, isothermal packed-bed reactors with linear and nonlin-ear reaction in the solid phase, and nonisothermal packed-bed reactor with first-order exothermic reaction in the solidphase. They have also examined the convergence of the ef-fective models derived to infinite order and obtained analyti-cal criteria for the existence of the low dimensional or effec-tive models. We review here some interesting new results forthe case of isothermal nonlinear reaction and nonisothermalexothermic first-order reaction in a packed-bed catalytic re-actor.

For the isothermal model, the unsteady state species bal-ance for fluid and solid phases that includes mass transferbetween the phases and a nonlinear reaction in the solidphase, and excludes diffusion in the fluid phase of a packed-bed catalytic reactor, is given by

e]Cf

]t852u

]Cf

]z2kcav~Cf2Cs!, ~7.3a!

~12e!]Cs

]t85kcav~Cf2Cs!2R~Cs!, ~7.3b!

respectively, subject to the following boundary and initialconditions:

Cf5Cf 0@z50, Cf5Cf0~z!, Cs5Cs

0~z!@t850.~7.3c!

Defining

tc5L

u, tm5

1

kcav

, t r5Cf 0

R~Cf 0!,

t5avkct8, j5z

L, ci5

Ci

Cf 0

, i 5s, f ,

~7.4!

Da5tc

t r

5R~Cf 0!L

Cf 0u, pm5

tm

tc

5u

Lkcav

,

Dapm5tm

tr

5pmDa, r ~cs!5R~Cf 0cs!

R~Cf 0!,

the model equations~7.3! in nondimensional form may bewritten as

e]cf

]t52cf1cs2pm

]cf

]j, ~7.5a!

~12e!]cs

]t5cf2cs2pmDar~cs!, ~7.5b!

cf51@j50, ci5ci0~j!@t50. ~7.5c!

This simple model contains three characteristic times,namely, the residence~convection! time (tc), the exchangeor interphase mass transfer time (tm), and the reaction time(t r). When the interface mass transfer time is small com-pared to convection time and the reaction time, i.e.,pm,1and pmDa5Dapm,1, the solid and fluid concentrationswould be close, and it is possible to derive a single effectiveequation for the average concentration using the center mani-fold theorem. The authors show that the effective equationfor the average concentration,Cm@5eCf1(12e)Cs#, toleading order is

]Cm

]t81ue

]Cm

]z5De

]2Cm

]z22Re~Cm!, ~7.6!

where

ue5uF112e~12e!R8~Cm!

kcavG , R8~Cm!5S ]R

]CmD ,

De5u2~12e!2

kcavF11

3e~122e!R8~Cm!

kcavG ,

Re~Cm!5R~Cm!F12e2R8~Cm!

kcavG .

It is interesting to note that the effective model thus derivedis different from the standard pseudohomogeneous modelused in the literature. It has extra terms that come as correc-tions to convection, source, and dispersion terms of the stan-dard pseudohomogeneous model and they depend on the re-action parameters. For the special case of first-order reaction,the effective model was derived to infinite order, and Fig. 10shows the dependence of the correction factor to the disper-sion constant~ratio of the dispersion constant with reactionto that without reaction! on reaction parameter (Dapm).

Balakotaiah and Dommeti~1999! also studied the gen-eral coupled heat and mass transfer problem with first-orderexothermic reaction in the solid phase with negligible diffu-sion of mass and heat in the solid and fluid phases. Themathematical model is given by

e]Cf

]t852u

]Cf

]z82kcav~Cf2Cs!, ~7.7a!

~12e!]Cs

]t85kcav~Cf2Cs!2R~Cs ,Ts!, ~7.7b!



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

er fcp fF ]Tf

]t81u0

]Tf

]z8G5hav~Ts2Tf !, ~7.7c!

~12e!rscps

]Ts

]t852hav~Ts2Tf !

1~2DHR!R~Cs ,Ts!, ~7.7d!

Cf5Cf 0@z850, Cf5Cf0~z8!, Cs5Cs

0~z8!@t850,

~7.7e!

Tf5Tf 0@z850, Tf5Tf0~z8!, Ts5Ts

0~z8!@t850,

~7.7f!

and may be put in nondimensional form,

e]cf

]t52pm

]cf

]j2cf1cs , ~7.8a!

