278
Ind. Eng. Chem. Fundam. 1982, 21 278-284
Pruess, S. A., University of New Mexico Mathematics Department, discusdons at University of British Columbia, Vancouver. B.C., Canada, Aug 15-19. 1980. Serth, R. h.Int. J . Numer. Meth. Eng. 1975, 9 , 691. Sleicher, C. A. AIChE J . 1959, 5 , 145. Sleicher, C. A. AIChE J . 1960, 6 , 529 Wehner, J. F.; Wllheim, R. H. Chem. Eng. Sci. 1956, 6 , 89.
Whitaker, S. I n d . Eng. Chem. Fundam. 1980, 79, 210. Wiiburn, N. P. Ind. Eng. Chem. Fundam. 1964, 3 , 189.
Received for review June 9, 1981 Revised manuscript received April 21, 1982 Accepted April 30, 1982
Optimal Catalyst Activity Profiles in Pellets. 1. The Case of Negligible External Mass Transfer Resistance Masslmo Morbldelll and Albedo Servlda Istituto di Chimica Fisica, Elettrochimica e Metallurgia, Politecnico di Milano, 20 133 Milano, Italy
Arvlnd Varma Deparfment of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
The optimal catalyst activity distribution in an isothermal pellet, in which a bimolecular Langmuir-Hinshelwood reaction occurs under steady state conditions, is determined for the case of negligible external mass transfer resistance. I t is shown that the effectiveness factor is maximized by concentrating all the active catalyst in a precise location in the pellet. Analytic expressions for this location, as a function of all the Involved physicochemical parameters, are given for three pellet geometries: slab, cylinder, and sphere. Numerical computations are reported to analyze the case where the active catalyst is deposited in a step manner, centered around the optimal location. The effect of the step-width on catalyst effectiveness is investigated in this case. Finally, the sensitivity of the catalyst performance to the step location is studied in order to provide guidelines for practical applications of the results obtained.
1. Introduction It is increasingly being recognized that catalyst pellets in which the catalyst activity is not uniform throughout the pellet can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniformly distributed. This recognition has been both theoretical (cf. Kasaoka and Sakata, 1968; Minhas and Carberry, 1969; Shadman-Yazdi and Petersen, 1972; Corbett and Luss, 1974; Smith and Carberry, 1975; Smith, 1976; Villadsen, 1976; Becker and Wei, 1977; Nystrom, 1978; Ernst and Daugherty, 1978) and experimental (cf. Michalko, 1966; Summers and Hegedus, 1978; Hegedus et al., 1979). In the absence of poisoning and under isothermal conditions, it is evident that if the reaction rate increases with reactant concentration, then it is best to concentrate the active catalyst near the external surface of the pellet. If, however, for a range of reactant concentrations, the reaction rate decreases as the reactant concentration increases, then it would be intuitively proper to locate the active catalyst in a zone within the pellet interior. This latter abnormal reaction rate behavior is exhibited by bimolecular Langmuir-Hinshelwood kinetics when one reactant is strongly adsorbed on the active catalyst site; examples include CO and hydrocarbon oxidation over noble metal catalysts (cf. Voltz et al., 1973) as in automotive exhaust catalysis. Becker and Wei (1977) confirmed this intuitive result by numerically computing effectiveness of flat slab catalyst pellets for the case of CO oxidation over platinum. They showed that except in cases of severe diffusional limitations, pellet activity is increased by locating the active catalyst in an inner core of the pellet. 0196-4313/82/1021-0278$01.25/0
It is worth noting that with the use of suitable site-blocking agents, activity profiles of this type along with prescribed widths of the active zone can indeed be realized (cf. Michalko, 1966; Summers and Hegedus, 1978; Hegedus et al., 1979). The question that naturally arises is: given a set of diffusion and reaction parameters, and a fixed total amount of active catalyst, precisely how should the catalyst be distributed within the pellet, so that the pellet activity is maximized? We address this question in the context of bimolecular Langmuir-Hinshelwood kinetics, when the coreactant not strongly adsorbed is in excess (such as for CO or hydrocarbon oxidation over noble metals, in the presence of excess oxygen), for pellets of three different shapes. Part 1 deals with the case of negligible external mass transfer resistance, while the effect of external mass transfer is treated in part 2 (Morbidelli and Varma, 1982). 2. The Basic Equations For an isothermal Langmuir-Hinshelwood reaction occurring in a symmetric porous pellet with nonuniform distribution of active catalyst, the steady-state mass balance is
