Bifurcation analysis of a two-dimensional catalytic monolith reactor model

*Corresponding author. Tel.: #1-713-743-4318; fax: #1-713-743-4323.E-mail address: [email protected] (V. Balakotaiah).

Chemical Engineering Science 56 (2001) 1435}1442

Bifurcation analysis of a two-dimensional catalyticmonolith reactor model

Nikunj Gupta�, Vemuri Balakotaiah��*, David H. West��Department of Chemical Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4792, USA

�The Dow Chemical Company, 2301 N. Brazosport Blvd., Freeport, TX 77541, USA

Abstract

We present a complete bifurcation analysis of a general steady-state two-dimensional catalytic monolith reactor model thataccounts for temperature and concentration gradients in both axial and radial directions and uses Danckwerts boundary conditions.We show that the ignition/extinction characteristics of the monolith are determined by the transverse Peclet number (P"R�u� /¸D

�,

ratio of transverse di!usion to convection time) and the transverse Thiele modulus (��"2Rk

�(¹

�)/D

�, ratio of transverse di!usion to

reaction time). When ��;1, ignition occurs at P values of order B��

�and the monolith behaves like a homogeneous reactor with

simultaneous ignition/extinction of the surface and the #uid phase. However, when ��<1, surface ignition occurs very close to the

inlet (or for very short residence times corresponding to large values of P) to a maximum surface temperature of B/(¸e�)� (a"1/2 for

#at velocity and a"2/3 for parabolic velocity pro"le) while the #uid-phase conditions are still close to the inlet values. In this fastreaction, mass transfer controlled regime, the #uid temperature reaches the adiabatic value (and the mean exit conversion is close tounity) only when the P values are of order unity or smaller. We show that the behavior of the monolith is bounded by two simpli"edmodels. One of them is the well-known convectionmodel and the second is a new model which we call the short monolith (SM) model.The SMmodel is described by a two-point boundary value problem in the radial coordinate and has the same qualitative bifurcationfeatures as the general two-dimensional model. We also show that when the #uid Lewis number is less than unity (¸e

�(1), there exist

bifurcation diagrams of surface temperature versus residence time containing isolated solution branches on which the surfacetemperature exceeds the adiabatic temperature. Finally, we present explicit analytical expressions for the ignition, extinction andhysteresis loci for various models and also for the #uid phase conversion and temperature in the fast reaction (mass transfercontrolled) regime. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Monolith reactor; Catalytic monolith; Bifurcation; Ignition and extinction; Catalytic combustion; Surface reaction

1. Introduction

Catalytic monoliths are used in automobile converters,oxidation of VOCs, power generation, partial oxidationreactions and selective removal of NO

�from exhaust

gases. The monolith reactor is a large number of small,long tubes (in parallel) through which the reacting gas#ows. The catalyst is deposited on the wall of the mono-lith reactor either as a porous washcoat layer or on thewall of the duct. The reactant in the gas stream is trans-ported to the surface by transverse di!usion and iscarried forward by convection and axial di!usion,

thus producing concentration gradients in both axial andradial directions. Previous studies of monoliths used sim-pli"ed models such as two-dimensional plug-#ow model(or with a parabolic velocity pro"le but without axialdi!usion or conduction) (DamkoK hler, 1937; Heck,Wei & Katzer, 1976; Young & Finlayson, 1976) or aone-dimensional two-phase model with empirical (ortheoretical) correlations for the heat and mass transfercoe$cients (Hegedus, 1975; Heck et al., 1976; Dommeti,Balakotaiah & West, 1999). The predictions of suchsimpli"ed models were found to be vastly di!erent. Forexample, the two-dimensional plug-#ow model wasfound to predict a unique steady-state for all values of theparameters while the one-dimensional two-phase modelpredicts multiple solutions even at zero residence time(when the asymptotic values for the Sherwood andNusselt numbers are used). In this work, we compare the

0009-2509/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 3 6 8 - 7

bifurcation features of these commonly used models withthose of a general two-dimensional model that includesdi!usion and conduction in both axial and radial direc-tions. We also elucidate the ignition/extinction behaviorof the monolith in the global parameter space and pres-ent analytical solutions for the #uid-phase conversionand temperature in the practically important fast reac-tion regime. Due to space limitations, we present heremainly the results and omit details.

