Optimum activity distribution in a catalyst pellet for a complex reaction

OPTIMUM ACTIVITY DISTRIBUTION IN A CATALYST PELLET FOR A COMPLEX REACTION A. L. CUKIERMAN, M. A. LABORDE and N. 0. LEMCOFF* PINMATE, Departamento de Industrias, Ciudad Universitaria, 1428 Buenos Aires, Argentina (Received 2 August 1982; accepted 16 March 1983)

Abstract-An analysis of the influence of the non-uniform distribution of the active species in a catalyst pellet on the effectiveness factor and point yield for the van de Vusse reaction network is carried out. The effect of diierent parameters on the poiat yield for a single pellat and on the integral yield for a fixed bed reactor is studied, and the optimum activity distribution

is determined.

INTRODUCTION

A non-uniform activity distribution can improve the performance of porous catalysts, increasing the selectivity or the average rate of reaction as compared with a uniformly active catalyst having the same total amount of active material. A series of studies[l-131 has been published in the recent years in which the influence of the non-uniform distribution of the intrinsic catalytic activity of a pellet on its effectiveness factor, selectivity and/or deactivation is analyzed. In all of these studies a single pellet was considered, and few of them[2,3,5,8] dealt with a complex reaction. There appear to be no study of the performance of a catalytic reactor which operates with non-uniform activity pellets. In the present work, the intluence of the non-uniform activity on the effectiveness factor of a spherical pellet for a complex reaction is analyzed. Likewise, the maximum yield of the intermediate species at the exit of a fixed bed reactor is evaluated for the optimum distribution of catalytic activity in the pellet.

(3) with the following

boundary r=O

condition

C,=l

c,=o

(41

and where r = k,‘L/u, kZ1= k,D/k,D, k,l = 2k,°C,Jk,“. The normalized activity distribution function is given by

k=kO+”

(5)

where .$ is the dimensionless radius and a, the activity distribution parameter, indicating the steepness of the activity profile. In the present work ct will be considered positive, that is the activity increases towards the pellet surface. The material balances for components A and B within the spherical pellet are, in dimensionless form:

MATHSMATICAL FORMLJLATION

A fixed bed reactor containing spherical catalyst ticles with a non-uniform activity distribution is sidered. In addition, isothermal system, steady state, flow and negligible external mass transfer resistance assumed. The van de Vusse reaction network is sidered

parconplug arc con-

+~%,2(0+kd,0Z)=O

h,’ g

B

(6)

8 - h&#,) = 0

(7)

h, = Rd/(klo/D,+), h, = Rq(/(kzO/D& 0 = CA/C.& 4 = CL/Cl and CA and CL represent, respectively, the concentrations of A and R inside the pellet. The boundary conditions are: where

k3

A+A-D.

(8) for the reactor in terms of dimensionless variables are given by: The mass balance

equations

-2=

q,C_+(l+k,,C,)

(2)

*Author to whom correspondence should be addressed. Visiting Department of Chemical Engineering. University of

Professor,

Houston, TX 77004, U.S.A. 1977

[=l

e=1

The effectiveness

factors

$=C,/C*. are evaluated

(9) from

1978

A. L. 3 kz,CsICA

Q = h1’(1a+3 = l- k,,C,/C...

CUKIERMAN

y ( d5 1 <‘,

‘(0-k,,$~)5”+~d5 Io

(11)

The point yield is defined, at any point in the reactor, as Y, _

’

dC, _ 1- k,,CJCA

dCA

l+k,,C..,

ra

1)~

(12)

and the overall yield as C

(13)

Y=&

the concenwhere C,. and Ca, represent, respectively, trations of A and B at the reactor exit. Differential equations (6) and (7) together with boundary conditions (8) and (9) are solved by means of an orthogonal collocation scheme[l4]. Between six and nine collocation points were used, according to the value of Ihe Thiele modulus. To obtain the effectiveness factor, a Gauss quadrature was applied to eqns (10) and (11). The balance equations for the reactor were integrated numerically by means of a third-order Runge-Kutta method. In order to obtain the maximum yield, a Fibonacci search was used[lS].

RESULTS AND DISCUSSION

Effect

of the non-unijonn

activity

distribution

on the

yield

et al.

