185
Znd. Eng. Chem. Res. 1995,34, 185-191
Adsorption of Branched and Cyclic Paraffins in Silicalite. 2. Kinetics CBlio L. Cavalcante, Jr. and Douglas M. Ruthven* Department of Chemical Engineering, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Intracrystalline diffusivities in silicalite have been measured by the gravimetric sorption uptake method for several branched and cyclic Cg paraffins (2-methylpentane73-methylpentane72,2dimethylbutane, 2,3-dimethylbutane7methylcyclopentane, and cyclohexane). The zero-length column (ZLC) method was also used to check and validate some of the results. Branching and cyclization of saturated hydrocarbon structures lead to significant changes in critical molecular diameter and hence to differences in kinetic behavior. The following general trend was observed for the diffusivities: linear > single-branched > double (ternary C)-branched > cyclic paraffins > double (quaternary C)-branched. Steric hindrance was most pronounced for compounds with a quaternary carbon atom, such as 2,2-dimethylbutane. The kinetic data conform well to a one-dimensional (slab) diffusion model, reflecting the anisotropic diffusion behavior characteristic of the silicalite/ZSM-5 structure.
Introduction In the previous paper (Cavalcante and Ruthven, 1995) we reported equilibrium data for the CSparaffin isomers in silicalite; we report here kinetic data for the same sorbates (2-methylpentane (2MP), 3-methylpentane (3MP), methylcyclopentane (MCP), cyclohexane (cycloCS), 2,2-dimethylbutane (22DMB), and 2,3-dimethyl butane (23DMB). The molecular dimensions of these species are summarized in Table 1 of part 1. Since the critical molecular diameters are all close t o the free diameter of the silicalite channels (-6.0 A), large steric effects are t o be expected. In this regime, sometimes referred to as "hindered diffision" diffisivities are small ( < 10-lOcm"s-l) and small differences in critical diameter are accompanied by large changes in diffusivity while molecules exceeding a certain critical diameter are effectively excluded. Such a diffusional cutoff has been reported for both zeolite A (Breck, 1974) and silicalite (Harrison et al., 1984).
using an Edwards MK2 Diffstack high vacuum pump. Following an incremental step change in sorbate pressure, the increase of sample weight with time was followed by means of the data acquisition system. The collected data were later manipulated by available spreadsheets and graphics software to obtain and analyze the fractional uptake (mJm-1 versus time curves. For an isothermal system in which the uptake rate is controlled by intracrystalline diffusion, and there is only a small change in the adsorbed phase concentration (D,can be assumed as constant), the solution for the transient diffusion equation for a spherical particle in terms of the uptake of sorbate by the solid assumes the well-known form given by Crank (1975): "t 9-90 -_--
mw
Qm-90
- ~ - - x6 ; e x-p 1 --
2 n=l n
[ n2y] (1)
A simplified and convenient solution for short times is
Theory and Experimental Section Gravimetric uptake measurements were made using a high-sensitivity microbalance (Cahn 2000 RG) in order to minimize the quantity of sample required for the diffusion measurements. Even though the sample weights were generally about 10 mg, great care was still required to minimize the intrusion of extracrystalline resistances. To confirm intracrystalline diffusion control, measurements were repeated, under similar conditions, with different sample quantities and configurations and with different zeolite crystal size fractions. The major advantages of this method are its simplicity and the simultaneous determination of the equilibrium isotherm. The electrobalance was connected through a data acquisition board ADC-1 to a Macintosh personal computer, with data acquisition software ADControl. The microbalance weight measurements were accurate to about mg. Sorbate partial pressures were monitored through a Datametrics transducer with a range of 0-100 Torr absolute. The zeolite sample was normally precalcined at 500 "C, and prior to each experimental run, the sample was degassed by heating a t 400 "C under a vacuum of approximately 10-4-10-5 Torr, 0888-5885I95/2634-0185$09.0~l0
(Kiirger and Ruthven, 1992):
It can be seen from eq 2 that, for short times, a plot of mJm.. versus the square root of time should be linear in the initial part of the curve and the slope yields the difisivity D,. For long times, the solution is of the form (Kiirger and Ruthven, 1992):
(3) It is evident from eq 3 that a plot of In (1 - mJmd versus time will approach a straight line in the long time region, and the slope of this line yields directly the intracrystalline diffusion coefficient. These simple patterns of behavior can be obscured by heat effects or when the sample contains a wide distribution of crystal size, but they nevertheless provide a reasonable basis for the analysis of transient uptake curves. The short time response is less susceptible to thermal effects or
0 1995 American Chemical Society
186 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 the distribution of adsorbent particle size, so it was mainly used in this study. Calculations for the long time region plot were performed mainly to check the consistency of the diffusion model. For slab-shaped particles, the complete solution to the diffusion model is (Crank, 1975)
m, mm
-
4 -Qo
-
4”-40
The short time solution is
-_m”
&
1
’
1
F 1:
’
20
0
1
.
