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Ind. Eng. Chem. Res. 1997, 36, 3416-3420
Optimal Catalyst Activity Profiles in Pellets. 11. The Case of Multiple-Step Distributions Roberto Baratti and Viera Feckova† Dipartimento di Ingegneria Chimica e Materiali, Universita´ degli Studi di Cagliari, Piazza D’Armi, 09123 Cagliari, Italy
Massimo Morbidelli* Laboratorium fur Technische Chemie, ETH Zentrum, CAB C40, Universita¨ tstrasse 6, CH-8092 Zurich, Switzerland
Arvind Varma* Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
For reactions where the rate varies linearly with catalyst loading, for any performance index such as effectiveness, selectivity, or yield, the optimal active element distribution is a step function, where the number and location of the steps depend upon the specific problem under study. In all cases reported to date in the literature, single-step distributions have been found to be optimal. In this work, an example involving two parallel reactions is presented, where a multiple-step distribution is optimal. 1. Introduction Given a fixed amount of the active element, the problem of its optimal distribution in a catalyst pellet has received a general solution which can be summarized as follows (Wu et al., 1990): For any catalyst performance indexse.g., effectiveness, selectivity, or yieldsand for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a nonisothermal pellet, with finite external heat- and mass-transfer resistances, the optimal catalyst activity distribution is a Dirac-δ function. It has recently been discussed (Baratti et al., 1993) that this result is physically consistent only in the limit of low catalyst loadings. At high loadings, large local concentrations of the active element (e.g., Pt, Pd, Ag, etc.) lead to the formation of large macrocrystals and consequently to loss of specific active surface area. In order to obtain results of wider applicability, Baratti et al. (1993) reformulated the problem by taking as the optimization variable the local concentration of the catalytically active element, µ(s). This was assumed to be bounded:
0 e µ(s) e R
(1)
where the upper bound R can be regarded, for example, as the saturation concentration. If the relationship between the active element concentration and the reaction rate is linear (which is generally true for sufficiently low catalyst loadings), the optimization problem again has a general solution (Baratti et al., 1993). This occurs because the problem is then linear with respect to the optimization variable, µ(s). Thus, by applying the Maximum Principle, it can be shown that the solution is given by a step function, where µ(s) is equal to either one or the other of the two bounds in eq 1, i.e., zero and R. The number and location of the steps is given by the sign changes of the * Authors to whom correspondence should be addressed. † Present address: Department of Chemical and Biochemical Engineering, Faculty of Chemical Technology, Radlinskeho 9, 81247 Bratislava, Slovak Republic. S0888-5885(96)00619-7 CCC: $14.00
so-called switching function, F(s), whose evaluation depends upon the specific problem under examination and, in general, can only be pursued numerically. Several examples of application of these results have been reported by Baratti et al. (1993). In all cases involving a linear relationship between the active element concentration and the reaction rate, a single-step distribution was found to be optimal. This is in agreement with earlier results by Morbidelli et al. (1982), who suggested for practical purposes to approximate the optimal Dirac-δ function with a single-step distribution centered at the same location and of width not larger than about 5% of the pellet dimension. In this work, an example where the optimal distribution is given by a multiple-step function is considered. This also allows us to investigate the transition of the shape of the optimal distribution from the step function type to the Dirac-δ mentioned above, as R increases approaching infinity. Moreover, the example shows that such optimal distributions of the active catalyst are indeed possible, particularly in the case of complex reacting systems. This is a warning to not limit the analysis to single-step optimal distributions, since further improvements in the process performance may be achievable with multiple-step distributions. 2. The Reacting System Let us consider two parallel, irreversible reactions occurring in a symmetric, porous and nonuniformly active catalyst pellet: 1
2
A 98 B and A 98 C
(2)
whose dimensionless reaction rates are given by the Langmuir form
fi(u,v) )
umi exp{γi(1 - 1/v)} [1 + σiu exp{δi(1 - 1/v)}]2
i ) 1, 2
(3)
In the case of negligible external heat- and masstransfer resistances and equal heats of reaction, Prater’s © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3417
with BCs:
q)0
at s ) 0
q ) 1/(n + 1)
at s ) 1
(12a) (12b)
3. The Optimization Problem The optimal distribution, µ(s), can be determined through the Maximum Principle using the variational procedure developed by Baratti et al. (1993). For brevity, we summarize here only the main results of this analysis. The Hamiltonian is given by the following linear expression with respect to µ:
H ) Fµ(s) + λ1g(s) s-n Figure 1. Global reaction rate, F, as a function of reactant concentration, u, for various κ values: (a) 0.08; (b) 0.06; (c) 0.05.
