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Counterdiffusion of p-Xylene/Benzene and p-Xylene/o-Xylene in Silicalite Studied by the Zero-Length Column Technique Stefano Brandani,† Mohamad Jama, and Douglas M. Ruthven* Department of Chemical Engineering, University of Maine, Orono, Maine 04469-5737
This paper describes the extension of the zero-length column (ZLC) method of measuring intracrystalline diffusion to the study of counterdiffusion. Theoretical ZLC response curves are derived for two simple models considering chemical potential gradient as the driving force for transport with the intrinsic mobilities either constant or inversely dependent on total loading. Experimental response curves for benzene-p-xylene and p-xylene-benzene in silicalite are shown to be consistent with the first of these models. The response curves conform quantitatively to the response predicted from this model using the measured pure component diffusivities. However, the behavior of the benzene-o-xylene and p-xylene-o-xylene systems does not conform to the simple model. The retardation caused by o-xylene is substantially greater than that predicted from the single-component diffusivity data. A simple microdynamic explanation is suggested to account for this apparent anomaly. Most industrial applications of zeolites, either as catalysts or as selective adsorbents, involve binary or multicomponent diffusion, but experimental studies of intracrystalline diffusion have focused mainly on the easier problem of diffusion in single-component systems. The basic theoretical framework for diffusion in a binary system was set out in a seminal paper by Habgood and co-workers in 1966.1 Experimental studies of co- and counterdiffusion of benzene-toluene in silicalite were reported by Qureshi and Wei2 while somewhat similar studies, carried out by the FTIR method, were reported by Niessen and Karge.3 Co- and counterdiffusion studies of aromatics in a silicalite membrane were reported by Baertsch et al.4 and by van den Broeke,5 who also presented a more detailed analysis of the NiessenKarge data based on application of the generalized Stefan-Maxwell equation. Some of these results were reported by Krishna and Wesselingh6 in their review article. During the last several years Ka¨rger7,8 and McCormick9 and their co-workers applied PFG NMR methods to the study of binary diffusion in zeolites. Such methods are, however, applicable only where molecular mobilities are relatively high. The diffusivities encountered in the sterically hindered regime are generally too small to measure by NMR and it is with such systems that some of the more important effects are encountered. The results from these studies show that, in a binary system, the mobility of the faster diffusing species is in general reduced by the presence of a slower moving species whereas the mobility of the slower species is not significantly affected. When the molecules are sufficiently small that they can pass freely within the pore system (CH4-CF4, and CH4-Xe in silicalite or CH4C2H4 in NaY), the effect is relatively modest, but when the molecules cannot pass freely within the channel * To whom correspondence should be addressed. Phone: (207)581-2283. Fax: (207)581-2323. E-mail: druthven@umche. maine.edu. † Permanent address: Chemical Engineering Department, University College London, Torrington Place, London WC1E 7JE, United Kingdom.
