Selectivity and Deactivation of Diffusion-Limited Reactions in a Pore

gasket or a simple pore tree, using numerical or approximate solutions, respectively. The selectivity, in a system of a fast and slow simultaneous rea...
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Ind. Eng. Chem. Res. 1999, 38, 3261-3269

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Selectivity and Deactivation of Diffusion-Limited Reactions in a Pore-Fractal Catalyst Moshe Sheintuch Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, Israel 32000

One of the main results of studies into reaction and diffusion in a pore-fractal “catalyst” is the existence of an intermediate low-slope asymptote, in the plot of log(rate) vs log k, which separates the known asymptotes of kinetics- and diffusion-controlled rates. In that domain the fractal catalyst is more active than a catalyst of uniform pores having similar average properties. Here we consider the selectivity and deactivation processes in pore-fractals such as the Sierpinski gasket or a simple pore tree, using numerical or approximate solutions, respectively. The selectivity, in a system of a fast and slow simultaneous reactions, can be markedly different than that in a uniform-pore catalyst. Specifically, slow undesired reactions of higher order can be effectively suppressed in such a system. Three mechanisms of deactivation are considered and are characterized by curves of rate vs average activity: Pore-fractal catalysts are less sensitive to uniform deactivation or to deactivation of the inner pore generations, since in the intermediate domain the rate is only weakly dependent on k; such catalysts are more sensitive to deactivation of the outer pore generations. Introduction The problem of diffusion and first-order reaction on a fractal porous catalyst has been considered by several authors since its formulation by Sheintuch and Brandon (1989). One of the objectives of such a study is the possible optimization of catalyst activity, capitalizing on the observation that many natural and physiological systems that supply reactants to reactive sites, like blood vessels or natural trees, are built in a self-similar branching structure. Studies have shown that mass fractals differ in their behavior from pore fractals. One of the main results of reaction and diffusion in pore fractals is the existence of an intermediate low-slope asymptote, in the plot of log(rate) vs log k, which separates the known asymptotes of kinetics- and diffusion-controlled rates (with rate ∼ k or k0.5, respectively). This feature was obtained by exact (Giona et al., 1996), approximate (Sheintuch and Brandon, 1989), or numerical results (Mougin et al., 1996), using a first-order reaction and only one strictly hierarchial loopless pore tree, as well as in simulations (Gavrilov and Sheintuch, 1997) of diffusion and mth-order (m > 0) reaction in a two-dimensional fractal porous catalyst of a Sierpinski gasket type (Figure 1). In the intermediate domain the fractal-catalyst activity is higher than that of the porous catalyst of uniform pores having similar porosity and surface area. The purpose of this article is to consider two other important aspects of reaction engineering, those of selectivity and deactivation, in pore-fractal catalysts. The qualitative and quantitative differences between fractal and uniform-pore catalysts suggest that the concentration field is markedly different and consequently the reaction rates, as well as the selectivity due to competition between such reactions, are expected to be crucially dependent on the concentration field. We verify this suggestion by a numerical simulation and by an analytical approximation. In a system of fast low-

Figure 1. Gasket with one-dimensional pores, used as a model catalyst.

order and slow high-order reactions, the rate of the latter can be effectively suppressed. A catalyst may lose activity by poisoning, i.e., the diffusion of a poisonous component, by deactivation (e.g., due to coking) through an undesired product that is produced either by the reactant or by the product, by pore blockage, and by other mechanisms. Each of these mechanisms can be characterized and analyzed, but we will limit our analysis to simple cases that can be described by one or no additional parameters. Slower deactivation is expected in a pore fractal since in its intermediate domain the rate is only weakly dependent on k; similarly the pore fractal is going to be less

10.1021/ie990003o CCC: $18.00 © 1999 American Chemical Society Published on Web 08/18/1999