relationship between the dimensionless reactant concentration, u, and the dimensionless temperature, v, leads to
v ) 1 + β(1 - u)
(4)
This relation allows one to represent the overall dimensionless rate of consumption of reactant A as a function of only one variable, say u, i.e.
F(u) ) f1[u,1+β(1-u)] + κf2[u,1+β(1-u)]
η)
∫01µ(s) F(u) sn ds
n+1 F(1)
(6)
while satisfying the constraint given by the definition of µ(s):
∫01µ(s) sn ds ) 1
(n + 1)
(7)
and the reactant mass balance within the pellet. For convenience, the latter is considered in the form:
du ) g(s) s-n ds
(8)
dg ) φ2µ(s) F(u) sn ds
(9)
with BCs:
u)1
at s ) 1
(10a)
g)0
at s ) 0
(10b)
Moreover, the constraint (7) can be rewritten as follows:
dq ) µ(s) sn ds
where F is the so-called switching function, given by:
F ) [(1 + λ2)φ2F + λ3]sn
(11)
(14)
The Lagrange multipliers, λi, in the above equations are defined as follows:
dλ1 ∂H ∂F )) -φ2(1 + λ2)µ(s) sn ds ∂u ∂u
(15)
dλ2 ∂H )) -λ1s-n ds ∂g
(16)
dλ3 ∂H ))0 ds ∂q
(17)
(5)
This is a very convenient feature of this problem which we have chosen in order to illustrate the optimization procedure; however, this feature is not necessary to give the specific results reported in this paper. The behavior of the global reaction rate, F(u), is shown in Figure 1 for three values of the dimensionless parameter κ and the other parameters fixed as β ) 0.3, γ1 ) 20, γ2 ) 35, σ1 ) 8, σ2 ) 0.02, δ1 ) 0, δ2 ) 50, and m1 ) m2 ) 2. These values were chosen for illustrative purposes, so as to clearly separate the two maxima of the function F. We seek the active element dimensionless concentration, µ(s), which maximizes the overall effectiveness factor:
(13)
with BCs:
λ1 ) 0
at s ) 0
(18a)
λ2 ) 0
at s ) 1
(18b)
The solution of the optimization problem is then given by the distribution µ(s) which maximizes the Hamiltonian (13), while satisfying the constraints given by ODEs (8), (9), (11), and (15)-(17) together with the BCs (10), (12), and (18). Since the Hamiltonian is linear with respect to µ(s), the solution is straightforward (cf. Ray, 1981) and leads to the following step function:
µ)R
when F > 0
µ)0
when F < 0
(19)
where the switching function F is given by eq 14. It should be noted, as mentioned above, that the number and location of the steps of the optimal distribution depend upon the specific form of F(s), whose evaluation requires the complete solution of the problem (usually possible only numerically) for each particular reacting system under consideration. 4. The Optimal Distribution 4.1. Example 1. The results obtained through the variational method are best utilized by performing the optimization using a direct numerical method (see Baratti et al., 1993, for details) and limiting the search only to distributions of the step-function type. In particular, we first consider single-step distributions of height R. In this case, the problem reduces to a single parameter optimization, i.e., the optimal location of the step, since its width is fixed by condition (7). Next, we consider two-step distributions, both of height R, where the optimization parameters become three: the locations of the two steps and the fraction of active element in each of them. In the present case, we do not need to
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Figure 2. Optimal effectiveness factor, η (continuous curve), and corresponding fraction of active catalyst in the outer step, X (dashed curve), as a function of R; φ ) 3; κ ) 0.06.
