Heat and mass transfer coefficients in catalytic monoliths

Chemical Engineering Science 56 (2001) 4771–4786www.elsevier.com/locate/ces

Heat and mass transfer coe#cients in catalytic monolithsNikunj Gupta, Vemuri Balakotaiah ∗

Department of Chemical Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4792, USA

Received 19 October 2000; received in revised form 5 March 2001; accepted 27 March 2001

Abstract

We analyze the classical Graetz problem in a tube with an exothermic surface reaction and show that the heat(mass) transfercoe#cient is not a continuous function of the axial position and jumps from one asymptote to another at ignition=extinction points.We show that the steady-state heat(mass) transfer coe#cient is not a unique function of position in parameter regions in which theGraetz problem with surface reaction has multiple solutions. We also analyze the more general two-dimensional model (with axialconduction=di9usion included and Danckwerts boundary conditions) and show that for ;xed values of the reaction parameters, theheat(mass) transfer coe#cient has three asymptotes. Unlike the Graetz problem, in this case the heat(mass) transfer coe#cient isalways ;nite and bounded at the inlet and is given by a new asymptote. We present analytical expressions for all three asymptotesfor the case of =at and parabolic velocity pro;les.It is also shown that in catalytic monoliths, ignition=extinction may often occur in the entry region and hence the local transfer

coe#cients and not the average values proposed in the literature determine the ignition=extinction behavior. Finally, we use thenew results to develop and analyze an accurate one-dimensional two-phase model of a catalytic monolith with position dependentheat and mass transfer coe#cients and determine analytically the dependence of the ignition=extinction locus on various designand operating parameters. ? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Heat transfer; Mass transfer; Surface reaction; Monolith; Graetz problem

1. Introduction

Mathematical models of convection with di9usion andsurface reaction in more than one spatial dimension areoften approximated by using the concept of an e9ectivemass and heat transfer coe#cient between the bulk =uidphase and the surface. This reduces the dimension ofthe model (by eliminating the transverse coordinates)and the resulting two-phase models are much easier tohandle. The e9ective heat and mass transport coe#cientsthat appear in the reduced (low dimensional) models areoften expressed in dimensionless form in terms of thewell-known Sherwood and Nusselt numbers. In reactionengineering applications, it is common to use a constantvalue for Sherwood and Nusselt numbers to approximatefor the transport gradients between the bulk and the

∗ Corresponding author. Tel.: +1-713-743-4318; fax: +1-713-743-4323.

E-mail address: [email protected] (V. Balakotaiah).

surface. This constant value corresponds to the asymp-totic value reached for the case when the longitudinaldimension is su#ciently large. However, using this ap-proximation and ignoring the dependence of the transfercoe#cients on velocity or position and reaction param-eters may lead to erroneous prediction of the ignitionand extinction points for exothermic surface catalyzedreactions.In this work, we determine the heat and mass transfer

coe#cients in a tube with exothermic surface catalyzedreactions with parabolic as well as a =at velocity pro-;le. [The parabolic pro;le case corresponds to fully de-veloped laminar =ow or developing =ow with very largeSchmidt and Prandtl numbers (Sc=Pr=∞) while the=at velocity pro;le case corresponds to developing =owwith Sc=Pr=0. These two limits give upper and lowerbounds on the transfer coe#cients for the case of de-veloping velocity pro;le with ;nite Schmidt and Prandtlnumbers.] We derive analytical expressions for the Sher-wood and Nusselt numbers in various regimes (short andlong distances from the inlet) and analyze how the trans-

0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0009-2509(01)00134-8

4772 N. Gupta, V. Balakotaiah / Chemical Engineering Science 56 (2001) 4771–4786

port coe#cients change with reaction and =ow parame-ters. For the case of an exothermic surface reaction, wecharacterize the behavior of these transport coe#cientsand show the parametric dependence of the transition be-tween various regimes. We use these results to developand analyze an accurate low-dimensional (two-phase)model of the monolith with position dependent heat andmass transfer coe#cients and determine the ignition andextinction loci as a function of various design and oper-ating variables.

2. Mathematical models

In this section, we present the mathematical modelsused to derive the transport coe#cients. We consider acylindrical tube on the surface of which a single ;rst-orderexothermic reaction occurs. We assume that the physicalproperties (such as the density, heat and mass di9usiv-ities) remain constant. We also assume azimuthal sym-metry (this assumption may not be valid in some casesas discussed in the last section). With these assumptions,the steady-state two-dimensional model in dimensionlessform is given by

f(�)@c@z=1P

(1�@@�

(�@c@�

))+1Pe@2c@z2

; (1)

f(�)@�@z=LefP

(1�@@�

(�@�@�

))+LefPe

@2�@z2

(2)

with boundary conditions:

1Pe@c@z=f(�)(c − 1); Lef

Pe@�@z=f(�)� at z=0; (3)

@c@z=0;

@�@z=0 at z=1; (4)

@c@�=0;

@�@�=0 at �=0; (5)

@c@�=− �2s

2R̂(c; �);

@�@�=�2s2

BLef

R̂(c; �) at �=1; (6)

where R̂(c; �) is the dimensionless reaction rate given by

R̂(c; �)= c exp(

�1 + �=�

): (7)

The function f(�) is used to represent the =ow conditionsinside the channel. For the case of =at velocity pro;lef(�)=1 whereas when we have fully developed laminar=ow inside the channel, f(�)=2(1 − �2): The variousdimensionless groups appearing in the above equation are

de;ned below:

z=xL; �=

rR; �= �

(T − T0T0

); c=

CC0;

Pe=uLDm

; �2s =2Rks(T0)Dm

; P=uR2

LDm;

�2 =PPe=(RL

)2; �=

ERgT0

; B= �(−MH)c0�fCpfT0

;

Lef=kf

�fCpfDm; Da=

�2sP=2ks(T0)LRu

: (8)

Here, (c; �) are the dimensionless concentration andtemperature of the =uid and z and � are the axial and radialcoordinates, respectively. The square of surface Thielemodulus, �2s , is the ratio of transverse(radial) di9usiontime to the reaction time (�2s is also referred to as theDamkNohler number for surface reaction). The axial Pecletnumber (Pe) is the ratio of axial di9usion to the convec-tion time and the transverse Peclet number P is the ratioof radial di9usion time to convection time (P= 1

2� Re Sc,where � is the aspect ratio, Re is the Reynolds number andSc is the Schmidt number). The DamkNohler number, Da,is the ratio of convection to reaction time. The parame-ter B is the dimensionless adiabatic temperature rise, � isthe dimensionless activation energy and the =uid Lewisnumber (Lef) is the ratio of heat to mass di9usivities.It should be noted that the above model includes

conduction=di9usion in both the axial and radial direc-tions and uses the more realistic Danckwerts boundaryconditions at the inlet (z=0) instead of the more com-monly used Dirichlet boundary conditions (c=1, �=0at z=0). The latter are based on the assumption that theconcentration and temperature gradients do not propa-gate upstream of the inlet (z=0). This assumption maynot be valid for fast reactions or on the ignited branches.The isothermal version of the above model is obtained

by dropping the energy balance equation (Eq. (2)) andsimplifying the boundary condition, Eq. (6) to

@c@�=− �2s

2c: (9)

We now discuss two limiting forms of the above modelfor the case of � → 0 (or Pe → ∞) and � → ∞ (orPe→ 0) which are simpler to understand. The ;rst case,which has been analyzed extensively in the prior litera-ture, corresponds to the case of an in;nitely long tube (theclassical Graetz problem with a surface reaction) whilethe second case (� → ∞) corresponds to a very shorttube. As we shall show later, these two simpler modelsbound the behavior of the more general model presentedabove for all values of the parameters. These models arealso very useful in determining the asymptotic behaviorof the heat and mass transfer coe#cients and bounds onthe Sherwood and Nusselt numbers.

