2066
Ind. Eng. Chem. Res. 1991,30,2066-2074
Role of Adsorbate Interactions in Fixed Bed Reactor Design with Multicomponent Chemisorption? Asha S. Datar, Sudarshan D. Prasad,* and Lakshmangudi K. Doraiswamy Physical Chemistry Division, National Chemical Laboratory, Pune 41 1 008, India
The influence of adsorbate interactions in molecular pair formation and consequently on surface rates and space time values has been demonstrated. The mean selectivities are also shown to appreciably depend on the nature of adsorbate interactions, viz., attractive or repulsive. Refinement in accounting for adsorbate interactions is shown to cause significant variation in the selectivity predictions for a parallel reaction. Multicomponent chemisorption with interaction effects is investigated for the first time with a view to explore the applications in reador design strategies. The Fowler-Guggenheim model for two-component adsorption is employed for analyzing a series reaction. The space time values for the optimal production of the intermediate have been computed for seven combinations of the interaction energies. Criteria for predicting the trends in space time values in comparison with the Langmuir model (no interactions) have been derived. Adsorbate interactions are shown to be of considerable relevance in fixed bed reactor design, even in multicomponent chemisorption.
Introduction The influence of adsorbate interactions in the formation of several surface phases (inter alia also interconversions among them) has been well documented (Ertl, 1983,1985; Woodruff et al., 1983) by workers in the field of surface science. Its application in the analysis of surface reactions is only now beginning to be appreciated by kineticists (Barteau et al., 1981; Gland et al., 1982; Silverberg et al., 1985; Zhdanov, 1981). It has been shown earlier that these interaction effects can cause drastic changes in the values of operating variables needed to realize a set selectivity pattern (Bhat et al., 1984). An earlier work analyzed the effects of adsorbate interactions in determining activity profiles of the fixed bed reactor, in particular the space time needed to achieve a desired degree of conversion or mean selectivity (Doraiswamy and Prasad, 1987). In that (first) approach only rough rate expressions incorporating interaction effects were used (Bhat et al., 1984). In the present work, however, considerable refmement of the rate models has been attempted, and even though only the simple plug-flow reactor (PFR) model is employed, we believe that the trends in reactor behavior will reflect the subtleties of interaction effects. We also address the question as to what degree of refinement is needed in treating interaction effects to enable prediction of reactor performance with the desired degree of confidence. To make the treatment tractable and not to mask the real intent of the work, we analyze two simple systems: (a) a parallel reaction network and (b) a series reaction network. More complex networks can be broken into a combination of these two basic types. The parallel reaction network differs from the series in one major way: in the former, methods of one-component-adsorption theory are adequate, while in the latter a minimum of two adsorbate concentrations is involved, and hence multicomponent-adsorption theory is an essential prerequisite. There is almost a total absence of literature dealing with interaction effects in multicomponent chemisorption, especially concerning reactor design strategies, and in this paper we attempt a n analysis of this problem.
Formulation The two reaction schemes chosen in the present work are presented in Figure 1.
* To whom correspondence should be addressed. NCL Communication No. 4729.
These two examples constitute, respectively, cases of parallel and series reaction networks. In both these reaction schemes adsorption equilibrium between & and &, and B, and B,, is assumed. Surface reactions are assumed to be slow in comparison to the rates of adsorption and desorption such that adsorption isotherms relate the gasand surface-phase concentrations. In scheme I only one surface concentration enters into the kinetic rate expressions, so one-component-adsorption isotherms are adequate for the kinetic treatment. For scheme 11,two surface concentrationsare involved (if we assume that the product C, is rapidly desorbed), and multicomponent-adsorption isotherms are required.
Parallel Reactions: Selectivity For analyzing the one-component-adsorptionsystems, we employ two models: (1)the Fowler-Guggenheim (FG) model (Fowler and Guggenheim, 1939; Hill, 1956), in which random distribution of adsorbate molecules is assumed, and (2) the quasi-chemical (QC) model (Fowler and Guggenheim, 1939; Hill, 19561, in which random distribution of adsorbate molecular pairs is assumed. It can be shown that interaction effects are best manifested for second-order (biomolecular) surface reactions in which a pair of occupied nearest-neighbor sites are needed for the ensuing catalytic reaction. If we denote an occupied site by the number 1 and a vacant site by the number 0, then the reaction rate for a second-order surface rate is given by rs = k@ll (1) where p l l denotes the probability of observing a molecular pair. This form of the rate model ignores the change in activation energy of surface reaction due to adsorbate interactions. In other words, on an energy level diagram an activated complex and the reactant molecular pair A'A and AA are shifted to the same extent due to adsorbate interactions such that the activation energy for surface reaction is virtually unaffected. If this assumption does not hold true, we get a different rate law (Silverberg et al., 1985). In the FG model format, since random distribution of the adsorbate molecules is assumed even in the presence of interactions, the equation for the pair probability p l r , eq 1, retains the simple form r, = k,e2 (2) The rate of the first-order reaction is given by rt = k# (3)
0888-5885/91/2630-2066$02.50/00 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 2067
0
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Scheme Il
Scheme I
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Po
Figure 1. Schematic representation of parallel and series reaction network.