2. Mathematical models

We consider a cylindrical tube on the surface of whicha single exothermic "rst-order reaction occurs. We as-sume constant physical properties and azimuthal sym-metry (the validity of this assumption is discussed later).Without any further assumptions, the general steady-state two-dimensional monolith model is given by thefollowing equations and boundary conditions:

f (�)�c

�z"

1

P�1

��

�c

��#

1

Pe

��c�z�

, (1)

f (�)��z

"

¸e�

P �1

��

��#

¸e�

Pe

��z�

, (2)

1

Pe

�c

�z"f (�)(c!1),

1

Pe

��z

"f (�)(�!��) at z"0, (3)

�c

�z"0,

��z

"0 at z"1, (4)

�c

��"0,

��

"0 at �"0, (5)

�c

��"!

��2

c exp(�/(1#�/�)),

��

"

B��

2¸e�

c exp(�/(1#�/�)) at �"1. (6)

The function f (�) is used to represent the local #owconditions inside the channel. For the case of #at velocitypro"le f (�)"1 whereas when we have laminar #ow in-side the channel, f (�)"2(1!��). The various dimen-sionless groups appearing in the above equations arede"ned below:

z"

x

¸

, �"

r

R, c"

C

C�

, �"��¹!¹

�¹

��,

�"

E

R¹�

, P"

u� R�

¸D�

, Pe"

u� ¸D

�

, ��"��

¹�!¹

�¹

��,

¸e�"

k�

��c��

D�

, B"

�(!�H)C�

��C

��¹

�

, ��"

2Rk�(¹

�)

D�

.

Here, � and z represent radial and axial coordinates andc and � are the dimensionless concentration and temper-ature, respectively, and �

�is the dimensionless inlet #uid

temperature. The square of the transverse Thielemodulus, ��

�, is the ratio of transverse di!usion time to

the reaction time. The transverse Peclet number (P) is theratio of transverse di!usion time to the convection time.The axial Peclet number (Pe) is the ratio of axial di!usiontime to convection time. The parameter B is the dimen-sionless adiabatic temperature rise, � is the dimensionlessactivation energy and the #uid Lewis number (¸e

�) is the

ratio of heat to mass di!usivities. It should be noted thatthe above model includes conduction/di!usion in boththe axial and radial directions and uses the more realisticDanckwerts boundary conditions at the inlet (z"0) in-stead of the usual Dirichlet boundary conditions(c"1, �"�

�at z"0) used in the prior literature. The

latter assumes that the concentration and temperaturegradients cannot propogate upstream of the inlet. Intuit-ively, it can be seen that this assumption is not satis"edfor the case of fast reactions (or on the ignited branches).Thus, our general two-dimensional model is a set of twocoupled elliptic partial di!erential equations.Various simpli"ed versions of the above model have

been developed and studied extensively in the literature.These include the one-dimensional two-phase model andthe commonly used two-dimensional convection model(with plug #ow or parabolic velocity pro"le but withoutaxial di!usion or conduction). In the prior literature, theconvection model was treated as an initial value problemand hence only one solution was found. Our analysisshows that the convection model de"nes a di!erential-algebraic system (with index in"nity) and has an in"nitenumber of solution pro"les with a well-de"ned ignitionand extinction point. Our attempts to describe the quali-tative behavior of the general two-dimensional modellead us to formulate another simple and useful model forthe case when PeP0 (�"aspect ratio"R/¸PR,short tube). We shall refer to this model as the shortmonolith (SM) model. It may be shown that the behaviorof the general two-dimensional model is bounded bythese two models for all values of parameters. Moreimportantly, for the case of non-isothermal reaction, theSM model retains all the qualitative features of the gen-eral model and gives the bifurcation behavior correctly.We now present the two limiting models and analyze thesteady-state features of these limiting models as well asthe general model de"ned by Eqs. (1}6)

2.1. The two-dimensional convection model (PePR)

This model was "rst proposed by DamkoK hler (1937)and has been used extensively in the prior literature(Heck et al., 1976; Young & Finlayson, 1976). It ignoresthe axial conduction (di!usion) and uses Dirichletboundary conditions at the inlet. It is de"ned by the

1436 N. Gupta et al. / Chemical Engineering Science 56 (2001) 1435}1442

Fig. 1. Bifurcation diagrams of mean #uid (��) and surface temperature

(��) vs. axial position (z/P) for B"5.0, ¸e

�"0.2 and di!erent values of

Thiele modulus.