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sensitive to poisoning since the large pores may be poisoned while the small ones, which contribute most of the activity, are still active. We present numerical and analytical results: The selectivity was calculated for a two-dimensional gasket (Figure 1), whose cross-section is a Cantor set, made of narrow pores (aspect ratio of 0.1) with five generations of pores (see Gavrilov and Sheintuch (1997) for details); the intermediate domain was found to be wider with more generations, but the number of generations is limited due to computational difficulties. These calculations are compared with analytical results that were obtained in a loopless pore tree of simple hierarchy in which the main pore branches out into identical smaller pores, etc.; the reaction and diffusion problem was recently studied in such an object showing the existence of an intermediate asymptote in which the rate is independent of k or of the number of pore generations (Sheintuch, 1999). The relation between the two objects is discussed below. Deactivation results were calculated only for the simple pore tree. The gasket differs from the pore tree on three counts: it contains closed loops (i.e., all points can be reached and supplied from the surface by several paths); it is characterized by mixed hierarchies with pores of size ri bifurcating into smaller pores of size ri+1, ri+2, ..., rN; and the surface supplies the main pores as well as smaller and more frequent pores. To show why the results of the simple pore tree apply to the gasket note that, while the loops are important when diffusion resistance is small, under diffusion-control conditions the loops can be ignored as all pores are supplied through the shortest possible path. This is evident in the concentration field solved for diffusion and reaction in the Sierpinski gasket (Figure 1; Gavrilov and Sheintuch, 1997). The gasket can be viewed then as a series of scaled-down pore trees that are supplied by the surface, and the solution of the single large tree can be summed up for all its scaled-down images to derive the approximation for the gasket. At low k’s the gasket can be modeled as self-nesting catalytic squares, which is quite similar to a simple pore tree. The intermediate asymptote described above applies to pore fractals; studies of reaction and diffusion in mass fractals such as DLA (diffusion-limited aggregate) or CCA (cluster-cluster aggregate) do not exhibit this asymptote (Sheintuch and Brandon, 1989; Coppens and Froment, 1995; Elias-Kochav et al., 1991; Rosner and Tandon, 1994). The problem of diffusion toward a reactive corrugated fractal surface has been studied extensively. Several scaling laws have been derived and numerically verified for this problem (Meakin and Sapoval, 1991; Sapoval et al., 1993; Gutfraind and Sheintuch, 1992). While these surfaces, having wide two-dimensional pores, can be argued to represent porous catalysts, such structures cannot fold a large surface area unless one draws narrow and long (i.e., onedimensional) pores, and that is the approach taken here. Both structures were tested numerically by Gavrilov and Sheintuch (1997). This distinction holds for threedimensional structures: to fold a large surface area, we need to describe them by one-dimensional cylindrical pores rather then by three-dimensional caves, suggesting that the results derived here apply for threedimensional pore-fractal objects when the pores are long and narrow.

Previous studies of selectivity in a system with several reactions employed mass or surface fractals for their simulation: Meakin (1986) studied the selectivity of the following reaction system, which incorporated adsorption and uni- and bimolecular reactions, using Monte Carlo simulations and the DLA as the catalytic object: (i) A + s f As; (ii) As f B; and (iii) A + As f C. Coppens and Foment (1996), in their sudy of catalytic re-forming of naphtha, considered the selectivity problem during diffusion and reaction on a surface fractal. Several chemical engineering applications on fractal solids were reviewed by Giona (1997). Selectivity Here we study the effect of the fractal pore structure on the selectivity of a system that incorporates two parallel reactions of orders m and l:

A f P1

r1 ) -k1cm

A f P2

r2 ) -k2cl

(1)

To reduce the number of parameters, we assume also that the first reaction is much faster than the second, so that we can use the reaction-diffusion concentration field of the former to calculate the overall rate of the latter. This applies to reactions of very high (or very low) selectivities, which is the typical situation in many commercial systems. We will define, therefore, the selectivity relative to that of a kinetics-controlled system as

S≡

k1r2 〈cl〉 ) m k2r1 〈c 〉

(2)

where 〈.〉 denotes averaging over the catalytic object. Numerical Solutions. In a recent article (Gavrilov and Sheintuch, 1997) we calculated the concentration field of the Sierpinski gasket catalyst, subject to a fixed concentration at its boundaries, assuming that it catalyzes an mth-order reaction. The calculation was conducted in the following way: In a gasket with narrow pores the pore network can be described as a set of onedimensional pores of different sizes. A one-dimensional concentration gradient can be assumed in each pore and can be solved as a function of its end-point concentrations. The concentration values at the pore crossings can be determined by the flux balances for all crossings. Such a model allows one to solve the field for objects with a reasonable number of generations (four or five in the present case) and a reasonable number of pores per side. The problem is reduced then to solving a system of linear (m ) 1) or nonlinear algebraic equations. For the linear case we used the usual procedures for solving a large set of linear equations, while in the case of nonlinear kinetics (m ) 2 or 3; see below) we wrote the solution for each pore segment, after linearizing the rate expression around the average concentration in the pore, and iterated the procedure until convergence. For each concentration field (with m ) 1, 2, or 3) we calculated the selectivity according to eq 2 as follows: The concentration field in a pore segment of length L