N. Gupta, V. Balakotaiah / Chemical Engineering Science 56 (2001) 4771–4786 4773

2.1. The short tube model (�→ ∞)

When the characteristic time for longitudinal di9usionis much smaller compared to that for transverse di9usion,convection and reaction (� � 1; Pe�1); we can ignorethe axial gradients in the monolith and simplify the modelby integrating or averaging in the axial direction. Thisaxially lumped model is given by[1�dd�

(�dcd�

)]+ P[f(�)](1− c)=0; (10)

[1�dd�

(�d�d�

)]− PLef

[f(�)]�=0 (11)

with boundary conditions:

dcd�=0;

d�d�=0 at �=0; (12)

dcd�=− �2s

2R̂(c; �);

d�d�=�2s2

BLef

R̂(c; �) at �=1:

(13)

It should be pointed out that this model cannot be ob-tained as a limiting case of the general model if Dirich-let boundary conditions are used at the inlet (Hayes &Kolackzkowski, 1994). It can be shown that using theDirichlet boundary conditions leads to some inconsisten-cies when axial di9usion is included in the model.

2.2. Two-dimensional convection (in6nitely long tube)model (�→ 0)

This model was ;rst proposed by DamkNohler (1937,Chap. 2) and has been studied extensively in the priorliterature (Heck, Wei, & Katzer, 1976; Young & Fin-layson, 1976; Tronconi & Forzatti, 1992). It ignores theaxial conduction (di9usion) terms and uses Dirichletboundary conditions at the inlet and hence represents thelimiting case of � → 0. It is de;ned by the followingdi9erential-algebraic system of equations:

f(�)@c@"=1�@@�

(�@c@�

); (14)

f(�)@�@"=Lef�

@@�

(�@�@�

); (15)

c=1; �=0 at "=0; (16)

@c@�=0;

@�@�=0 at �=0; (17)

@c@�=− �2s

2R̂(c; �);

@�@�=�2s2

BLef

R̂(c; �) at �=1:

(18)

Without reaction or when Eq. (18) is replaced by a Dirich-let or Neumann boundary condition, this model reducesto the classical Graetz problem.

3. Sherwood number for a single isothermal reaction

In this section, we solve the various models for thecase of isothermal ;rst-order reaction and give analyticalexpressions for the Sherwood number for parabolic and=at velocity pro;les. We also analyze the asymptotic be-havior of the Sherwood number for each of these cases.The Sherwood number is de;ned by

Sh=2kcRDm

=−2@c=@�|�=1cm − cs ; (19)

where cm is the mixing-cup concentration given bycm=

∫ 10 2�f(�)c(�) d� and cs(= c(�=1; z)) is the sur-

face concentration.

3.1. The short tube model (�→ ∞)

For the case of isothermal ;rst-order reaction with =atvelocity pro;le a closed-form solution can be obtainedfor the model de;ned by Eqs. (9), (10) and (12) (Bal-akotaiah, Gupta, & West, 2000). Using this solution, weget the following expression for the Sherwood number:

Sh=1

I0(√P)

2√PI1(

√P)− 1=P

: (20)

Here, I0 and I1 are the modi;ed Bessel functions of orderzero and one, respectively. It should be noted that theSherwood number for this model is independent of theposition (due to axial averaging) and reaction parameter(Thiele modulus, �s) and depends only on the transversePeclet number, P. The two asymptotes of Eq. (20) aregiven by

Sh=

{2√P for P → ∞;

8 for P → 0(21)

with a transition around P=16.For the case of parabolic velocity pro;le, the solution

of Eqs. (9), (10) and (12) as well as the Sherwood numbermay be expressed as an in;nite sum involving the Graetzfunctions. It may be shown that

Sh(P)=

∑∞j=1

&2j �2j(

1 +&2jP

)

∑∞j=1

�2j(1 +

&2jP

)(1− &2j

�2s

) ; (22)

where

�j=∫ 1

0�f(�)!j(�) d� (23)


Fig. 1. Dependence of the Sherwood number on the transverse Pecletnumber for the short tube model.

and &2j and !j(�) are the eigenvalues and eigenfunctionsof the self-adjoint eigenvalue problem

1�f(�)

dd�

(�d!d�

)=− &2!; 0¡�¡ 1; (24)

!′(0)=0; (25)

!′(1) +�2s2!(1)=0; (26)

∫ 1

0�f(�)!i(�)!j(�) d�= *ij: (27)

It appears from Eq. (22) that the Sherwood number de-pends on �2s . However, after some rescaling (and usingthe Green’s function of the linear operator in Eq. (10)with Dirichlet boundary condition), it can be shown thatit is independent of the Thiele modulus (as in the caseof =at velocity pro;le). The asymptotic behavior of thefunction de;ned by Eq. (22) may be found more easilyby analyzing Eqs. (9), (10) and (12) for small and largevalues of P. (For large P, the solution can be expressed interms of Airy functions while for small P, a regular per-turbation expansion gives the solution.) It can be shownthat the asymptotes are given by

Sh=

25=331=3+(2=3)

+( 13 )P1=3 = 2:31P1=3 for P → ∞;

48=11 for P → 0(28)

with a transition around P ≈ 6:We plot in Fig. 1 the Sherwood number as a function of

P−1 for both =at and parabolic velocity cases. As we shallshow later, these curves de;ne the upper(lower) boundon the Sherwood or Nusselt number of the general model

(Eqs. (1)–(6)) at the exit(inlet). It should be pointed outthat Eq. (20) and the large P asymptote given by Eq. (28)are new analytical results.

3.2. Convection (long tube) model (�→ 0)

The Sherwood=Nusselt number dependence on theGraetz number, Gz (=4="=4P=z); predicted by the con-vection model has been discussed in detail in the heattransfer literature, especially for the case of laminar =owor Graetz problem (Carslaw & Jaeger, 1959; Levich,1962; Kays & Crawford, 1966; Knudsen & Katz, 1979;Shah & Bhatti, 1987, Chap. 3). Here, we present somenew intermediate asymptotes and give a summary of thevarious asymptotic regimes.For the case of an isothermal ;rst-order reaction with

=at velocity pro;le, the Sherwood number can be writtenexplicitly as

Sh(")=

∞∑j=1

�2se−&2j "(

&2j +�4s4

)∞∑j=1

e−&2j "(

&2j +�4s4

)(�2s&2j

− 1) ; (29)

where &j is jth root of the characteristic equation

2&jJ1(&j)=�2s J0(&j) (30)

and J0 and J1 are Bessel functions of order zero and one,respectively.Unlike the short tube model, the Sherwood number

for this model depends on the Thiele modulus (reactionparameter) as well as ": We now look at the asymptotesof Eq. (29) for the case of slow (�s�1) and fast (�s �1) reaction. Taking the limit �s → 0, Eq. (29) can besimpli;ed to the following form:

Sh(") =1∑∞

i=1 (1− e−&2i ")=&2i

=1

18 −

∑∞i=1 e

−&2i "=&2i

(J1(&i)=0; &i ¿ 0): (31)

The two asymptotes of Eq. (31) are given by

Sh=

√/"

for "→ 0;

8 for "→ ∞:(32)

Taking the limit �s → ∞, the expression for Sherwoodnumber given by Eq. (29) could be written in the follow-ing simpli;ed form :

Sh(")=∑∞

i=1 e−&2i "∑∞

i=1 e−&2i "=&2i

(J0(&i)=0) (33)

and the two asymptotes for this case are given by

Sh=

2√/"

for "→ 0;

5:783 for "→ ∞:(34)


For any large but ;nite value of �s, Eq. (29) exhibitsthree di9erent asymptotes de;ned by

Sh=

√/"

for 0¡"� 4�4s;

2√/"

for "� 4�4s; "�1;

5:783 for "→ ∞:

(35)