(C2H4-C2H6 or aromatics in silicalite), the effect is much more dramatic because, in these systems, transport is effectively controlled by the slower moving species. The results reported in the present study conform to this same general pattern. In a previous paper we noted the possibility of extending the ZLC (zero-length column) technique to the study of counterdiffusion in large zeolite crystals.10 The purpose of the present paper is to explore this idea in greater detail, to present a more detailed theoretical model for the ZLC response for a counterdiffusion system, and to report the results of an experimental ZLC study of counterdiffusion of p-xylene/benzene and p-xylene/o-xylene in large silicalite crystals. Theoretical. The ZLC method depends on following the desorption of a sorbate from a pre-equilibrated adsorbent sample under known conditions of purge flow rate and temperature. To distinguish it from variants of the experimental technique, the simple experiment in which the sample is equilibrated with sorbate and desorbed with an inert carrier (usually He) is referred to as “normal ZLC” or NZLC. For a linear system of spherical adsorbent particles the desorption curve is given by
c c0
∞
) 2L
∑
exp(-βn2Dt/R2)
n)1[β
2 n
+ L(L - 1)]
(1)
where βn is given by the roots of
βn cot βn + L - 1 ) 0
(2)
R is the mean equivalent radius of the adsorbent particles and
L)
1 F R2 3 KVs D
(3)
In the long time region only the first term of the summation is significant, so eq 1 approaches an asymptote defined by
10.1021/ie990691b CCC: $19.00 © 2000 American Chemical Society Published on Web 02/17/2000
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c ) c0 β
2 1
2L exp(-β12Dt/R2) + L(L - 1)
(4)
It is evident that a plot of ln(c/c0) versus t will yield directly both the equilibrium parameter (KVc) and the diffusional time constant (R2/D). The linearity requirement is fulfilled in NZLC experiments at low sorbate concentrations within Henry’s Law region and, under these conditions, the derived diffusivity corresponds to D0, the limiting transport diffusivity at zero loading while K represents the dimensionless Henry’s constant. The linearity requirement is also fulfilled for a tracerexchange system (TZLC) in which the sample is preequilibrated with an isotopically labeled stream of sorbate and then desorbed by purging with a stream containing the same sorbate at the same concentration but in unlabeled form.11 In this type of experiment the sorbate loading remains constant and the counterdiffusion exchange of the labeled species is followed using a species-sensitive detector such as an on-line MS. For such a system the equilibrium constant K is replaced by the equilibrium concentration ratio q*/c and the diffusivity is the tracer (or self-) diffusivity at the relevant loading level. The issue of how this simple analysis must be modified when the NZLC experiment is carried out under nonlinear conditions has been addressed in recent papers by Brandani12 and Brandani, Jama, and Ruthven.13 In the third type of ZLC experiment (countercurrent ZLC or CCZLC) the adsorbent is pre-equilibrated with component A and then purged with a carrier stream (usually He) containing component B at a specified partial pressure. Because the equilibrium adsorption isotherms for A and B will generally be different, in this type of experiment there will be a net flux either into or out of the adsorbent particle (in contrast to the tracerexchange experiment where there is no net flux or the NZLC experiment where there is a desorptive flux). The magnitude of this flux will depend on the binary adsorption isotherm as well as on the mobilities of the two species. A more elaborate mathematical model is therefore needed to interpret the experimental response curves. We consider a system in which the two components (A and B) are adsorbed in accordance with the binary Langmuir model:
qA bAcA ) ; θA ) qs 1 + bAcA + bBcB
bBcB θB ) 1 + bAcA + bBcB (5)
The basic flux equation, assuming the chemical potential gradient to be the driving force, is
Ji ) -
D0i ∂µi q RT i ∂r
(6)
and for a binary Langmuir system with an ideal vapor phase this is equivalent to
Ji ) -D0iqsθi
[
]
1 ∂θi h 1 ∂θ θi ∂r 1-θ h ∂r
(7)
where θ h ) θA + θB and the intrinsic diffusivity (D0i) is assumed to be independent of concentration.