where k ( x ) is the local rate constant, along with boundary conditions (BCs) dC dX
=o;
0 1982 American Chemical Society
x = o
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 279
where the external mass transfer resistance has been neglected, and n = 0, 1 , 2 for the infinite slab, infinite cylinder, and sphere, respectively. The rate constant density function a(x), defined as the ratio between the local rate constant k ( x ) and its volume averaged value (Corbett and Luss, 1974; Wang and Varma, 1980) a(x) = k ( x ) / R (3) can be introduced in eq 1 leading to the following expression of the right-hand side
RC
(1
(4)
+ KC)2a(x)
By applying the volume average operator to eq 3, it follows by definition that
-
If the volume average rate constant k is held fixed, then the amount of active catalyst used with different pellet shapes (i.e., the value of n), and with different activity profiles (Le., the form of a ( x ) ) is the same. Introducing the following dimensionless quantities u = C/Cf; s = x/R; 42 = R R2/[De(l a)']; u = KCf; f ( u ) = (1 u)2u/(l UU)' (6)
+
+
+
eq 1 and 2 reduce to
L[ul = 42f(u)a(s)
(7)
du/ds = O ; s = 0
(84
u=l; s = l
(8b)
with the BCs
where
Dimensionless Concentmtion,~
Figure 1. Shape of the rate function f ( u ) = (1 + u)*u/(l+ various values of u.
for
The shape of the rate function f(u) depends on the parameter u and is shown in Figure l. In particular, f(u) has a maximum at
u, = 1/u (14) which can be derived from eq 6 imposing f'(u,) = 0, and noting that f"(u,) < 0. Being the dimensionless concentration, 0 I u I 1; it then follows that the value of u, is in the range of interest only for u 2 1. If u I 1, the dimensionless Langmuir-Hinshelwood kinetics reaches its maximum value in the range 0 Iu I 1at u = 1. This can be seen noticing that for u < 1, f'(u) > 0 for u c [0,1]. Summarizing, in the range 0 Iu I1 max f ( u ) = f(u,) = M (15) U
(9) Furthermore, eq 5 leads to 1 J1u(s)sn ds = n + l From the definition of the effectiveness factor, it follows that
where ; u>1 u, = l/u; M = (1+ u ) ~ / ~ ufor u, = 1; M = 1; for u I 1 From the expression of the objective function q given eq 12, it is then apparent that
q = (n
+ 1)J
1
1
f(u)a(s)snds I(n + 1 ) M l a(s)s" ds 0
(16) in which substituting eq 7 and 10 yields
which using eq 10 gives q I M
The optimization problem consists in the evaluation of the activity density function a(s), which maximizes the effectiveness factor q, under the constraint given by eq 10. Thus summarizing, we wish to obtain a(s) such that max rl Ms), 4 S ) l (13) ab)
where a(s) is subject to condition (lo), while u(s) can be evaluated, as a function of a(s), through eq 7 and 8. 3. Optimal Rate Constant Density Function The examined problem has the typical structure of a variational problem. However, as will be shown shortly, the employment of the classic variational method (cf. Schechter, 1967) is not successful. Therefore the solution will be pursued through an analytical analysis of the involved functions.
Thus whatever be the value of the rate constant density function a(s), the effectivenessfactor q can never be greater than M, using eq 15 this means q 5 (1+ u ) ~ / ~ ufor ; u>1 q I1; for u I1 (18) It is then apparent that if an expression of the function a(s) exists for which q = M , this expression will constitute the solution of the present optimization problem. This expression actually exists, and it is given by a(s) = 6(s - S)/(n 1)k (19)
+
where 6(s - S) is the Dirac 6 function and S is that value of s where the rate function f(u) reaches its maximum value; i.e., u(S) = u, where u, is given by eq 15. That this specific a(s) is indeed the optimal one can be proved by substituting eq 19 in eq 12 to give
280
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 = c .-..