~12e!]cs

]t5cf2cs2Dapmr ~cs ,ys!, ~7.8b!

e]yf

]t5Lep~ys2yf !2pm

]yf

]j, ~7.8c!

~12e!

a

]ys

]t52Lep~ys2yf !1bDapmr ~cs ,ys!, ~7.8d!

cf51.0@j50, ci5ci0~j!@t50, ~7.8e!

yf50@j50, yi5yi0~j!@t50. ~7.8f!

The nondimensional variables and parameters in the aboveequation are defined as

tm51

kcav

, t r5Cf 0

R~Cf 0!, t5avkct8,

j5z8

L, ci5

Ci

Cf 0

, i 5s, f ,

Da5tC

tr

5R~Cf 0!L

Cf 0u, pm5

tm

tC

5u

Lkcav

, ~7.9!

Dapm5tm

tr

5pmDa, r ~cs!5R~Cf 0cs!

R~Cf 0!,

b5~2DHR!Cf 0

r fcp fTf 0

, Lep5tm

th

5h

r fcp fkc

,

where tm and th are the characteristic times for interphasemass and heat transfer, respectively,t r is characteristic timefor reaction,Da is the Damko¨hler number which is the ratioof characteristic time for convection to reaction,Dapm is theparticle mass Damko¨hler number and is the ratio of inter-phase mass transfer time to reaction time, Lep is the particleLewis number and is the ratio of interphase mass transfertime to heat transfer time, andb is the nondimensional heatof reaction.

When the characteristic times for interphase heat andmass transfer are small compared to convection and reactiontimes, the two-phase model@Eq. ~7.8!# can be reduced to thepseudohomogeneous model. Balakotaiah and Dommeti~1999! used center manifold theory to show that the effective~or pseudohomogeneous! model to orderpm for average con-centration and temperature is

]Cm

]t81uec

]Cm

]z81uC

]Tm

]z85De

]2Cm

]z822Re , ~7.10a!

rmcpm

]Tm

]t81uetr fcp f

]Tm

]z81uc~2DHR!

]Cm

]z8

5ke

]2Tm

]z821~2DHR!Re , ~7.10b!

where

Cm5eCf1~12e!Cs ,

Tm5er fcp fTf1~12e!rscpsTs

rmcpm

,

f5a

Lep~12e1ae!2, f152e~12e!,

f25e~12e!~11f!, f35e2,

f45e2abf, f552bf, s5~12e1ae!

a,

De5u2~12e!2

kcav

, ke5~r fcp f!

2u2~12e!2

hav~12e1ae!2,

C5f2

kcavS ]R

]TD , uc52uf2

kcavS ]R

]CD ,

uec5uF11f1

kcavS ]R

]CD G ,

FIG. 10. Dependence of the effective diffusivity of the pseudohomogeneousmodel on the particle Damko¨hler number.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

uet5uF12f1f~2DHR!

r fcp fkcavS ]R

]TD G ,

Re5R~Cm ,Tm!F12f3

kcavS ]R

]CD1e2af

~2DHR!

r fcp fkcavS ]R

]TD G .

The rigorously derived pseudohomogeneous model, Eq.~7.10!, has several new features compared to the standardpseudohomogeneous models used in the literature. For ex-ample, the new model has cross-coupled convection terms,corrections to the convective velocities and the reaction rateterm. The effective velocities (uec anduet) and the effectivereaction rate (Re) depend on the derivative of the reactionrate w.r .t. concentration and temperature. For exothermicreactions (DHR,0,]R/]T.0) with monotone kinetics~re-action rate increasing with concentration,]R/]C.0), theeffective velocity (uec) exceeds the average velocity (u),while uet is smaller thanu. Finally, if the effective model isderived to orderpm

2 , the effective diffusivity and conductiv-ity will also show dependence on the derivative of the reac-tion rate.

VIII. DISCUSSION

We have summarized in this article, the bifurcation fea-tures of some well known chemical reactor models describedby partial differential equations, with emphasis on convec-tion dominated systems. We now discuss briefly some topicsof current interest which have not been included in this re-view.

