16
Ind. Eng. Chem. Fundam. 1985,2 4 , 16-27
Selectivity over Unifunctional Multicomponent Catalysts with Nonuniform Distribution of Components Dady B. Dadyburjor’ Department of Chemical Engineering & Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York 12 18 1
Expressions describing the activity and selectivity are derived for a unifunctional multicomponent catalyst consisting of a more active phase distributed in a less active but not inert matrix. Series and parallel reaction sets are considered separately. For each reaction set, various distributions of the more active phase are evaluated under changing condffions of catalyst size and coke level. “Constrained” optimal distributions of the more active phase in the less active matrix are obtained for every such case. Values of parameters used are consistent with hydrocarbon cracking over zeolite/silica-alumina catalyst.
Introduction A unifunctional multicomponent (UFMC) catalyst may be defined as one containing two or more components, all with qualitatively the same catalyst functionality. In other words, all the components catalyze the same reaction, but the rates of reaction are different. Catalytic selectivity, reactant and product diffusivities, and resistance to deactivation are also different for the individual components. Only two components are considered in the present work. Here a catalytically more active component is distributed in a matrix which, while less active than the first component, is not inert. An example of a UFMC catalyst is the typical zeolite/silica-alumina catalyst used for catalytic cracking. The zeolite component has a higher activity, better selectivity, lower diffusivity, and increased resistance to deactivation by coking. The composite appears to have improved qualities of activity, selectivity, and coke resistance, compared to either component used individually. The advantages of distributing an active catalyst uniformly over a catalytically inactive support have been discussed by Ruckenstein (1970) and Varghese and Wolf (1980). In general, the much higher diffusivities in the inert support (compared to the active component) and the small size of the active component distributed over the support lead to much higher effectiveness factors both for fresh and for deactivated catalysts. The improvement over the one-component catalyst is particularly striking if deactivation is of the pore-mouth type. A preliminary analysis of nonuniform distributions of an active component over a support was carried out by Becker and Wei (1977). For a fresh catalyst on an inert support, the active component was optimally placed at the surface of the support, whereas for a pore-mouth deactivated catalyst the center location was optimal. They concluded that, for a catalyst subject to deactivation, some intermediate placement would be optimal. Other nonuniform distribution profiles have been evaluated for activity by Minhas and Carberry (1969), Shadman-Yazdi and Petersen (1972), and Corbett and Luss (1974), among others. In all these cases, the support is assumed to be catalytically inert. The problem of defining an optimal distribution profile was raised independently by Morbidelli et al. (1982) and by the present author (Dadyburjor, 1982), but for two *Department of Chemical Engineering, P. 0. Box 6101, West Virginia University, Morgantown, WV 26506. 0196-4313/85/1024-0016$01.50/0
different physical situations. Morbidelli et al. (1982) considered the support to be catalytically inactive, in which case the optimum distribution is of the Dirac delta function type. In the situation considered by Dadyburjor (1982), the high-diffusivity support is not inert to the reaction catalyzed by the other component, i.e., the catalyst is of the UFMC type. In this case, some surprising results are obtained. The lowest overall reaction rate is obtained when the more active material is placed at the surface of the pellet, while distributing the more active material uniformly in the pellet leads to a higher overall reaction rate than placing the more active material at the center of the pellet. Further, an optimum distribution of the more active component in the less active matrix yields an overall reaction rate even higher than the small size asymptotic value. For a fresh catalyst, the optimum distribution is a monotonically decreasing function of distance from the pellet center. When diffusion limitations are increased, either by increasing the pellet size or by simulating the effect of coking, the optimum distribution contains a maximum value not a t the center of the pellet. The location of the maximum moves toward the outside surface with increasing diffusion limitations, up to a point. With a further increase in diffusional limitations, the maximum of the optimum distribution moves back toward the pellet center. Comparing these effects with those of Becker and Wei (19771, it is clear that if the diffusivities of the components are sufficiently different, even a relatively small catalytic activity in the support can lead to qualitatively different results. Changing the distribution of the more active component may lead to changes in the overall selectivity of the UFMC catalyst that are even more important and interesting than those observed in the overall activity. For the case of an inert support and no diffusion inside the active component entity, the effects of nonuniform distribution on selectivity have been considered by Corbett and Luss (1974) and by Ernst and Daugherty (1978). The case of the UFMC catalyst is considered in the present work. Although the physical situation is based, as before, on hydrocarbon cracking over zeolite/silica-alumina catalyst, no attempt will be made here to model this very complicated reaction network. Instead only the two cases of a general series reaction set and a general parallel reaction set are considered. However, intrinsic rate constants and diffusion coefficients used in these two cases will be assigned values consistent with those expected for reactants, intermediates, and products arising from hydrocarbon 1985 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
17
r---(cl
(bl
(d ) (el Figure 1. Schematic of distributions considered in spherical geometry: (a) pure annulus; (b) active surface; (c) active center; (d) dilute annulus; (e) uniform. Shaded regions denote more active (zeolite) component; unshaded areas represent less active (silica-alumina) matrix component.