*
1
I
‘
80
60
40
]
100
Time (min)
(5)
where 1, is the half-thickness of the slab and, for long times,
(6) By comparing eqs 3 and 6 , it may be seen that the intercept of the long time region plot at time zero provides an indication of the main path of diffusion in the crystal; either three-dimensional (spherical)or onedimensional (slab). The intercept will be close t o 8/n2 for the one-dimensional model and close to 6/n2for the spherical model. For most of the systems studied throughout this work, the one-dimensional model appeared to be the more appropriate. This is to be expected if diffusion occurs predominantly through the straight channels. An illustration of the procedure used is shown in Figure 1 for one specific run. In a gravimetric experiment the equilibrium isotherm is determined simultaneously with the kinetic data so the “corrected” diffusivity (Do)can be easily obtained by means of the Darken equation (Kiirger and Ruthven, 1992):
(7) The ZLC method was also employed, mainly to confirm some of the gravimetric results. Its main advantages are the use of very small amounts of sorbents (1-2 mg) for the intracrystalline diffusion measurements and a relatively large inert purge flow rate through the system, thus minimizing extraneous heat and mass transfer effects. The experimental procedure and theoretical analysis have been previously described elsewhere (Eic and Ruthven, 1988; Voogd et al., 1991; Hufton and Ruthven, 1993). In general, good agreement was observed between the gravimetric and ZLC data. The silicalite samples used for these studies were the same as those used in part 1; their main dimensions are summarized in Table 1. Scanning electron micrographs of two of the samples (Figure 2) show the typical elongated prismatic form of the relatively well formed silicalite crystals with a relatively narrow size distribution. Some of the samples show significant twinning, but the evidence from the diffusion rate measurements suggests that this has little effect since the time
0.01 20
0
60
40
80
Time (min)
Figure 1. Uptake curve for cyclohexane in silicalite (crystal size 66 x 66 x 223 pm) at 17 Torr,300 “C, showing conformity with the one-dimensional diffision model. Table 1. Silicalite Samples and Average Crystal Sizes
X sample A B C D E
Olm) 66 25 95 45 1.9
Y bm) 66 35 95 45 1.9
2 Olm) 223 65 265 95 2.4
eq rad Olm) 43 18 60 27 1.0
half-thick Olm) 33.0 17.5 47.5 22.5 0.95
constants were similar to those from untwinned samples of comparable size.
Results and Discussion The one-dimensional slab model, with the characteristic length taken as the half-thickness of the crystal, was employed for all systems. It was observed, in general, that this model provides a more adequate representation of the experimental data than the isotropic spherical model, which was also tested. This is in agreement with the anisotropic nature of silicalite, and with the conclusions reached in earlier studies with other sorbates (Ruthven et al., 1990; Ruthven et al., 1991; Hong et al., 1991; Car0 et al., 1992). Diffisivity values for diffision in the transverse direction were reported to be about 4-5 times higher than for diffision in the longitudinal direction (Hong et al., 1991). As with the equilibrium results in part 1, the diffusivities are first presented for each structural group of sorbates and comparisons are then drawn between the different groups.
Ind. Eng. Chem. Res., Vol. 34, No. 1,1995 187
t
1.00E.11
1
.
0.0
.
.
I
.
.
I
3.0
2.0
1.0
. 5.0
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i
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.+
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.
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I
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0.0
.
, .
.
I
, . I
.
3.0
2.0
5.0
4.0
6.0
9 (4LJI)
Figure 3. (a) Effktive and (b) corrected diffisivities for cyclohexane in sample A showing dependence on concentration(filled symbols, desorption;open symbols, adsorption).
Figure 2. Scanning electron micrographs for samplesA (top)and B (bottom).