It should be emphasized that the Sherwood number in theentry region always approaches the value for constant =ux(√/=") and this behavior is independent of the reaction

parameter (�s). The Thiele modulus determines the zoneof validity of this constant surface =ux asymptote. Thus,for large �s, there are two asymptotes in the entry region.This fact has not been recognized in the prior literatureand is important in determining the ignition=extinctionbehavior of the convection model for the nonisothermalcase.The existence of two asymptotes in the entry region

and the transition between them can be seen more clearlyby examining the simpli;ed version of the convectionmodel for short distances. Analytical solution of the heatconduction problem in a semi-in;nite region with radia-tion boundary condition (Carslaw & Jaeger, 1959) maybe used to show that for "→ 0,

Sh=�2s exp (�

4s "=4) erfc(�

2s√"=2)

1− exp(�4s "=4) erfc(�2s√"=2)

; (36)

where erfc is the complementary error function. The twoasymptotes of Eq. (36) are

Sh=

√/"

for "�4s�4; "�1;

2√/"

for "�4s � 4; "�1(37)

with the transition between them occurring at "�4s ≈ 4:For "� 1, the expression given by Eq. (29) simpli;es

to

Sh=&21�

2s

�2s − &21; (38)

where &1 is the ;rst root of Eq. (30). The two asymptotesof Eq. (38) are Sh=8 (for �s → 0) and Sh=5:783 (for�s → ∞) with a transition around �s of unity.For the case of parabolic velocity pro;le, the Sherwood

number may be expressed as

Sh(")=

∞∑j=1

!j(1)2

&2je−&

2j "

∞∑j=1

!j(1)2

&4je−&

2j "

(1− &2j

�2s

) ; (39)

where !j(�) is de;ned by Eqs. (24)–(27). For �s → 0,Eq. (39) may be simpli;ed to (the Graetz problem with

constant =ux)

Sh(")=1

1148 −

∑∞j=1!j(1)2=&

2j e

−&2j "; (40)

where &j ¿ 0 and !j(�) are the eigenvalues and eigen-functions with Eq. (26) replaced by !′(1)=0 [A table ofvalues of &j and !j(1) is given by Shah & Bhatti, 1987].Similarly for �s → ∞, Eq. (39) simpli;es to (Graetzproblem with constant surface temperature)

Sh(")=

∑∞j=1!

′j(1)

2=&2j e−&2j "∑∞

j=1!′j(1)2=&4j e

−&2j "; (41)

where &j and !j(�) are the eigenvalues and eigenfunc-tions with Eq. (26) replaced by!(1)=0: The two asymp-totes of Eq. (40) are given by

Sh=

25=3+(2=3)32=3"1=3

=2:065"1=3

for "→ 0;

48=11 for "→ ∞:(42)

(The small " asymptote corresponds to the classical Lev-eque solution with a constant =ux and can be obtainedmore easily by similarity transform, see Slattery, 1999.)Similarly, the two asymptotes of Eq. (41) are given by

Sh=

25=3

+(4=3)32=3"1=3=1:709"1=3

for "→ 0;

3:656 for "→ ∞:(43)

(Again, the small " asymptote corresponds to the classi-cal Leveque solution and can be obtained by similaritytransform, see Levich, 1962.) Fig. 2 shows the boundson Sherwood number for this case and also an interme-diate case with �s=40: As in the =at velocity case, for

Fig. 2. Dependence of the Sherwood number on " for the convectionmodel with =at velocity pro;le.


any ;nite �s; Sh(") has three asymptotes given by

Sh=

25=3+(2=3)32=3"1=3

=2:065"1=3

for 0¡"� 8�6s;

25=3

+(4=3)32=3"1=3=1:709"1=3

for8�6s

�"�1;

3:656 for "→ ∞:

(44)

[Note: the transition between the two small " asymp-totes occurs when �2s =Sh(") is of order unity. Since Sh(")varies as "−1=3 for parabolic pro;le, the region of valid-ity of the constant =ux asymptote is smaller than thatof the =at velocity case for the same �s value. An ex-plicit expression for the Sherwood number similar to thatgiven by Eq. (36) is not available for the classical Lev-eque problem with the more general boundary conditiongiven by Eq. (9). Formal solutions can be written down interms of contour integrals as shown by Apelblat (1980)and Basic and Dudukovic (1991). However, these formalsolutions are not transparent and do not show the transi-tion between the constant =ux and constant concentrationasymptote at "=32(+(2=3))2=9�6s ≈ 8=�6s as given byEq. (44). Numerical solutions for the corresponding heattransfer problem have been presented by Shah and Bhatti(1987, Chap. 3) for various values of the Biot number.Pancharatnam and Homsy (1972) present a series solu-tion for the surface concentration and numerical resultsfor the local Sherwood number as a function of the Graetznumber. Again, these numerical solutions do not showthe transition between the two entry region asymptotes.]It should be mentioned here that for a given P, the Sher-wood curve as shown in Fig. 2 is valid from "=0 (inlet)to 1=P (exit).Thus, the convection model (�→ 0) predicts that Sher-

wood number depends on " as well as �s: The maxi-mum di9erence between the two extreme cases of largeand small Thiele modulus for parabolic velocity pro;leis about 20% for any value of ". For the case of =atvelocity pro;le the maximum di9erence is about 57%for "�1 and 38% for " � 1: Fig. 3 shows the exactbounds on the Sherwood number for this model for thecase of parabolic velocity pro;le and also shows a gen-eral case (;nite Thiele modulus). Figs. 2 and 3 show thatwhen �s � 1, the transition between the two entry re-gion asymptotes occurs at " values given by Eqs. (35)and (44), respectively.

3.3. General 2-D model

The full two-dimensional model for the case of singleisothermal reaction is given by Eqs. (1), (3)–(5) and (9).For the case of =at velocity pro;le analytical solution tothis model is obtained by taking the inner product witheigenfunctions of the transverse di9usion operator. Theresulting second-order equation is similar to the homo-

Fig. 3. Dependence of the Sherwood number on " for the convectionmodel with parabolic velocity pro;le.

geneous axial dispersion model and can be solved easily.The solution may be expressed as

c(z; �) =∞∑j=1

2J1(&j)J0(&j�)&j[J 20 (&j) + J

21 (&j)]

×[

02je02j+01jz − 01je01j+02jz02je02j − 01je01j − 01j02j=Pe(e02j − e01j)

];

(45)

where

01j=Pe2+

√Pe2

4+(PeP

)&2j ; (46)

02j=Pe2

−√Pe2

4+(PeP

)&2j (47)

and &j is jth root of the characteristic equation given byEq. (30). The mean and exit surface concentrations onproper substitution and reduction are given by

cm(z) =∞∑j=1

�4s&2j [�4s =4 + &2j ]

×[


];

(48)

cs(z) =∞∑j=1

�2s[�4s =4 + &2j ]

×[


]:

(49)


Using these expressions, the Sherwood number for thiscase can be written explicitly as

Sh(z; P; Pe; �2s)=

∞∑j=1

�2sfj(z)(&2j +

�4s4

)∞∑j=1

fj(z)(&2j +

�4s4

)(�2s&2j

− 1) ; (50)

where

fj(z)=02je02j+01jz − 01je01j+02jz

02je02j − 01je01j − 01j02jPe

(e02j − e01j): (51)

This form of the solution is not convenient for thenumerical computation of the Sherwood number whenPe¡ 1. For small Pe values a second form of the solutionmay be obtained by ;rst solving an eigenvalue problemfor the convection–di9usion operator in the axial direc-tion. This approach gives the solution in the form

c(�; z) =∞∑n=1

�n(z)ePe z=2

×[1− �2s I0(

√P0n�)

[�2s I0(√P0n) + 2

√P0nI1(

√P0n)]

];

(52)where

�n(z) =82n

0n(Pe2 + 4Pe+ 42n)(53)