Equation 7 shows that the flux expression in a binary system contains two terms, one as a result of the movement of molecules of type A and the other arising from the overall flow. Combining this expression with the differential mass balance yields the following expression for the local adsorption/desorption rate:
[
]
2 θi ∂2θ 1 ∂θi ∂ θi 2 ∂θi h h 2 ∂θ ) 2 + + + + 2 Ri ∂τ ξ ∂ξ 1 θ h ξ ∂ξ ∂ξ ∂ξ θi h ∂θ h 2 1 ∂θi ∂θ (8) + 2 ∂ξ 1 θ h ∂ξ ∂ξ (1 - θ h)
()
This is equivalent to the expression derived by Round, Newton, and Habgood1 for the uptake of a binary mixture in an adsorbent particle. This formulation was used more recently by van de Broeke5,6 to account for the experimental data obtained by Niessen and Karge3 for the adsorption/desorption behavior of benzene/pxylene in silicalite crystals. The overall mass balance for the ZLC system reduces to
|
∂θi ∂ξ
ξ)1
+
|
θi ∂θ h 1 - θ ∂ξ
ξ)1
+ Li(Ci - C0i) ) 0
(9)
with the dimensionless parameters defined by
τ)
D*t R
, ξ) 2
Di r FR2 , Li ) , Ri ) , Ci ) bici, R 3KiVsD0i D* Ki ) biqs (10)
and
θi
|
ξ)1
)
Ci ; 1 + CA + CB
Ci )
θi 1-θ h
|
ξ)1
(11)
In the special case in which the equilibrium isotherms for A and B are the same, we have
L i D j Rj ) ) Lj Di Ri
(12)
The numerical solution of this set of nonlinear partial differential equations, subject to the initial and boundary conditions τ e 0, θA ) θ0A, and θB ) 0, yields the ZLC response in the form cA/c0A versus τ (for specified values of the parameters LA, LB, and D0A/D0B). Such solutions were generated using the gPROMS numerical simulation program. A representative set of such curves, for bA ) bB, is shown in Figure 1 for the desorption of A from a sorbent pre-equilibrated with A and desorbed with a stream containing B at the same partial pressure. It is also possible to generate the response curves for the adsorbing component (B) but because, in practice, the response for component B cannot be measured, this is of lesser interest. In this model it is assumed that the intrinsic diffusivities (D0A and D0B) are independent of loading (or flux). An alternative model, originally suggested by Barrer and Jost,14 assumes that the mobility is proportional to the free space within the micropores or, in the Langmuir model, the fraction of unoccupied sites [D0 ) h )]. D00(1 - θ
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0.2 is minor (Figure 2). The differences are more pronounced for the first model. However, even for the first model, differences in the slope of the long time tail become important only for θ > 0.5. When component B diffuses faster than A (D0B/D0A > 1), the effect on the rate of desorption of A is minor, even at high loading. However, when B diffuses more slowly, we see that, at high loadings, the desorption of A is substantially retarded. The shape of the curves for the two models is also different in that the curves derived from the second model do not show the characteristic double curvature which can be seen in some of the curves derived from model 1 (particularly when the curves are plotted over an extended concentration range). Experimental Section Figure 1. Theoretical ZLC response curves for desorption of component A in a countersorption (CCZLC) system calculated from eqs 8 and 9 with C0A ) bAc0A ) C0B ) bBc0B. For each value of θ0 (0.1, 0.5, and 0.8) the desorption curves are calculated for D0A/ D0B ) 0.2, 1.0, and 5.
The experiments were carried out using the same system as described previously.1 The single-component NZLC response was measured for o-xylene and mxylene at several different flow rates and temperatures using a column containing large silicalite crystals (equivalent spherical radius ) 25 µm). In the countercurrent diffusion measurements the sample was equilibrated initially with deuterated p-xylene and desorbed with a helium stream containing either m-xylene or o-xylene at the same partial pressure. Results and Discussion
Figure 2. Theoretical ZLC response curves for a countersorption (CCZLC) system calculated from eqs 13 and 14 with bc0A ) bc0B. For each value of θ0 (0.1, 0.5, and 0.8) the desorption curves are calculated for D0A/D0B ) 0.2, 1.0, and 5.0.