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3263

and width r, with known end-point concentrations (say cA and cB), is

c(x) ) C1 cosh(φx/L) + C2 sinh(φx/L) - R (cB + R) - (cA + R) cosh φ ; sinh φ m-1 c* (3) R) m

C1 ) (cA + R); C2 )

where the Thiele modulus is (with k ) k1)

φ)

[

]

2km(c*)m-1L2 Dr

1/2

(4)

D is the molecular diffusivity and c* is the reference point for linearization. The various moments are

〈c〉 ≡

φ cosh φ - 1 + C2 -R ∫0Lc dx/L ) C1 sinh φ φ sinh2 φ + φ sinh(2φ) - 2φ (C12 + C22) 4φ

〈c2〉 ) -R2 - 2R〈c〉 + C12 + C1C2

〈c3〉 ) -R3 - 3R2〈c〉 -3R〈c2〉 + 3 sinh φ + sinh3 φ sinh3 φ + C1C22 + C13 3φ φ cosh3 φ - 1 -3 cosh φ + 2 + cosh3 φ C12C2 + C23 φ 3φ (5)

(

)

The selectivity was calculated from eq 2 after summingup the various moments (eq 5) over all segments of the pore structure. Figure 2 presents the selectivity dependence on k in a five-generation gasket and its comparison with a square “catalyst” with uniform pores having the same size, surface area, and total porosity. The following features are noted: (a) Both particles exhibit the expected asymptotes of S ) 1 at very slow rates and S ) (m + 1)/(2l - m + 1) at very fast reactions, which are typically limited to the pores branching from the surface; obviously S < 1 (>1) when the main reaction is of smaller (higher) order. This result can be easily derived from the classical reactiondiffusion problem. (b) Over a wide domain of parameters the two objects admit quite different selectivities with the ratio between the two values as high as 2; this intermediate domain of k’s overlaps the intermediate domain in the plot of rate dependence of k, in which the fractal catalyst is more reactive than the uniform-pore one. (c)Within this intermediate domain the fractal catalyst exhibits, when m < l, a higher selectivity at lower k’s and lower selectivity at higher k’s; the situation is reversed when m > l. (For a first-order main reaction, Figure 2a,b, the selectivity is higher in the domain approximately described by -12 < log k < -8 and is lower in the domain of -8 < log k < -2.). Recall that our selectivity definition (S) refers to the slow reaction, so that in most situations we are interested in suppressing S. Indeed, the side reaction is suppressed in the latter domain, and we show below that this suppression becomes more pronounced in catalysts with a larger number of pore-tree generations.