×[cos(

√2nz) +

Pe2√2nsin(

√2nz)

]; (53)

0n=2nPe+Pe4

(54)

and 2n is the nth positive root (eigenvalue) of

cot√2n=

√2nPe

− Pe4√2n: (55)

The Sherwood number can be expressed as

Sh(z; P; Pe; �2s)=

∞∑n=1

�n(z)

√P0nI1(

√P0n)[

�2s2I0(√P0n) +

√P0nI1(

√P0n)

]

∞∑n=1

�n(z)2

I0(√P0n)− 2I1(

√P0n)√P0n[

�2s2I0(√P0n) +

√P0nI1(

√P0n)

]: (56)

We now look at the various asymptotes of Eqs. (50)and (56) for the case of slow (�s�1) and fast (�s �1) reaction. Taking the limit �s → 0, Eq. (50) can be

simpli;ed to the following form:

Sh(z; P; Pe) =1∑∞

i=1 [1− fi(z)]=&2i

=1

18 −

∑∞i=1 fi(z)=&

2i

(J1(&i)=0; &i ¿ 0):

(57)

While Eq. (57) is valid for all Pe, the sum convergesslowly for Pe→ 0: It is easily seen that Eq. (57) reducesto Eq. (31) in the limit Pe→ ∞: For Pe¡ 1; we take thelimit �s → 0 of Eq. (56) to get the following simpli;edversion:

Sh(z; P; Pe)

=∑∞

n=1 �n(z)∑∞n=1 �n(z)[I0(

√P0n)=2

√P0nI1(

√P0n)− 1=P0n]

:

(58)

For Pe → 0; this expression reduces to that of the shorttube model. Taking the limit �s → ∞, the expression forthe Sherwood number given by Eq. (50) could be writtenin the following simpli;ed form:

Sh(z; P; Pe)=∑∞

i=1 fi(z)∑∞i=1 fi(z)=&

2i

(J0(&i)=0): (59)

Similarly, for Pe¡ 1; the expression for Sherwood num-ber in the limit �s → ∞ is given by

Sh(z; P; Pe)

=2∑∞

n=1 �n(z)√P0nI1(

√P0n)=I0(

√P0n)∑∞

n=1 �n(z)[1− 2I1(√P0n)=

√P0nI0(

√P0n)]

: (60)

It can be easily shown that these expressions reduce tothe ones given by the convection model and the shorttube model in the limits of Pe→ ∞ and 0; respectively.

It follows from Eqs. (56), (58) and (60) that for Pe→0, Sh depends only on P while for Pe→ ∞, Sh dependsboth on " and �2s : For any ;nite Pe (or aspect ratio, �), Sh


depends on z; P; Pe and �2s :While the convection modelpredicts an in;nite Sherwood number at the inlet (z=0);the SM model gives a constant value (from inlet to exit)which depends only on the transverse Peclet number, P:Our analysis of Eqs. (50) and (56) shows that Sh(z) is;nite for any z and is bounded between the values pre-dicted by the convection and SM models. It has three(and not two) de;ning regimes as a function of position.Analysis of Eqs. (50) and (56) shows that the Sherwoodnumber (for any ;nite Pe) at the inlet (z=0) is ;nite andis independent of reaction parameters. It may be shownthat the asymptotic behavior near the inlet (for the caseof =at velocity pro;le) is given by

Sh(06 z¡Pe−1)=

2√PPe=

2RuDm

for Pe� 1;

ShSM for Pe�1;(61)

where ShSM is the Sherwood number given by the shortmonolith model (Eq. (20)). It can be inferred from Eq.(61) that for the case when Pe is small (¡ 1), the Sher-wood number from inlet to exit is nearly constant and isgiven by the SM model. Also, in this case (Pe¡ 1) theSherwood number is independent of the reaction param-eters. The other two asymptotes for the case of ;nite Pecan be summarized as follows:

Sh(Pe−1¡z¡ 1)=

{ShCon(z= "P) for Pe� 1;

ShSM for Pe�1;(62)

where ShCon is the Sherwood number given by the con-vection model and is dependent on " and the reactionparameters. The above results are obvious and need nofurther explanation. We show in Fig. 4, a schematic plotof Sherwood number as a function of position for thecase of ;nite Pe and also the two limits of Pe → 0 and∞. For any ;nite Pe, Sh(z) is bounded by two limit-ing curves corresponding to �2s =0 and∞ (Fig. 4b). ForPe→ 0, these two curves coincide. In addition, the Sher-wood number becomes independent of position (Fig. 4a).For Pe→ ∞ the two curves are farthest from each otherand the Sherwood number depends on z=P as well as �2s(Fig. 4c). Thus, for practical calculations it may be as-sumed that the mass transfer coe#cient is nearly constant(independent of position and �2s) if Pe�1. However, forPe � 1, we have to use the three asymptotic regimes asshown in Fig. 4b. [Remark : Eq. (61) states that the masstransfer coe#cient near the inlet is equal to the averagevelocity for Pe� 1.]For the case of parabolic velocity pro;le, an analytical

solution of the general model may be obtained in termsof the con=uent hypergeometric function (for the � de-pendence). We do not pursue it here since the results areanalogous to the =at velocity pro;le case and we havealready given expressions for the limiting models.

Fig. 4. Bounds on the Sherwood number given by the general modeland the two limiting models.

It should be noted that if the =ow is laminar but thevelocity pro;le is not developed at the entrance to themonolith, the entry region Sherwood number can varyin a much wider range (as the dependence changes from2(P Pe)1=2 to 2:31(P Pe)1=3. The ratio of these two limitsis 0:866( UuR=Dm)1=3 which is about 4 for typical operatingconditions). Since ignition=extinction may often occurat the inlet, this distinction is very important in manypractical situations. The above estimate shows that if the=ow pro;le is not known at the entrance to the monolith,the uncertainty in the Sherwood number can be in therange 100–400%.Since the isothermal problem (with linear kinetics) has

a unique solution, the Sherwood number is a continuousand unique function of ". This is no longer the case forthe nonisothermal case discussed in the next section.

4. Sherwood and Nusselt numbers for an exothermicreaction

For the case of an exothermic reaction, both the Sher-wood and Nusselt numbers are required in the two-phasemodels for a complete characterization of the system.There exist several studies in the literature to character-ize the behavior for the nonisothermal case, especiallyfor the convection model (Hegedus, 1975; Young & Fin-layson, 1976; Heck et al., 1976; Groppi, Belloli, Tron-


coni, & Forzatti, 1995; Hayes & Kolackzkowski, 1994).A lot of analysis has been done in the past to character-ize the behavior for the nonisothermal case. Some of theavailable results are misleading since they do not accountfor multiple solutions of the nonisothermal model. In thissection, we present a comprehensive analysis of the be-havior of the transport coe#cients in the nonisothermalcase and clarify some earlier results in the literature.We start this section by de;ning the Nusselt number,

Nu=2hRkf

=2@�=@�|�=1�s − �m ; (63)

where �s is the surface temperature and �m is the mixingcup temperature computed in the same manner as themixing cup concentration.