Fig. 2. A schematic bifurcation diagram of the convection model for¸e

�"1, B"10, �"R and �

�"5.

following coupled equations:

f (�)�c

�z"

1

P�1

��

�c

��, (7)

f (�)��z

"

¸e�

P �1

��

�� (8)

with boundary conditions:

c"1, �"��at z"0, (9)

�c

��"0,

��

"0 at �"0, (10)

�c

��"!

��2

c exp(�/(1#�/�)) at �"1, (11a)

��

"

B��

2¸e�

c exp(�/(1#�/�)) at �"1. (11b)

As stated earlier, this model was treated as an initialvalue problem in the prior literature but it de"nes anindex in"nity di!erential-algebraic system. We also notethat in the limit ��

�P0 and PP0 (��

�/P"Da is "nite),

this model reduces to the homogeneous PFR model.

2.2. The short monolith (SM) model (PeP0)

The short monolith model which has been developedrecently by Balakotaiah, Gupta and West (2000) assumesthat the characteristic time for longitudinal di!usion ismuch smaller compared to transverse di!usion and reac-tion (PeP0 or �<1) and ignores the axial gradientswithin the monolith. This model is given by a two-point

boundary value problem in the radial coordinate:

1

��d

d��dc�d��#Pf (�)(1!c� )"0, (12)

1

��d

d��d�Md��!

P

¸e�

f (�)(�M !��)"0, (13)

dc�d�

"0,d�Md�

"0 at �"0, (14)

dc�d�

"!

��2

c� exp(�M /(1#�M /�)),

d�M��

"

B��

2¸e�

c� exp(�M /(1#�M /�)) at �"1. (15)

Here, (c� , �M ) are the axially averaged concentration andtemperature, respectively. Note that when ��

�P0 and

PP0 with Da"��/P "nite, this model reduces to the

homogeneous CSTR model.Hereafter, we take �

�"0 (feed and reference temper-

ature are equal), �"R(positive exponential approxima-tion) and analyze the bifurcation features of the abovemodels by taking the residence time (Da or 1/P) as well as��as bifurcation parameters. We plot the surface temper-

ature (��"�(�"1)) as well as the mean #uid (or mixing

cup) temperature (��), where �

�(z)"�

�2� f (�)�(�, z) d� as

a function of the bifurcation parameter.

3. Analysis of the two-dimensional convection model

We de"ne a new axial coordinate "z/P and writeEqs. (7) and (8) as

f (�)�c

�"

1

��

�c

��, (7a)

N. Gupta et al. / Chemical Engineering Science 56 (2001) 1435}1442 1437

f (�)��

"

¸e�

� ��

�c

��. (8a)

As stated earlier, the model de"ned by Eqs. (7a), (8a),(9}11) was treated as an initial value problem in the priorliterature and hence only a unique solution was found forall values of the parameters.Fig. 1 shows the bifurcation diagrams of surface and#uid-phase temperatures versus obtained using thisinitial value approach for ¸e

�"0.2, B"5, �

�"0,

f(�)"1 and several ��values. When �

�(�H

�"

2�2e�� exp(!2�¸e�!2¸e

�), the #uid and the

surface temperatures are close to each other andthe monolith behaves like a homogeneous tubularreactor. Both the #uid and solid (surface) temperaturesshow simultaneous ignition to the adiabatic value (B)and the surface temperature does not exceed theadiabatic temperature. However, when �

�'�H

�the surface ignites to a maximum possible temperature ofB/(¸e

�)�, (where a"1/2 for #at velocity pro"le and

a"2/3 when #ow is laminar) while the #uid temperaturerises slowly to the adiabatic value. After the surfaceignites, the monolith is in the mass transfer controlled(fast reaction) regime.A careful examination of the local heat balance equa-

tion (11b) shows that it can have multiple solutions for��over a range of values (the right- and left-hand sides

of Eq. (11b) correspond to local heat generation and heatremoval rates, respectively). Whenever Eq. (11b) has mul-tiple solutions, the axial derivatives ��/� and �c/� arenot uniquely determined. It is due to this reason theconvection model is an index in"nity di!erential-alge-braic system. We show in Fig. 2 a true schematic bifurca-tion diagram of the convection model for ¸e

