Membranes Maxwell-Stefan equations

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... 

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). 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

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 ... 

Ghosh, U. K., Pradhan, N. C., Adhikari, B. (2006). Separation of water and o-chlorophenol by pervaporation using HTPB-basedpolyurethaneurea membranes and application of modified Maxwell-Stefan equation,... 

Diffusion in ZSM-5 supported membranes was studied from ambient to 723K. The Maxwell-Stefan equations were used for the interpretation of the results. From these, micropore diffusion coefficients and apparent diffusion activation energies were obtained. The results were described by ... 

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

Transport of components within the porous membrane can be expressed using the dusty gas model, which is based on the Maxwell-Stefan equations for multi-component molecular diffusion [8] ... 

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. 

A new approach is the application of chemometrics (and neural networks) in modeling [73]. This should allow identification of the parameters of influence in solvent-resistant nanofiltration, which may help in further development of equations. Development of a more systematic model for description and prediction of solute transport in nonaqueous nanofiltration, which is applicable on a wide range of membranes, solvents and solutes, is the next step to be taken. The Maxwell-Stefan approach [74] is one of the most direct methods to attain this. 

For zeolite membranes, the separation of water and alcohol molecules can be explained by strong interactions between the water molecules and ionic sites in the zeolite crystal lattice and the partial sieving achieved by the zeolite channels (Shah, Kissick, Ghorpade, Hannah, Bhattacharyya, 2000). Macroscopic transport equations describing the mass transfer through such composite membranes are often Maxwell— Stefan based (Krishna van Den Broeke, 1995). Wee, Tye, and Bhatia (2008) listed several zeolite materials for the dehydration of alcohols, such as silicalite or mordenite. Most of these materials were supported by an a-Al203 porous support membrane. 

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). 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

Another model referred to in the literature as a diffusion model [50] is similar in nature to the BFM, but is derived by assuming the membrane can be modelled as a dust component (at rest) present in the fluid mixture. The equations governing species transport are developed from the Stefan-Maxwell equations with the membrane as one of the mixture species. The resulting equation for species i is identical to Eq. (4.4) [50], thus the BFM and this diffusion model are equivalent. 

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

Adopting the dusty gas model(DGM) for the description of gas phase mass transfer and a Generalized Stefan-Maxwell(GSM) theory to quantify surface diffosim, a combined transport model has been applied. The tubular geometry membrane mass balance is givoi in equation (1). 

One of the early mechanistic models for a PEM fuel cell was the pioneering work of Bemardi and Verbrugge [45, 46]. They developed a one-dimensional, steady state, isothermal model which described water transport, reactant species transport, as well as ohmic and activation overpotentials. Their model assumed a fully hydrated membrane at all times, and thus calculated the water input and removal requirements to maintain full hydration of the membrane. The model was based on the Stefan Maxwell equations to describe gas phase diffusion in the electrode regions, the Nemst-Planck equation to describe dissolved species fluxes in the membrane and catalyst layers, the Butler Volmer equation to describe electrode rate kinetics and Schlogl s equation for liquid water transport. 

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