4.1. Short tube model (�→ ∞)

For the case of =at velocity pro;le, an analytical solu-tion may be given for the model de;ned by Eqs. (10)–(13). It may be shown that (Balakotaiah et al., 2000)

c(�)=1−(√

LefB

I1(√P=Lef)

I1(√P)

× �sI0(√P=Lef)

)I0(

√P�); (64)

�(�)=

(I0(√P=Lef�)

I0(√P=Lef)

)�s; (65)

where the surface temperature, �s= �(1) is given by thenonlinear algebraic equation

�2s (B− a1�s) exp(

�s1 + �s=�

)− a2�s=0 (66)

with

a1 =

√LefI1(

√P=Lef)

I0(√P=Lef)

I0(√P)

I1(√P); (67)

a2 =2√PLefI1(

√P=Lef)

I0(√P=Lef)

: (68)

The mean =uid phase conversion and temperature areexpressed as

xm=�sBa2P; �m=Bxm: (69)

Using Eqs. (64)–(69) we can determine the Nusselt andSherwood numbers explicitly. The Sherwood number forthe nonisothermal case is identical to the one given by the

Fig. 5. Schematic bifurcation diagram of surface temperature andSherwood (Nusselt) number as a function of transverse Peclet numberfor the short tube model (�s=10; B=5; Lef =0:2).

isothermal case. The Nusselt number can be expressed as

Nu=1

I0

(√PLef

)√

PLef

I1

(√PLef

) − PLef

: (70)

It can be seen that the expression for Nusselt number is thesame as that for the Sherwood number with P replaced by(P Le−1f ). The asymptotic behavior of Eq. (70) is givenby

Nu=

2

√PLef

for P → ∞;

8 for P → 0:

(71)

It should be noted that while the short tube model pre-dicts Sh and Nu to be continuous and unique functionsof P, there exist multiple solutions for Eq. (66). Weshow in Figs. 5 and 6 schematic bifurcation diagrams of�s versus P−1 as well as the variation of Nu with P−1

for B=5; Lef=0:2 and two values of �s. For �s=10,the bifurcation diagram of �s versus P−1 is S-shapedwith an ignition and extinction point (Fig. 6). Both thesepoints correspond to P values in the entry region (P−1

Lef ¡ 0:0625). For �s=0:1, the bifurcation diagramhas an isolated solution branch, in addition to a lowerS-shaped branch (Fig. 5). Here, the isola (and the twoextinction points) occur in the entry region while theignition and extinction points of the S-shaped curve arein the asymptotic regime (constant Nu value). Finally, itshould be noted that all the three solutions marked A–Cin Figs. 5 and 6 have the same Sh and Nu values. Also,the Sherwood and Nusselt numbers vary continuously


Fig. 6. Schematic bifurcation diagram of surface temperature andSherwood (Nusselt) number as a function of transverse Peclet numberfor the short tube model (�s=0:1; B=5; Lef =0:2).

(and do not jump) as P goes through ignition=extinctionvalues. As we shall see in the next section, both these ob-servations are no longer valid for the convection model(�→ 0 or Pe→ ∞) or any ;nite value of Pe (�¿ 0).The fact that Sherwood number remains unaltered with

inclusion of Arrhenius kinetics is explained by its inde-pendence on the reaction parameters. The nonisothermalreaction can be interpreted as isothermal reaction withexponential variation of Thiele modulus with tempera-ture. Since the Sherwood number is independent of Thielemodulus it should remain the same in either case. Thesame reasoning applies to the Nusselt number indepen-dence on �2s , B, Lef and �.The asymptotes for the parabolic velocity case are

given by

Nu=

{2:31( P

Lef)1=3 for P → ∞;

48=11 for P → 0:(72)

4.2. Convection model (�→ 0)

Several e9orts were made in the past to determine theheat and mass transfer coe#cients using the nonisother-mal convection model. To explain the di9erence betweenthe earlier approaches and ours, we begin this section witha well-known result from the literature. Fig. 7 shows anumerically computed bifurcation diagram of the surfacetemperature and Sherwood(Nusselt) number as a func-tion of " (=4=Gz, where Gz is the Graetz number) forthe nonisothermal (single reaction) convection model us-ing the initial value approach as has been done by severalauthors in the past. In these approaches, the convectionmodel was treated as an initial value problem. It was dis-cretized in the transverse direction and the resulting ordi-nary di9erential equations were integrated to determinethe variation of temperature, concentration or Sh as a

Fig. 7. Bifurcation diagram of surface temperature and Sherwood(Nusselt) number as a function of " for the convection model treatedas an initial value problem (�s=1; B=5; Lef =0:2). The maximumin Nu(Sh) is spurious and is due to Gibbs phenomenon.

function of axial position. It may be seen that in this plotthe Sherwood number goes through a sharp peak at someaxial position (corresponding to the ignition of surfacereaction) and then settle down to a di9erent asymptote.Similar results have also been shown for nonmonotonickinetics (Young & Finlayson, 1976; Heck et al., 1976).In this section we show that the solution determined bythis approach is only one of the many possible solutionsof the convection model. In addition, we show that the lo-cal maximum that appears in the Sh versus " plot shownin Fig. 7 is due to numerical inaccuracies associated withthe Gibbs phenomenon that occurs in the Fourier seriesrepresentation of a discontinuous function (see Arfken,1985). The actual �s versus " plot has a jump discontinu-ity (due to ignition of the surface) which is approximatedby a continuous pro;le in the initial value approach. Simi-larly, the true Sherwood number decreases monotonicallywith a discontinuous jump (drop) to a lower asymptoteand does not have a peak as shown in Fig. 7. We explainthe reasons for these jumps below.The nonisothermal case is fundamentally di9erent from

the isothermal case in the sense that the local heat bal-ance given by Eq. (6) can have multiple solutions at someaxial position depending on the surface temperature atthat position. As shown elsewhere (Gupta & Balakota-iah, 2001; Gupta, Balakotaiah, & West, 2001), the non-isothermal convection model can have an in;nite numberof discontinuous solution pro;les corresponding to igni-tion and extinction of the surface at di9erent axial po-sitions. All these pro;les cannot be obtained by treatingthe nonisothermal convection model as an initial valueproblem (IVP) as has been done in the past. Due to thenonlinear boundary conditions, the nonisothermal con-vection model is not an initial value problem but is actu-


ally an index in6nity di;erential-algebraic system andhence may have in6nite number of solutions. These so-lutions should be computed carefully (using a numericalmethod that can treat discontinuous pro;les e.g., integralequation approach) to get the right qualitative (and quan-titative) features. The ignition=extinction of the surfaceleads to a discontinuous jump in the values of �s and cs aswell as =uxes inside the channel. Our analysis shows thatat these ignition=extinction points the Sherwood and Nus-selt numbers also go through a discontinuous jump. Weelucidate this behavior here brie=y using schematic bifur-cation diagram and schematic plots of Sherwood=Nusseltnumber. A detailed bifurcation analysis of the convec-tion model can be found elsewhere (Balakotaiah et al.,2000; Gupta & Balakotaiah, 2001). We summarize heresome results that are useful in determining the depen-dence of Sh=Nu numbers on ". It can be shown that theignition=extinction point can lie either in the entry or theasymptotic regime depending on the value of the Thielemodulus. This transition Thiele modulus is of the orderunity (for f(�)=1) and may be expressed as

�∗s =2

√2e−1 exp(−2√Lef + 2Lef): (73)

When �s¡�∗s , ignition=extinction occurs in the asymp-

totic (large ") regime and the =uid and surface tempera-tures stay close to each other like in a homogeneous reac-tor. Both the =uid and solid (surface) temperatures showsimultaneous ignition to the adiabatic value (B) and thesurface temperature does not exceed the adiabatic temper-ature. However, when �s¿�∗

s , ignition=extinction oc-curs in the entry region (small " values) and the surfaceignites to a maximum possible temperature of B=(Lef)a

(where a=1=2 for =at velocity pro;le and a=2=3 forparabolic pro;le), while the =uid temperature rises slowlyto the adiabatic value. This ignition/extinction also leadsto a transition in the heat(mass) transfer coe#cient as weexplain next.We show in Fig. 8 a true schematic bifurcation dia-

gram of the convection model for Lef=1; B=10 and�s=5 and �=∞. If the convection model is treated as aninitial value problem, only one solution (correspondingto the latest possible ignition) represented by the curveOCHIDL is found. In addition, the discontinuous jumpfrom I to D is approximated as a continuous curve (whichalso leads to an inaccurate calculation of the local Sher-wood and Nusselt numbers due to the Gibbs phenomenonexplained earlier). The local heat balance equation hasmultiple solutions for " values between points C and I.The surface temperature can jump to a higher value atpoint C and the pro;le OCEF corresponds to the “earliestignited solution”. In between the late and early ignitionpro;les, there can be an in;nite number of solution pro-;les corresponding to ignition between points C and I.One such pro;le OCHJK is shown in Fig. 8. In Fig. 9,we show a schematic plot of Sherwood=Nusselt numbercorresponding to the bifurcation diagram shown in Fig. 8.