cracking reactions. Assignment of these values is discussed in a later section. For each reaction set, the Analysis section obtains expressions for the selectivity expected for distributions of the UFMC catalyst illustrated in Figure 1. Note that the active-surface and active-center distributions contain a band of pure, more active material at the outer surface and a t the center, respectively. The pure annulus and dilute annulus distributions are similar to one another, except that in the pure annulus case the annulus contains only the more active component, while the dilute annulus contains a uniform mixture of active component and matrix. The uniform distribution is a special case of the dilute annulus. While it is possible to write the differential equations defining an optimum distribution function for maximum selectivity in each of the reaction sets considered, the evaluation of such a function in the general case appears impractical for either reaction set. Instead, a “constrained” optimum is obtained for each set by computing inner and outer values of the dilute annulus corresponding to the largest value of the selectivity.
Analysis For mathematical tractability, we replace the circular geometries of Figure 1with infinite flat plates. Then, as in the work of Dadyburjor (1982), all microspheres of the more active component a given distance from the center of the matrix sphere are replaced by a single stratum of more active component a related distance from the center of the matrix flat plate. (Such a transformation requires that the microspheres be much smaller than the matrix sphere, and therefore that the stratum thickness be much smaller than the thickness of the matrix plate. These conditions are met in zeolite/silica-alumina catalysts.) For the active center, active surface, and pure annulus distributions, the annulus of finite thickness containing only the more active component is replaced by two strata of finite thickness containing only the more active component at the appropriate distance from the centerline of the matrix plate. 1. Parallel Reaction Set. First consider the set of reactions given by A
h
B
>k
c
9
(la)
Henceforth B is the desired product. First-order reaction rates are assumed, with kB and kc the intrinsic rate constants for the formation of the two products. Then an intrinsic rate constant for A, also first order, can be written
kA
kB
+ kc
(1b)
Figure 2. Pure annulus distribution in flat plate geometry. 6 and y define the limits of the more active component (shaded region, coordinate xn). Inner and outer regions contain only matrix component (unshaded, coordinates xi and x , respectively).
For a flat plate, the overall rate of loss of A per unit volume of composite is
where k A is defined as the activity, DA represents the effective diffusivity of A, x is the thickness coordinate, and subscript C denotes a condition at the external surface of the composite. LC is the half-thickness of the composite flat plate. Similarly,the selectivity, the ratio of the overall rate of formation of B to the overall rate of loss of A, is given by r
We now evaluate rates and selectivities for the parallel reaction set over various distributions of the composite catalyst. a. Pure Annulus. The pure annulus distribution in flat-plate geometry is illustrated in Figure 2. The UFMC catalyst contains a volume fraction E of more active material, which takes up the region between /3Lc and yLc (i.e., y 0 + E ) on both sides of the centerline. Material balances for reactant A and product B in the three regions of Figure 2, the usual boundary conditions a t the center dAi _ - o = - a i a t xi= 0 (34 dxi dx i and at the outside surface A. = Ac a t xo = Lc
(3b)
Bo = Bc at no = Lc (3c) and matching conditions at the interfaces for the concentrations and fluxes of A and B yield the profiles of A and B. Then from eq 2 can be obtained the required expressions for the activity and selectivity for the parallel reaction set for the pure annulus distribution. The expressions are given in Table I, in terms of Thiele moduli +Am, and other nondimensional groups defined in Table 11. It is worth noting the two special cases of the pure annulus distribution. The “active surface” distribution is analogous to the “egg shell” of Becker and Wei (1977) and results when 0 1- e and when y 1 in the expressions of Table I. The “active center” distribution is analogous to the “egg yolk” of Becker and Wei (1977), and results when P = 0 and y 1 E in the expressions of Table I. The special cases will be considered along with the general case in the Results and Discussion section to follow. b. Dilute Annulus. The dilute annulus distribution of a UFMC catalyst in flat plate geometry is illustrated in Figure 3. The regions between the centerline and PLC, and those between yLc and the external surface, on either side of the centerline, contain only the matrix component.