Figure 2. Plot of SIand SA,vs log po for repulsive interaction for the case of the QC model. bo = 1 X 108 Torr; Q = 19.5 kcal/mol.
Within the framework of both FG and QC models, the rate expression for the first-order reaction remains unchanged and is given by eq 3. The pair probabilities pll, however, depend sharply on whether the FG model or the more refined QC model is employed. In addition, computation of the surface coverage e is all that is required for the computation of pair probabilities for the FG model. For the QC model, however, the following set of three simultaneous equations is needed for the computation of the pair probabilities pll, plo,and poo,once the surface coverage is computed: (4) 2Pll + PlO = 28 + pl0 = 2(1- e) (5) 2pzpo0
probabilities are much lower when repulsive forces are operative; an exactly opposite effect is manifested for attractive interactions. In a plug-flow reactor, a concentration profile exists from the inlet to the exit; hence surface concentrations and pair probabilities vary from point to point. An average selectivity can be defined as
= 4 exp(w/kr) (6) pll,plotand poodenote site pairs contain two, one, and no adsorbate molecules, respectively, and w denotes the interaction energy of a pair of adsorbed molecules. The computation of surface coverage is a straightforward numerical exercise, the details of which may be found in our previous work (Doraiswamy and Prasad, 1987). As has been shown, a single expression suffices to relate the surface coverage to pressure, for both the FG and QC models: PlO2/o31lPo0)
P = bo exp(-Q/Rr)3(8/(1 - 6)) exp(zd/kr) P = bo exp(-Q/Rr)g(Ne/(1 - 8 ) )
(7)
where g(e) = [(2 - 2e)/(o
+ 1 - 2e)lz
8 = I1 - 460 - e)(l - exp(-w/kT)))0.6
(8b)
(8c) As before Q, bo, and z denote the heat of adsorption, entropy change factor, and coordination number, respectively. The rate of the monomolecular surface reaction which is proportional to the surface coverage does not depend to any significant degree on whether the FG or QC model is used. However, in sharp contrast to this, the pair probabilities pll depend appreciably on the choice of the model. These observations can be neatly summarized if we define two selectivities: (9) SI = k # / ( k # + kg,,) (10) s2 = k$ll/(k$ll + ka) where Sland S2denote the selectivities with respect to the products of the first- and second-order reactions, respectively. The pair probabilities are largely dependent on the type of interaction. In comparison to the case where interactions are neglected (Langmuir model), the pair
where X, and Xo denote the exit and inlet conversions and SIand S2the mean selectivities, respectively. An alternative way of defining selectivities involves computing the mean rates of the first- and second-order reactions. Thus
i,,=
where
s,3gll
dX/(X, - Xo>
and iis denote the mean rates and S A 1 = Ff/(rf + Fa) SA2 = 1 - S A 1
(14)
(15)
(16) In actual computation it hardly matters whether we employ the definition (eqs 11 and 12) or eqs 15 and 16, for the selectivity differences within the purview of both definitions are only marginal. Only the repulsive interactions are considered; the same results hold good also for attractive interactions and hence are not shown in Figure 2. While our prime goal has been the analysis of the selectivity behavior, the space time needed for achieving a desired mean selectivity (at fixed exit and inlet conversion levels) is of equal importance. This space time T is readily computed as
With reference to scheme I, Figure 1,we immediately see that the total rate is the s u m of the second- and fmt-order rates and hence eq 17 can be readily arrived at. Figure 3 implies that the mean selectivity depends mainly on the choice of the inlet pressure po for a fixed exit conversion for both attractive and repulsive interactions. As the pressure increases, SI falls and 3, (as it is equal to 1 - SI)increases monotonically. Similarly, the
2068 Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991
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Figure 5. Plot of selectivity ratio &/S, vs inlet pressure po for the QC model with repulsive interactions.
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Figure 3. (a) Variation of selectivity S , with inlet pressure for the FC model for both attractive and repulsive interactions. (b) Variation of selectivity 3, with inlet pressure for the QC model for both attractive and repulsive interactions. 0
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Figure 6. Comparison of selectivity ratios for both the FG and QC models with repulsive interactions.