(d) The latter effects are more pronounced when the main reaction is of second- or third-order (m ) 2 or 3); in these cases the effects are more pronounced also with the five-generation gasket than with the four-generation one (not shown), while with linear kinetics the curves are quite similar. We try now to qualitatively account for these observations: The pore-fractal system should be superior to its uniform-pore counterpart: the narrow pores, in which most if the reaction is conducted, are characterized by a small effective diffusivity but also by a shorter length. They are supplied by larger and longer pores, and consequently the diffusion resistance in both levels is similar. Within the intermediate domain the N-generation catalyst amplifies the seletivity achieved in a single pore, as is quantitavely demonstrated below. Analytical Solution. To verify that the numerical results described above are general and to find out whether the selectivity can be even further enhanced or suppressed in objects of a larger number of generations, we derive an analytical solution and study it numerically. That calls for analysis of a relatively simple object, and to that end we choose a branching pore tree with a simple hierarchy, in which pores of a certain size branch-out only to the next smaller size pores. In a recent article (Sheintuch, 1999) we derived the exact or approximate solution (for a first- or mth-order reaction, respectively) in several catalytic geometries that resemble the gasket (Figure 1). We studied first the branching pore tree with simple hierarchy, and we recall here its main results before extending the analysis to competition of two reactions. The gasket differs from the pore tree on several counts, as it contained closed loops and mixed branching, as we discuss later along with the implications of these differences. We consider now a pore tree, in which the trunk bifurcates into smaller pores and the branching proceeds, through N + 1 generation, into smaller and smaller pores. The trunk of size r0 bifurcates into branches of size r1, which in turn split into branches of r2, and the branching process proceeds to smaller pores characterized by ri, Li, Di at level i, and with 2ni smaller pores branching (per unit length) from the ith pore. Assume now that at the large pores n0 ) A (a certain constant), and the smaller pores are scaled down versions with n1 ) A/δ and ni ) A/δi. Since the pores of size ri+1 are a scaled-down version of those of size ri, by a factor of δ, then ri ) r0δi , Li ) L0δi. We consider two diffusion regimes: in the case of molecular diffusion Di ) D0, while in the Knudsen diffusion regime the diffusivity is proportional to the pore-size, Di ) D0δi. To find the relevant parameters for the pore tree (A, δ) consider now a gasket (Figure 1) of size L0, which is composed of n2 self-similar scaled-down gaskets of size δ. Since there are n - 1 trunks of size r0 per side, we find

δ)

1 - (n - 1)r0/L0 n

(6)

The fractal dimension of the gasket is Ds ) -2 ln(n)/ln δ, and it approaches 2 as the pores becomes narrower. There are (n - 1) pores bifurcating from the trunk in the section of size δL0, and their density is A ) (n 1)/L0δ.

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Figure 2. Selectivity (S) dependence on rate coefficient (k) for reaction and molecular diffusion in a five-generation Sierpinski gasket (O) and in a uniform-pore catalyst (+) of similar size, area, and porosity for fast/slow reactions of order 1/2 (a), 1/3 (b), 2/1 (c), 2/3 (d), 3/1 (e), and 3/2 (f).

To portray the rate dependence on the rate constant, we employ the recursive procedure explained in Shei-

ntuch (1999): The rate in every pore level can be written in the following form

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RN ) cN-1[2kDNrN]1/2tghφN; φN ≡ [2kLN2/DNrN]1/2 RN-j ) cN-j-1[DN-jrN-j(2k +

φN-j ≡

2nN-jRN-j+1/cN-j)]1/2tghφN-j

[

]

(2k + 2nN-jRN-j+1/cN-j)LN-j2 DN-jrN-j

1/2

;

nN-j )

n-1 (7) L0δN-j+1

When all pores are diffusion-controlled and only the reaction contribution of the smallest pores is significant, then in a molecular-diffusion regime we find for the intermediate domain (Sheintuch, 1999; A ) (n - 1)/L0δ)

R ) cb[2D0r0A[2D0r0A[2D0r0A... [2kDNrN]1/2]1/2]1/2]1/2 ) cb (2D0r0A)(1/2) + (1/4) + (1/8) ... -(N+1)

(2k)2 R f cb(2D0r0A)

(8)

and a similar asymptote exists in a Knudsen diffusion regime. For a first-order reaction the rate is linear with cb, the bulk concentration, so we set all ci ) 1 without loss of generality. Equation 7 can be written in the dimensionless recursive form

[

ZN-j ) [Kδp(N-j) + ZN-j+1δ(p-1)(N-j)]1/2tgh (Kδp(N-j) + 2

4(n - 1)

ZN-j+1δ(p-1)(N-j))

δ2

δ2(N-j)(1-p)

]

1/2

(9)

after redefining the variable R and the parameter k in the dimensionless form

Ri ≡

2D0r0(n - 1) L02δ2 (10) Zi; K ≡ k L0δ 2D0r0(n - 1)2

Computation of the overall rate (Z0) from eq 9 (Sheintuch, 1999) exhibits the three asymptotes described above: The domain of the intermediate asymptotes (Z ) 1 in the new coordinates, see below) is wide and evident in the molecular diffusion regime, while in the Knudsen regime it is shorter. This domain becomes wider, as expected, in gaskets with more generations. To show that such an asymptote exists for eq 9, note that when all pores are diffusion-controlled the tgh(...) term approaches unity, and when the reaction rate in the (N - j) pore (the term multiplying K) is small in comparison with rates in smaller pores, eq 9 is reduced to ZN-j2 ) ZN-j+1 in the molecular diffusion case; hence, Z f 1 in the intermediate domain. For a detailed analysis, see Sheintuch (1999). Now the concentration profile in pore i, subject to a known ci+1 concentration at one end and sealed on the other, is