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
18
Table I. Expressions for the Pure Annulus Distribution
Table 11. Parameters for Activities and Parallel Reactions Thiele Moduli @Ad
LC(hAd/DAd)”*
QAm
LC(hAm/DAm)“*
LC(~AZ/DA~)‘”
OAP ~
= V Lz(kAz/DAz)”z
A
@B” = L z ( h ~ z / D ~ z ) ’ ’ 2
More Active Component-Less Active Matrix Ratios $Ad
@AmD,, 3
~
GAdDAd
smaller of the length scales and is smaller than a differential element in the other length scale. The smaller length scale is used to obtain concentrationsand “overall” reaction rates in a typical stratum of more active material. This information is then used in the larger length scale to evaluate terms in the material balances for the annular region. While less approximate techniques are possible (see for example Neogi and Ruckenstein, 1980; Neogi, 1983), the present analysis has been used previously (Dadyburjor, 1982) and yields results adequate to our needs. Using the smaller length scale, consider first a single stratum of the more active component of half width L, (see Figure 3, inset). At the center of the stratum
-dAV- - 0 = - at x,= B
V
0 (54 dx, dx, At the stratum surface, the concentrations are set equal to those in the annulus a t that point, i.e.
A, = A,
at x , = L,
(5b)
B, = B,
at x , = L,
(5c)
This is reasonable, provided that L, uniform > active center. However, earlier work (Dadyburjor, 1982) indicates that for activities, the distributions should be ranked as: uniform > active center > active surface. Hence if we wish to optimize for the yield of the desired product (i.e., the product of the activity and the selectivity) an intermediate distribution of more active component will be required. In the present work we do not seek such an optimal distribution. Instead, we postulate that such a distribution can be approximated by a dilute annulus distribution, and
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Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
0 I1
I
-k
I
0 010
E2
1
1
I
0 100
0 150
1
I
0045
0 095
0 145
0 050
0 015
-ke, -
00
0 02
005
010
020
050
100
Lc
Figure 6. Yield for the parallel reaction set as a function of size Lc, cm. Other parameter values are as in Table V. Curves c, 8, and u are as in Figure 5. Dash-dotted curve m represents the largest yield from a dilute annulus distribution, i.e., the constrained maximum. Corresponding constrained optimal values of p and y change slightly with Lc. The inset shows the constrained optimal distribution, m, for Lc = 0.1 cm. The active surface distribution, s, is also shown for comparison.
we seek values of 0and y that maximize the value of the yield for this distribution. We term this the constrained optimum distribution. The results are shown in Figure 6. As expected from the rankings above, the yield corresponding to the uniform distribution is greater than that for the active center or active surface distributions. However, all these curves are superseded by that for the constrained optimum distributions, curve m. Note that we are not restricted to a single set of values for p and y throughout curve m, but whatever results in the largest yield for each value of Lc. However, the values chosen for p are only weakly dependent on Lc while the value for y is unchanged. The inset of Figure 6 shows the constrained optimum distribution corresponding to Lc = 0.1 cm Ca, = 0.89; ym = 1.00). By comparison with the active surface distribution, also shown, the constrained optimum can be considered to correspond to a “dilute active surface” distribution. Increasing Lc increases the values of 0, slightly (to 0.90 for Lc = 1.01 cm); i.e., the dilute region is moved closer to the external surface and made less dilute. In other words, for large diffusivity resistances, the improvement in the selectivity for surface distributions tends to overshadow the loss in activity. c. Effect of Coke. No general model exists at present that can satisfactorily quantify the effect of coke level on rate constants and diffusivities inside catalysts, although some specific efforts have been reported; see, for example, Butt (1972). However, as was pointed out by Dadyburjor (1982), it is reasonable to assume that upon coking the UFMC catalysts, the internal mass transfer (or intrinsic reaction) is more significantly affected in that phase, matrix or zeolite, where the diffusivity (or rate constant) has the larger value. Consequently, DAm and kAZwere decreased to simulate the effect of coking. To determine the effect of coking on the selectivity of the parallel reaction set, these two parameters are again decreased in the present work. Further, since kB, >> kcz, a decrease in kBz plays a more significant role, consistent with our premise
Figure 7. Selectivity for the parallel reaction set as some rate constant and diffusivity parameters are decreased to simulate coking. Other parameter values are as in Table V. Curves c, s, and u are as in Figure 5. D,m 10-2 IO1
--
- - -- - - 10--- -- - -10-1 il 0
0-3
t
I I
O 8‘150
0
too
Od50
0dso
kA r
Figure 8. Yield for the parallel reaction set. Coking is simulated as in Figure 7. Other parameter values are as in Table V. Curves c, s, m, and u are as in Figure 5. Also shown are values of (curve 0,) and y (curve 7,) corresponding to the constrained optimal distribution.
above. Consequently, following eq l b , the entire decrease in k h is attributable to a loss in kBz;Le., kcz is assumed unchanged. Finally, DBm is also assumed to decrease during coking, and its loss in magnitude is such that the ratio DAm/DBmis unchanged during the process. The changes in selectivity and yield under these conditions are shown in Figures 7 and 8, respectively. The results are not inconsistent with those of Figures 5 and 6, where the mass transfer resistances were changed by increasing Le. For selectivity, the active surface distribution remains the most favorable, and the active center case the least favorable, as the effect of “coking‘ increase in Figure 7. In contrast to Figure 5, however, the selectivity here decreases with increased “coking”. This is to be expected, since the model used assumes that while kBzdecreases, kcZ is unchanged. For relatively “uncoked” catalysts, the uniform distribution results in a greater value of the yield
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
1 0 .