This leads to the rate expression
[
] [
]
∂2θi 2 ∂θi h 1 ∂θi ∂2θ h 2 ∂θ ) (1 - θ h) 2 + + θi 2 + R ∂τ ξ ∂ξ ξ ∂ξ ∂ξ ∂ξ
(13)
in place of eq 8 with the ZLC mass-balance expression
(1 - θ h)
|
∂θi ∂ξ
ξ)1
+ θi
∂θ h + Li(Ci - C0i) ) 0 ∂ξ
(14)
in place of eq 9. Representative response curves calculated numerically from this model using gPROMS are shown in Figure 2. In comparing different ZLC response curves, it is important to remember that the “effective” diffusivity of the desorbing component is proportional to the slope of the long time linear asymptote of the plot of ln c/c0 versus t (see eq 4). The families of response curves derived from these two models, for similar parameters, are significantly different, although there are some common features. Not surprisingly, the effect of a difference in diffusivity between the two diffusing species becomes apparent only at relatively high loading. With the second model, even at θ ) 0.8, the variation in the response over the range 5 > DA/DB >
Single-Component Diffusion of o-Xylene and m-Xylene. Representative single-component NZLC response curves for o-xylene and m-xylene at various flow rates are shown in Figure 3 b-d. The rapid initial drop in concentration to a low level followed by a slow tail is characteristic of a slow diffusion system. The asymptotic slope of the response curves for o-xylene and m-xylene are essentially independent of flow rate, as expected for a diffusion-controlled system. Under comparable conditions the responses for o-xylene and m-xylene are very similar and quite different from the p-xylene response which is shown in Figure 3a. The p-xylene response drops more slowly in the initial region and more rapidly in the long time region, which is as expected for a faster diffusing sorbate. It is not possible to derive quantitatively reliable diffusivity values from the o- and m-xylene response curves since, because of the low concentration levels, the long time asymptotes are not well-defined. Nevertheless, the trends with purge flow rate and temperature appear to be reasonably consistent so approximate values can be estimated on the basis of eq 4. The results of such an analysis are summarized in Table 1. The derived diffusivity values for o-xylene are approximately consistent with the gravimetrically measured values reported by Richard et al.15 Diffusivities for m-xylene appear to be somewhat larger (but still substantially smaller than those for p-xylene). Equilibrium capacities for o-xylene and m-xylene appear to be somewhat smaller than those for p-xylene. CCZLC for Benzene/p-Xylene and Benzene/oXylene. Our previous study of diffusion of benzene and p-xylene showed that the NZLC and TZLC responses for benzene are qualitatively consistent with the behavior of an “ideal” system in which transport behavior follows the normal rules.10 This conclusion is strengthened by a more detailed quantitative analysis of the CCZLC data.
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Figure 3. Comparison of NZLC desorption curves for o-, m-, and p-xylene. (a) p-Xylene at 200 °C, 5 Torr, and 10 cm3/min purge flow rate (replicate experiments). (b) o-Xylene at 200 ° and 5 Torr, showing the effect of the purge flow rate. (c) m-Xylene at 175 °C and 5 Torr, showing the effect of the purge flow rate. (d) m-Xylene at 5 Torr and 5 cm3/min purge flow rate, showing the effect of the temperature. Table 1. Analysis of Response Curves for o-Xylene and m-Xylene sorbate T (°C)
o-xylene 200
m-xylene
200
150
175
175
200
flow rate 3.5 10 5 5 10 5 (cm3/min) a 140 130 105 Kapp D/R2 5 × 10-4 5.3 × 10-4 5 × 10-4 8 × 10-4 7.6 × 10-4 9.3 × 10-4 -1 (s ) D × 109 3.0 3.3 3.0 5.0 4.8 5.6 (cm2 s-1) 9 b 3.5 3.5 1.2 2.3 2.3 3.5 D0x × 10 (cm2 s-1) a Approximate value of K estimated from intercept assuming a linear isotherm. b Diffusivity for o-xylene/silicalite from gravimetric measurements15.
Table 2. Parameters for Theoretical Response Curves Shown in Figure 4a
benzene p-xylene
D0/R2 (s-1)
KVs (cm3)
Bz at 6 Torr PX at 5 Torr
4.5 × 10-4 8.5 × 10-4
5.1 7.5
λ ) 0.45 λ ) 0.45
Bz at 58 Torr λ ) 0.86
a Values are derived from single-component measurements at 130 °C.1 F ) 19 cm3/min at column conditions.