Cyclic Paraffins. Diffisivity data at different temperatures were obtained for cyclohexane (cyclo-c6) and methylcyclopentane (MCP). A representative gravimeti ric uptake curve for cyclohexane in silicalite is shown in Figure la, for particular conditions of temperature, sorbate pressure, and crystal size. The theoretical curve, calculated according to eq 4 with the experimental value of diffusivity, is also shown; it is evidently in good agreement with the experimental data. In Figure lb, a plot of I n (1 - mJm,) versus time is shown, confirming the superiority of the slab diffision model in preference to the spherical model, since the intercept is closer to 8/n2(slab model) than to 6/n2(spherical model). The variation of diffusivity with loading, at several different temperatures, is shown for cyclohexane in Figure 3a for a crystal of 66 x 66 x 223 pm (sample A). No significant trend with increasing loading is observed. calcuThe corrected intracrystalline diffusivities (DO), lated from the effective diffusivities and the slopes of the isotherms reported in part 1, according to Darken's equation (eq 7), are shown in Figure 3b. A slightly decreasing trend with adsorbed phase concentration may be observed for the corrected diffisivities, which is not unusual for zeolite systems (Kiirger and Ruthven, 1992). It can also be seen that, at zero loading, the effective diffisivities are quite similar to the corrected diffusivities, as is to be expected since, at
1
L
-,
1.00E-08
WuelrL(I0J
0
1.00€.09-:
A
x
I.OOE-lO-.
3
8
0
l.OOE-ll-l
0 X
X X
c
1.00E-141.5
2.5
2.0
D
lOOO/r
Figure 4. Arrhenius plot showing temperature dependence of intracrystalline diffisivities for cyclohexane in silicalite.
low concentrations, the correction fador d In p/d In q approaches unity (Henry's law region). Plotting the corrected diffisivities DO versus the inverse of the absolute temperature (Figure 4), a straight line is observed, following the Arrhenius law:
For cyclohexane, the energy of activation, estimated from the slope of this plot, is approximately 12.8 kcaV
188 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
\
Table 2. Activation Energies for Diffusion of Cyclohexane in SilicaliteIZSM-5
E, (kcal/mol) 12.8 15.5 12.1 8.1
this study Xiao and Wei (1992) Chon and Park (1988) Wu et al. (1983)
Table 3. Comparison of Diffusivity Data for 3-Methylpentane and 2-Methylpentane
-.
1.00E-09
1.00E-10-
DO(cm2/s) temp silicalite ("C) (95 x 95 x 265 bm)" 3MP 2MP
23 30 50 80 150
1.54 x 1.20 x 3.71 x 1.59 x 2.10 x
0
ZSM-5 d = 6 bmb 6 = 12 bmc 1.23 x
lo-@
1.88 x 4.70 x 1.53 10-9 1.28 x
a This study. Kulkarni and Anthony (1991). Xiao and Wei (1992). From extrapolation.
mol. Experimental data from other authors (Chon and Park, 1988; Wu et al., 1983; Xiao and Wei, 1992) are also included in Figure 4 and Table 2. In general, the results are within what may be considered as reasonable agreement for data for different samples and from different laboratories. The results of Wu et al. (19831, for 2 pm crystals of silicalite, suggest a much lower energy of activation (Table 21, but this was derived from only two experimental points (at 100 and 200 "C). Their diffusivity value at 100 "C, calculated using a spherical diffusion model, shows good agreement with our results (extrapolated to that temperature), but their value at 200 "C is somewhat lower than our value for that temperature. Chon and Park (1988) report about the same energy of activation (12.1 kcal/mol) for cyclohexane diffusion in small crystals of ZSM-5 (with thickness 0.3-0.8 pm), but somewhat lower diffusivity values than we obtained, when extrapolated to their temperature range (75- 115 "C). For evaluation of their data, they used the same one-dimensional diffusion model as was used in our calculations. An interesting point from Chon and Parks results is that methylcyclohexane appears to diffuse slightly faster than cyclohexane. This difference may be within the experimental scatter or it may be a real effect arising from an orienting effect of the methyl group attached to the bulky cyclic ring. Diffusivities for 1,4-dimethylcyclohexanewere found to be about two orders of magnitude higher than the values for cyclohexane, which may support the idea of an orienting effect of methyl groups attached t o a ring. Our diffusivity results are about an order of magnitude higher than the values reported by Xiao and Wei (1992) for a ZSM-5 sample (crystals of about 12 pm diameter), especially at lower temperatures, but their reported energy of activation is much higher than our value (see Table 2). Some of the difference in diffusivity values may be ascribed to the use of different diffusion models in the analysis of the experimental uptake curves. In general, the slab model yields diffusivity values about 4 times higher than the values obtained from the spherical model, when applied t o the same set of experimental data. Also, the results reported by other authors were obtained using zeolite samples with smaller crystals than we used in the present study. However, the relatively large differences observed in activation energies would not be explained by the choice of diffusion
1.00E-13 1.6
,
, 1.8
.