consider a larger number of steps. This can be understood readily from Figure 1, where it is seen that the overall reaction rate, F(u), exhibits two maxima. Thus, we can expect at most two locations within the pellet where the catalyst is best utilized, i.e., those where the reactant concentration, u, attains the values corresponding to the maxima of F(u). This is, of course, not true, in general, in practical applications where the involved reacting systems can be more complex than the simple example selected here for illustrative purpose.
In all cases, the optimization results from the direct numerical method were confirmed by evaluating the sign changes of the switching function, F. Several optimization problems have been considered, by analyzing the shape of the optimal distribution which, as R f ∞, should approach the Dirac-δ function. A typical result is illustrated in Figure 2, where the optimal value of the effectiveness factor, η, is shown as a function of R (continuous curve). The corresponding optimal distributions, µ(s), are shown in Figure 3 for a few selected values of R (continuous curves) together with the corresponding profiles of the reactant concentration, u(s) (dashed curves). It appears that, for low values of R (R < 4.33), the optimal distribution is given by a single-step function (Figure 3a,b). As R increases, it becomes possible to concentrate more catalyst in a given support area. In particular, for the cases shown in parts c and d of Figure 3, the catalyst has been placed in two locations, which, as can be seen from the corresponding average values of u, are near to those where F(u) exhibits its maxima. In these cases the optimal distribution is given by a twostep function. When R is further increased (R > 7.40), the optimal distribution again takes the form of a single-step function (Figure 3e,f), since R is now large enough so that all the catalyst can be located in the vicinity of the point where its utilization is maximized, i.e., where F(u)
Figure 3. Optimal active element distribution, µ (continuous curve), and reactant concentration, u (dashed curve), within the catalyst pellet; φ ) 3; κ ) 0.06.
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Figure 4. Optimal effectiveness factors for inner and outer single steps as functions of R; φ ) 3; κ ) 0.08.
exhibits the larger of its two maxima. In the present case (Figure 1), this corresponds to the lower value of u and hence to the inner one of the two steps noted above. As R approaches infinity, the optimal distribution remains of the single-step type, while its width decreases approaching a Dirac-δ function located where F(u) exhibits the highest of its maxima, which in this case corresponds to the ideal value of η ) 4.06. The same transition can also be observed by considering the values of the fraction of the active element in the outer step, X, shown in Figure 2 as a function of R. The two discontinuity points at about R ) 4.33 and 7.40 correspond to the transition from one- to two-step optimal distributions and vice versa, respectively. It should be noted that the evolution of the optimal distribution as a function of R does not always imply a two-step function. In particular, for the case of κ ) 0.08 in Figure 1, the optimal distribution involves only the more common single-step function, located where u = 0.76, which approaches the Dirac-δ as R f ∞. In this case, the values of F at the two maxima are essentially the same (see Figure 1). Thus one expects that, as R f ∞, Dirac-δ functions located where either u = 0.2 or u = 0.76 should give the same value of η. For this reason, the performance of locally optimum distributions, with the catalyst placed in a single step where u = 0.2 (inner step) or u = 0.76 (outer step), was studied as a function of R. These results, shown in Figure 4, indicate that the outer step provides higher η values and that, as expected, the performance of the two steps becomes indistinguishable as R f ∞. It is worth noting that in the example discussed above we have considered two reactions, both exhibiting a Langmuir rate expression but with rather different values of the involved kinetic parameters. The latter is needed to obtain the overall reaction rate which reaches a local maximum at two different values of the reactant concentration (cf. Figure 1). This observation leads to the general conclusion that, in practical applications, the optimal active catalyst distribution can exhibit the multiple-step form only in cases where the involved reactions have rather different kinetics. Since we are considering here the case where the catalyst is the same, this implies that the two reactions should exhibit different mechanisms, i.e., involving the breakage of different chemical bonds. An example would be the isomerization and hydrogenation of an unsaturated hydrocarbon. 4.2. Example 2. The features discussed above in the context of Example 1 can also be obtained in a variety of different situations. As a further example, consider the case of two parallel exothermic reactions: one with
Figure 5. (a) Global reaction rate, F, as a function of reactant concentration. (b) Optimal effectiveness factor, η (continuous curve), and corresponding fraction of active catalyst in the outer step, X (dashed curve), as a function of R; φ ) 2.
