Maxwell membrane permeation

The generalized Maxwell-Stefan equation provides a rational basis for the analysis of sorption rate measurements and membrane permeation in multi-component systems. For a binary Langmuir system ... 

Using the Maxwell-Stefan theory, we will closer investigate the influence of isotherm inflection on the diffusivity. When we assume that the Maxwell-Stefan diffusion coefficient (sometimes also called the corrected diffusion coefficient) is independent of the loading, the loading dependence of the conventional Pick diffusion coefficient will be completely determined by the adsorption isotherm. We will demonstrate that on the basis of mixture isotherms we can predict the membrane permeation efficiency without having to know the diffusion coefficients exactly. 

Mixed-matrix membranes have been a subject of research interest for more than 15 years [28-33], The concept is illustrated in Figure 8.10. At relatively low loadings of zeolite particles, permeation occurs by a combination of diffusion through the polymer phase and diffusion through the permeable zeolite particles. The relative permeation rates through the two phases are determined by their permeabilities. At low loadings of zeolite, the effect of the permeable zeolite particles on permeation can be expressed mathematically by the expression shown below, first developed by Maxwell in the 1870s [34],... 

Krishna and Paschek [91] employed the Maxwell-Stefan description for mass transport of alkanes through silicalite membranes, but did not consider more complex (e.g., unsaturated or branched) hydrocarbons. Kapteijn et al. [92] and Bakker et al. [93] applied the Maxwell-Stefan model for hydrocarbon permeation through silicalite membranes. Flanders et al. [94] studied separation of C6 isomers by pervaporation through ZSM-5 membranes and found that separation was due to shape selectivity. 

A theory of gas diffusion and permeation has recently been proposed [56] for the interpretation of experimental data concerning molecular-sieve porous glass membranes. Other researchers [57,58], on the basis of experimental evidences, pointed out that a Stefan-Maxwell approach has to be preferred over a simple Pick one for the modeling of mass transfer through zeolite membranes. 

The application of the Maxwell-Stefan theory for diffusion in microporous media to permeation through zeolitic membranes implies that transport is assumed to occur only via the adsorbed phase (surface diffusion). Upon combination of surface diffusion according to the Maxwell-Stefan model (Eq. 20) with activated-gas translational diffusion (Eq. 12) for a one-component system, the temperature dependence of the flux shows a maximum and a minimum for a given set of parameters (Fig. 15). At low temperatures, surface diffusion is the most important diffusion mechanism. This type of diffusion is highly dependent on the concentration of adsorbed species in the membrane, which is calculated from the adsorption isotherm. At high temperatures, activated-gas translational diffusion takes over, causing an increase in the flux until it levels off at still-higher temperatures. 

Application of the Maxwell-Stefan equations to permeation through zeolitic membranes was done by Kapteijn et al. [50,56] and Krishna and van den Broeke [57]. Kapteijn showed that both the temperature and occupancy dependence of the steady-state /i-butane flux can accurately be described by Eqs. (20), (24), and (25) [56]. The advantage of using the Maxwell-Stefan description is that it is able to describe both occupancy and temperature... 

The same model was applied to permeation of lighter hydrocarbons (C1-C3) through the silicalite-1 membrane [50]. In the case of methane, ethane, and ethene, some concentration dependence of the Maxwell-Stefan diffusivity was observed. This can be caused either by the importance of interfacial effects, which are not taken into account, or by the contribution of activated-gas translational diffusion to the net flux. The diffusivities calculated from these permeation experiments were, however, in rather good agreement with diffusivity values from the literature, which implies that these zeolitic membranes could also be a valuable tool for the determination of diffusion coefficients in zeolites. 

Figure 16a Permeation flux of n-butane ( ) through a silicalite-1 membrane as a function of the feed partial pressure of /i-butane (T = 300 K, P oi = 100 kPa). Included are the calculated Fickian ( ) (Eq. 5) and Maxwell-Stefan ( ) (Eq. 20) diffusivities. (Adapted from Ref. 56.)...

Figure 17 Simulations of transport of a two-component mixture (pi, P2 = 50 kPa) across a zeolitic membrane using the Fickian and Maxwell-Stefan descriptions (Eqs. 5 and 21, respectively). Permeate partial pressures are taken to be zero. The following parameters were used = 0.01 kPa", ...

The temperature dependence of the methane permeation through a silicalite membrane, showing a maximum and a minimum as a function of temperature (Fig. 3 [14]), can not be predicted by using the Maxwell-Stefan description for surface diffusion only. Such a maximum and minimum in the permeation as a function of temperature can be predicted only when the total flux is described by a combination of surface diffusion and activated-gas translational diffusion (Fig. 15). 