�"1,

B"10 and ��"5. If the convection model is

treated as an initial value problem, only one solution(corresponding to the latest possible ignition) representedby the curve OCHIDL is found. In addition, thediscontinuous jump from I to D is approximated as acontinuous curve (which also leads to an inaccuratecalculation of the local Sherwood and Nusselt numbers).The local heat balance equation (11b) has multiplesolutions for values between points C and I. Thesurface temperature can jump to a higher value at pointC and the pro"le OCEF corresponds to the `earliestignited solutiona. In between the late and earlyignition pro"les, there can be an in"nite numberof solution pro"les corresponding to ignition betweenpoints C and I. One such pro"le OCHJK is shownin Fig. 2.The exact values corresponding to the ignition and

extinction points (I and E) can be found only numer-ically. However, excellent approximations may be ob-tained by using an equivalent two phase model.Multiplying Eqs. (7a) and (8a) by 2�, integrating andusing the boundary conditions, the convection model

may be written in an alternate form

dc�

d"!��

�c�exp�

��

1#��/��"!Sh()(c

�!c

�), (16)

d��

d"B��

�c�exp�

��

1#��/��"¸e

�Nu()(�

�!�

�), (17)

c�

"1, ��

"0 at "0, (18)

where Sh() and Nu() are the local Sherwood and Nus-selt numbers de"ned by

Sh()"!2

�c

�� (c

�!c

�), Nu()"

2��

(��!�

�). (19)

Eqs. (16}18) de"ne the two-phase plug-#ow model of themonolith (with position-dependent heat and mass trans-fer coe$cients). It should be noted that if exact expres-sions can be substituted for Sh() and Nu(), then themodel de"ned by Eqs. (16}18) is identical to the two-dimensional convection model. It can be shown that(Gupta and Balakotaiah, 2001) the Sherwood and Nus-selt numbers vary in a range given by

f�())Sh())f

�(), (20a)

f�(¸e

�))Nu())f

�(¸e

�), (20b)

where

f�()"

1

��

[1!e�� /��], (J

�(�

�)"0, �

�'0) (21a)

"��

for ;1,

8 for <1,(21b)

f�()"

��

e��

��

e�� /��

�

, (J�(�

�)"0), (22a)

"�2

��for ;1,

5.783 for <1.(22b)

The functions f�() and f

�() are schematically shown in

Fig. 3. In the same "gure, we also show how theSherwood and Nusselt numbers jump from one asym-ptote to the other at the ignition and extinction points. Ingeneral, on the low temperature slow reaction branch(��

�e��;1) OCHI, the upper bound gives very accurately

the exact Sherwood or Nusselt number, while the lowerbound applies to the ignited branch (��

�e��<1). We can

now use these to obtain accurate expressions for theignition and extinction points of the convection model.


Fig. 3. Variation of the Sherwood and Nusselt numbers along thesolution branches shown in Fig. 2.

Eliminating c�and c

�from Eqs. (16) and (17), we

obtain

d��

d"¸e

�Nu()(�

�!�

�), (23a)

Nu()(��!�

�)"

��

¸e��B!�

�!¸e

�

Nu()Sh()

(��!�

�)�

�exp��

1#��/��. (23b)

We note that Eq. (23b) is same as the local heat balanceEq. (11b) and for �

�'�H

�the ignition and extinction

points are on the small asymptote and are given by

(��)

��"

e��¸e�

(B!�¸e�), (24)

(��)

��"��

B

�¸e�

!1�exp�1!B

�¸e��. (25)

When ��(�H

�, ignition/extinction occurs in the region

of high (constant Sherwood/Nusselt numbers) andonly if

B'4¸e�!2#ln�

8

��,B

. (26)

When B'B, it may be shown that

(��)��

"

8e��¸e�

(B!¸e�), (27a)

(��)��

"8�B

¸e�

!1�exp�1!B

¸e��. (27b)

When ��(�H

�and B(B

, the two-dimensional convec-

tion model behaves like a homogeneous plug-#ow reac-tor and shows no ignition or extinction but may exhibitparametric sensitivity.

