Stefan approach, Maxwell

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. 

The Stefan-Maxwell equations (12.170 and 12.171) form a system of linear equations that are solved for the K diffusion velocities V. The diffusion velocities obtained from the Stefan-Maxwell approach and by evaluation of the multicomponent Eq. 12.166 are identical. 

Many practical adsorption processes involve multicomponent systems, so the problem of micropore diffusion in a mixed adsorbed phase is both practically and theoretically important. Major progress in understanding the interaction effects has been achieved by Krishna and his coworkers through the application of the Stefan-Maxwell approach. The diverse patterns of concentration dependence of diffusivity that have been observed for many systems can, in most cases, be understood on this basis. The reader is referred, for details, to the review articles cited in the bibliography. 

Stefan-Maxwell expression of diffusive fluxes and a Darcy expression of convective ones, was frequently employed, especially for the IMR with separate feed of reactants [30,48,49]. Such studies clearly indicated that the use of the Stefan-Maxwell approach to diffusion has to be preferred over a simple Pick one, especially when large pressure differences are imposed across the membrane. 

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. 

For the mathematical description of the component transport through a porous membrane there are two modeling approaches common. The first is the so-called extended Pick model (EFM), which can be applied to describe the transport of diluted, nonadsorbable gases in mesoporous membrane materials at low pressure (Veldsink et al., 1995 Papavassiliou et al., 1997 Al-Juaied et al., 2001). The second, the more general dusty gas model, is based on the Stefan-Maxwell approach for multicomponent diffusion (Mason et al., 1967 Krishna and WesseUngh, 2000). Both models require knowledge of the above-mentioned membrane properties (e.g.. Bo, c/t). Because for a specific membrane material these parameters are a priori not predictable, they have to be determined experimentally. Typical membrane materials and characteristic transport parameters used in this work are Hsted in Table 5.1. 

Typical examples where this extended analysis is necessary are conditions of possible back diffusion of components from the tube to the shell side. Additionally, more concentrated mixtures, Knudsen and/or surface diffusion may require the application of the Stefan-Maxwell approach in order to describe the mass transport in the porous medium (Krishna and Wesselingh, 1997). 

This form is the form suggested by the dusty gas theory, and will be formally proved in the context of Stefan-Maxwell approach in Chapter 8. 

To illustrate the Stefan-Maxwell approach in the analysis of the Loschmidt diffusion tube, we apply it to the experimental results obtained by Arnold and Toor (1967). The system is a ternary mixture containing methane, argon and hydrogen. The tube length is 0.40885 m and the two sections of the tube are equal in length. The operating conditions are ... 

To illustrate the solution procedure of this Stefan-Maxwell approach, we apply it to the experimental results of Duncan and Toor (1962). The two well-mixed bulbs used have volumes of 7.8 x 10 and 7.86 x 10 m, respectively, and the capillary tube has a length and a diameter of 0.086 m and 0.00208 m, respectively. The operating conditions are 35 and 101.3 kPa. Duncan and Toor (1962) used a ternary system of hydrogen, nitrogen and carbon dioxide in their experiment. We use the following numerical values to denote the three components used in their experiment 1-hydrogen 2-nitrogen and 3-carbon dioxide. 

We have shown the Stefan-Maxwell approach in solving for combined bulk and Knudsen diffusion in binary and ternary systems. Now we would like to present the analysis of a multicomponent system, and will show that the analysis can be elegantly presented in the form of vector-matrix format. 

Stefan-Maxwell Approach for Bulk-Knudsen Diffusion in Complex System of Capillaries... 

We have seen in Section 8.6 that the analysis using the Stefan-Maxwell approach is readily carried out for the case of a simple capillary, namely a uniformly sized capillary. In this section we will extend the analysis to more complex pore networks and will consider the three cases ... 

In the last sections, you have learnt about the basic analysis of bulk flow, bulk flow and Knudsen flow using the Stefan-Maxwell approach. Very often when we deal with diffusion and adsorption system, the total pressure changes with time as well as with distance within a particle due to either the nonequimolar diffusion or loss of mass from the gas phase as a result of adsorption onto the surface of the particle. When such situations happen, there will be an additional mechanism for mass transfer the viscous flow. This section will deal with the general case where bulk diffusion, Knudsen diffusion and viscous flow occur simultaneously within a porous medium (Jackson, 1977). 

To formulate the Stefan-Maxwell approach for surface diffusion, we will treat the adsorption site as the pseudo-species in the mixture, a concept put forwards by Krishna (1993). If we have n species in the system, the pseudo species is denoted as the (n+l)-th species, just like the way we dealt with Knudsen diffusion where the solid object is regarded as an assembly of giant molecules stationary in space. We balance the force of the species i by the friction between that species i with all other species to obtain ... 

Stefan-Maxwell approach for bulk-Knudsen diffusion in complex. . 487... 

A model, frequently referred to as dusty-gas model [1-3], can be used to describe multi-component diffusion in porous media. This model is based on the Stefan-Maxwell approach for diluted gases which is an approximation of Boltzmann s equation. The pore walls are considered as consisting of giant molecules ( dust ) distributed in space. These dust molecules are treated as the n+l-th pseudo-species in a n-component gaseous mixture. The dust particles are kept fixed in space, and are treated like a gas component in the Stefan-Maxwell equations. This model analyzes the transport problem by distinguishing three separate components 1) diffusion, 2) viscous flow and 3) structure of the porous medium. 

Stefan-Maxwell Approach A more rigorous approach to multispecies diffusion effects is known as the Stefan-Maxwell diffusion model and should be used for higher order models seeking the greatest accuracy. The Stefan-Maxwell equation for multicomponent diffusion flux in the x direction is shown as... 

BanaL F. A., Abu Al-Rub, F., Jumah, R., and Al-Shannag, M. (1999a). Application of Stefan-Maxwell approach to azeotropic separation by membrane distillation. Chem. Eng. J. 73, 71. 

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