ci(x) ) ci+1

cosh(φi(1 - x/1)) cosh φi

(11)

for a first-order reaction. The space-averaged moments are

tghφi 〈ci〉 ) ci+1 ≡ ci+1ηi(φi); φi sinh(2φi) + 2φi 〈ci2〉 ) ci+12 ≡ ci+12ξi(φi); 2 4φi cosh φi 2 sinh(3φ i)/3 + 6 sinh φi ≡ ci+13ξi(φi) (12) 〈ci3〉 ) ci+13 8φi cosh3 φi Now, in the general case of a fast mth-order and a slow lth-order competing reactions the selectivity (as defined in eq 2) is

S)

L0(〈c0l + B(〈c1l + B(〈c2l + ... + B〈cNl〉)〉)〉)〉) L0(〈c0m + B(〈c1m + B(〈c2m + ... + B〈cNm〉)〉)〉)〉)

(13)

where B ) 2(n - 1) is the ratio of total surface area in two successive generations; the averaging has to be conducted from the smallest pores (with c ) cN) to successively higher pores while accounting for the contribution of all the connected smaller ones. For m ) 1 and l ) 2 or 3 this expression can be written as

S)

ξ0(1 + B...(1 + BξN-2(1 + BξN-1(1 + ΒξN)))...) η0(1 + B...(1 + BηN-2(1 + BηN-1(1 + BηN)))...) (14)

where ξi are the average concentrations defined in eq 12 for second- or third-order reactions. The selectivity in the competition between a fast firstorder and a slow second- or third-order reactions, computed for the branching pore tree (for an 8- or 10generation gasket with r0 ) 0.05 and n ) 4, 5 or 6, Figure 3), exhibits the features obtained numerically for the gasket: (a) S approaches unity for very small k’s and 1/2 or 1/3 for very large k’s. (b) In the intermediate domain S passes through a minimum of very small values; the minimal S is smaller with a larger number of generations (N), and it appears in both kinetic cases and with either diffusion regime. This may be an important result, as typically the slow reaction is the undesired one, suggesting that it can be suppressed to very small values in fractal objects. To explain the latter result, note that if all levels of the pore tree are diffusion-controlled, then in eq 14

ξN ) sηN; ξj ) sηj; s ) 1/l

(15)

where s is the asymptote for each pore level. In the limit of large B, where the small pores contribute most of the activity,

S f sN+1

(16)

Thus, the side reaction can be suppressed when N is sufficiently large and the desired reaction is of a higher order (m < l). The solution of a uniform-pore object is not plotted here, but typically it will show a monotonic behavior with k, as demonstrated in Figure 2. As explained in the Introduction the gasket differs from the pore tree as it contains closed loops, it is characterized by mixed hierarchies, and the surface supplies the main pores as well as smaller and more frequent pores. Under diffusion-controlled conditions the

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Figure 3. Selectvity dependence on K, computed for a simple loopless pore tree of N ()8 (a,b,e,f) or 6 (a,b)) generations and n ) 3 (*), 4 (+), or 5 (O), with molecular (e,f) or Knudsen (a-d) diffusion, a first-order main reaction, and a second- (a,c,e) or third-order (b,d,f) slow reaction.

loops can be ignored as all pores are supplied through the shortest possible path. The gasket can be viewed then as a series of scaled-down pore trees that are connected to the surface, but the summation over trees of different sizes, which admit the intermediate domain over different K values, leads to averaging, and the features of the gasket solution are not as evident as those of the pore tree. Thus, the approximate selectivities calculated for the simple pore tree (Figure 3) cannot be quantitavely compared with the numerical selectivities calculated for the gasket (Figure 4), but the similarities in behavior and in asymptotic values were pointed out above. Deactivation: A vast literature dealing with deactivation and poisoning processes (e.g., Froment and Bischoff (1986)) uses a wide array of mechanisms; the outcome of applications of these mechanisms to a pore fractal will depend on the nature of the process leading to loss of activity. Here