Figure 4 shows the ZLC desorption curves for benzene at 130 °C and a 19 cm3/min purge rate for several different experiments. In the first series of runs (a) the sample was pre-equilibrated with benzene at about 6 Torr and desorbed with a purge stream containing either the same partial pressure of deuterated benzene (TZLC) or 5 Torr partial pressure of either o-xylene or p-xylene (CCZLC). The TZLC response for benzene and
Figure 4. CCZLC response curves for benzene/p-xylene and benzene/o-xylene (benzene is the desorbing species) at 130 °C and 19 cm3/min purge flow rate. In series (a) the sample is equilibrated at about 6 Torr benzene and desorbed with a purge stream containing either 6 Torr deuterated benzene (TZLC) or 5 Torr p-xylene or o-xylene (CCZLC). In series (b) the sample is equilibrated with benzene at 58 Torr and desorbed with either He (NZLC) deuterated benzene at 58 Torr (TZLC) or o- or p-xylene at 5 Torr (CCZLC). The lines show theoretical response curves calculated, according to model 1, with the parameters given in Table 2.
the CCZLC responses for benzene-p-xylene, under these conditions, are essentially identical. This is consistent with the theory presented above (Figure 1) since
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Figure 5. Comparison of CCZLC desorption curves for p-xylene vs o-xylene (175 °C, 5 Torr, 30 cm3/min purge flow rate) with NZLC and TZLC responses under similar conditions. Parts (a) and (b) show replicate experiments with fresh and “aged” adsorbent samples.
both the equilibria and the diffusivities for benzene and p-xylene at 130 °C are similar and, at these partial pressures, the loading is in the range of 50-60% of saturation for p-xylene and 40-50% for benzene. Under these conditions the TZLC and CCZLC responses should be very similar. The tail of the CCZLC response for benzene versus o-xylene is slightly slower which is again consistent with the simple theory for the situation in which the diffusivity of the adsorbing species is much smaller than that of the desorbing species. The second series of experiments (b) was carried out at higher benzene loading levels (58 Torr benzene partial pressure) corresponding to about 85% of saturation. The asymptotic slopes of the NZLC and TZLC curves for benzene and CCZLC curve for benzene/pxylene are all similar, showing essentially similar diffusivities. Furthermore, the slopes and therefore the diffusivities are essentially the same as those for the lower loading experiments, which is as expected since
it was shown previously that the benzene diffusivity is essentially independent of loading.10 The difference in the intercepts is due to differences in the equilibrium arising from competitive adsorption. The theoretical (solid) lines shown in Figure 4 are calculated from the simple counterdiffusion model (D0A and D0B independent of loading) using the parameters derived from our earlier measurements of tracer and transport diffusion of benzene and p-xylene in the same silicalite crystals (see Table 2).10 These curves provide quantitative a priori predictions of the benzene-p-xylene CCZLC response at both 6 and 58 Torr benzene partial pressures, thus confirming the validity of the simple diffusion model. The slow diffusion of o-xylene makes it difficult to obtain accurate equilibrium data. However, the available information from earlier gravimetric measurements15,16 suggests that the equilibrium isotherms for o-xylene and p-xylene are quite similar. The Henry
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Figure 6. Comparison of CCZLC desorption curves for p-xylene vs o-xylene (150 °C, 5 Torr, and 30 cm3/min purge flow rate) with NZLC and TZLC responses under similar conditions.