It should also be pointed out that when ¸e�(1, there

can be isolated solution branches in a bifurcation dia-gram of �

�versus . A complete analysis of the steady-

state behavior of the convection model (including theextension of the above cited results for parabolic velocitypro"le) is given elsewhere (Gupta & Balakotaiah, 2000).It can be shown that in the mass transfer limited

regime the exit mean conversion is given by

x�

"��

��

4

��

(1!e�� ), (28)

where �is the jth zero of the Bessel function J

�(�). The

two asymptotes of Eq. (28) are

x�

"1 for P;1, x�

"

4

��Pfor P<1 (29)

with the transition occurring at P"16/�+5.1. Thus, toachieve high conversion in the #uid phase, we needP(5 and B, ��

�values such that an ignited branch

exists.

4. Analysis of the short monolith (SM) model

In this section, we determine the bifurcation behaviorof the short monolith model for �PR and #at velocitypro"le. As we show later, the SM model describes all thequalitative features of the full two-dimensional model.Eqs. (12}15) may be solved analytically to obtain the

following pro"les:

c� (�)"1!��¸e

�B

I�(�P/¸e

�)

I�(�P/¸e

�)

�M�

I�(�P)�I� (�P�), (30)

�M (�)"I�(�P/¸e

��)

I�(�P/¸e

�)�M�, (31)

where I�and I

�are modi"ed Bessel functions and the

surface temperature �M�is given by the algebraic equation:

�M�a�!(B!a

��M�)��

�exp(�M

�)"0, (32a)

a�"

2�P¸e�I�(�P/¸e

�)

I�(�P/¸e

�)

, a�"a

�

I�(�P)

2�PI�(�P)

. (32b)

The corresponding #uid-phase mean conversion andtemperature are given by

x�

"

�M�

B

a�P, �

�"Bx

�. (33)

It can be shown that the boundary between unique andmultiple solutions (hysteresis locus) of Eqs. (32) isgiven by

B"4a�,B

, ��

�"

a�e��

a�

,��. (34)


Fig. 4. Bifurcation diagram of mean and surface temperature versusThiele modulus for B"5, ¸e

�"0.2 and di!erent values of transverse

Peclet number.

Fig. 5. Classi"cation diagram for ¸e�"0.2. Solid curve is the hyster-

esis locus while the dashed curves are the isola (22) and the maxima(-.-.-.-) locus.

Thus, for any given P, there is a unique solution if eitherB(B

or �

�'�

�. The two asymptotes of Eqs. (34)

correspond to the homogeneous CSTR (PP0,B"4, ��

�/P"e��) and the mass transfer limited regime

with strong radial gradients (P<1, B"4�¸e�,

��"2e��P). The transition between the two asym-

ptotes occurs at ��"�H

�"2e��. When B'B

, the bi-

furcation diagrams of ��versus ��

�is S-shaped with an

ignition and extinction point. The ignition-extinction lo-cus (bifurcation set) may be written in parametric form

B"

a��

��!1

, ��"

a�(�

�!1)e��a�

, 1)��(R. (35)

The ignition locus corresponds to 1)��(2 while the

extinction locus is obtained for 2)��(R. For B'B

,

the following explicit expressions may be obtained for theignition and extinction loci:

(B��)��

"e��a�, (��

�)��

"

a�B

a��

exp�!

B

a��. (36)

Fig. 4 shows a bifurcation diagram of ��and �

�versus

��for B"5, ¸e

�"0.2 and two di!erent P values. As in

the case of the convection model, ��and �

�are close to

each other when P;1. However, when P<1 only thesurface reaches high temperatures and the monolith op-erates in the mass transfer controlled regime.It may be shown that in this mass transfer controlled

regime (��<1) the mean exit conversion is given by

x�

"

2I�(�P)

�PI�(�P)

. (37)

The two asymptotes of Eq. (37) are x�

"1 for P;1 and

x�

"2/�P for P<1 with a transition at P"4. Com-paring these with the asymptotes of the convectionmodel, we see that the di!erences between them is small(in the mass transfer controlled regime). The major di!er-ence appears in the prediction of the range of ��

�values

over which multiple solutions exist. As seen earlier, forP;1, this range may be zero for the convection model(ignition/extinction may not exist). The two asymptotesof the ignition locus for the SM model are given by

B��

P"�

e�� for P;1,

2e��¸e

�P

for P<1,(38)

while the extinction asymptotes are given by

��

P"�

(B!1) exp(1!B) for P;1,

2

�P�B

�¸e�

!1�exp�1!B

�¸e�� for P<1.