For values of a smaller than LY*,Ne is greater than iVA, while for (Ygreater than (Y*, the opposite is found. At the maximum it is verified that q, is greater than q.+ The actual value of OLat the maximum, a*, will depend on the rates of increase of ?A and qe with a. In certain conditions a minimum in the point yield is also found, but at values of (I lower than a*. Let’s analyze first the effect of different parameters on the point yield when the effectiveness factors are unity. It can be seen from eqn (12) that the point yield will increase as C,, k2, and k,, decrease. The effect of C, will depend on the ratio of the rates of reactions 1 and 3. When the second order reaction of disappearance of A is important, Yi decreases as CI increases, but when reaction 3 is not very important, Yti increases with C,. E&t of the system parameters on the ejectiveness tors, the maximum point yield and LX*

fac-

The influence of the parameters of the system on the optimum value of the activity distribution parameter a* was analyzed. However, the variation of u* with the value of the parameters depends on the relative increase rate of the effectiveness factors with (Y, namely on the relative values of N, and NA. As Ne increases, the value of OL*at which eqn (1.5) is verified also increases, while the opposite is expected as Na increases. Therefore, an analysis of the effect the parameters and a have on the effectiveness factor was carried out simultaneously. Since ala does not depend on C, it is obvious that a variation of C, will only affect 7B (Fig. 1). It can be seen that as C, increases, NB also increases and therefore the maximum point yield is expected to shift to higher values

From the analysis of eqn (12) it follows that the point yield Y, is related to the value for negligible diffusional resistance, Y,, through

I.0

I’

3

Yi -= Y;,

-.9s ?a

(14)

Therefore, an increase in the yield with respect to the case when chemical control exists can be expected when the effectiveness factor qB is greater than qa. When a =0 the effectiveness factors are those of a uniformly active catalyst pellet. As LY increases both effectiveness factors increase. Shadman-Yazdi and Petersen[8] found the opposite behavior, but they did not normalize the activity distribution function and therefore they compared pellets with different amounts of the active species. When the activity distribution parameter is very high the effectiveness factors tend to unity and the system behaves as if it were homogeneous. At intermediate values of (Ya maximum in the point yield can be expected. At that point

__--

70

__--0---__L__ ------------

h -2 --

a=4

--_ \

2

0

0.4 ‘? 0.3

NBl 01.= N&* where

(15) p.25

Fig. 1. Influence of Cg and a on the eff&iveness factors (CA = 1.O,k>, = OS, ksl= 1.0).

Optimum activity distribution in a catalyst pellet of 01 (Fig. 2). We can also observe that higher vahtes of Nn correspond to the region where the influence of CR on nB is greater. As C.., varies both effectiveness factors will vary but a greater effect on nA can be expected, specially at high Cn (Fig. 3). Therefore, the maximum point yield shifts towards lower values of a (Fig. 2). The effect of k,, on the maximum point yield is similar to that of C, and it will not affect n*. It can be seen that as k2, increases the maximum point yield is found at higher values of a (Fig. 4). The influence of k,, on nA is greater since q~ is only affected indirectly (Fig. 5). At low values of k,,, the ratio N.JNa is less than 1, but as k,, increases, it passes through a maximum. Therefore, 01* shows a minimum at intermediate values of kS, (Fig. 4).

1979

20 -

-

h,=5

---’

hl=Z

a* 15 -

IO -

_,B..i; 5-

O-

0.01

I 20 a=

0.05

1

0.5

I

I

---

-

15 -

\

ca=oo,/“’

h,‘2

h:

I :

I \ \ \

0 0.25

I h,=5

c, .I.0

IO -

5

I

2

5

The maximum yield shifts towards higher values of a as the Thiele modulus increases, because the effect on nR is greater and higher ratios NB/NA are obtained. We can also see in Fig. 6 that the influence of the Thiele modulus on the point yield depends on the value of (Y, but the point yield at OL*increases with h,.

:

L. ____lz=-C_-I_=I I 0.5 0.75

/I/

t H’ I 1.0

CA Fig. 2. Effect of Cn and Cg on the optimum activity distribution parameter (k2, = 0.5, kll = 1.O).

0.5 0.25

3 h3l Fig. 4. Influence of k~ and k,l on the optimum activity distribution parameter (CA = 1.0, CB = 0.01).

0.5

CB

0.1

% ---. % 0.5

t 0.75

I

CA

Fig. 3. Effectiveness factors 9~ and na (CB = 0.01, k21= OS, b, = 1.0).