Fig. 8. A schematic bifurcation diagram of surface temperature versus" for the nonisothermal convection model.

Fig. 9. A schematic plot illustrating the variation of theSherwood=Nusselt number with " for the nonisothermal convectionmodel.

The upper and the lower curves are the one correspond-ing to �s → 0 and ∞, respectively. In the same ;gurewe also show how the Sherwood and Nusselt numbersjump from one asymptote to the other at the ignition andextinction points.It is clear from the above discussion that the Sher-

wood (Nusselt) number for the convection model is nota continuous function of ". Equivalently, the diagram ofSh versus " is also multivalued and discontinuous. Forthe case when the axial Peclet number (Pe) is ;nite, thebifurcation diagram of �s (z=1) versus Da (or 1=P) isqualitatively similar to those for the two limiting mod-els (Gupta & Balakotaiah, 2001). The main di9erence isthat while multiple solution pro;les may exist (for a ;xedPe, P, �2s ; B; Lef and �) each of them is a continuouspro;le in the axial coordinate (z or "). This is mainlydue to the inclusion of axial di9usion in the conservationequations. However, each solution pro;le gives rise to


a di9erent Sherwood(Nusselt) pro;le and all these pro-;les are bounded by the limiting curves de;ned earlier.To explain further, the Sherwood(Nusselt) number pro-;le for the ignited solution would be the one given bythe general model for the case of large equivalent Thielemodulus ((�2s)eq=�

2s e

�s , �s=B for the ignited solutionand " � 1). Thus, due to nonlinearity of the boundaryvalue problem which gives rise to multiple solutions, wealso get multiple solutions for the transfer coe#cients.Fortunately, for wall catalyzed reactions, this jump is nottoo large (typically 20–60%) and it is still possible todescribe the system behavior using a low dimensionalmodel.

5. Two-phase models with transfer coe"cients

In this section, we present a low dimensional(two-phase) model for catalytic monoliths which usesthe mass and heat transfer coe#cients discussed above.Most studies in the past were done using constant heatand mass transfer coe#cients thereby assuming a con-stant ;nite resistance between the solid and gas phaseregardless of the =ow conditions inside the channel.From our analysis we note that such an approximationcan grossly overestimate the transport resistances forthe case of high =ow rates (or near the inlet). We usethe general two-dimensional monolith model to derivean accurate two-phase model with transport resistancesrepresented by the Sherwood and Nusselt numbers.

5.1. Formulation of a general two-phase model

An equivalent two-phase (or two-mode) model can berigorously derived for the model described by Eqs. (1)–(6) using the local mass and heat transfer coe#cients be-tween the phases. To formulate this model we start withEqs. (1)–(6) and integrate them in the radial direction bytaking the mixing-cup averages, cm=

∫ 10 2�f(�)c(�) d�

and �m=∫ 10 2�f(�)�(�) d�. For the case of laminar =ow

we also need the area average concentration and temper-ature given by Uc=

∫ 10 2�c(�) d� and

U�=∫ 10 2��(�) d�. It

can be noticed that for =at velocity pro;le (f(�)=1),the mixing-cup and area average properties are the same(i.e., cm= Uc; �m= U�). Averaging Eqs. (1) and (2), we get

− 1P

∫ 1

02�[1�@@�

(�@c@�

)]d�

+∫ 1

02�[f(�)

dcdz

− 1Ped2cdz2

]d�=0; (74)

−LefP

∫ 1

02�[1�@@�

(�@�@�

)]d�

+∫ 1

02�[f(�)

d�dz

− LefPe

d2�dz2

]d�=0; (75)

which reduce to

− 1P

(2�@c@�

∣∣∣∣1

0

)+dcmdz

− 1Ped2 Ucdz2

= 0; (76)

−LefP

(2�@�@�

∣∣∣∣1

0

)+d�mdz

− LefPe

d2 U�dz2

= 0: (77)

There are two ways of substituting for =ux at the surfacein above equations. The direct approach is to use theboundary conditions given by Eqs. (5) and (6). Anotherway is to use the de;nition of Sherwood and Nusseltnumber which gives

1Ped2 Ucdz2

− dcmdz

− Sh(z)P

(cm − cs)=0; (78)

LefPe

d2 U�dz2

− d�mdz

+Nu(z)Lef

P(�s − �m)=0: (79)

[Here, we have replaced cs= c(1; z) and �s= �(1; z)for obvious reasons.] Note that the Sherwood andNusselt numbers in the above equation need to beexact for this two-phase model to give exact repre-sentation of the general two-dimensional model (i.e.,Sh= Sh(z; P; Pe; Lef; �2s ; B; �) and Nu=Nu(z; P; Pe;Lef; �2s ; B; �)):Also, we can use the de;nition of the Sherwood and

Nusselt numbers with the boundary conditions given byEqs. (78) and (79) to get the following algebraic equa-tions:

Sh(cm − cs)− �2s cs exp(

�s1 + �s=�

)=0; (80)

Nu(�s − �m)− BLef

�2s cs exp(

�s1 + �s=�

)=0: (81)

Similarly, we can average out the boundary conditionsgiven by Eqs. (3) and (4) to get the following BCs forthe two-phase model:

1Ped Ucdz= cm − 1; Lef

Ped U�dz= �m at z=0; (82)

dcmdz

=0;d�mdz

=0 at z=1: (83)

Eqs. (78)–(81) with the boundary conditions (82) and(83) de;ne the one-dimensional two-phase model rep-resentation of the general two-dimensional model. Forthe case of =at velocity pro;le, Uc= cm and the two-phasemodel described above is complete and exact, given theexact Sherwood and Nusselt numbers. When the velocitypro;le is parabolic, Uc is di9erent from cm but our analy-sis shows that for heterogeneous reactions Uc and cm stayclose to each other and hence Uc can be replaced by cmin the two-phase model. (This is no longer valid if thereis also a homogeneous reaction. In this case the trans-fer coe#cients are known to depend on the reaction pa-rameters. For example, for a fast reaction in the liquidphase, the gas–liquid mass transfer coe#cient is given bykc=

√kDm, see Danckwerts, 1970.)


To complete the formulation of the two-phase model,we need to give explicit formulas for the local Sher-wood and Nusselt numbers, i.e., Sh(z; P; Pe; Lef; �2s ; B; �)and Nu(z; P; Pe; Lef; �2s ; B; �): However, such explicitformulas require a complete solution of the fulltwo-dimensional model, which is not possible (exceptfor some special cases as seen earlier). Thus, we needto introduce further approximations to complete theformulation of the two-phase model.In most of the past work, asymptotic (constant) Sher-

wood and Nusselt numbers were used in the two-phasemodels. More recently, it has been proposed to use an em-pirical interpolation formula for �2s dependence (Brauer& Fetting, 1966; Groppi et al., 1995). Our analysis hereshows that use of constant values can grossly overesti-mate the transport resistance. In addition, the empiricallyproposed position and parameter dependent transport co-e#cients do not account for the multivaluedness or thediscontinuous jumps in the values of the heat and masstransfer coe#cients. For the position dependency of trans-port coe#cients, most researchers in the past have arguedthat the one for the fast reaction case should be the obvi-ous choice as it would give the least error in calculationof the transport limited (high conversion) branch.From our analysis of the convection model we note that

the transport coe#cients near the inlet always go to theconstant =ux asymptote (�s → 0) regardless of the reac-tion parameters. It is only after ignition that they jumpfrom the constant =ux asymptote to the constant surfaceconcentration asymptote (�s → ∞). Hence, if one is in-terested in predicting the ignition=extinction behavior ac-curately, the transport parameters corresponding to theconstant =ux case (or the upper bound) should be used.Also, it should be noted that using this single asymptoteis convenient and would always give all the qualitativefeatures correctly but might give some error in the nu-merical values of the state variables (temperature or con-version) on the ignited branch. However, using the fastreaction asymptotic values for the transport coe#cientsgives an erroneous ignition point and hence the entire ig-nited branch.Thus, for the practical case of Pe � 1, the best

two-phase model is obtained if we use

Sh(z)=

{2√PPe; 0¡z¡ 1

Pe ;