(39)

We note that large P asymptotes of the two models di!er

only by a factor 2/��"1.128.When the residence time (1/P) is taken as the bifurca-

tion variable, the behavior of the monolith becomescomplex. Now, isolated solution branches can exist withhigh surface temperatures. Fig. 5 shows a classi"cationdiagram in the (B,�

�) plane for ¸e

�"0.2. This classi"ca-

tion diagram is obtained by plotting the hysteresis andisola loci along with the maxima locus de"ned by

��"

8¸e�

1!¸e�

exp(!B). (40)


Fig. 6. Bifurcation diagram of mean and surface temperature forB"5, ¸e

�"0.2 and di!erent values of Thiele modulus.

Each region of this diagram corresponds to a di!erenttype of bifurcation diagram of �

�versus 1/P (or Da). In

region (i), there is a unique steady-state for all values ofthe residence time. In region (i-a) the surface temperatureis a monotonic function of Da while in (i}b) it goesthrough a maximum and the surface temperature goesabove the adiabatic temperature. In region (ii) the bifur-cation diagram is S-shaped while in region (iii) thereexists an isolated high-temperature branch (in additionto the S-shaped low-temperature branch). Some numer-ically computed bifurcation diagrams are shown inFig. 6.The above results can be extended for the case of

parabolic velocity pro"le. It may be shown that themaximum surface temperature is higher and the region ofmultiple solutions is larger than the #at velocity case for¸e

�(1 while the converse is true for ¸e

�'1.

5. Bifurcation analysis of the general two-dimensionalmodel

Before we analyze the general two-dimensional model,we consider the special case in which ¸e

�"1 and

f (�)"1. (In addition to ��"0 and �PR.) We can

eliminate the concentration using the invariant to obtaina single equation for the temperature pro"le.

��z

"

1

P�1

��

��#

1

Pe

��z�

, (41a)

1

Pe

��z

!�"0 at z"0,��z

"0 at z"1, (41b)

��

"0 at �"0,��

"

��2(B!�)e� at �"1. (41c)

We have already analyzed the behavior of the abovemodel in the two limits of PeP0 (SM model) andPePR (convection model). When P<1, the hysteresislocus is independent of Pe and is given by B"4. Mul-tiple solutions exist in some range of ��

�values provided

B'4 and P<1. In addition, the ignition and extinctionpoints are also independent of Pe (di!ering by atmosta factor of 1.128). Thus, it may be concluded that whenP<1, the SMmodel accurately describes the behavior ofthe full two-dimensional model.When P;1 (and ��

�;1), the model de"ned by Eqs.

(41) reduces to the homogeneous Danckwerts model:

1

Pe

d��

dz�!

d��

dz!Da(B!�

�)e��"0, (42a)

1

Pe

d��

dz!�

�"0 at z"0,

d��

dz"0 at z"1, (42b)

where Da"��/P. Then two asymptotes of the hysteresis

locus of this model are given by (Hlavacek & Hofmann,1970; Balakotaiah, 1996)

B"4, Da"e��, ��

"2 for PeP0, (43a)

Pe"

2

Be��, Da"

1

B, �

�"1 for PePR. (43b)

For any "nite value of P, the hysteresis locus of Eq. (41) inthe (B,Pe) plane is bounded by the two curves B"4(corresponding to Pe"0) and Pe"(2/B)e�� (corre-sponding to Pe"R). Thus, the region of multiple solu-tions is the smallest for the homogeneous Danckwertsmodel.When 1;Pe((2/B)e��, the ignition and extinction

loci of Eq. (42) are given by

(Da)��

+

1

B, �

Da

Pe��

+(B!1)e��. (44)

Similarly when Pe;1, the ignition and extinction loci ofEq. (42) are given by

(Da)��

+

e��

B, (Da)

��+(B!1)e��. (45)

The above observations can be extended to the case of¸e

�O1. Once again, we note that when P<1, the SM-

model gives an accurate description of the full two-dimensional model.Fig. 7 shows a numerically computed bifurcation set in

the ��}B plane for P"Pe"5 (�"1) and

P"0.1, Pe"5 (�"0.14). In both cases, the B values atthe hysteresis point is only slightly higher than 4. Asexpected, as P increases, the region of multiple solutionsmoves to higher values of ��

�.