’
’
’
’
I
t t
’
’
I
-3
I I
’
23
: /
0.5
L-----J 0 50
O-8.00
I
0.95
P Figure 9. Selectivity for the series reaction set when kBE > kBm. Parameters are as in Table V. Dashed line: pure annulus distributions, ,8 varied from 0 to 0.95. Pointa c and s correspond to active center and active surface distributions, respectively. Continuous lines: dilute annulus distributions for various values of 18 as a function of y. Point u represents a uniform distribution. Dashdotted line: largest value of selectivity for dilute annulus distribution as a function of 8. The corresponding y curve as a function of ,8 is a horizontal line at y = 1. Point m represents the Constrained optimum distribution and lies at the maximum value of the dashdotted line.
than does the active surface case in Figure 8. However, a t increasing levels of ”coking”, the yield from the active surface distribution is greater, an effect which is not observed in Figure 6, perhaps because the diffusional resistance is not increased sufficiently in the latter. As in Figure 6, constrained optimum distributions of the dilute annulus type are shown in Figure 8, one for every “coke” level. At low and intermediate levels of “coking”, the constrained optimum is a “dilute surface” distribution. Particularly a t the intermediate levels, much greater yields are found than those for the uniform distribution. As the *coke” level increases, the constrained optimum approaches the active surface distribution. Since the activity of the “coked” catalyst is low, the change in the activity by changing the distribution is insignificant, and the greater selectivity of the active surface distribution results in a greater yield for this type of distribution. 4. Series Network, k g , > kgmand kD, > kDm.As mentioned in subsection 2 above, two sets of values have been given in Table V for the rate constants of the desired reaction in the series reaction network. The set considered here is such that the rate of formation of the desired product is larger in the zeolite than in the matrix. Such a situation simulates the formation of hydrogen transfer products and their subsequent conversion to coke precursors and coke. a. Effect of Distribution. First consider the effects of changing distributions in the nominal catalyst described in Table V. The results are shown in Figure 9. For the pure annulus distributions, the selectivity is greatest for the active center case, and it decreases slightly as the annulus is moved toward the outside surface. For the parameter values used, the matrix is almost inert for the B, compared to the zeolite. Hence the reaction A conversion of reactant A is expected to be minimal for the active center distribution. The low conversion results in
-
002
005
010
020
050
100
Lc
Figure 10. Selectivity for the series reaction set ( k >~kB,)~ with increasing catalyst size, Lc, in cm. Other parameter values are as in Table V. Curves c, m, a, u represent distributions as in Figure 9. Curves 8, and ym describe the optimum location of the dilute annulus corresponding to curve m.
a high selectivity for this case, since the desired product B tends to react further, to form the undesired product D, a t a rate proportional to the concentration of B. The selectivity for dilute annulus distributions as functions of y for two values of /3 are also shown in Figure 9. In general these curves lie above that for the pure annulus distribution, presumably because of the greater ability of the desired product B to diffuse out of the smaller sized zeolite stratum in the dilute annulus case and thus be unable to react further in the zeolite. In the matrix, diffusion of B would be relatively rapid and the subsequent reaction to D would be much slower, compared to the situation in the zeolite. For perhaps the same reason, the selectivity increases with y and 8, i.e., as the annulus moves closer to the surface, at least for values of /3 that are not too large. As the value of 8 increases beyond a certain value, however, the selectivities decrease. This is most clearly seen in the dash-dotted line, which plots the largest selectivities for the dilute annulus distribution at different values of 8. The values of y corresponding to these values everywhere corresponds to 1.0; i.e., we have a “dilute active surface” distribution. For large /3 values, the decreasing selectivites are probably due to an increase in diffusion resistance as zeolite strata are forced closer together. The maximum of the dash-dotted curve, point m, corresponds to the constrained optimal distribution for maximum selectivity in the dilute annulus distribution. It is interesting to note that the distribution corresponding to point m in Figure 9 is almost identical with that in Figure 6, inset, corresponding to the maximum yield for a parallel reaction network with the same desired product. b. Effect of Catalyst Size. The effect of increasing Lc is shown in Figure 10. For all but the very large sizes, the uniform distribution results in a better selectivity than do the active center or active surface cases, and this is expected from the results of Figure 9. For the large sizes, the active surface distribution is the more favorable and the active center case is least favorable, perhaps due to the increased diffusion resistance through the zeolite. All three distributions are overshadowed by the constrained optimal case a t all values of the size, and the improvement increases with increasing size, at least in the range of sizes
24
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985 t
k
50
80 I
7,
_______-
Dz
10 I
-
-
------a,
_ / - -
'6,
Figure 11. Selectivity for the series set ( k g z > kg,) as some rate constant and diffusivity parameters are decreased to simulate coking. Other parameter values are as in Table V. Labels c, m, s, u, p,, ym are as in Figure 9.
