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Ind. Eng. Chem. Res. 1997, 36, 1882-1886
GENERAL RESEARCH Difference of Diffusivities in Zeolites Measured by the Non-Steady-State and the Steady-State Methods Wugeng Liang,* Songying Chen, and Shaoyi Peng Institute of Coal Chemistry, The Chinese Academy of Sciences, Taiyuan, Shanxi 030001, People’s Republic of China
Diffusion coefficients reported for zeolites have ranged over a wide span of magnitude depending on methods of measurement. We have analyzed determinations made by various investigators and methods on a number of zeolites and guest molecules. When immobilization of a fraction of molecules is taken into account, the corrected diffusivities are consistent regardless of method and a unifying picture of diffusivities, valid for catalytic processes, is obtained. Introduction Catalytic and separation processes employing zeolite molecular sieves can be significantly influenced by the diffusivities of the participating molecules. Indeed, the magnitudes and differences in their diffusivities can be the controlling parameters in shape-selective catalysis (Weisz and Frilette, 1960; Chen and Weisz, 1967; Weisz, 1980; Csicsery, 1984) and in separation processes (Ruthven, 1984). During the past decades, many experimental determinations of diffusion coefficients in zeolites have been reported. Most of them have employed non-steady-state methods such as the gravimetric measurement of uptake, chromatography, or the zero length column (ZLC) procedure. Some methods have involved a steady state, such as some NMR techniques (Karger and Ruthven, 1989; Karger and Ruthven, 1992) or diffusivities derived from the catalytic utilization factor (Haag et al., 1980; Garcia and Weisz, 1993a). In some systems, such as methane-4A (Allonneau and Volino, 1986; Yucel and Ruthven, 1980a), propane5A (Karger and Ruthven, 1981), and n-butane-5A (Karger and Ruthven, 1981; Yucel and Ruthven, 1980b), good agreement exists between the diffusivities obtained by the two kinds of method. However, in some other systems, for example, benzene-NaX (Eic et al., 1988; Karger and Pfeifer, 1987), xylenes-NaX (Karger and Pfeifer, 1987; Goddard and Ruthven, 1986), and n-butane-NaX (Doelle and Riekert, 1977; Ruthven and Doetsch, 1976; Karger et al., 1980), a wide span of magnitude differences exist in experimental diffusivities obtained by the two kinds of method. There has been no reason to assume these differences result from any experimental inaccuracies (Karger and Ruthven, 1989, 1992). Various attempts to explain the differences have failed to produce a universal resolution. Traditionally, the non-steady-state methods use Fick’s second law to simulate the uptake curves to obtain the transport diffusivities (Crank, 1975). Fick’s classical second law is valid when all the molecules that exist in the given system are participating in the concentration * Author to whom correspondence should be addressed. Present address: Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6. FAX: 403-492-2881. E-mail:
[email protected]. S0888-5885(96)00493-9 CCC: $14.00
gradient. However, it has been pointed out (Weisz, 1967; Weisz and Hicks, 1967) that when adsorption occurs during transport within a microporous structure, there can exist a large number of immobilized molecules in any volume element at any moment of time. Since these do not contribute to the concentration gradient of the mobile molecules, it is necessary to revise the classical second law of Fick accordingly (Weisz, 1967; Weisz and Hicks, 1967). This has been illustrated to apply to the transport into a zeolite in a study of aromatic molecules in zeolite ZSM-5 (Garcia and Weisz, 1990; Garcia and Weisz, 1993b). In this work, the diffusivities obtained for a number of zeolites and molecular species are corrected using the revised diffusion equation to obtain the actual diffusivities. By comparing these diffusivities with diffusivities obtained by the steady-state method, the effect of the accumulation of the immobile molecules on diffusion in zeolites is further demonstrated. Analysis of the Two Kinds of Experimental Method Non-Steady-State Methods. The non-steady-state methods for measuring intracrystalline diffusivity depend on measuring the flux into or out of a zeolite crystal under well-defined boundary conditions. The diffusivity is then calculated by matching the experimental flux or uptake rate to the appropriate theoretical solution derived from the classical Fick’s diffusion equations (Crank, 1975). When in the pores of a zeolite some or all molecules immobilized at internal sites (Garcia and Weisz, 1990; Weisz, 1995), the molecules transported into the pore of the zeolite should equal the total molecules including both the molecules in the mobile state and the molecules in the immobilized state; as a result, the mass balance for a spherical particle is
(
D
)
∂c ∂q ∂2c 2 ∂c + ) + 2 r ∂r ∂t ∂t ∂r
(1)
where ∂c/∂t refers to the mobile molecules in the pore, ∂q/∂t to the immobile molecules in the pore, and D is the actual molecular diffusivity. © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1883
With
∂q ∂q ∂c ) ∂t ∂c ∂t
(2)
we obtain
D
(
) (
)
∂q ∂c ∂2c 2 ∂c + ) 1+ 2 r ∂r ∂c ∂t ∂r
(3)
wherein ∂q/∂c reflects the nature of the adsorption isotherm, if the local adsorption equilibrium is attained rapidly compared to the time scale of the diffusion (uptake) process (Crank, 1975). For the case of a linear isotherm, using Henry’s adsorption constant, K, we have ∂q/∂c ) K, including K ) qf/c0, where qf is the final uptake amount for the applied concentration c0. Equation 3 then takes the form
(
)
D ∂c ∂2c 2 ∂c + ) 2 1 + qf/c0 ∂r r ∂r ∂t
(4)
Figure 1. Comparison of the apparent diffusivity, the actual diffusivity, and the NMR diffusivity of benzene over NaX at 403 K and under different adsorbate concentrations (GM, results obtained by gravimetric method; NMR, results obtained by NMR method; NM, the actual diffusivity obtained after correction of GM or ZLC; ZLC, results obtained by the zero length column method).