where u, and M are both defined as a function of u in eq 15. The evaluation of B must be performed separately for u I 1 and u > 1. 3.1. u I 1. If u I1, then u, = 1, and recalling from eq 8b that u = 1 at s = 1, it is readily concluded that S = 1. From eq 15 and 20, 7 = 1 for this case. 3.2. u > 1. If u > 1then some more computations are necessary in order to evaluate S . Let us first consider the range 0 I s < S. Substituting eq 19 in eq 7 gives L [ u ]= 0 (21) which when integrated using the boundary condition 8a yields u1 = A (22) where A is the remaining integration constant. Similarly in the range s < s I1, eq 7 leads again to eq 21, which must now be integrated with the other boundary condition 8b. The resulting concentration profile is defined from the solution of the equation u&) = 1
+ BS's-" 1
ds;
s < s I1
(23)
The shape of u&) is then different for the three examined geometries, and in particular it can be represented as uz(s) = 1 + B*,(s) (24) where J / n ( ~= ) s - 1; for n = 0 = Ins; for n = 1 = 1 - (l/s); for n = 2 (25) The remaining unknowns A , B, and S can now be evaluated. From the continuity of the concentration profile, u1 = u2 at s = S; thus from eq 22 and 24 A = 1 B+,(s) = u($) (26) Furthermore, from the definition of S itself, which requires u, = u(S), it follows that u(S) = l / u , and then substituting this in eq 26 gives
+
Finally, from eq 15 and 20, the value of the effectiveness factor is 7 = (1 + u)2/4u (28) which substituted along with eq 24 in eq 12 leads to
Using eq 25, 27, and 29 it is then possible to evaluate for each geometry the corresponding value of s
.
c
e
.
u
.
0
0" 05u c "7
v)
E " -
s
//
02
-
04 06 Dimensionless Posifion,s
08
I
Figure 2. Dimensionless concentration profiles within the catalyst pellet with the optimal activity distribution.
From inspection of eq 30 it appears that while for the infinite cylinder (n = 1) and the sphere (n = 2), 0 IS I 1 for all values of 4 2 0 and u L 1, for the infinite slab (n = 0) the value of S can become negative. This situation occurs for low values of 4, and in particular for (31)
Naturally, a value of s < 0 is physically unrealistic. However, it is possible to prove in a straightforward manner that if the active catalyst is concentrated in the point sl,with s1 > S , the resulting concentration profile is characterized by u(sl) > u(s),and as a consequence also by v1 < q. From this it follows that if the value of S calculated through eq 30a is negative, then the optimum catalyst distribution is to concentrate all the active catalyst at the point S = 0. The resulting value of the effectiveness factor will in this case be smaller than M , defined by eq 15, but it is still the maximum obtainable for the given values of u and 4. Note that this result is consistent with the fact that also for the two other geometries, from eq 30b 0 at fixed u, S 0. and 30c it follows that as 4 It is worth pointing out that although the examined problem has the typical structure of a variational problem, the employment of the classic variational method cannot be successful. In fact the solution of the problem is a Dirac 6 function which does not belong to the function class considered by the variational method. This is a typical solution in which this method fails because the function which maximizes the given functional (12) does not also make it stationary. In conclusion, the expression of the rate constant density function a(s) which maximizes the effectiveness factor is the Dirac 6 function given by eq 19. In it appears the parameter .? which for u I1 is S = 1, while for u > 1 it is given as a function of and $I for each geometry by eq 30. The corresponding value of the effectiveness factor is (1 + ~ ) ~ / ( 4for u ) u > 1, and 1 for u I1, for all the geometries considered-except for the slab for sufficiently small 4 when u > 1. For illustration, the concentration profiles within the catalyst pellet for the three examined geometries are shown in Figure 2. Note that the value of the effectivenessfactor, given by the previously obtained optimum activity distribution, is actually larger than all those given by the various step-distributions used by Becker and Wei (1977).
-
-
4. Application of the Obtained Solution
From the analysis developed above, it follows that in order to maximize the effectiveness factor of a catalyst pellet, it is necessary to locate all the active catalyst exactly at a specific point S. Its position depends upon the reaction parameter u, and it is on the surface of the particle if u 5 1, while it moves toward the center if u > 1 depending
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
281
and the smallest value of S, for a given 4, is always given by the a = 3 curve. This particular behavior is due to the definition of the Thiele modulus that we have adopted in eq 6, in which the parameter a is present. If we now introduce a “clean” Thiele modulus 4o defined as 402 = kR2/D, so that 42 = and we substitute it in eq 30, we obtain 4:/(1
+
4(a - 1) ,q=1--
for n = 0
(324
402
for n = 2
-Thiele
Modulus, +o
Figure 3. Optimal location of the active catalyst, c as a function of the Thiele modulus for various values of the adso_rptionconstant parameter, u. -Thiele modulus defined as (a) @ = [ k R 2 / D , ] 1 / 2 / (+1 a); (b) $0 = [kR2/D,]’/2.