we employ the simplest possible mechanism in order to consider when and whether the pore-fractal catalyst is superior to the uniform-pore one. Three mechanisms of activity distributions are considered. The simplest model of a slow deactivation assumes a uniform poisoning process and relates the overall degree of deactivation in a reaction-diffusion process (ψ ) rate/ initial rate) to the degree of change in the rate coefficient (a ) 1 - k(t)/kinitial), using steady-state solutions as the deactivation process is typically very slow. In a uniformpore object ψ ) (1 - a) when diffusion resistance is absent, while ψ ) (1 - a)1/2 when this resistance is controlling. We turn now to the pore tree. Three domains were described above for a pore fractal, and each domain is charactarized by a different ψ(a) dependence: In the intermediate one, where the rate is weakly dependent on k, the overall degree of deactivation (ψ) is only weakly dependent on a. In the other two asymptotes, ψ ) (1 - a) for small k’s and ψ ) (1 - a)1/2 for large k’s. As k declines, due to uniform deactivation,

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Figure 4. Residual rate (ψ ) rate/initial rate) vs degree of deactivation (a), in three mechanisms of deactivation (uniform, a,b; outer pore, c,d; and inner pore, e,f) and medium or high initial activity (K0 ) 10-4 (a,c, e) or 1 (b,d, f) computed for molecular diffusion and a first-order reaction in a simple loopless pore tree of N ()8) generations and n [)3 (*), 4 (+), or 5 (O)]. Similar results were obtained for Knudsen diffusion.

the system crosses between these domains of different asymptotes. The boundaries of these asymptotes vary with diffusion regime (p ) 1 or 2 for molecular or Knudsen diffusion, respectively) and with the number of generations. The whole ψ(a) curve was computed using the procedure described above (eq 9), showing that the pore fractal is less susceptible to deactivation than the uniform-pore objects (Figure 4). The overall ψ(a) curve varies from a linear curve, when the initial value (K0) is sufficiently small to place it in the reactioncontrolled asymptote (not shown), to a deactivation insensitive at small a’s followed by a large drop at high degrees of deactivation, when the initial activity is within the intermediate asymptote (Figure 4a, K0 ) 10-4, p ) 1) or even within the diffusion-controlled asymptote(Figure 4b, K0 ) 1, p ) 1). In deactivation due to the formation of an undesired product (like coke) the rate of deactivation is related to the concentration field in the catalyst. When the un-

desired product is produced mainly by conversion of the reactant, deactivation will be faster at the pore mouths and at the large pores which supply the reactant, while when the product leads to deactivation, the loss of activity will be faster in the pore ends and in the smaller pores where most of the reactant is consumed. To simplify the problem, we study the effect of complete deactivation of successive pore generations and to that end distinguish between inner-generation and outergeneration deactivation. In the former case we compute Z(0) as before (eq 9) with k(i) ) 0 for the inner layers (M < i e N) and k(i) ) k for the outer layers (i e M); the corresponding inactive fraction is 1 - a ) (BM+1 1)/(BN+1 - 1) with B ) 2(n - 1). In the latter case k(i) ) k for M < i e N and k(i) ) 0 for i e M and the corresponding area is 1 - a ) (BN+1 - BM+1)/(BN+1 1). While the ψ(a) curves are not continuous (Figure 4), they represent the effect of deactivation without adding additional parameters. A more realistic model, with