considered here. The constrained optimal distribution moves marginally toward the outside surface with increasing size, but qualitatively retains its "dilute active surface" nature. c. Effect of Coking. The model used to simulate coking in this reaction set is consistent with that used in the parallel reaction set. Again the most significant effect is assumed to occur for the parameters with larger values. Hence kBz and itDz are decreased while the corresponding matrix rate constants are unchanged, and D b and DB, are decreased while the corresponding zeolite diffusivities are unchanged. As before, the decreases in the rate conY 050
0 05
t
stants are linear, from the corresponding fresh zeolite values to those corresponding to the matrix, while the diffusivities are decreased exponentially, from the corresponding fresh matrix values to those corresponding to the zeolite. The results are shown in Figure 11 to be qualitatively the same as in Figure 10. As before, at increasing levels of "coking", the constrained optimal distribution moves closer to the surface. A t very high levels of "coke", the constrained optimum actually coincides with the active surface distribution, which was not apparent in the range of resistances used in Figure 10. Further, as the rate constants in the zeolite decrease, the selectivity for the active surface distribution actually increases slightly and passes through a maximum before decreasing with a further increase in "coking". 5. Series Network, kB, > kBzand kD,> k b Under these conditions, the rate of formation of the desired product is greater in the matrix than in the zeolite, while the rate of the subsequent reaction to the undesired product is greater in the zeolite than in the matrix. Such a situation can be expected to be of limited practical importance and is treated here for the sake of completeness. It can be considered to correspond to the formation of 0-scission products during the catalytic cracking process, and their subsequent conversion to coke precursors and coke. a. Effect of Distribution. For the nominal sized, "fresh" UFMC catalyst, the selectivities for various pure annulus distributions are shown by the dashed line in Figure 12. Since in the zeolite the rate of formation of B is slow while the rate of its conversion to (undesired) D is relatively rapid, and since diffusivitiesthrough the zeolite are relatively small, it is clear that there will be a significant change in selectivity as the pure annulus is moved from the center (most selective) to the surface (least selective), and this is in fact shown to occur in Figure 12a. At a very low value of 0, the dilute annulus distributions give rise to lower selectivities than does the pure annulus distribution for the same value of 8, further indicating the importance of minimizing the reactions in the zeolite under
1
a
\ \
\ \ \
O
I
\
\
\
\ \
\ \
9 ' \ > 000L '
'
'
0 50
'
'
'
095
OOd"
*
'
'
'
0 50
'
'
'
09
P Figure 12. Selectivity for the series reaction set when k g , > k ~ Parameters ~ . are as in Table V. (a) Dashed line: pure annulus distribution with points c and s corresponding to active center and active surface distributions, respectively. Continuous lines: dilute annulus distributions for various values of @ as a function of y. Point u represents the uniform distribution. (b) Dashed line: largest value of selectivity for dilute annulus distribution as a function of @.The largest value, point m, represents the constrained optimum under these conditions. Continuous line: values of y corresponding to largest selectivity a t a given value of p.
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985
25
Figure 13. Selectivity for the series reaction set ( k > ~ke,)~with increasing catalyst size, Lc,in cm. Other parameters are as in Table V. (a) c, m, s, u represent distributions as in Figure 12. (b) Location of dilute annulus for constrained optimum distribution.