also on the Thiele modulus value. This result is consistent with the well-known fact that for fwst-order irreversible kinetics, a = 0, and the optimal choice is to locate all the active catalyst on the support surface, where the value of the reactant concentration is the highest. In order to investigate the influence of the parameter a on the optimal location S,let us consider for a fixed 4 value and for the spherical geometry the function as/&. From eq 30c it is readily derived that as/au > o for 3 < 0 < a
ai/aa < o
for 1 < 0 < 3
-
-
-
q502/[402
+ 12(a -1)]
(32~)
From these equations it readily follows that, for a fixed +o, increasing u values lead to decreasing s values, starting froms = 1at u = 1until.3-0 when u-+ a. Thisis shown in Figure 3b, where for the spherical pellet, the S vs. 42 curves for various a values have been plotted. In conclusion then, for a given value of the Thiele modulus 40, the optimal location of the active catalyst moves from the surface to the center of the pellet as the adsorption constant parameter a increases. This is p h y s i d y reasonable because increased u causes the maximum of the reaction rate to occur at smaller u (see Figure 1). Thus for a fixed 4o (i.e,, fixed relative intrinsic reaction and diffusion rates), these reduced u values are encountered at smaller S. Again for the slab case, negative values of s must be replaced by s = 0.
5. Step-Distribution of Active Catalyst From a practical point of view, it may not be possible (and from catalyst dispersion and sintering viewpoints, not desirable) to locate all the active catalyst at one specific location. We now examine the more realistic case in which the catalyst is present only in a narrow region centered about the previously determined optimal point S. In other words, we now consider a porous pellet with a distribution of the active catalyst, given by the following density distribution a(s) = 0; s < s1 or s = a;s1 s2 (33)
+
Since &/du = 0 at u = 3, it follows that a = 3 is a minimum for the function S = S(a). Furthermore, from eq 30c, it is also apparent that S = 1 at a = 1 and also S -,1for a a. Therefore it can be concluded that for a given 4 value, there always exist two values of u E [l,m], which give rise to the same optimal location S. Moreover for a fxed 4, the optimal location of the active catalyst moves from the surface (S = 1 at a = 1) to the interior of the pellet as a increases from 1 to 3 (d.?/aa < 0), and then for a greater than 3 it comes back to the surface (ds/da > 0), which is reached for a a (S -,1). It might be worth repeating, as discussed earlier, that S = 1 for all a 5 1. This same behavior is also exhibited by the infinite cylinder and the infinite slab. In the latter case, if a negative value of .? is obtained through eq 30a, it must be replaced by S = 0 as outlined in the previous section. In Figure 3a the values of S vs. 4 for various a are shown for the case of a spherical catalyst pellet. These curves, at a fixed u, have the expected shape. In particular for 4 0, S 0, while .? -, 1 for 4 -, a. As previously illustrated, each curve is characterized by two values of a, +
S=
where s1 = S - A, s2 = S A, and a is a constant which can be evaluated using condition (10) as follows cy
= l / [ s 2 n + l - Sln+l]
(34)
It can be noticed that when A -,0, this expression of a(s) becomes identical with the one obtained from the optimization problem given by eq 19. The concentration profile within the particle in this case can be evaluated through the solution of eq 7 and 8, in which the rate constant density function given by eq 33 can be introduced. The integration range 0 < s < 1, can then be divided into three regions: (a) 0 < s < sl, where a(s) = 0 and then the solution of eq 7 with the BC (8a) leads to u1 = uo
(35)
where uois an integration constant, which also represents the dimensionless concentration value at the symmetric axis; (b) s2 < s < 1,where again a(s) = 0 but the BC (8a) is replaced by (8b), and then the concentration profile is given by u3
=1
+ b~,(s)
(36)
282
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
-
0. I
IO
I Thiele Modulus, @
Modulus,+'
-Thiele
Figure 4. Effectiveness factor, q, vs. Thiele modulus, @2, for a slab pellet with a step-distribution of active catalyst, characterized by a half-thickness value of A = 0.01.