I
.
2.0
, 2.2
.
. 2.4
6
IMx)/T
Figure 5. Arrhenius plot showing the comparison between diffisivity values for different crystal sizes (cyclohexane in silicalite).
model and may suggest the intrusion of extraneous resistances to mass transfer for the smaller crystals. To confirm the absence of extraneous effects (such as heat transfer or extracrystalline resistance to mass transfer) in our experiments, measurements were repeated with other crystal sizes: 95 x 95 x 265 pm (sample C) and 1.9 x 1.9 x 2.4 ,um (sample E). The results are compared in Figure 5, from which it is clear that the diffusivities obtained with the larger crystals are quite consistent. However, as is commonly observed for other systems, the diffusivity for cyclohexane in the smaller crystals of silicalite is about 1order of magnitude lower than the values measured for the large crystals. It is not clear if this difference is due to a mass transfer resistance at the crystal surface (which would be more important for smaller crystal sizes). The other possibility, the intrusion of heat transfer resistance, does not seem plausible, since the diffision of cyclohexane in silicalites is very slow so the process should be almost isothermal. As can be seen from Figure 1, a typical uptake measurement takes about 1 h, even a t the highest temperature employed (300 "C). It is worth pointing out that our results for the smaller crystals are in relatively good agreement with the other authors' data, previously referred to, which were also obtained with small crystals. The variation with loading of both effective and corrected diffusivities for methylcyclopentane in silicalite sample C (95 x 95 x 265 pm) versus sorbate loading is shown in Figure 6 for 150, 175, and 205 "C. The effective diffusivities appear to be almost constant at the higher temperature, but show an increasing trend at lower temperatures. However, as with cyclohexane the corrected diffusivities show a regular decrease with increasing solid phase concentration. The activation energy calculated according to the Arrhenius equation (eq 8) is approximately 11.7 kcaymol, which is essentially the same as the value obtained for cyclohexane. This is somewhat surprising since the critical molecular diameter of methylcyclopentaneis significantly smaller than that of cyclohexane and, indeed, the actual diffusivities for methylcyclopentane are about two orders of
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 189 205 C
1.00E-08
..
10
0
0 0
u -
0
[3 1 7 5 C
n
-k
.
-
* 4 AgA-
150
'AA
A
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c
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,
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o
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0
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t a
.
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q
1
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,
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,
,
,
, , ,
,
2.0
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6.0
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(wg)
Figure 6. (a)Effective and (b)corrected diffisivities for methylcyclopentane in silicalite (sample C) showing concentration dependence.