the present set of conditions. The dilute annulus selectivities are only slightly dependent on 8 and y, due to the small dimensions of the strata, but appear to increase generally with decreasing p. At low values of p, the selectivity decreases with increasing y, while the opposite behavior may be observed a t larger values of p. For the nominal case, therefore, the active center distribution appears to be the most favorable in terms of the selectivity. This can also be seen by comparing point c in Figure 12a with point m in Figure 12b. The largest selectivities for dilute annulus distributions in Figure 12b correspond to the pure annulus curve in Figure 12a at small values of p. Note also that y values corresponding to the largest selectivity for dilute annulus distributions in this region are equal to the sum of /3 and E . At large values of 8, the dilute annulus distribution tends toward a “dilute surface” distribution, Le., y = 1. b. Effect of Catalyst Size. As expected from the preceeding discussion, Figure 13a indicates that a t all values of Lc, the selectivities for this reaction set decrease in the order: active center > uniform > active surface. The constrained optimal distribution, curve m, however shows some unusual trends, and these are best seen in Figure 13b. Around Le = 0.1 cm, the constrained optimum distribution is in fact the active center, as could be expected from Figure 12. For much lower values of Le, a “dilute surfacen type distribution is preferred. Apparently the dilute annulus selectivity for relatively large values of p and y increases much more rapidly than that for the active center case, as Le decreases. Further, the overall rate of loss of A (not shown) as Le decreases is found to increase for both distributions, but to a marginally greater extent for the active center case. Consequently, the improvement in the selectivity for the dilute surface distributions (compared to the active center) is due to a greater relative decrease in the undesired (B D) reaction in the former distribution. This is perhaps due to a greater relative increase in the diffusion and reaction of B with decreasing mass transfer in the active center distribution. It should be noted that the actual relative improvement in the constrained optimal distribution (compared to the active center case) is slight at the low values of Le, and the uniform distribution selectivity is not far behind the other two. -+
When the values of Lc are increased above the nominal value, the active center distribution is superseded by a pure annulus distribution moving toward the surface; consequently, the latter type becomes the constrained optimal distribution. Increasing the mass transfer resistance decreases the overall rate of loss of A in the active surface distribution to a larger extent than in the active center. This can be seen in the increase in selectivity for the active surface distribution in Figure 13a, while the active center and uniform (dilute annulus) selectivities decrease drastically. This behavior results in the pure annulus selectivities being greater than those of the dilute annulus and passing through a maximum a t an intermediate value of p. The corresponding value of /3 increases with Le as the selectivity of the active surface distribution increases. c. Effect of Coking. The model for coking used here is consistent with those for the other reaction set: kBm and kD, are decreased while kg, and kD, are left unchanged; and the matrix diffusivities DAm and DBmare decreased while the zeolite diffusivities DAzand DBz are left unchanged. The effects, shown in Figure 14, are similar to those obtained when the resistance is increased by increasing Le in Figure 13. As in that case, for Lc greater than the nominal value, the constrained optimum distribution is of a pure annulus type, with the annulus moving from the center toward the surface as the level of “coke” increases. At relatively high “coke” levels, as matrix and zeolite parameters approach each other, all the selectivities approach one another. Hence the apparent change toward a dilute annulus distribution in that region is not significant.
Summary and Conclusions The overall activity and the selectivity for series reactions and parallel reactions have been investigated for unifunctional multicomponent (UFMC) catalysts containing a less diffuse, more active component distributed in a more diffuse, less active (but not inert) matrix. Two major types of distributions have been considered, the pure annulus case and the dilute annulus case. Further, the active surface and active center distributions, special cases of the pure annulus type, and the uniform distribution, a special case of the dilute annulus type, were explicitly considered. In addition, constrained optimum distribu-
26
Ind. Eng. Chem. Fundam., Vol. 24, No. 1, 1985 EO
zD'-
IO
50
1
I
= rate constants in dilute annulus: for A, eq 7c; for B in parallel reactions, eq 7d; for B in series reactions,
k A d ; k&.kDd