Figure 6. Effectiveness factor, q, vs. Thiele modulus, @2, for a slab pellet with a step-distribution of active catalyst, characterized by a half-thickness value of A = 0.001.
where # n ( ~ )depends on the geometry and is given by eq 25; (c) s1 < s < s2;in this region a(s) = a and eq 7 reduces to
with the BCs
I
sphere\,
du2 dul u2 = u1 and - = -; at s = s1 ds ds u2 = u3 and
du2 ds
dua ds
-= -;ats
= s2
(384 (38b)
Note that two of the four boundary conditions above reported are actually related to the two external zones,and in fact they will be used to define the two integration constants appearing in eq 35 and 36. Equations 38a and 38b can then be rewritten as follows
01 10-3
'
'"',.,'
'
u2 = 1
+ b#,(s)
du2 = 0; ds
du2 d$n and - = b-; ds ds
at s = s1 at s = s2
(394
.
'
I
I
'
10-1
0.5
I
4 I
n
/I
161..
(39b)
It is now necessary to solve the system of eq 37,39a, and 39b for the unknowns U2(S), b, and u,,. Unfortunately, an analytical solution cannot be obtained, and so a numerical procedure is necessary. Since the system under consideration can exhibit multiplicity, in order to determine all the possible solutions, it is convenient to fix the value of uoleaving the Thiele modulus 4 as an unknown. That is, for a given set of u, sl,s2, and uo,the complete problem is solved, giving in particular the corresponding value of 4. If this value is different from the desired one, then a new trial on uois selected. This procedure is iterated until the desired value of 4, within a tolerance of lo", is obtained. The solution of eq 37,39a, and 39b with respect to the unknowns u2(s),b, and 4, was obtained through the orthogonal collocation method (Villadsen and Michelsen, 1978). In applying this method to eq 37 with the boundary conditions (39a), with N collocation points, a nonlinear algebraic system of N equations results. This system can then be solved together with the remaining eq 39b, allowing the simultaneous evaluation of all the ( N + 2) unknowns, i.e., u2(s)a t the N collocation points, b and 4. For most of the calculations performed, eight collocation points were sufficient to provide the required accuracy with respect to 4. However, for large values of 4, due to the steep slope of the concentration profile, a larger number of collocation points was required.
"
+A Figure 6. Maximum values of the effectiveness factor, vu, for a stepdistribution of the active catalyst centered about the point 8, as a function of the half-thickness A. qmn is the optimum value given by eq 20. I
u2 = uo and
'
10-2
cylinder
---A
Figure 7. Values of the Thiele modulus @,, corresponding to the qu values shown in Figure 6. @- is the value corresponding to qm=, evaluated by eq 30.
Finally, once the solution is achieved, the effectiveness factor is readily evaluated using eq 12
which from eq 36 and 25 gives
v=-
(n + U b 42
(41)
In Figures 4 and 5, for a given set of A, > 1,and S, the values of the effectiveness factor as a function of 4 are reported. It can be seen that for decreasing values of A the value of qu, that is of the maximum of the r] vs. 4 curve,
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
283
n=O u ~20, A.0.025
-Thiele
Modulus, b2
Figure 9. Effectiveness factor, q , vs. Thiele modulus, 6,curves for a slab pellet with a step-distribution of active catalyst, centered about various positions S within the pellet.
0.1-
'
increases and approaches the limit value predicted by eq 15 and 20. This fact is more evident in Figures 6 and 7 where, for given a and S, the values of qu, and of the corresponding &, are reported as a function of A, for the three geometries examined. It is quite clear that as A 0, both qu and & approach the theoretical values predicted through eq 20 and 30. Moreover, it is interesting to note that also for values of A as large as -0.025, the value of q is virtually indistinguishable from the optimal one related to A = 0. This means that if the active catalyst is distributed in a region occupying as much as 5 % of the characteristic pellet dimension [half-thickness ( n = 0), radius ( n = 1, 2)], but centered at the specific point S, the effectiveness factor is in practice still maximized. Furthermore, it appears from Figure 6 that while the value of q depends on the geometry for particles with uniform catalyst distribution (A = 0.5) this is not so for the optimum q value, which is a function only of u. Therefore, the advantage which can be taken through a proper distribution of the active catalyst in the porous support is greater for the spherical geometry than for the others.