magnitude larger than the values for cyclohexane. Although the critical molecular diameter plays a key role in determining the intracrystalline diffisivity and activation energy it is evidently not the only significant factor. Chon and Park (1988) suggested that a methyl group, attached to a saturated ring as in MCP, exerts a beneficial orienting effect, and our results appear to support this idea. Single-Branched Paraffins. Diffusion measurements were carried out for 2-methylpentane (2MP) and 3-methylpentane (3MP) in silicalite crystals with dimensions 95 x 95 x 262 pm. The objective was to determine the effect of the methyl group location in the main chain on the diffusivity. The effective and corrected diffusivities again showed a slight variation with adsorbed phase concentration (Cavalcante, 19931, as observed previously for the cyclic compounds. The corrected diffusivities are plotted against reciprocal temperature in Figure 7. It is observed that the temperature dependence of the diffusivities conforms well to the Arrhenius law (eq 8). Diffusivity values are, as expected, considerably higher than for the cyclic paraffins, reflecting the smaller critical diameters of these molecules (see Table 1 of part 1). It can be seen that diffusion of 2MP is on average about twice as fast as 3MP, with a lower diffusional activation energy (11.0 kcdmol for 2MP vs 13.8 k c d mol for 3MP). This may be explained by the longer tail in the 2MP molecule, providing a somewhat more favorable orientation for the molecule through the silicalite channels as compared to 3MP, a less asymmetric molecule. Diffusion data for 2MP in ZSM-5 have been reported by Xiao and Wei (1992). Their values compare quite
Figure 7. Arrhenius plot showing temperature dependence of intracrystalline dfisivities of 2-methylpentane and 3-methylpentane in silicalite. Table 4. Diffusivities (cm2/s)for 2,3-Dimethylbutanes in Silicalite for Different Samples and Experimental Methods temp ("C) sample uptake ZLC 110 E 7.0 x 10-l' 150 E 5.0 x D 2.8 x 200 E 2.2 10-9 245 250 280
D A A
8.4 x 1.1 10-9
2.3
10-9
8.1
10-9
D
A
2.7 x 10-9
well with our results, as may be seen from Table 3. It is remarkable and somewhat gratifying that this good agreement was observed for samples with quite different crystal sizes. Table 3 also shows results for 3MP in HZSM-5 at a much lower temperature (Kulkarni and Anthony, 1991). Extrapolation of our data to that temperature shows a difference of about 2 orders of magnitude. "his may due to the extrapolation over such a large temperature range (-75 OC) or to the difficulties of measuring reliable diffusivities at that low temperature, where the intrusion of external (bed diffusion) resistance will be greater. Double-Branched Paraffins. Gravimetric experiments were performed for 2,3-dimethylbutane (23DMB) and 2,2-dimethylbutane (22DMB) with samples A (66 x 66 x 223 pm) and E (1.9 x 1.9 x 2.4 pm), at temperatures between 100 and 250 "C. The individual results including the concentration dependence of the measured effective and corrected diffusivities are available elsewhere (Cavalcante, 1993). Some experiments were also made by the ZLC method for 23DMB in sample D (45 x 45 x 95 pm), confirming the earlier gravimetric results. The results for the ZLC runs are summarized in Table 4, along with the gravimetric uptake results. The good agreement of the results is better seen in Figure 8, which shows the temperature dependence of the diffusivities of 23DMB in the different silicalite samples, together with the values obtained for 22DMB. As expected, the diffusivity values for the double-branched paraffins are much lower than those for the single-branched compounds, due to the increased
190 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
-.
1 OOE-08
X
22DMB-S.mple E
3
B
Y
Cp
1 OOE-09-
. 100E.100
i I 0OE.l I
25
2.0
100E.12 1 5
I WE.12
IOOOIT
Figure 8. Arrhenius plot showing temperature dependence of intracrystalline diffisivities of 2,3-dimethylbutane and 2,2-dimethylbutane in silicalite. Table 5. Diffusivities and Activation Energies of Dimethylbutanes in Silicalite/ZSM-5 this study temp DO ("C) (cm2/s) 22DMB 23 1 x 150 9 x 23DMB
150 4.5 x 200 2.5 10-9
22DME
literature
Ea Do (kcal/mol) (cm2/s) 17.4 6x 6x 1 10-13d 13.8 2.5 x 2.7 x
Ea (kcaumol) 16.7 15.8 14.4 12.4
a Extrapolated from Figure 8. * From Kulkarni and Anthony (1991). From Xiao and Wei (1992). From Post et al. (1983). e From Voogd and van Bekkum (1991).
steric constraint caused by addition of one more methyl group to the main paraffin chain. Most interestingly, it is observed that the diffisivities for 23DMB are about 2 orders of magnitude larger than the values for 22DMB. This is clearly due t o the quite different critical diameters of 23DMB (5.8 A) and of 22DMB (6.3 A). These dimensions, being so close to the silicalite pore openings, clearly dominate the diffusion process through the channels. The quaternary carbon, which is present in the 22DMB molecule, severely restricts the passage of that molecule through the narrow silicalite channels. From Figure 8, activation energies can be estimated, using eq 8. In Table 5 a comparison of the diffusivity and activation energy values obtained in this study with values previously reported for similar systems (Xiao and Wei, 1992; Post et al., 1983; Voogd and van Bekkum, 1991; Kulkarni and Anthony, 1991) is shown. All results are in relatively good agreement, except for the value reported by Kulkarni and Anthony (1991) at 23 "C for comparison with which the values from the present study had to be extrapolated over a large temperature range. Again, it should be noted that some minor discrepancies can be attributed to the use of the spherical diffusion model for the other reported values.