constants differ by no more than a factor of about 2 and the saturation capacities are almost the same. In the absence of more precise data we therefore assume that the equilibrium parameters for o-xylene and p-xylene are identical. The dotted and broken lines shown in Figure 4 are calculated on this basis from the simple counterdiffusion model assuming Dox/Dpx ) 0.1 and 0.01, respectively. Within this range the predicted CCZLC response curves are relatively insensitive to the diffusivity ratio. (At the higher loading the dashed and dotted lines are almost coincident with the benzenep-xylene predictions.) The predicted CCZLC response curve for benzene-o-xylene is slightly slower than that for benzene-p-xylene, but this curve does not match the experimental response which, particularly at the higher benzene loading level, is substantially slower. The effect of the slowly diffusing o-xylene in obstructing desorption of the faster moving benzene is evidently more pronounced than the model suggests. Matching the experimental CCZLC curves for benzene-o-xylene requires unrealistic values for the equilibrium parameters. We therefore conclude that, in contrast to the benzene-pxylene system, the simple diffusion model cannot properly represent the behavior of the benzene-o-xylene system. This is consistent with the observations of Baertsch et al.4 who, in permeation measurements with a silicalite membrane, found that the effect of o-xylene in retarding the permeation of p-xylene was greater than expected on the basis of single-component diffusion data. The anomolous behavior of o-xylene, which was originally reported some years ago by Beschmann et al.,17 is further confirmed by the form of the CCZLC response curves for the p-xylene-o-xylene system (Figures 5-7). CCZLC for p-Xylene/o-Xylene (or m-Xylene). Representative CCZLC response curves (p-xylene desorbing, o- or m-xylene adsorbing) are shown in Figures 5-7. Also shown for comparison are the NZLC and TZLC response curves for p-xylene under similar conditions. The response curves show remarkable consistency at all temperatures. It is clear that the effective intracrystalline diffusivity for p-xylene under counterdiffusion conditions, as measured by the slopes of the long time asymptotes, is almost the same as the self-diffusivity derived from the TZLC response. Diffusivity values for p-xylene derived from the slopes of the long time asymptotes for the CCZLC experiments (versus oxylene) are compared on an Arrhenius plot in Figure 8
with the NZLC and TZLC diffusivity data. As is evident directly from the response curves, the CCZLC and TZLC diffusivities are essentially the same and substantially smaller than the transport diffusivities derived from the NZLC experiments. In the initial region (first 20 s) the CCZLC response curves at a given flow rate are practically coincident with the NZLC and TZLC curves. This represents the washout period during which the cell is purged (with some desorption of p-xylene from the surface of the crystals). In the long time region the CCZLC curves approach a linear asymptote with a slope that is almost the same as that of the TZLC curves. However, while the NZLC and TZLC curves conform to the simple monotonic form predicted from simple diffusion models, the CCZLC curves show an abrupt drop in concentration at about 25 s, reminiscent of a percolation threshold. We do not fully understand the reason for this behavior, although the pattern was quite reproducible. It is evident that the behavior of o-xylene cannot be accurately described by the simple diffusion model. It appears that, at a certain critical uptake of o-xylene, which is reached after about 25 s, the fast path for desorption of p-xylene (presumably corresponding to diffusion through the straight channels) is shut off so that the remainder of the desorption process occurs at the same slower rate as that for tracer exchange. Various hypotheses can be constructed to explain these observations. For example, the sharp transition may correspond to the point at which all the straight channels are essentially blocked by at least one o-xylene molecule. Tentative support for this hypothesis again comes from the studies of Baertsch et al.4 who showed that the permeation rate data are consistent with singlefile diffusion behavior in which the flux is controlled by the slowest component. However, to prove or disprove such a hypothesis would require a much more detailed experimental study. Conclusions The experimentally observed behavior of the benzenep-xylene system in countercurrent ZLC measurements is quantitatively consistent with the predictions of the simple counterdiffusion model for a binary Langmuir system in which the flux is determined by the gradient of chemical potential and the corrected diffusivities (D0A and D0B) are independent of loading. This model provides an excellent prediction of the behavior of the binary system from the single-component kinetic and equilibrium parameters. o-Xylene and m-xylene diffuse much less rapidly than benzene or p-xylene. In contrast to the benzene-pxylene system, which conforms closely to the simple theory, the behavior of the benzene-o-xylene system is not quantitatively consistent with the simple counterdiffusion model. The effect of o-xylene in retarding the desorption of p-xylene is substantially greater than that predicted from the single-component diffusion and equilibrium data. Our earlier study10 showed an important qualitative difference between the (single-component) diffusional behavior of benzene and p-xylene. Benzene shows essentially ideal behavior in which the transport diffusivity at low loading coincides with the tracer selfdiffusivity. In contrast, for p-xylene, the self-diffusivity, although independent of loading, is substantially smaller than the transport diffusivity (see Figure 8). This
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Figure 7. Comparison of CCZLC desorption curves for p-xylene vs o-xylene (130 °C, 5 Torr, and 30 cm3/min purge flow rate) with NZLC and TZLC responses under similar conditions with (a) fresh and (b) aged adsorbent samples.