-
6. Sensitivity of Catalyst Performance to Step Location Let us now investigate the sensitivity of the value of the effectiveness factor to the active catalyst position if. This analysis appears to be of crucial importance due to the inevitable difficulties encountered in the location of the catalyst step, centered on a very precise point. In Figure 8 the values of q, for a slab with fixed A and a, as a function of the catalyst position s are reported, for different values of 4. For each 4 value, the optimal location Sopthas been calculated by eq 30a and is reported in the same figure. It can be observed that if the optimal value is
0 < sop,< 1 [which is satisfied when a > 1and, only for the slab, if the value of 4 respects also the condition (31)]then the system exhibits two completely different types of behavior de-
pending on the active catalyst location, in particular depending on whether s < sop,or s > s,,,. Specifically if the center of the active catalyst zone is located in s < so,,,then the actual effectiveness factor is lower than its maximum value. The extent of this difference is proportional to the difference (sop?S). However, if the location of the active catalyst is s > sopt,then the value of q undergoes a dramatic decrease. The reason for this is more clearly illustrated in Figure 4 ~ for different positions S of the 9. In it the ~ - curves active catalyst are shown. We again refer to a slab with fixed A and a. It is apparent that as S increases, the ,lr42 curve moves toward larger values of @. Now, for a given value of 42 and from eq 30a its corresponding optimal location Soopt,the optimal point on the ~ - curve 4 ~ is the maximum in these curves. Therefore, if a value S > Sopt is considered, the point of the new q-& curve, corresponding to the same value of 42,is in the lower branch, and then the value of q is drastically reduced. On the other hand, if a value S < Soptis chosen, then the new ~4~ curve admits, always for the same 4, a possible steady state on the higher branch for which the effectiveness factor q is only slightly lower than the optimal one. It is therefore important in practice to locate the catalyst in a position s which never exceeds the optimal value, while an error in the other direction (5 < would have less serious consequences. Finally, from Figure 9, it is also possible to note that the effect of A on the maximum value of q is actually affected by the value of S. In particular, as s 1 the loss in effectiveness factor due to the nonzero value of A is larger. It should be evident from the results presented herein that in a fixed-bed reactor where the reactant concentration changes along the tube length, the value of a will also change. The optimal catalyst activity distribution within the pellet will therefore vary with axial position in the reactor, to accomodate the changing a. Since it would not be practical to make pellets with more than one or two types of activity distributions, the question which naturally arises is: what is the activity profile within the pellets so that the reactor performance is optimized? This question has been answered for a plug-flow reactor and forms the content of part 3 of this series (Morbidelli et al., in preparation).
-
Acknowledgment
M.M. thanks the Italian Consiglio Nazionale delle Ricerche (Progetto Finalizzato Chimica Fine Secondaria) for financial support during the course of this study. A.V. is grateful to the Department of Energy for partial support of this work, as part of a joint research program between Ford Motor Company and the University of Notre Dame. We thank Professor C. Citrini of Politecnico di Milano for
Ind. Eng. Chem. Fundam. 1982, 21, 284-289
284
stimulating discussions concerning this work. Nomenclature
a = activity distribution function C = reactant concentration De = effective diffusion coefficient f =-function defined by eq 6 k , k = local and volume averaged reaction rate constant K = adsorption rate constant
M = parameter defined by eq 15 n = integer characteristic of pellet geometry; = 0 for infinite slab, = 1 for infinite cylinder, = 2 for sphere R = characteristic pellet dimension; half-thickness (n = 0), radius ( n = 1, 2) s = x/R s = dimensionless optimal location of the active catalyst, defined in eq 19 u = C/Cf V , = pellet volume x = distance from center of pellet Greek Letters A = half-thickness of the step-distribution (33) 6 = Dirac .. 6 function 7 = effectiveness factor 9, $o = Thiele modulus; 4 = 40/(l + a), +o = ( R 2 k / D , ) 1 / z u = KCP