first-order kinetics and low activation energy (thus with optimal active catalyst location close to the pellet external surface) and the other with Langmuir kinetics and high activation energy (and hence with optimal catalyst location closer to the pellet center). The corresponding dimensionless rate of the overall consumption of reactant A, F(u), is then likely to exhibit two local maxima, as is indeed the case for the following values of the kinetic parameters: β ) 0.4, γ1 ) 5, γ2 ) 25, σ1 ) 0, σ2 ) 8, δ1 ) δ2 ) 0, m1 ) m2 ) 1, and κ ) 0.038, as shown in Figure 5a. By repeating the same analysis as for the previous example, it can be found that, as the parameter R increases, the optimal activity distribution undergoes the same transitions, from single to double and back to single-step distribution. In particular, from the values of the optimal effectiveness factor shown in Figure 5b as a function of R, it appears that, for R < 6.7 and R > 14, the optimal catalyst distribution is given by a single-step function located where the overall reaction rate achieves its maximum values, i.e., approximately u ) 0.6 and u ) 0.1, respectively. On the other hand, for R values intermediate between the above two limits, the optimal activity distribution is given by a two-step function. 4.3. Discussion. In both of the above examples, we have considered optimization of the catalyst with respect to its efficiency, i.e., the overall rate of consumption of the reactant. Of course, similar arguments apply also in the case where other performance indices of the catalyst are optimized, such as selectivity, yield, or lifetime in the presence of deactivation. It is worth noting that, depending upon the particular performance index selected, the resulting optimal active catalyst distribution in the pellet is different (cf. Baratti et al., 1993). As an illustration, in Example 2 above, it is immediately seen that if we wish to maximize the yield of the first-order reaction, in the limit where R f ∞, we need to concentrate the entire catalyst at the location where the rate of this reaction is maximum. From Figure 5a, we see that this occurs toward the pellet external surface (i.e., where u = 0.6). On the other hand, if we wish to maximize the yield of the reaction
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exhibiting the Langmuir kinetics, we should locate the active catalyst toward the pellet interior (i.e., where u = 0.1). As R decreases, we have the usual transition from the Dirac-δ distribution to a step distribution, which remains, however, a single step and more or less centered about the same location. This behavior is indeed rather different from the case where catalyst effectiveness is maximized, where as discussed above, as R increases the optimal step distribution moves from the pellet exterior toward its interior and in an intermediate region becomes a two-step distribution. 5. Concluding Remarks In this work, we present the first example of a reacting system for which the optimal catalyst distribution has the form of a multiple-step function. On physical grounds, multiple steps arise from global reaction rates which exhibit multiple maxima, along with boundedness of the active element concentration. If the upper bound, R, is sufficiently low, it is not possible to accommodate all the catalyst in the most desirable location (i.e., in the vicinity of the global maximum of the reaction rate), requiring the catalyst to be spread away from it with consequent loss of efficiency (cf. Baratti et al., 1993). In some cases, such as the one illustrated in Figures 2 and 3, it is then best to locate the catalyst in two separate steps (i.e., in the vicinity where the two local maxima of the reaction rate occur). However, when the upper bound on the catalyst concentration increases (or, equivalently, the overall catalyst loading decreases), it always becomes possible to accommodate all the catalyst