Table IV ~ =? rate constants averaged over a single zeolite stratum in dilute annulus for: A, eq 6b; B in parallel reactions, eq 6d; B in series reaction, Table IV k&d = rate constant for series product formation in dilute annulus, Table IV E,, = parameter in overall rate of series product formation in zeolite stratum of dilute annulus, Table IV L,; L, = half thickness of: composite, zeolite stratum PA; FB = volume-averagedreaction rate over UFMC catalysts for: A; B FA,"; FB," = reaction rates averaged over a single zeolite stratum in dilute annulus for: A, eq 6a; B in parallel reactions, eq 6c SB= selectivity for formation of B, eq 2b SB,pu,d; SB,pup = selectivity in parallel reaction set: dilute annulus, ?able 111; pure annulus, Table I SB,ser d; S B ser = selectivity in series reaction set: dilute annuhs, ?able 111; pure annulus, Table I ud,l.p; vd,l.* = parameters in selectivity for series reaction over dilute annulus distributions, Table IV Up,l4;V,,l, = parameters in selectivity for series reactions over pure annulus distributions, Table IV x = thickness coordinate &; = parameters in Table 11 for parallel reactions over: dilute annulus distributions; pure annulus distributions Yd; .Yp = parameters in Table Iv for series reactions over: dilute annulus distributions; pure annulus distributions Greek Letters @ = inner limit of annulus, Figure 2 6d; 6,; 6, = parameter in selectivity for series reactions, Table IV t = overall volume fraction of zeolite in UFMC catalyst t d = volume fraction of zeolite in dilute annulus region of UFMC catalyst, eq 4 y = outer limit of annulus, Figure 2 4Ad; 4+; 4 ~ 7 , Thiele modulus of A in: dilute annulus region; matrix region; pure annulus region; Table I1 4 ~ 4 ~ ~;4~~ ~ =; Thiele moduli in zeolite stratum of dilute annulus dlstribution for: A, Table II; B in parallel set, Table 11; B in series set, Table IV +Bd; hBm; q5Bp = Thiele modulus of B in: dilute annulus region; matrix region; pure annulus region; Table IV $Ad, $A, = ratio of matrix to zeolite properties for A in: dilute annulus; pure annulus; Table I1 JiW, qBp= ratio of matrix to zeolite properties for B in: dilute annulus; pure annulus; Table IV Jim = ratio of reactant to product properties in matrix, Table IV Subscripts a = annular region of UFMC pellet, Figure 2 C = external surface of pellet c = active center distribution d = dilute annulus distribution i = inner (matrix) region of UFMC pellet, Figure 2 m = matrix component o = outer (matrix) region of UFMC pellet, Figure 2 p = pure annulus distribution par = parallel reaction network, eq 1 s = active surface distribution ser = series reaction network, eq 8 u = uniform distribution v = smaller length scale coordinate used in zeolite stratum of dilute annulus distribution w = larger length scale coordinate used in dilute annulus of UFMC catalyst z = zeolite component Registry No. Silica, 7631-86-9; alumina, 1344-28-1. Literature Cited
k ~ " kBv; ; k
,
~
10-1
10-2
CDBw
lo-'
10-4
Figure 14. Selectivity for the series reaction set ( k > kB,) ~ ~as some rate constant and diffusivityparameters are decreased to simulate coking. Other parameter values are as in Table V. Curves c, m, s, u represent distributionsas in Figure 12. Curves 6, and Y~ define the location of the dilute annulus in its constrained optimal position.
tions were obtained in terms of the optimum location of the dilute annulus. The distributions correspond to the largest value of the selectivity or of the yield (activity X selectivity) for the parallel reaction set, or to the largest value of the selectivity for the series reaction set. (Optimum distributions for activity have been considered in Dadyburjor, 1982.) Constrained optimal distributions give rise to significantly larger values of the selectivity than those obtained for a uniform distribution. Using numerical values of parameters characteristic of gas-oil cracking over zeolite/silica-alumina catalysts, improvements are shown to range from 10% to 100% with respect to the uniform distribution selectivity. Under the present conditions, the constrained optimal distributions reduce to the active surface case for the selectivity of the hydrogen-transfer product in the parallel reaction set, and to the active center case for the selectivity of the &scission product in the series reaction set. For the selectivity of the hydrogen-transfer product in the series reaction set (where coke precursors are the product of further reaction), the so-called "dilute surface'! distribution represents the constrained optimum. Here the zeolite is placed in small particles near the external surface. For the yield of the same (hydrogen transfer) product in the parallel reaction set (where the p-scission product is formed in the simultaneous reaction), qualitatively the same dilute annulus distribution represents the constrained optimum. It appears then that a "dilute surface" distribution is a likely candidate for a catalyst to maximize the amount of hydrogen transfer products from in the f d gas-oil cracking reaction set. Work in this direction is currently underway. Nomenclature Did; D = diffusivity of compound I (= A$): in dilute annulus, eq ya,b; in catalyst component j (= m,z), Table V k ~k ,~ ,kc, . k~ = rate constants for: disappearance of A, form_ation_of B, formation of C, formation of D f ~k ~;, dk;A , , = volume-averaged activity of UFMC catalyst: eq 2a; Table 111; Table I
x,
Becker, E. R.; Wei, J. J . Catel. 1977, 4 6 , 372. Butt, J. B. I n "Chemical Reaction Engineering", Bischoff, K. B., Ed.; Adv. Chem. Ser. 1972, No. 109.
27
Ind. fng. Chem. Fundam. 1985, 2 4 , 27-32 Corbett. W. E., Jr.; Luss. D. Chem. f n g . Sci. 1974, 29, 1473. Dadyburjor, D. B. AIChE J . 1982, 28, 720. Ernst, W. R.; Daugherty, D. J. AIChE J . 1978, 24, 935. Mlnhas, S.; Carberry, J. J. J . Catal. 1969, 74, 270. MorbMelll. M.: Servlda. A,: Varma. A. Ind. €nu. - Chem. Fundam. 1982, 27, 278. Neogl, P. AIChE J . 1983, 28, 498. Neogl, P.; Ruckenstein, E. AIChE J . 1980, 26, 787. Ruckenstein, E. AIChE J . 1970. 76, 151.