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Figure 4. Plot of selectivity ratio S2/& vs inlet pressure po for the FC model with repulsive interactions. total rate increases and hence the space time
T decreases. On the other hand, at large T , we have higher Thus, a trade-off has to be worked out between maximal selectivity and minimal space time. In Figure 4, the ratio of selectivities is plotted against inlet pressure for the FG model for three values of the interaction parameter w, a t constant values of the exit conversion X, and of the ratio of kinetic constants k,. There is appreciable divergence between the three curves, especially at intermediate values of the inlet pressure po. Normally, in the absence of interaction forces, the second-order rates show a much stronger de endence on should pressure than the first-order processes and increase with pressure. With increasing u (u> 0) notice
sl.
(I2/s1)
how the slopes decrease appreciably. Notice that at higher values of w (say, w = 3) the repulsive forces retard the second-order surface rates in comparison to the firsborder processes so much that the variation with pressure is very sluggish and the ratio of selectivities shows an approximately logarithmic dependence on the inlet pressure PO. (SP/Sl)FG = c1 + c2 PO (18) The influence of refinement in accounting for interaction forces and their overwhelming role in the determination of selectivities are illustrated in Figure 5. The sophisticated QC model weights interaction forces strongly. Notice the appreciable curvature of (#,/SI)plots for the QC model in comparison to the FG model, espetjall for the hi hest value of the interaction parameter w , (S2/ > (&/ l ) ~ at all pressures. This is readily apparent from Figure 6, wherein the (S2/s1)ratio is plotted as a function of the inlet pressure. At intermediate pressures, (s2/s1) constitutes only 10-20% of the corresponding FG mo% (for w = 3 kcal/mol). At high pressures, the divergence between the models diminishes as is evident from Figure 6. Thus the effecta of refinement in accounting for adsorbate ordering are appreciable, and even though increased computation is involved, the more refiied QC model has to be invoked in computing mean selectivities, especially when repulsive forces are operative. We now proceed to investigate the role of kinetics associated with surface reactions. As the ratio of the kinetic constants is varied 100-fold (k,= 0.1-lo), (s,/s,) varies almost in direct proportion. This is obvious from Figure
J
%
,
Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 2069
!I 0.4
Figure 7. Plot of normalized selectivityratio (S,/&)/k, for both the FG and QC models with repulsive interactions.
LO9 Po
Figure 9. Plot of selectivity ratio against inlet pressure po for the QC model with attractive interactions. I
Figure 8. Plot of selectivity ratio vs inlet pressure po for the FG model with attractive interactions.
7, wherein the ratio of selectivities (sz/&)is normalized by dividing the ratio of the kinetic constants k, and plotted as a function of the inlet pressure. The normalized ratios (&/g,)/k, are very close to each other, demonstrating that the kinetic constants have only a lower order effect in comparison to LO, especially for repulsive interactions. What could be the behavior of attractive forces? Firstly since attractive forces favor clustering and association of adsorbate molecules, one would expect opposite trends for selectivity with respect to the products of the second-order reaction. Figures 8 and 9 illustrate, respectively, the behavior of the FG and QC model selectivity ratios. Both show instabilities in the slope, a telltale nature of the self-acceleratory behavior of adsorption when attractive forces are operative. Alao,the most obvious feature is that the pressure range has been shrunk to only 2 orders of pressure (almost a reduction lo4) in comparison to the repulsive forces. The value of w / k T = -1 is the boundary of the phasetransition region for the FG model, while a lower value (more negative, w / k T = -1.386) is needed to ensure phase
1
0
-0.8
I
0
I
Figure 10. Comparison of selectivity ratios for the FG and QC models with attractive interactions.
transition in theadsorbed phase for the QC model (for a square lattice with coordination number z = 4). Correspondingly, the slopes are slightly steeper for the FG model in comparison to the QC model. But, in order to avoid the complexities due to phase transitions (we will have tribranched curves), we choose values of ( w / k T )just below or on the boundary of the phase-transition region. Figure 10 further illustrates the role of refinement in accounting for the interaction forces: here / (SZ/S1)~a is plotted as a function of inlet pressure. %r attractive forces this ratio is >1in contrast to the repulsive
(s2/sl)
2070 Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991 1.0
pair probabilities and three surface concentrations, Le., two of the adsorbate and one of the vacant sites). Even though this can be done, it is too complex a procedure to be used in PFR design calculations. Hence we restrict ourselves to the FG model for two-component systems, for both repulsive and attractive interactions. In the following analyais, we codme ourselves to the case when adsorption equilibrium prevails between the adsorbed and gas phases for both components A and B. Accordingly, surface reactions are assumed to be much slower than the actual rates of adsorption and desorption. Assuming the surface reactions 4 B, C, are first order with respect to the surface concentrations, we can readily write down the equations for fractional conversion XA and XB directly: dXA/dr = k16A (19)
1
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dXB/dr = kleA - k2$
1
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-0.5
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1 .o
1.5
(s2/sl)/k, for both
Figure 11. Plot of normalized selectivity ratio the FG and QC models with attractive interactions.
forces (wherein it is