F. Kapteijn, W.J.W. Bakker, G. Zheng, J. Poppe, and J.A. Moulijn, Permeation and separation of light hydrocarbons through a silicalite-1 membrane application of the generalized Maxwell-Stefan equations, Chem. Eng. J. 57 145 (1995). 

Fig. 14 Separation of C2H6/CH4 mixtures by permeation through a silicalite membrane, a Flux b selectivity. Continuous lines show the predictions of the Maxwell-Stefan model (Eq. 44) based on single-component diffusivities (Dqa> F>ob) with Dab from the Vignes correlation (Eq. 46). Dotted lines show predictions from the simplified Habgood model in which mutual diffusion effects are ignored (Eq. 45). From van de Graaf et al. [53] with permission...

Figure 6.4 Pure gas transport data at 25 °C of membranes AF1600 (O), AFl 6 350 30 fD), AF16 80 15 (A), AF16 80 30 (U), AF16 80 40 ( 0), silicalite-1 (O) as derived from literature data (see text), and predictions of the Maxwell model fora AF16/MFI30% membrane ( ) (a) Pure gas steady state permeability vs kinetic diameter of the permeating molecules (b) gas/methane separation factor (c) gas diffusion coefficients from time-lag experiments vs kinetic diameter (d) gas solubility vs the e/k Lennard-Jones parameter...

Defect-free membranes comprising zeolites and amorphous glassy perfluoropolymers can be prepared by modifying the surface of the filler. The pure gas permeation experiments of a series of Teflon AF 1600 membranes with various amounts of 80 and 350nm silicalite-1 crystals cannot be interpreted on the basis of the Maxwell model, but are compatible with a model in which a barrier to transport exists on the zeolite surface and a lower density polymer layer surrounds the crystals. With a small zeolite size (80nm) the low density layers around the crystals may coalesce and form percolation paths of lesser resistance and less selectivity. Silicalite-1 crystals improve the CO2/CH4 selectivity of Hyflon AD60X, and drive the N2/CH4 selectivity beyond the Robeson s upper bound. It also turns out that the presence of silicaUte-l crystals, like fumed silica, promote the inversion of the methane/butane selectivity of Teflon AF2400 in mixed gas experiments. 

For the multi-component permeation system in zeolite membranes, the mass transfer can be described using the general Maxwell-Stefan equations [10,11]... 

Experimental and Maxwell model predicted permeation properties of ZIF-8/polyimide mixed matrix membrane. The straight line is the upper bound trade-off between selectivity and permeability in the Robson plat for polymer membranes. The filled dot ( ) is an estimate for the pure ZIF-8 membrane. From Zhang B, Dai Y, Johnson JR, Karvan O, Koros WJ. High performance ZIF-8/6FDA-DAM mixed matrix membrane for propyiene/propane separations. J MembrSci 2012 389 34-42, with permission. 

Recent work of Van de Graaf et al. [209] and Kapteijn et al. [210] has shown that for diffusion of binary mixtures in Silicalite, the complete Maxwell-Stefan formulation, equation 5.14, tciking interchange into account provides a much better description of binary permeation experimental results across a Silicalite membrane than with a model ignoring the interchange mechanism (portrayed by D12). 

Intra-crystaUine permeation through a zeolite membrane can be described using different approaches (Krishna, 2006). For example, in the Fickian approach, the concentration gradient is the driving force through the zeolite membrane whereas in the Maxwell-Stefan (MS) approach the gradient of the thermodynamic potential is the driving force. The MS approach... 

The description of the separation of multicomponent mixtures requires a more complex approach, for example by using Maxwell-Stefan methodology. However, the real membrane often assumes a more complex structure, in which, beside the microporous zeolite layer, the mesoporosity of the intra-crystalline-defects and of the underlying support can play an important role, especially when the capillary condensation phenomenon can occur, as in the case of the permeation of vapour. Kondo and Kita (Kondo and Kita, 2010) attempted an interpretation of the dehydration process by including narrow non-zeolitic pores into the support. The water molecules in the feed selectively adsorbed in zeolite pores are then transported to the non-zeolitic pore, where they are released in the permeate side of the membrane. 

For the transport of gas mixtures, the generalised Maxwell-Stefan equation (Krishna and WesseUngh, 1997) has been widely adopted to describe multi-component diffusion. Although quantitative descriptions of gas diffusion in various microporous or mesoporous ceramic membranes based on statistical mechanics theory (Oyama et al., 2004) or molecular dynamic simulation (Krishna, 2009) have been reported, the prediction of mixed gas permeation in porous ceramic membranes remains a challenging task, due to the difficulty in generating an accurate description of the porous network of the membrane. 

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