There is a substantial amount of recent research on aclass of reactor models described by integro-differentialequations. Such models arise when there is global couplingor when integral constraints are placed on the spatially vary-ing state variables. Most of the published work on thesemodels dealt with determining specific types of solutions bynumerical computation. The second class of models that areof interest are those for which the linearization has a con-tinuous spectrum@dim(Ker(L ))5`, whereL is the linearoperator#. In the context of chemical reactors and reactingflows, this could occur when the reaction is fast. A summaryof the bifurcation techniques for analyzing the generic fea-tures of these two classes of models would be of interest.

Many partial differential equations describing chemicalreactors and reacting flows are often solved by discretizingthe model equations using standard techniques. When thesolutions exhibit boundary layers in space~or time!, and lo-cation of these is not knowna priori ~since it may depend onthe reaction parameters such as activation energy!, a largenumber of discretization points have to be used to avoidspurious solutions created by numerical dispersion. Such dis-crete models usually have a very high dimension. For ex-ample, 2D problems typically lead to discrete models of di-mensionN'104, while 3D models giveN'106 or morediscrete equations. For such models, it is impractical to usethe standard bifurcation formulas~given in Sec. II!, since

they require the evaluation of the Jacobian~requiring N2

derivatives!, eigenvectors, and second and third Fre´chet de-rivatives~requiringN3 andN4 derivatives!. Numerical tech-niques need to be developed to analyze such large scale bi-furcation problems.

Finally, the idea of using dynamical systems concepts todevelop accurate low dimensional models of reactors andreacting flows is very promising. However, it is not clearhow to derive the proper initial and boundary conditions thatare to be used with the low dimensional models and how todetermine the region of validity~convergence! of these mod-els. It is hoped that this review will stimulate further researchon these as well as other bifurcation problems related tochemical reactors and reacting flows.

ACKNOWLEDGMENT

This work was partially supported by a grant from theRobert A. Welch Foundation.

Alexander, R.~1990!. ‘‘Spatially oscillatory steady states of tubular chemi-cal reactors,’’ SIAM~Soc. Ind. Appl. Math.! J. Math. Anal.21, 137–153.

Aris, R., and Amundson, N. R.~1958!. ‘‘An analysis of chemical reactorstability and control—I. The possibility of local control, with perfect orimperfect control mechanisms,’’ Chem. Eng. Sci.7, 121–131.

Balakotaiah, V., and Luss, D.~1982!. ‘‘Structure of steady-state solutions oflumped parameter chemically reacting systems,’’ Chem. Eng. Sci.37,1611–1623.

Balakotaiah, V., and Luss, D.~1983!. ‘‘Multiplicity features of reactingsystems: Dependence of the steady-states of a CSTR on residencetime,’’ Chem. Eng. Sci.38, 1709–1721.

Balakotaiah, V., and Luss, D.~1984!. ‘‘Global analysis of the multiplicityfeatures of multi-reaction lumped parameter systems,’’ Chem. Eng. Sci.39, 865–881.

Balakotaiah, V., Luss, D., and Keyfitz, B. L.~1985!. ‘‘Steady state multi-plicity analysis of lumped-parameter systems described by a set of alge-braic equations,’’ Chem. Eng. Commun.36, 121–147.

Balakotaiah, V., and Luss, D.~1986!. ‘‘Steady-state multiplicity features oflumped parameter chemical reacting systems,’’ inDynamics of Nonlin-ear Systems, edited by V. Hlavacek~Gordon and Breach, New York!.

Balakotaiah, V., and Luss, D.~1988!. ‘‘Global mapping of parameter re-gions with a specific number of solutions,’’ Chem. Eng. Sci.43, 957–964.

Balakotaiah, V., and Chang, C.~1995!. ‘‘Dispersion of chemical solutes inchromatographs and reactors,’’ Philos. Trans. R. Soc. London, Ser. A351, 1–37.

Balakotaiah, V., and Dommeti, S. M. S.~1999!. ‘‘Effective models forpacked-bed catalytic reactors,’’~to appear in Chem. Eng. Sci.!.

Balakotaiah, V., Christoforatou, E. L., and West, D. H.~1999!. ‘‘Transverseconcentration and temperature nonuniformities in adiabatic packed-bedcatalytic reactors,’’~to appear in Chem. Eng. Sci.!.