160
1.80
2.00
2.20
2.40
2.60
2.80
3.00
3.20
0
lO0Orr
Figure 9. Arrhenius plot showing comparison between corrected diffisivities for linear, singly-branched, doubly-branched, and cyclic paraffins in silicalite.
Conclusions A general overview of the intracrystalline diffisivities in silicalite for the c6 cyclic, single-branched, and double-branched paraffins is shown in Figure 9, which also includes the data from earlier studies with linear paraffins. It can be seen that diffusivities decrease substantially with an increase in the critical molecular diameter, from the linear paraffin t o the doublebranched isomers. This is better illustrated in Figure 10, where the diffusivities at 150 "C are plotted against the critical molecular diameter, showing the cutoff point at a diameter in the region of the silicalite channel dimensions (5.5-6.0 A). Steric hindrance was more pronounced for compounds with a quaternary carbon atom, such as 2,2-dimethylbutane. Cyclohexane, which has a larger critical molecular diameter than 2,2-dimethylbutane, diffuses more rapidly. However, the cyclohexane structure is more flexible and, as a result, the steric hindrance is evidently less pronounced. In general, the following trend for the diffusion of (26 paraffins in silicalite was observed: linear single-branched > double (ternary C)-branched > cyclic > double (quaternary C)-branched paraffins. Another important observation from the results plotted in the form of Figure 9 is the possibility of separating the various isomers commonly present in a Cg refinery stream on the basis of their kinetic behavior. For example, the separation of linear and single-branched paraffins from the cyclic and double-branched compounds would certainly be attractive for increasing the antiknock properties of automotive fuel, without the use of aromatic compounds, which have been heavily regulated by environmental agencies. It is evident from Figures 9 and 10 that the use of silicalite could be appropriate for that purpose, since its channel dimen-
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 191 1.00E-06
go = adsorbed phase concentration at time zero, % glg
I R
= gas constant r, = crystal radius, cm
T = absolute temperature, K t = time, s
1.00E-07
2MP 0
5-
Literature Cited
nC6
I.WE-08
3 I U
I.OOE-09-.
n
cyclo
C6
I.OOE-lO-.
0
1.00E-ll-.
0 22DMB
4.0
5.0
Molccule Critical Diamcler (A)
Figure 10. Variation of corrected diffisivity (at 150 “C) with critical molecular diameter.
sion is quite similar to the “cutoff’ point for single- and double-branched compounds. The diffisivity difference between the slowest diffusing single-branched paraffin studied ( 3 M p )and the fastest double-branched (2,3DMB) is still about 1 order of magnitude, which should be sufficient to yield an attractive selectivity factor for an industrial separation of a stream enriched in doublebranched and cyclic compounds. The above conclusion is based on simple singlecomponent static measurements at relatively low sorbate concentrations. No account has yet been taken of the presence of other components, nor has the concentration level been raised to the range of interest in any practical application. In general, the single-component measurements provide a good indication of the possibility of achieving an effective separation. However, multicomponent measurements at higher concentrations are obviously required before proceeding to any possible industrial application of these ideas.
Acknowledgment The silicalite crystals were kindly supplied by Prof. David Hayhurst (University of South Alabama) and Union Carbide. Financial support from CAPES (Brazil) and NSERC (Canada) is gratefully acknowledged.
Nomenclature D = Fickian transport diffisivity, cm2/s
DO= corrected transport diffisivity, cm2/s Do*= pre-exponential factor in Arrhenius law (eq 8) D, = intracrystalline difisivity, cm%
De = effective diffusivity, cm2/s E, = energy of activation, kcdmol K = dimensionless Henry‘s law equilibrium constant 1, = crystal half-thickness (one-dimensional slab model), cm mt = mass adsorbed at time t, g m, = mass adsorbed at time t,, g p = sorbate partial presure, Torr q = adsorbed phase concentration, % glg
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Received for review January 21, 1994 Revised manuscript received August 25, 1994 Accepted September 1, 1994@ IE940039A Abstract published in Advance ACS Abstracts, November 1, 1994. @
