Maxwell-Stefan concept

Maxwell-Stefan Concept. In substance, the resistance to diffusion of component A through a solution of A and B is assumed to be proportional to the relative velocity of A with respect to B, Ua — Ub] to the distance dl through which the diffusion occurs and to the number of molecules of A and B in the path of the diffusion, in turn proportional to the concentrations ca and cb. The resistance must be overcome by a concentration gradient in the direction of diffusion dcA- Thus,... 

A more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann s equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Pick s formulation with the aid of several examples. The MaxweU-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 

Define the concepts Maxwell-Stefan (MS) diffusivity and thermodynamic factor. 

The same set of transport mechanisms learnt in Chapter 7 is again considered in Chapter 8, but is dealt with in the framework of Maxwell-Stefan. This is the cornerstone in dealing with multicomponent diffusion in homogeneous media as well as heterogeneous media. We first address this framework to a homogeneous medium so that readers can grasp the concept of friction put forwards by Maxwell and Stefan in dealing with multicomponent systems. Next, we deal with diffusion of a multicomponent mixture in a capillary and a porous medium where continuum diffusion, Knudsen diffusion as well as viscous flow can all play an important role in the transport of molecules. 

In this chapter, we will re-examine these processes, but from the approach developed by Maxwell and Stefan. This approach basically involves the concept of force and friction between molecules of different types. It is from this frictional concept that the diffusion coefficient naturally arises as we shall see. We first present the diffusion of a homogeneous mixture to give the reader a good grasp of the Maxwell-Stefan approach, then later account for diffusion in a porous medium where the Knudsen diffusion as well as the viscous flow play a part in the transport process. Readers should refer to Jackson (1977) and Taylor and Krishna (1994) for more exposure to this Maxwell-Stefan approach. 

We have presented in this chapter a systematic approach of Maxwell-Stefan approach in dealing with flow in homogeneous media as well as inside a capillary or porous media. The approach using the concept of friction is elegant, and it puts the various flow mechanisms under the same framework. For large pore space, the Maxwell-Stefan approach is comprehensive, and it is able to describe experimental results, such as the uphill diffusion, which the traditional Pick s law approach can... 

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 ... 

The separation factor of these organic polymer membranes is typically located in a moderate range, of around 5 and 10, but rarely higher than 20. As a rule of thumb and proven by recent publications, the membrane selectivity can be approximated as the product of the adsorption selectivity and diffusion selectivity [2]. This chapter provides a wealth of information on diffusion inside micro- and mesoporous structures using concepts and ideas that originate from Maxwell and Stefan. A molecular-level understanding of diffusion in a variety of materials such as zeolites, MOFs, covalent organic frameworks (COFs), carbon nanotubes, and cylindrical silica pores is provided with the aid of extensive data sets of molecular... 

Krishna (2000), starting from the fundamental concepts of Maxwell J. C. and Stefan X, describes the conhguration diffusion, also called surface diffusion, within zeolite crystals by means of the following set of equations, namely the generalized MS approach (Kapteijn et al, 2000 Krishna, 2000). Nevertheless, this set of equations can also be used to describe diffusion in other nanostructured materials such as carbon nanotubes ... 

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