Carr, J. ~1981!. Applications of Center Manifold Theory~Springer, NewYork!.

Dangelmayr, G., and Armbruster, D.~1986!. ‘‘Steady-state mode 1 interac-tions in the presence ofO~2! symmetry and in non-flux boundary valueproblems, inMultiparameter Bifurcation Theory~American Mathemati-cal Society, Providence, RI!.

Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sand-stede, B., and Wang, X.~1997!. ‘‘Auto 97: Continuation and bifurcationsoftware for ordinary differential equations’’~Cal. Inst. Tech., Pasadena,CA!.

Dommeti, S. M. S., Balakotaiah, V., and West, D. H.~1999!. ‘‘Analyticalcriteria for validity of pseudohomogeneous models of packed-bed cata-lytic reactors,’’ ~to appear in Ind. Eng. Chem. Res.!.

Field, R. J., and Burger, M.~1985!. Oscillations and Travelling Waves inChemical Systems~Wiley, New York!.

Golubitsky, M., and Keyfitz, B. L.~1980!. ‘‘A qualitative study of thesteady-state solutions for a continuous flow stirred tank chemical reac-tor,’’ SIAM ~Soc. Ind. Appl. Math.! J. Math. Anal.11, 316–339.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

Golubitsky, M., and Schaeffer, D. G.~1985!. Singularities and Groups inBifurcation Theory~Springer-Verlag, New York!.

Gray, P., and Scott, S. K.~1990!. Chemical Oscillations and Instabilities~Clarendon, Oxford!.

Hlavacek, V., and Hoffmann, H.~1970!. ‘‘Modeling of chemical reactorsXVI—Steady state axial heat and mass transfer in tubular reactors: Ananalysis of the uniqueness of solutions,’’ Chem. Eng. Sci.25, 173–185.

Hudson, J. L., and Ro¨ssler, O. E.~1986!. ‘‘Chaos and complex oscillationsin stirred chemical reactors,’’ inDynamics of Nonlinear Systems, editedby V. Hlavacek~Gordon and Breach, New York!.

Jensen, K. F., and Ray, W. H.~1982!. ‘‘The bifurcation behavior of tubularreactors,’’ Chem. Eng. Sci.37, 199–222.

Khinast, J., and Luss, D.~1997!. ‘‘Mapping regions with different bifurca-tion diagrams of a reverse-flow reactor,’’ AIChE. J.43, 2034–2047.

Khinast, J., Gurumoorthy, A., and Luss, D.~1998!. ‘‘Complex dynamicfeatures of a cooled reverse-flow reactor,’’ AIChE. J.44, 1128–1140.

Kubicek, M., and Marek, M.~1983!. Computational Methods in BifurcationTheory and Dissipative Structures~Springer, New York!.

Kuramoto, Y. ~1984!. Chemical Oscillations, Waves, and Turbulence~Springer, New York!.

Lovo, M., and Balakotaiah, V.~1992!. ‘‘Multiplicity features of adiabaticautothermal reactors,’’ AIChE. J.38, 101–115.

Marek, M., and Schreiber, I.~1991!. Chaotic Behavior of DeterministicDissipative Systems~Cambridge University Press, Cambridge!.

Matros, Y. S., and Bunimovich, G. A.~1996!. ‘‘Reverse flow operation infixed bed catalytic reactors,’’ Catal. Rev. Sci. Eng.38, 1–68.

Mercer, G. N., and Roberts, A. J.~1990!. ‘‘A center manifold description ofcontaminant dispersion in channels with varying flow properties,’’SIAM ~Soc. Ind. Appl. Math.! J. Math. Anal.50, 1547–1565.

Morbidelli, M., Varma, A., and Aris, R.~1987!. ‘‘Reactor steady-state mul-tiplicity and stability,’’ in Chemical Reaction and Reactor Engineering,edited by J. J. Carberry and A. Varma~Marcel Dekker, Inc., New York!.

Olsen, R. J., Schmidt, L. D., and Aris, R.~1994!. ‘‘The effect of flowvelocity on ignition and extinction in homogeneous-heterogeneous com-bustion,’’ Combust. Sci. Technol.99, 377–392.