ShCon(z=P); 1Pe ¡z¡ 1;

(84)

where ShCon is the Sherwood number for the convec-tion model given by Eq. (31). Similar expressions maybe written for the Nu(z) and for parabolic pro;le case.(As stated earlier, for Pe¡ 1; which may be possible inlaboratory reactors or in gauze converters, constant val-ues for Sh and Nu given by the SM model may be used.Alternatively, if we want to decouple the transport ef-fects from kinetics, we should design laboratory reac-tors with Pe¡ 1. This can be realized by having a short

catalytic section sandwiched between a fore and aftersections.)

5.2. Ignition=extinction behavior of the two-phasemodel

We develop here approximate analytical expres-sions for the ignition=extinction points of the fulltwo-dimensional model using the equivalent two-phasemodel. We consider here the practical case of laminar=ow with Pe¿ 103 and assume �=∞ and Lef=1 (Themore general case is treated in Gupta & Balakotaiah,2001).With these assumptions, the equivalent two-phase

model takes the form

1Ped2�mdz2

− d�mdz

+Nu(z)P

(�s − �m)=0; (85)

Nu(z)(�s − �m)= �2s (B− �m)e�s[1 + �2se�s =Sh(z)]

; (86)

cm=1− �mB; (87)

cs=cm

[1 + �2se�s =Sh(z)](88)

with boundary conditions1Ped�mdz

− �m=0 at z=0; (89)

d�mdz

=0 at z=1 (90)

and

Nu(z)= Sh(z)=

2:31(PPe)1=3; 0¡z¡

0:71Pe

;

2:065(Pz)1=3;

0:71Pe

¡z¡ 0:1P;

4:364; z¿ 0:1P:(91)

We now consider various limiting cases of this model anddetermine the ignition and extinction loci with �2s as thebifurcation variable (Note that the number of parametersin the reduced model given by Eqs. (85)–(91) is thesame as that in the full two-dimensional model). First,we note that when the tube radius is very small (�2s →0; P → 0), the above model reduces to the homogeneousDanckwerts model

1Ped2�mdz2

− d�mdz

+Da(B− �m)e�m =0; (92)

1Ped�mdz

− �m=0 at z=0; (93)

d�mdz

=0 at z=1; (94)

where Da=�2s =P. It may be shown that (Balakota-iah, 1996) this model has multiple solutions when


1�Pe¡ (2=B)eB−2. The ignition and extinction loci ofthis homogeneous model may be expressed as

Daig ≈ 1B;(DaPe

)ex= (B− 1)e1−B: (95)

As �s increases, transverse gradients become signi;cant.It may be shown that when �s¡�∗

s ; ignition=extinctionoccurs in the region of high " (constant Sh and Nu values)and only if B exceeds a critical value (corresponding tothat at a hysteresis point) given by

B¿Bh; P=4:364(Bh − 1) exp(2− Bh): (96)

When Eq. (96) is satis;ed, the following expressions maybe derived for ignition=extinction point:

(�2s)ig ≈4:364e−1

B; (97)

(�2s)ex ≈ 4:364(B− 1)e1−B: (98)

When �s¿�∗s (fast reaction), the ignition=extinction

point moves from the fully developed to the entry re-gion. By analyzing the local heat balance equation (86),it may be shown that ignition=extinction occurs only ifB¿ 4. When this condition is satis;ed, the ignition pointmay be determined by ignoring reactant consumptionand rewriting Eq. (86) as

Nu(z)�s=�2sBe�s : (99)

Di9erentiating Eq. (99) w.r.t. �s and eliminating �s givesthe ignition point as

eB�2s ≈

2:31(PPe)1=3; 0¡z¡

0:71Pe

;

2:065(Pz)1=3;

0:71Pe

¡z¡ 0:1P:(100)

Similarly, the extinction locus may be obtained as

�2seB−1

B− 1 ≈

2:31(PPe)1=3; 0¡z¡

0:71Pe

;

2:065(Pz)1=3;

0:71Pe

¡z¡ 0:1P:(101)

As �2s increases, the ignition=extinction locus movesfrom the inlet to the entry region. For example, whenB¿ 4 and B�2s ¿ 0:85Pe1=3; ignition occurs at the inlet(z¡ 0:71=Pe).Thus, there are four regimes of ignition=extinction (ho-

mogeneous, fully developed regime, entry or developingregime and inlet regime) in a catalytic monolith. Substi-tution of the numerical values (or examination of Eqs.(95), (97) and (100)) shows that the ignition point movesfrom the homogeneous regime to the inlet regime as theinlet gas temperature (or channel radius) is increased.Numerical simulations reported by Oh (1993) con;rmthis (though this work uses the fully developed asymp-totic values for Sh and Nu and hence misses the correctfunctional dependence of ignition on various design andoperating parameters).

6. Conclusions and discussion

The main contribution of this work is the clari;cationof the asymptotic behavior of the local Sherwood andNusselt numbers for surface catalyzed reactions in mono-liths. For the commonly used Graetz model, we haveshown that the local heat and mass transfer coe#cientsare neither continuous nor unique functions of the ax-ial coordinate. In general, they depend on z (position),P (transverse Peclet number), Pe (axial Peclet number),Lef (the =uid Lewis number) as well as on the reactionparameters (�2s ; B and �) and jump from one asymptoteto another at ignition=extinction points. Only in the limitof the axial Peclet number going to zero, they become in-dependent of position and reaction parameters and dependonly on P and Lef. For any ;nite Pe¿ 1; the transportcoe#cients may fall in one of three asymptotic regimes.We have given explicit formulas for the transport coe#-cients in each of these regimes.It should be pointed out that the local Nusselt and

Sherwood numbers cannot be calculated accurately usingthe nonisothermal model and a numerical method basedon the initial value approach or discretization due to theGibbs phenomenon associated with discontinuous func-tions. Integral equation methods or other methods that arebetter suited should be used to calculate the discontinu-ous pro;les (Rosner, 2000).This work also shows that commonly used low-

dimensional models such as the one-dimensionaltwo-phase model cannot predict the qualitative behaviorof the full PDE model unless we use position (and Pe,P and Lef) dependent transport coe#cients. The mostaccurate two-phase model is obtained when we use Shand Nu values corresponding to the constant surface =uxmodel.The model used in this work neglects conduction in

the solid phase. Inclusion of the solid phase conductiondoes not change the behavior of Sh(z): However, it isknown that the Nusselt number is in=uenced by the wallthickness and conductivity (Trevino, Becerra, &Mendez,1997).We have analyzed the in=uence of wall conductionusing the convection model with a constant wall =ux. Forthis case the model with wall conduction is given by

f(�)@�@z=LefP1�@@�

(�@�@�

); 0¡z¡ 1; 0¡�¡ 1;

(102)

�=0 at z=0; (103)

@�@�=0 at �=0; (104)

@�@�(1; z)=

12+ &

d2�(1; z)dz2

at �=1; (105)

d�(1; z)dz

=0 at z=0; 1; (106)


Fig. 10. Variation of the Nusselt number with position when wallconduction is included.

