Stefan Approach

Krishna, R. and Wesselingh, J.A. 1997. The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52, 861-911. 

Schneider R, Kenig EY, Gorak A. Dynamic modelling of reactive absorption with the Maxwell-Stefan approach. Trans IchemE 1999 77(part A) 633-638. 

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. 

A more rigorous approach for the combined effect of the various diffusion modes is based on the Maxwell -Stefan approach [36], for porous materials often referred to as the dusty gas model . 

Multicomponent mixtures are even more difBcult to tackle. The Maxwell-Stefan approach vnth the Yang correction gives the best results up to now [70], but this implies that the permeation, and hence separation behaviour cannot be predicted on the basis of one-component data only. Binary data are stil required. The transient permeation behaviour in Figure 18b, however, can be modeled qualitatively well, whereas the Fickian approach is unable to do so [83],... 

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]. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

In this textbook we have eoneentrated our attention on mass transfer in mixtures with three or more species. The rationale for doing this should be apparent to the reader by now multicomponent mixtures have characteristics fundamentally dijferent from those of two component mixtures. In fact, a binary system is peculiar in that it has none of the features of a general multicomponent mixture. We strongly believe that treatments of even binary mass transfer are best developed using the Maxwell-Stefan equations. We hope that this text will have the effect of persuading instructors to use the Maxwell-Stefan approach to mass transfer even at the undergraduate level. 

Maxwell-Stefan (dusty gas) approach by taking the membrane to be the additional component in the mixture. When the model is extended to account for thermodynamic nonidealities (what may be considered to be a dusty fluid model) almost all membrane separation processes can be modeled systematically. Put another way, the Maxwell-Stefan approach is the most promising candidate for developing a generalized theory of separation processes (Lee et al., 1977 Krishna, 1987). 

M.3 Carry out a literature survey on multicomponent diffusion in inorganic ceramic materials and examine whether it would be beneficial for researchers in this area to adopt the Maxwell-Stefan approach instead of the generalized Fick s law. A paper by Cooper (1974) is a good starting point for your survey. 

A qualified question is then whether or not the multicomponent models are really worthwhile in reactor simulations, considering the accuracy reflected by the flow, kinetics and equilibrium model parts involved. For the present multiphase flow simulations, the accuracy reflected by the flow part of the model is still limited so an extended binary approach like the Wilke model sufEce in many practical cases. This is most likely the case for most single phase simulations as well. However, for diffusion dominated problems multicomponent diffusion of concentrated ideal gases, i.e., for the cases where we cannot confidently designate one of the species as a solvent, the accuracy of the diffusive fluxes may be significantly improved using the Maxwell-Stefan approach compared to the approximate binary Fickian fluxes. The Wilke model might still be an option and is frequently used for catalyst pellet analysis. 

Krishna R, Wesselingh JA (1997) The Maxwell-Stefan Approach to Mass Transfer. Chem Eng Sci 52(6) 861-911... 

To establish the relationship between self- and transport diffusion it is necessary first to consider diffusion in a binary adsorbed phase within a micropore. This can be conveniently modeled using the generalized Maxwell-Stefan approach [45,46], in which the driving force is assumed to be the gradient of chemical potential with transport resistance arising from the combined effects of molecular friction with the pore walls and collisions between the diffusing molecules. Starting from the basic form of the Maxwell-Stefan equation ... 

R. Schneider, E. Y. Kenig, A. Gorak, Dynamic Modeling of Reactive Absorption with the Maxwell-Stefan Approach, Trans. IChemE, Part A, 1999,... 

Diffusion in Porous Media Maxwell-Stefan Approach... 

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. 

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