Planeaux, J., Jensen, K. F., and Fan, W. W.~1986!. ‘‘Dynamic behavior ofcontinuous stirred-tank reactors with extraneous thermal capacitance,’’in Reacting Flows: Combustion and Chemical Reactors~AmericanMathematical Society, Providence, RI!.

Poore, A. B. ~1973!. ‘‘A model equation arising from chemical reactortheory,’’ Arch. Ration. Mech. Anal.52, 358–388.

Razon, L. F., and Schmitz, R. A.~1987!. ‘‘Multiplicities and instabilities inchemically reacting systems—A review,’’ Chem. Eng. Sci.42, 1005–1047.

Rovinsky, A. B., and Menzinger, M.~1992!. ‘‘Chemical instability inducedby a differential flow,’’ Phys. Rev. Lett.69, 1193–1196.

Rovinsky, A. B., and Menzinger, M.~1993!. ‘‘Self-organization induced bydifferential flow of activator and inhibitor,’’ Phys. Rev. Lett.70, 778–781.

Schneider, F. W., and Munster, A. F.~1991!. ‘‘Chemical oscillations, chaos,and fluctuations in flow reactors,’’ J. Phys. Chem.95, 2130–2138.

Scott, S. K.~1991!. Chemical Chaos~Clarendon, Oxford!.Shvartsman, S., and Sheintuch, M.~1995!. ‘‘One- and two-dimensional spa-

tiotemporal thermal patterns in a fixed-bed reactor,’’ Chem. Eng. Sci.50, 3125–3141.

Song, X., Williams, W. R., Schmidt, L. D., and Aris, R.~1991!. ‘‘Bifurca-tion behavior in homogeneous-heterogeneous combustion: II. Computa-tion for stagnation-point flow,’’ Combust. Flame84, 292–311.

Stroh, F., Lovo, M., and Balakotaiah, V.~1990!. ‘‘On the number of steadystates of the nonadiabatic tubular reactor,’’ AIChE. J.36, 1905–1908.

Stroh, F., and Balakotaiah, V.~1991!. ‘‘Modeling of reaction-induced flowmaldistributions in packed beds,’’ AIChE. J.37, 1035–1052.

Subramanian, S., and Balakotaiah, V.~1994!. ‘‘Convective instabilities in-duced by exothermic reactions occurring in a porous medium,’’ Phys.Fluids 6, 2907–2922.

Subramanian, S., and Balakotaiah, V.~1995!. ‘‘Mode interactions inreaction-driven convection in a porous medium,’’ Chem. Eng. Sci.50,1851–1866.

Subramanian, S., and Balakotaiah, V.~1996!. ‘‘Classification of steady-stateand dynamic behavior of distributed reactor models,’’ Chem. Eng. Sci.51, 401–421.

Subramanian, S., and Balakotaiah, V.~1997a!. ‘‘Analysis and classificationof reaction-driven stationary convective patterns in a porous medium,’’Phys. Fluids9, 1674–1695.

Subramanian, S., and Balakotaiah, V.~1997b!. ‘‘Classification of steady-state and dynamic behavior of a well-mixed heterogeneous reactormodel,’’ Chem. Eng. Sci.52, 961–978.

Uppal, A., Ray, W. H., and Poore, A. B.~1974!. ‘‘On the dynamic behaviorof continuous stirred tank reactors,’’ Chem. Eng. Sci.29, 967–985.

Wiggins, S.~1990!. Introduction to Applied Nonlinear Dynamical Systemsand Chaos.

Yakhnin, V. Z., Rovinsky, A. B., and Menzinger, M.~1994!. ‘‘Differential-flow-induced pattern formation in the exothermicA→B reaction,’’ J.Phys. Chem.98, 2116–2119.

Yakhnin, V. Z., Rovinsky, A. B., and Menzinger, M.~1996!. ‘‘Wave phe-nomena in the packed-bed reactor: Their relation to the differential-flowinstability,’’ Can. J. Chem. Eng.74, 647–650.

Zeldovich, Ya. B., and Zysin, Y. A.~1941!. ‘‘On the theory of the heatrelease rate from chemical reactions in streams,’’ Zh. Tekh. Fiz.11,501–508.



130.216.129.208 On: Fri, 28 Nov 2014 00:08:58

Bifurcation analysis of chemical reactors and reacting flows

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