where

&=twRL2

(kwkf

)(107)

and tw(kw) is the wall thickness(conductivity). Fig. 10shows the numerically computed Nusselt number forf(�)=1 and P=0:01: For &=0; Nu(z) is given by theGraetz pro;le with constant =ux boundary condition.However, for any small but ;nite &; Nu(z) approachesthe value given by the Graetz problem with constanttemperature boundary condition for z → 0 and the con-stant =ux asymptote for z → 1: The & value determinesthe transition point between these asymptotes. Thus, weconclude that inclusion of wall conduction does not alterthe main conclusions of this work.The model analyzed here neglects hydrodynamic en-

trance e9ects. To account for the e9ects, one has to solvethe Navier–Stokes equations along with the convectivedi9usion equation and the energy balance. Thus, the lo-cal Sherwood and Nusselt numbers depend on the as-pect ratio (�), the Reynolds number (Re), the Prandtl(Pr) and Schmidt (Sc) numbers (instead of the prod-ucts P= 1

2�Re Sc and P=Lef=12�Re Pr):However, as ex-

plained in the introduction, the two limiting cases (=at andparabolic velocity pro;les) give upper and lower boundson Sh and Nu:This work ignores the heat losses from the monolith

channel (e.g. by radiation). These can be included bychanging the wall boundary conditions. We expect theNusselt (and Sherwood) number to be still bounded by thetwo limits corresponding to the constant =ux and constantwall temperature, respectively.A major assumption in this work is that the tempera-

ture and concentration pro;les are azimuthally symmet-ric. Our recent work has shown that for the case of fastreactions (�2s � 1) azimuthal symmetry may not exist(when B exceeds some critical value and P � 1): Thus,the two-dimensional solutions predicted by Eqs. (1)–(7)

are not stable to azimuthal perturbations (in some rangeof parameter values) and we have to consider the fullthree-dimensional partial di9erential equation model (theconcept of a transfer coe#cient is not useful in the de-scription of such systems). Mathematically speaking, thefast reaction case corresponds to the continuous spectrumfor which we cannot average to obtain a low-dimensionalmodel (except for special inlet or initial conditions). Theinadequacy of the traditional transfer coe#cient conceptbecomes even more apparent when there is a homoge-neous reaction or more than one surface reaction. In thelatter case, the Sherwood number of any intermediatespecies can vary over a wide range and take a di9erentvalue on each solution branch.Finally, it should be pointed out that the main con-

clusions of this study extend to other laminar as well asturbulent boundary layers with exothermic surface cat-alyzed reactions. In all these cases, the nonlinear surfaceboundary condition makes the mathematical model to bean index in;nity di9erential-algebraic system (when ax-ial di9usion=conduction in the =ow is ignored). Hence,the local heat and mass transfer coe#cients depend onthe reaction parameters and are discontinuous functionsat the limit=bifurcation points and multivalued. In addi-tion, in the fast reaction regime, there may be a largenumber of three-dimensional solutions coexisting withthe two-dimensional solutions!

Notation

B adiabatic temperature risec dimensionless reactant concentrationCpf speci;c heat capacityDm molecular di9usivityGz Graetz numberh local heat transfer coe#cient(MH) heat of reactionI0 modi;ed Bessel function of order zeroI1 modi;ed Bessel function of order oneJ0 Bessel function of order zeroJ1 Bessel function of order onekc local mass transfer coe#cientkf thermal conductivity of the =uidks(To) surface reaction rate constant at inlet conditionskw wall thermal conductivityL monolith lengthLef =uid Lewis numberNu Nusselt numberP transverse Peclet numberPe axial Peclet numberR tube radiusRg universal gas constantR̂ dimensionless reaction rateSh Sherwood numbertw wall thicknessT absolute temperature


Uu average =uid velocityx conversionz dimensionless axial distance

Greek letters

� aspect ratio (=R=L)" dimensionless axial coordinate (= xDm=R2 Uu)� dimensionless radial distance� density� dimensionless temperature� dimensionless activation energy�s transverse Thiele modulus

Subscripts

m meano inlet conditionss surface (or solid phase)f =uid phase

Acknowledgements

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

References

Apelblat, A. (1980). Mass transfer with chemical reaction of the ;rstorder: Analytical solutions. Chemical Engineering Journal, 19, 19.

Arfken, G. (1985). Mathematical methods for physicists. New York:Academic Press.

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. ChemicalEngineering Science, 55, 5367–5383.

Basic, A., & Dudukovic, M. P. (1991). A series solution for masstransfer in laminar =ow with surface reaction. A.I.Ch.E. Journal,37, 1341–1353.

Brauer, H. W., & Fetting, F. (1966). Sto9transport bei wandreaktionin einlaufgebiet eines stromungsrohres. Chemical EngineeringTechnology, 38, 30–35.

Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids.New York: Oxford University Press.

DamkNohler, G. (1937). In=uence of di9usion, =uid =ow and heattransport on the yield in chemical reactors. Der Chemie-Ingeniur,3, 359–485.

Danckwerts, P. V. (1970). Gas–liquid reactions. New York:McGraw-Hill.

Gupta, N., & Balakotaiah, V. (2001). Bifurcation analysis ofheterogeneous combustion in a catalytic monolith. ChemicalEngineering Science, submitted.

Gupta, N., Balakotaiah, V., &West, D. H. (2001). Bifurcation analysisof a two-dimensional catalytic monolith reactor model. ChemicalEngineering Science, 56, 1435–1442.

Groppi, G., Belloli, A., Tronconi, E., & Forzatti, P. (1995). Acomparison of lumped and distributed models of monolithiccatalytic combustors. Chemical Engineering Science, 50, 2705–2715.

Hayes, R. E., & Kolackzkowski, S. T. (1994). Mass and heat transfere9ects in catalytic monolith reactors. Chemical EngineeringScience, 49, 3587–3599.

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

Heck, R. H., Wei, J., & Katzer, J. R. (1976). Mathematical modelingof monolithic catalysts. A.I.Ch.E. Journal, 22, 477–483.

Kays, W. M., & Crawford, M. E. (1966). Convective heat and masstransfer. New York: McGraw-Hill, Inc.

Knudsen, J. G., & Katz, D. L. (1979). Fluid dynamics and heattransfer. New York: R. E. Krieger Publishing Company.

Levich, V. G. (1962). Physicochemical hydrodynamics. EnglewoodCli9s, NJ: Prentice-Hall.

Oh, S. H. (1993). Catalytic converter modeling for automotiveemission control. In E. R. Becker, & C. J. Pereira (Eds.), Computeraided design of catalysts. pp. (259–295). New York: MarcellDekker Inc.

Pancharatnam, S., & Homsy, G. M. (1972). An asymptotic solutionfor the tubular =ow reactor with catalytic wall at high Pecletnumbers. Chemical Engineering Science, 27, 1337–1340.

Rosner, D. (2000). Transport processes in chemically reactingsystems. New York: Dover.

Shah, R. K., & Bhatti, M. S. (1987). Laminar convective heat transferin ducts. In S. Katac, R. K. Shah, & W. Aung (Eds.), Handbookof single-phase convective heat transfer. New York: Wiley.

Slattery, J. C. (1999). Advanced transport phenomena. Cambridge:Cambridge University Press.

Trevino, C., Becerra, G., & Mendez, F. (1997). The classicalproblem of convective heat transfer in laminar =ow over a thin;nite thickness plate with uniform temperature at lower surface.International Journal of Heat and Mass Transfer, 40, 3577–3580.

Tronconi, E., & Forzatti, P. (1992). Adequacy of lumped parametermodels for SCR reactors with monolith structure. A.I.Ch.E.Journal, 38, 201–210.

Young, L. C., & Finlayson, B. A. (1976). Mathematical models ofmonolith catalytic converter. A.I.Ch.E. Journal, 22, 331–353.

Heat and mass transfer coefficients in catalytic monoliths

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