When theB, P and Pe are such that an ignited solutionexists and ��

�exceeds the value at the extinction point,

the monolith operates in the mass transfer controlledregime. The mean #uid-phase conversion in this regime


Fig. 7. Bifurcation set in the ��}B plane for Pe"5.0 and P"0.1 and

5 (¸e�"1).

depends only on P and Pe and may be obtained bysolving the two-dimensional isothermal model with zerosurface concentration. It may be shown that the twoasymptotes for mean #uid conversion in this mass trans-fer limited regime are given by

x�

"�1, P;1,

2

�Pg(Pe), P<1,

(46)

where g(Pe) is a rather weak function of Pe (Balakotaiah

et al., 2000) varying from 1 at Pe"0 to 2/�� ("1.128)for PePR. Thus, the conditions for obtaining highconversion (or nearly complete combustion) in themonolith may be summarized as B'B

, ��

�'(��

�)��

and P)4[g(Pe)]�. Here Band (��

�)��are the B value at

the hysteresis point and ��value at the extinction point,

respectively.We have also found a very surprising and new result

that when P<1 and ��e�<1, the two-dimensional

solution loses azimuthal symmetry giving rise to 3-Dsolutions having localized hot spots. This topic of patternformation will be pursued elsewhere.

6. Conclusions and discussion

In this work we have analyzed a two-dimensionalmodel of a catalytic monolith with a "rst-order exother-mic reaction. We have elucidated the steady-state behav-ior of the monolith in the entire parameter space andgiven analytical expressions for the ignition, extinctionand hysteresis loci. Most of the results presented here are

new and (to our knowledge) have not appeared anywherein the prior literature. Due to space limitations, we havenot been able to give the details of how these asymptoticresults were derived or their physical interpretation in thedesign of catalytic monoliths. This will be pursued indetail in forthcoming publications. Here, we give a briefinterpretation of the results for the case of catalytic con-verters for treating exhaust gases.The analysis in Section 3 shows that once the monolith

is in the mass transfer controlled regime, the e%uentconversion depends only on P. Thus, to achieve highconversions, P values should be of order unity (smallerP values may increase the pressure drop but do notincrease the conversion signi"cantly). The ignited branchexists for ��

�'(��

�)��. However, the initial value ap-

proach taken to analyze the convection model in theprior literature does not predict an extinction point andpredicts only one solution corresponding to late ignition.As shown in Section 3, for typical parameter values (e.g.,B"15, ¸e

�"1) this could lead to a design that is con-

servative by a factor 10 or higher. The analytical resultsgiven here could lead to a quick order of magnitudeestimation of the in#uence of various design and operat-ing parameters on the monolith behavior.

Acknowledgements

This work was supported by grants from the Robert A.Welch Foundation, the Texas Advanced TechnologyProgram and the Dow Chemical Company.

References

Balakotaiah, V. (1996). Structural stability of nonlinear convection}reaction models. Chemical Engineering Education, Fall, 234}239.

Balakotaiah, V., Gupta, N., & West, D. H. (2000). A simpli"ed modelfor analyzing catalytic reactions in short monoliths. Chemical En-gineering Science, 55(22), 5367}5383.

DamkoK hler, G. (1937). In#uence of di!usion, #uid #ow and heat trans-port on the yield in chemical reactors. Der Chemie-Ingeniur, 3(2),359}485.

Dommeti, S. M. S., Balakotaiah, V., & West, D. H. (1999). Analyticalcriteria for validity of psuedohomogeneous models of packed-bedcatalytic reactors. I&EC Research, 38, 767}777.

Gupta, N., & Balakotaiah, V. (2001). Bifurcation analysis of the plug#owmodel of a catalytic monolith. Chemical Engineering Science, inpreparation.

Heck, R. H., Wei, J., & Katzer, J. R. (1976). Mathematical modeling ofmonolithic catalysts. A.I.Ch.E. Journal, 22, 477}483.

Hegedus, L. L. (1975). Temperature excursions in catalytic monoliths.A.I.Ch.E. Journal, 21, 849}853.

Hlavacek, V., & Hofmann, H. (1970). Modeling of chemical reactors-XVI: Steady-state axial heat and mass transfer in tubular reactors:An analysis of uniqueness of solutions. Chemical EngineeringScience, 25, 173.

Young, L. C., & Finlayson, B. A. (1976). Mathematical models of themonolith catalytic converter. A.I.Ch.E. Journal, 22, 331.


Bifurcation analysis of a two-dimensional catalytic monolith reactor model

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