Shadman-Yazdl, F.; Petersen, E. E. Chem. Eng. Sci. 1972, 27, 227. Thomas, C. L.; Barmby, D. S. J . Catal. 1968, 72, 341. Varghese, P.; Wolf, E. E. AIChE J . 1980, 26, 55. Welsz, P. B. CHEMTECH 1973, 498.
Received for review October 17, 1983 Revised manuscript received April 20, 1984 Accepted April 23, 1984
Generalized Statistical Model for the Prediction of Binary Adsorption Equilibria in Zeolites Douglas M. Ruthven' and Francis Wong+ Depatiment of Chemical Engineering, Universky of New Bruns wick. Fredericton, New Brunswick, Canada €38 5A3
A simple generalized statistical thermodynamic representation of adsorption equilibrium isotherms is proposed and used as a basis for the prediction of binary equilibria from single-component isotherm data. Experimental single-component and binary equilibrium isotherms for sorption of cyclohexane-n heptane in 13X molecular sieves are presented, and the model is shown to provide a good representation of these data as well as of other binary equilibrium data available from the literature.
The problem of predicting binary or multicomponent adsorption equilibria from single-component isotherms is of great practical significance and has therefore attracted considerable attention. Among the more successful methods proposed are the ideal adsorbed solution theory (Myers and Prausnitz, 1965; Glessner and Myers, 1969) and the more recent extension of this approach to nonideal systems (Costa et al., 1981), the simplified statistical thermodynamic model (Ruthven, 1971; Ruthven et al., (1973), and the vacancy solution theory (Suwanayuen and Danner, 1980). A comparison of the predictions of several of these theories was presented by Danner and Choi (1978) and by Kaul(l982). The ideal adsorbed solution theory and the vacancy solution theory are generai classical thermodynamic approaches which do not depend on detailed physical models for the adsorbed phase. This is a significant advantage since an analysis of single component equilibrium data for zeolitic adsorbents reveals that none of the simple model isotherms is universally applicable. This has led Barrer and Coughlan (1968) and Kiselev (1968) to resort to the semiempirical virial expression to correlate equilibrium data. The virial isotherm is, however, of limited value in relation to the problem of predicting multicomponent equilibria since there is no obvious way to derive the required virial coefficients for a mixture from the single-component data. A simplified statistical model which depends on approximating the expressions for the configuration integral by simplified expressions involving the Henry constant and the molecular volume was suggested by Ruthven (1971) and extended to binary systems by Ruthven et al. (1973). The main assumptions of this model are as follows. (i) The adsorbed molecules are considered as confined within a particular cage of the zeolite with only relatively infrequent exchanges between molecules in neighboring cages. (ii) 'Irving Oil Ltd., Saint John, N.B. 0196-4313/85/1024-0027$01.50/0
The molecules within any particular cage are considered as delocalized and freely mobile within the free volume of the cage. (iii) Sorbateaorbate attraction is neglected. (iv) Sorbate-sorbate repulsion is accounted for by a reduction in the free volume of the cage. These assumptions appear reasonable, as a first approximation, for systems involving sorption of nonpolar molecules such as saturated hydrocarbons. For such systems, the model has been shown to provide a good representation of the single-component isotherms and a good prediction of the binary isotherms from the singlecomponent parameters (Ruthven and Loughlin, 1972; Loughlin et al., 1974; Ruthven, 1976; Holborow and Loughlin, 1977; Singhal, 1978; Danner and Choi, 1978). However, the assumptions upon which this model is based are clearly inappropriate for localized sorption. It is therefore not surprising that the model has been found to provide a poor prediction of the binary equilibrium isotherms for systems containing COz and/or C2H4(Holborow and Loughlin, 1977) since with these sorbates a significant degree of localization is to be expected. A more general semiempirical approach which retains the statistical approach but avoids the need to introduce a specific model for the adsorbed phase is therefore suggested as a means of correlating single-component isotherms and predicting binary equilibria from single-component data.
Theoretical Model We consider the system to be divided into a number of equivalent subsystems with each subsystem being statistically representative of the macrosystem. For a zeolitic adsorbent the subsystem may conveniently be taken as an individual cage within the framework. The grand partition function ( E ) for the system may then be written in the form E = 1 + Z I U + 22u2 + ... + 2,um (1) where 2, represents the configuration integral for an in0 1985 American Chemical Society
