Maxwell-Stefan approach, calculating

Single-stage simulations reveal that intermolecular friction forces do not lead to reverse diffusion effects, and thus the molar fluxes calculated with the effective diffusion approach differ only slightly from those obtained via the Maxwell-Stefan equations without the consideration of generalized driving forces. This result is as expected for dilute solutions and allows one to reduce model complexity for the process studied (143). 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

For non-ideal systems, on the other hand, one may use either D12 or D12 and the corresponding equations above (i.e., using the first or second term in the second line on the RHS of (2.498)). In one interpretation the Pick s first law diffusivity, D12, incorporates several aspects, the significance of an inverse drag D12), and the thermodynamic non-ideality. In this view the physical interpretation of the Fickian diffusivity is less transparent than the Maxwell-Stefan diffusivity. Hence, provided that the Maxwell-Stefan diffusivities are still predicable for non-ideal systems, a passable procedure is to calculate the non-ideality corrections from a suitable thermodynamic model. This type of simulations were performed extensively by Taylor and Krishna [96]. In a later paper, Krishna and Wesselingh [49] stated that in this procedure the Maxwell-Stefan diffusivities are calculated indirectly from the measured Fick diffusivities and thermodynamic data (i.e., fitted thermodynamic models), showing a weak composition dependence. In this engineering approach it is not clear whether the total composition dependency is artificially accounted for by the thermodynamic part of the model solely, or if both parts of the model are actually validated independently. 

The simplest approach is to calculate binary mass-transfer coefficients F.. from the corresponding empirical correlation, substituting the MS diffusivity D. for the Fick diffusivity in the Sc and Sh numbers. The Maxwell-Stefan equations are, then, written in terms of the binary mass-transfer coefficients. For ideal gas multicomponent mixtures and one-dimensional fluxes, they become... 

When mass-transfer rates are moderate to high, an additional correction term is needed in equations (6-101) and (6-102) to correct for distortion of the composition profiles. This correction, which can have a serious effect on the results, is discussed in detail by Taylor and Krishna (1993). An alternative approach would be to numerically solve the Maxwell-Stefan equations, as illustrated in Examples 1.17 and 1.18. The calculation of the low mass-transfer fluxes according to equations (6-94) to (6-104) is illustrated in the following example. 

A molecular dynamics approach can also be used to predict mixed gas diffusivities in microporous materials, at the expense of computation cost (e.g., Qureshi and Wei, 1990 Chitra and Yashonath, 1995 Trout et al., 1997 Snurr and Karger, 1997). The empirical correlation of Vignes (1966) for binary diffusivities in liquid solutions and also metallic alloys has been used extensively for calculating binary diffusivities, using the Maxwell-Stefan formalism for flux equations (e.g., Krishna, 1990). 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [109] or the average species velocity is used [148]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. Nevertheless, the given momentum flux formula (2.537) is not useful before the unknown average velocity after the collisions v has been determined. For elastic molecular collisions this velocity can be calculated, in... 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

We will reserve the treatment of multicomponent systems until Chapter 8, where the systematic approach of Stefan-Maxwell will be used. Now we take an example to illustrate the binary flux calculation when the bulk diffusion is operating. 

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