Highlighting pitfalls in the Maxwell–Stefan modeling of water–alcohol mixture permeation across pervaporation membranes
Introduction
Microporous zeolite membranes have been applied on an industrial scale for production of fuel grade ethanol by pervaporation [1]. For modeling permeation of water–alcohol mixtures across microporous membranes in pervaporation and dehydration applications, the fluxes Ni are commonly related to the chemical potential gradients ▿μi by use of the Maxwell–Stefan (M–S) equations [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]:where ϕ represents the fractional pore volume of the microporous crystalline material, and the concentrations ci are defined in terms of accessible pore volume of the crystalline microporous layer. The fluxes Ni are defined in terms of the cross-sectional area of the crystalline framework. The xi in Eq. (1) is the component mole fractions of the adsorbed phase within the microporous structures:
The Ði characterize species i – wall interactions in the broadest sense. The Ð12 are exchange coefficients representing interaction between components i with component j. At the molecular level, the Ðij reflect how the facility for transport of species i correlates with that of species j. Conformity with the Onsager reciprocal relations prescribesFormally speaking, the M–S equations (1) serve only to define the phenomenological coefficients Ð1, Ð2, and Ð12. In practice, a number of assumptions and simplifications are commonly invoked in the application of Eq. (1) to the modeling of membrane permeation. These assumptions are listed below:
- (1)
The Ð1 and Ð2, can be identified with the corresponding M–S diffusivity for unary diffusion, evaluated at the same total loading or occupancy θt:where ci,sat is the saturation loading. This has the implication that the Ði in the mixture should be independent of the partner species. Put another way, for water(1)–methanol(2) diffusion, the value of Ð1 at any given mixture occupancy θt should be the same as for water(1)–ethanol(2) diffusion.
- (2)
While the dependence of Ð1 and Ð2 on the total loading ct = c1 + c2, or occupancy θt, are commonly accounted for in modeling exercises [7], [21], these coefficients are invariably assumed to be independent of the adsorbed phase composition, xi.
- (3)
The exchange coefficient Ð12 are not easily accessible from experiments. It is most commonly modeled using an interpolation formula that is based on the Vignes [22] model for diffusion in liquid mixtures:For diffusion within microporous materials, the Ðii are the self-exchange coefficients, that are determinable from MD simulations of unary diffusion of both Ði, and the self-diffusivity, Di,self, by use of the following relationship:The ratio Ði/Ðii can be viewed as a measure of the degree of correlations for unary diffusion of species i. As illustration, Fig. 1a presents MD data for CH4, water, methanol, and ethanol diffusion in FAU at 300 K. The degree of correlations increases with loading, ci. This is because the number of vacant sites within the structure decreases with increasing loading. Consequently, the number of times a molecule has to return to recently vacated sites increases with ci; this accounts for increasing correlations. The degree of correlations also depends on the pore size, connectivity and topology. Fig. 1b presents a comparison of Ði/Ðii data for CH4 diffusion in FAU, MFI, and LTA. The intersecting channel structures of MFI experience the highest degree of correlations. Since experimental data on the exchange coefficients are rarely available, it is quite common in the literature to invoke the assumption that Ði/Ðii ≈ 1, because of the lack of adequate information on the degree of correlations.
- (4)
In some special cases a further simplification is invoked with respect to the exchange coefficient Ð12, when modeling permeation across LTA and DDR membranes [5], [7]. For such zeolites, correlation effects are commonly assumed to be of negligible importance the windows separating the cages are in the 0.35–0.42 nm size range, allowing only one molecule at a time to hop from one cage to the neighboring one.
The main objective of this work is to highlight the break-down of all four of the afore-mentioned assumptions when describing permeation of water–alcohol mixtures across microporous membranes. To achieve these objectives Molecular Dynamics (MDs) simulations of diffusion of water–methanol, water–ethanol, and methanol–ethanol, mixtures in FAU, and LTA zeolites were carried out. Diffusion in fluid mixtures, without the restraining influence of pore-walls, was also investigated in order to obtain a clear scientific picture of the underlying physico-chemical principles. The current work is an extension, and amplification, of previous publications [23], [24], [25] in which the significant influence of molecular clustering on diffusion characteristics in microporous materials have been underlined. A small portion of earlier MD simulation results [25], have been included in the present work in order to provide a comprehensive palette of data and concepts that will be useful to a practicing membrane technologist for modeling purposes.
The simulation methodologies, along with details of the force fields used are exactly as that described in our earlier publication [25]; this information is not repeated here.
Section snippets
The Vignes interpolation formula
Let us begin by examining the validity of the Vignes interpolation formula (5) for water(1)–methanol(2), and water(1)–ethanol(2) mixtures in the liquid phase. MD simulations for the two binary mixtures are shown in Fig. 2a. The Vignes formula is seen to severely overpredict the variation of Ð12 with x1 for both liquid mixtures. Available experimental data confirm the trend portrayed in the MD simulations [26], [27], [28]. The failure of the Vignes interpolation is traceable to strong hydrogen
Is Ði in the mixture same as for pure component?
Fig. 5 presents data on MD simulated values of M–S diffusivities, Ði, of (a) water, (b) methanol, and (c) ethanol in equimolar (c1 = c2) water–methanol, water–ethanol and methanol–ethanol mixtures in FAU at 300 K. Also shown are the pure component Ði determined at the same total mixture occupancy θt in water–methanol and water–ethanol mixtures.
From Fig. 5a it can be observed that the Ði of water in mixtures with alcohols are significantly lower than that for pure water. The reason for this is that
Can we assume un-coupled diffusion for LTA and DDR?
For zeolites such as LTA and DDR the commonly made assumption in practice is that the correlation effects are of negligible importance [5], [7], and for modeling purposes it is common to invoke Eq. (8). In order to test this assumption we carried out MD simulations to determine Ð1, Ð2, and Ð12 for equimolar water(1)–methanol(2) mixtures in LTA for a variety of loadings at temperatures of 300 K and 350 K; see Fig. 7.
The simulations show that Ð12 is intermediate in value between Ð1 and Ð2 and the
Conclusions
For water–alcohol mixtures, the hydrogen bonding between water and alcohol molecules is much stronger than for water–water, and alcohol–alcohol pairs. This leads to a break-down of some commonly made assumptions in the Maxwell–Stefan modeling of membrane permeation.
- (1)
The Maxwell–Stefan diffusivity, Ði, of either component in water–alcohol mixtures are lower than the corresponding values of the pure components. In practice we need to take account of the influence of mixture composition on the Ði
Acknowledgements
RK acknowledges the grant of a TOP subsidy from the Netherlands Foundation for Fundamental Research (NWO-CW) for intensification of reactors.
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