The classical Fick solution will therefore yield an apparent diffusivity Dapp according to
Dapp
(
)
∂c ∂2c 2 ∂c + ) 2 r ∂r ∂t ∂r
Dapp )
D 1 + qf/c0
(5) (6)
Thus the apparent transport diffusivity Dapp is directly obtained from the uptake curve when the immobilization phenomenon is ignored, while D is the actual transport diffusivity of the mobile molecules in the pores. For nonlinear adsorption isotherms, the correcting eq 6 is still applicable within a maximal error of 1.6 (Weisz, 1967, 1995; Garcia and Weisz, 1990). When there are immobile molecules in the pore of zeolite, the apparent diffusivity obtained by the non-steady-state method should therefore be corrected with eq 6 to obtain the actual molecular diffusivity. In some cases, the diffusivity reported from nonsteady-state method has already been corrected using Darken’s equation to correct for nonlinearity of the adsorption isotherm:
d ln q d ln c
(7)
1 Dcor ) Dapp 1 + Kc
(8)
Dcor ) Dapp For the Langmuir isotherm,
From eq 6,
1 + K + Kc 1 + Kc
(9)
D ) (1 + K + Kc)Dcor
(10)
D ) Dapp Then, we have
Therefore, for the corrected diffusivity with Darken’s equation, the actual diffusivity is obtained from eq 10. Steady-State Methods. The steady-state methods measure the molecular diffusivity in NMR techniques that measure the rate at which tagged molecules
Figure 2. Comparison of the apparent diffusivity, the actual diffusivity, and the self-diffusivity of xylenes over NaX at 443 K and low adsorbate concentration (GM, results obtained by gravimetric method; NMR, results obtained by NMR method; NM, the actual diffusivity obtained after correction of GM).
migrate after equilibrium conditions of sorption have been attained. The immobile molecules make no contribution to the signals obtained. Therefore, the diffusivity obtained by such NMR methods should not have to be corrected. This also applies to derivation of diffusivity from measurements of the catalytic effectiveness factor (Haag et al., 1980; Garcia and Weisz, 1993a) since these are obtained long after initial sorption equilibrium is attained. Results and Discussion Analysis of the Systems in Which Significant Differences in Diffusivities Exist. Experimental results indicate that the disagreements in the diffusivities measured by the two kinds of method exist in the following systems: (1) benzene-NaX (Eic et al., 1988; Karger and Pfeifer, 1987); (2) xylenes-NaX (Karger and Pfeifer, 1987; Goddard and Ruthven, 1986); (3) normal paraffins-NaX (Doelle and Riekert, 1977; Ruthven and Doetsch, 1976; Karger et al., 1980); (4) n-propanesilicalite (Caro et al., 1985; Eic and Ruthven, 1989). In the following, with the model presented above, the disagreements in these systems are discussed. Benzene-NaX System. Eic et al. (1988) have investigated the diffusion of benzene in larger NaX
1884 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 1. Systems in Which Apparent Diffusivities Are Consistent with NMR “Self-Diffusivities”a adsorbate
a
sorbent
methane
4A
propane
5A
n-butane
5A
CF4
5A
triethylamine
NaX
technique
T (K)
Dapp or Ds (cm2/s)
references
GM NMR GM NMR GM NMR GM NMR GM NMR
300 300 435 435 400 400 473 473 445 445
5 × 10-11 3.8 × 10-11 2 × 19-9 2 × 10-9 9 × 10-9 8 × 10-9 1.7 × 10-8