in the most desirable location, thus leading to a single-step distribution, which in the limit as R f ∞ then approaches a Dirac-δ function. An exception to this behavior is obtained for pathological situations which do not occur readily in practice, where the global reaction rate has multiple identical local maxima. In such cases, as discussed above, for large values of R, step distributions located where any of the maxima of the global reaction rate occur, give essentially the same catalyst performance. Finally, we note that the conclusions above are drawn for the case of a one-dimensional optimization problem; i.e., the global reaction rate is a function of only one variable. The qualitative behavior of the solution is expected to remain the same for more complex situations involving many reactions and components. In these cases, optimal distributions with more than two steps may indeed be possible. The practical implication of the results presented here is that, particularly when dealing with complex systems as is the case in most industrial applications, one should not assume a priori that the optimal active catalyst distribution has the form of a single step. This could, in fact, well be a multiple-step distribution, which could lead to significantly improved catalyst performance. Using the numerical strategy adopted in this work, the search for multiple-step optimal active catalyst distribution is relatively straightforward, and it can lead to substantial improvements in catalyst performance. Acknowledgment We gratefully acknowledge support by the CNRENEL Project “Interactions of energy systems with human health and environment”, Rome, Italy, and by a NATO Collaborative Research Grant.
Notation C ) reactant concentration, mol m-3 Ce ) weight fraction of active element C h e ) volume average weight fraction of active element De ) effective diffusion coefficient, m2 s-1 E ) activation energy, J mol-1 F ) global rate of reaction, defined by eq 5 f ) rate of reaction, defined by eq 3 g ) function, defined by eq 8 k0 ) reaction rate constant H ) Hamiltonian function K0 ) inhibition adsorption constant, mol-1 m3 n ) integer characteristic of pellet geometry; 0 for infinite slab, 1 for infinite cylinder, 2 for sphere q ) function, defined by eq 11 R ) characteristic pellet dimension; half-thickness (n ) 0), radius (n ) 1, 2), m Rg ) ideal gas constant, J mol-1 K-1 s ) dimensionless position T ) temperature, K u ) dimensionless reactant concentration, C/C0 v ) dimensionless temperature, T/T0 X ) fraction of active element in the outer step Greek Letters R ) upper bound of the dimensionless active element concentration β ) dimensionless heat of reaction parameter, (-∆Hr)DeC0/ (λeT0) γ ) dimensionless activation energy, E/RgT0 δ ) dimensionless heat of adsorption, ∆Ha/RgT0 ∆Ha ) heat of adsorption, J mol-1 ∆Hr ) heat of reaction, J mol-1 κ ) k02/k01 λe ) pellet thermal conductivity, J K-1 m-1 s-1 µ ) dimensionless active element distribution, Ce(s)/C he F ) switching function, defined by eq 14 σ ) K0C0 φ ) Thiele modulus, R [k01C0/De]1/2 Subscript i ) reaction number Superscript o ) bulk value
Literature Cited Baratti, R.; Wu, H.; Morbidelli, M.; Varma, A. Optimal catalyst activity profiles in pelletssX. The role of catalyst loading. Chem. Eng. Sci. 1993, 48, 1869-1881. Morbidelli, M.; Servida, A.; Varma, A. Optimal catalyst activity profiles in pellets. 1. The case of negligible external mass transfer resistance. Ind. Eng. Chem. Fundam. 1982, 21, 278284. Ray, W. H. Advanced Process Control; McGraw-Hill: New York, 1981. Wu, H.; Brunovska, A.; Morbidelli, M.; Varma, A. Optimal catalyst activity profiles in pelletssVIII. General nonisothermal reacting systems with arbitrary kinetics. Chem Eng. Sci. 1990, 45, 18551862.
Received for review October 4, 1996 Revised manuscript received May 31, 1997 Accepted June 18, 1997X IE9606193
X Abstract published in Advance ACS Abstracts, July 15, 1997.