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deactivation at a rate that is proportional to the concentration of the main reaction reactant or product, will lead to continuous ψ(a) curves with a mixed degree of deactivation of various generations, but with an overall similar nature. Deactivation of inner generations results in an object of fewer active generations: If the initially active object operates within the intermediate domain, then, as was demonstrated above, its overall rate is highly insensitive to the deactivation up to very small values of average activity (1 - a), conditions that push the object into the reaction-controlled regime. Consequently the ψ(a) curve approaches a step function for high K0 (Figure 4e,f). Deactivation of the outer generations, on the other hand, not only diminishes the number of active generations but also adds layers of transport resistance. Consequently the ψ(a) curve exhibits a sharp decline (Figure 4c,d) already with the first and second layer and then levels off at an almost constant value. At high degrees of deactivation the main contribution to activity is due to the small and active pores that lie close to the mainpore entrance. Concluding Remarks A growing list of theoretical studies into reaction and diffusion in fractal catalytic objects have demonstrated that their behavior may be quantitatively and even qualitatively different than that of objects with uniform pores. Thus, pore-fractal “catalysts” may exhibit an intermediate low-slope asymptote, in the plot of log(rate) vs log k, which separates the known asymptotes of kinetics- and diffusion-controlled rates. In that domain the fractal catalyst is more active than a catalyst of uniform pores having similar average properties. Here we showed that the selectivity, in a system of a fast and slow simultaneous reactions, can be markedly different than that in a uniform-pore catalyst. Specifically, slow undesired reactions of higher order can be effectively suppressed in such a system. Three mechanisms of deactivation are considered and are characterized by curves of rate vs average activity: Pore-fractal catalysts are less sensitive to uniform deactivation or to deactivation of the inner pore generations, since in the intermediate domain the rate is only weakly dependent on k; such catalysts are more sensitive to deactivation of the outer pore generations. Translating this information into a viable technology will require to show that three-dimensional unordered but pore-fractal objects, like the inverse cluster-cluster agreggate (inverse or negative CCA; see Elias-Kochav et al., (1991)), exhibit the same features as the ordered objects examined here. This task will require performing numerical experiments and deriving analytical results to show that the results obtained here for two-dimensional ordered structures apply to three-dimensional unordered objects. With low porosity and narrow pores one would expect to find the qualitative behavior described here. Specifically, three-dimensional porefractal structures, with long and narrow pores, should exhibit the same behavior as the two-dimensional structures considered here, as long as the same hierarchy is maintained. Pore fractals may be synthesized by finding preparation methods that spontaneously produce such structures, like leaching of one component after phase separation, or by building ceramic solids around polymeric mass fractals before calcination of the solid.

The superiority of the fractal pore-structure, over the uniform one, is expected to be more pronounced in diffusion regimes that show higher sensitivity to the pore diameter than the Knudsen diffusion regime considered here. Such a dependence, however, will hold for one or two generations and, posing such a problem, would involve a transition in diffusion regime, introducing more parameters and making the analysis more tedious. Thus, such an approach is not considered here. Of the many reaction schemes that should be considered to find out whether the fractal catalyst can yield better selectivities, the scheme with two consecutive reactions is probably the most elementary and simple. The better effective diffusivity that was unraveled for the fractal catalyst suggests that the intermediate species in the scheme A f B f C would be able to diffuse faster in such a system than in a uniform-pore catalyst, and higher selectivities are expected. Notation a ) degree of deactivation A, B ) geometric parameter C1, C2 ) coefficients in eqs 3 and 5 ci, cb ) pore and bulk reactant concentrations, respectively Di ) diffusivity Ds ) fractal dimension k, K ) dimensional and dimensionless rate constants L, Li ) pore length l, m ) reaction orders N ) number of pore generations ni ) density of smaller pores bifurcating from the larger pore n ) number of subgaskets in a gasket p ) index of diffusion regime ()1 or 2 for molecular or Knudsen regime) ri ) pore diameter r1, r2 ) reaction rates Ri ) reaction rate S ) selectivity defined in eq 2 Z ) dimensionless rate (Eqn. 9) x ) pore coordinate Greek Letters R ) parameter in Thiele modulus δ ) scale ratio in the self-similar geometry φi ) local Thiele moduli η ) effectiveness factor ξi ) second or third moment (eqn. 12) ψ ) ratio of rate after deactivation to initial rate subscripts i ) ith generation pore 0 ) large pore mol, Kn ) molcular or Knudsen diffusion regime, respectively

Acknowledgment This research was supported by Technion VPR FundNew York Metropolitan Research Fund and by the Du Pont Educational Aid Fund. Simulations were conducted by C. Gavrilov. Literature Cited Coppens, M.-O.; Froment, G. F. Diffusion and Reaction in a Fractal Catalyst Pore: II. Diffusion and First-Order Reaction. Chem. Eng. Sci. 1995, 50, 1.

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Received for review January 4, 1999 Revised manuscript received June 22, 1999 Accepted June 24, 1999 IE990003O