Efect of the system parameters on the maximum integral yield It has already been mentioned that qA and nB increase with CL Therefore the concentration profiles in a fixed bed reactor become steeper and the exit concentration of A decreases while that of B increases. These have opposite effects on the integral yield and a maximum may be expected. The effect of k,,, k,,, h, and 7 on the optimum value of the activity distribution parameter a* was analyzed and it is shown for different operating conditions in Figs. 7-9. It can be seen that the optimum value of a increases with kZ1, as it was found for the point yield(Fig. 7). The opposite is observed for the maximum integral yield and the corresponding exit concentration of A, since both rates of disappearance of A and B are higher. The former, due to the increase in a*, and the latter, due to the increase in (Y*, and k2,. As k,, varies the optimum value of a shows a minimum (Fig. 8). A similar behavior was found for a* when the objective function was the point yield. If we analyze the maximum point yield, we can see that it is smaller as k,, increases, although the exit concentration of A does not vary greatly, specially at high h, or low k,,. This is due to the combined effect of k,, and CX*.AL low values of k,,, the amount of A consumed tends to increase with k,,, but the opposite is found for n*. Therefore, the net effect on C,, is almost negligible. On the other hand, the

A. L. CUK~ERMAN et al.

1980

rate of production of B decreases with [Yand so does the yield. At high values of k,,, (Y* increases with k,, and therefore lower values of C, are. found. The yield still decreases. although slower. since now the relative rate of production of B i higher.

At very low values of 7 the optimum integral yield is found, as it can be expected, at the same value of (x as the maximum point yield for Ca = 1.0 and Cg + 0. AS the space time varies, the value of a* show a maximum (Fig. 9). This behavior can be explained in terms of the effect of the concentrations of A and B on a* (Fig. 2). At high

I.()-

?

j0.:

0.2

I

0.5

I

I

I

5

I

IO k3l

Fig. 5. Effect

of kll and a on the effectiveness

factors (CA = 1.0, CB = 0.01, kZ1 = 0.5).

Fig. 6. Influence of the Thiele modulus on the point yield (C, = 1.0, Ce = 0.01, kzl = 0.5, k,, = 5).

Optimum activity distribution

1981

in a catalyst pellet

I5 0*

J 7

IO

5

0 0.9 y,c, 0.7 0.5

0.5

0.3

0.4

I

I_

0.5

-

I.0

____

b, Fig. 7. Effect of the rate constant ratio kzl on a*, integral yield and exit concentration of ACT= 0.5, kit = 1.0).

0

I

CAa y

Fig. 9. Effect of the space time on a*, integral yield and exit concentration of A (kn = 1.0).

I

?

0.6 -

0.2 -

"m-I

__-

=h4 y

Fig. 8. Influence of the rate constant ratio ksl on cP, integral yield and exit concentration

of A(7 = 0.5).

values of 7 both C, and C8 decrease, since also the inte~ed~ate product tends to disappear. These have opposite effects on CL.but the former seems to predominate and the value of a* decreases. The amounts of A consumed and R produced increase at low space time at an almost constant ratio and hence, no net effect on the yield is found. On the contrary, the influence on C,, is significant. At high values of 7, u* decreases and a lower net rate of production of B determines lower values of the yield. The optimum value of (Y increases with the Thiele modulus (Figs. 8 and Q), similarly to what it was observed in the analysis of the point yield. Both the integral yield and the exit concent~tion of A at the m~imum generalty increase with the Thieie modulus, ~though the opposite is observed when values for the same (u are compared, especiaily at low v&es of a.

reaction rate constants ratio, dimensionless length of reactor, cm din n/da, dimensionIess radial coordinate, cm pellet radius, cm fluid velocity, cm/s integral yield, dimensionless point yield, dimensionless dimensionless axial coordinate, dimensionless Greek

symbols

o

activity distribution parameter, dimensionless n effectiveness factor, dimensionless B concentration of A inside the pellet, dimensiontess 6 FIR, dimensionless 7 space time, s + concentration of B inside the pellet, dimensionless

CONCLL%iONS

The non-uniform activity distribution in a catalyst pellet for a van de Vusse reaction network may generate a maximum in both the point yield and the integral yield. The activity distribution parameter at the maximum point yield increases as CB, k,, and h, increase and as CA decreases. A minimum is found as k,, varies. The maximum point yield seems to increase with the Thiele modulus. For ditTerent sets of values of the reaction parameters and space time, optimum activity distribution parameters, namely optimum concentration profiles of the active species in the catalyst have been evaluated. The value of fy* corresponding to the maximum integrai yield increases as kzt and h, increase and, as k3, and 7 vary, it presents a minimum and a maximum, respectively, Acknowledgemenrs-The authors acknowledge the financial support of the Consejo National de Investigaciones Cientificas y Tecnicas, Argentina. N.O.L. is grateful to the Department of Chemical Engineering, University of Houston for use of the computing facilities and the preparation of the manuscript.

c = c”IC.4, C’ C” D

NOTATION

dimensionless concentration in the reactor, dimensionless concentration in the pellet, mol/cm3 concentration in the reactor, mol/cm’ effective diffusivity, cm’ls Thiele modulus, dimensionless average reaction rate oonstant, s-’ or cm3/mol s

s

SUbSCripts e 0

exit inlet

*

optimum value

Srrpersctipt

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