qn = function defined by eq 25
Subscripts f = bulk value
max = value corresponding to the active catalyst optimum location
Literature Cited Becker, E. R.; Wei, J. J . Catal. 1977, 4 6 , 365. Corbett, W. E.,Jr.; Luss, D. Chem. Eng. Sci. 1974, 2 9 , 1473. Ernst, W. R.; Daugherty, D. J. A I C M J . 1978, 2 4 , 935. Hegedus, L. L.; Summers, J. C.; Schlatter, J. C.; Baron, K. J . Catal. 1979, 56, 321. Kasaoka, S.;Sakata, Y. J . Chem. Eng. Jpn. 1988, 1 , 138. Michalko, E. U S . Patent 3 259 589, 1966. Minhas, S.; Carberry, J. J. J . Cafal. I9B9. 14, 270. Morbiielli, M.; Varma, A. Ind. Eng. Chem. Fundam. 1982, following paper in this issue. Morbidelli, M.;Servida, A.; Varma, A., manuscript in preparation. Nystrom, M. Chem. Eng. Sei. 1978, 33, 379. Schechter, R. S . “The Variational Method in Engineering”; McGraw-Hill: New York, 1967. Shadman-Yazdi, F.; Petersen, E. E. Chem. Eng. Sci. 1972, 27, 227. Smith, T. G. Ind. Eng. Chem. Process D e s . D e v . 1976, 15, 388. Smnh, T. G.; Carberry. J. J. Can. J . Chem. Eng. 1975, 53, 347. Summers, J. C.; Hegedus, L. L. J . Catal. 1978, 5 1 , 185. Vllladsen, J. Chem. Eng. Sci. 1978, 31, 1212. Vllladsen, J.; Michelsen, M. “Solution of Dlfferentil Equation Modeis by Polynomial Approximation”: Prentice-Hall: Englewood Cliffs, NJ. 1978. Voitz, S. E.; Morgan, c . R.; Liederman, D.;Jacob, S. M. Ind. Eng. Chem. Prcd. Res. D e v . 1973. 12. 294. Wang, J. B.; Varma, A. Chem’.Eng. Sci. 1980, 35, 613.
Received for review July 13, 1981 Accepted March 25, 1982
Optimal Catalyst Activity Profiles in Pellets. 2. The Influence of External Mass Transfer Resistance Massimo Morbldelll‘ and Arvlnd Varma Department of Chemical Engineering, UniversHy of Notre Dame, Notre Dame, Indiana 46556
The problem of optimal activity distribution in an isothermal catalyst pellet with a Langmuir-Hinshelwood reaction, already studied in part 1, is now extended to cases involving finite resistance to external mass transfer. The same activity distribution, i.e., a Dirac 6 function, as in the case of negligible external mass transfer resistance, is also found optimal in this case. Analytic expressions for the optimal locations of the active catalyst as a function of the involved parameters (4, Bi, u), are given for three pellet geometries: slab (n = 0), cylinder (n = l), and sphere (n = 2). A general relationship, valid-for all geometries, between the normalized optimal location p = x/(Vp/Sx) and only one speclfic combination, ?! , of the normalized physicochemical parameters is also given.
1. Introduction
It is well known that improved catalyst performance can result by using a nonuniform catalyst activity distribution within a pellet. An important aspect of catalyst design is the determination, both theoretically and experimentally, of active catalyst distributions in the pellet support which improve either the conversion, or the selectivity, or if some poisoning occurs, the life of the catalyst itself. In part 1of this work, Morbidelli et al. (1982, preceding article in this issue) have determined the optimum activity distribution for an isothermal catalyst pellet in which a very common special case of bimolecular Langmuir-Hinshelwood kinetics, where one of the reactants is strongly adsorbed on the catalyst while the other one is in excess, takes place. In that analysis, external mass transfer re-
’On leave from Politecnico di Milano, Milano, Italy. 0 196-431 318211021-0284$01.25IO
sistance was assumed to be negligible. In this work, we examine the influence of external mass transfer limitations on the optimum activity distribution function, for three pellet geometries: infinite slab, infinite cylinder, and sphere. 2. Mathematical Formulation of the Problem For a symmetric porous support with a nonuniform distribution of active catalyst, the steady-state mass balance has the form (Morbidelli et al., 1982)
where n = 0, 1, 2 indicates the slab, cylindrical, and spherical geometry, respectively, and u ( x ) is the activity distribution function, defined as the ratio between the local rate constant, k ( x ) and its volume average value k. It must then satisfy the condition @ 1982 American Chemical Society