anomaly was tentatively ascribed to the difference in the lengths of these molecules. While benzene can exchange freely between the straight and sinusoidal channels, the longer p-xylene molecule can make this transition only with difficulty. This is essentially the theory suggested by Shen and Rees to account for the difference in the frequency response behavior of these species.18,19 In a transport experiment p-xylene therefore diffuses rapidly out of the system through the straight channels. However, in a tracer-exchange experiment, desorption is slower since the desorbing p-xylene is forced to move between the straight and sinusoidal channels to pass the incoming molecules. Benzene, which can move easily between the two-channel systems, impedes the desorption of p-xylene much less than does p-xylene itself. The CCZLC response curves for p-xylene desorbing against o-xylene are, in the long time region, almost identical to the tracer ZLC response under comparable conditions. This is quite consistent with the same conceptual model. o-Xylene blocks the desorption path
Figure 8. Arrhenius plot showing temperature dependence of corrected diffusivity (D0), tracer-exchange diffusivity, and countercurrent diffusivity for p-xylene and p-xylene/o-xylene in silicalite. The countercurrent diffusivity coincides with the tracerexchange diffusivity.
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through the straight channels, thus leading to a reduced p-xylene desorption rate similar to that observed earlier in a tracer-exchange experiment. However, the initial portion of the CCZLC curves for p-xylene-o-xylene shows a consistent deviation from the monotonic shape predicted from a simple diffusion model and observed for most systems. This together with the anomalous behavior of the benzene-o-xylene countercurrent system suggests that the behavior of o-xylene is not properly described by the simple diffusion model. Acknowledgment The financial support provided by Exxon-Mobil Cooperation and by the Petroleum Research Foundation, administered by the American Chemical Society, is gratefully acknowledged. Nomenclature bi ) Langmuir equilibrium constant for component i c ) sorbate concentration in gas phase li ) sorbate concentration of component i in gas phase c0 ) steady (initial) value of c C ) bc D ) diffusivity D0i ) limiting (corrected) diffusivity for component i D ) self-diffusivity F ) purge flow rate J ) diffusive flux Ki ) dimensionless Henry’s constant ()biqs) for component i L (Li) ) parameter defined by eq 3 qi ) adsorbed-phase concentration of component i qs ) saturation limit in Langmuir isotherm (eq 5) r ) radial coordinate R ) crystal radius R ) gas constant t ) time T ) absolute temperature Vs ) crystal volume Ri ) Di/Dy βn ) roots of eq 2 ξ ) r/R θi ) qi/qs (eq 5) θ h ) θA + θB µi ) chemical potential of component i τ ) dimensionless time variable (Dt/R2)
Literature Cited (1) Round, G. F.; Newton, R.; Habgood, H. W. Analysis of Surface Diffusion in a Binary Adsorbed Film. Sep. Sci. 1966, 1, 219.
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Received for review September 17, 1999 Revised manuscript received January 6, 2000 Accepted January 11, 2000 IE990691B
