The Generalized Maxwell-Stefan Equations

Insertion of any other reference mixture velocity in place of v in Eq. 2.3.8 will not alter the value of, an expected and pleasing result because only relative motions of the various species are important in the description of the diffusion process. In chemical engineering applications it is often convenient to choose the molar average mixture velocity w and so we may write 

Eliminating the molar average velocity u using Eq. 1.2.6 allows us to write Eq. 2.3.14 in terms of the relative velocities (u — Uj) as 

We may use Eq. 2.3.15 as the starting point for developing our constitutive relations rather than the conventional Eq. 2.3.8. 

The first postulate of irreversible thermodynamics is that the fluxes (or dependent variables) are directly proportional to the driving forces (or independent variables). [Actually, it may be shown that the assumption of local equilibrium follows from the assumption of a linear relation between the fluxes and driving forces (Truesdell, 1969).] If we take the di as dependent variables and the (m, — Wy) as independent variables we may, therefore, write 

Equations 2.3.17 are the generalized Maxwell-Stefan (GMS) relations and the are the Maxwell-Stefan diffusion coefficients we encountered earlier. These equations are more useful when expressed in terms of the molar fluxes 

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In dilute electrolyte systems, the diffusional interactions can usually be neglected, and the generalized Maxwell-Stefan equations are reduced to the Nemst-Planck equations (B3) ... 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations... 

In the generalized Maxwell-Stefan equations, chemical potential gradients, which are the thermodynamic forces, are linear functions of the diffusion flows... 

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 GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10 and 1 x 10 , respectively. 

The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tiibenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24. 

An Application of the Generalized Maxwell-Stefan Equations—The Ultracentrifuge... 

In mixtures of electrolytes the generalized Maxwell-Stefan equations... 

The generalized Maxwell-Stefan equations may be written in terms of these dilfusion fluxes as... 

We have shown that the Nernst-Planck equation is only a limiting case of the generalized Maxwell-Stefan equations. Nevertheless, many ionic systems of interest are dilute and the Nernst-Planck equation is widely used. 

For liquid mixtures, effective diffusion coefficients may be defined using the generalized Maxwell-Stefan equations (Lightfoot and Scattergood, 1965). 

Kubaczka, A. and Bandrowski, J., An Explicit Approximate Solution of the Generalized Maxwell-Stefan Equations for the Multicomponent Film Model, Chem. Eng. Commun., 95, 89-97 (1990). 

For dilute gas mixtures we may employ the linearity postulate in irreversible thermodynamics to obtain the transport fluxes for heat and mass. The fundamental theory is examined in chap 2 and we simply refer to the expressions (2.456) and (2.457). Moreover, a particular form of the generalized Maxwell-Stefan equations, i.e., deduced from (2.298) in chap 2, is given by ... 

These relations are called the generalized Maxwell-Stefan equations and are the inverted counterparts of the Pick diffusion equations (2.281). These two descriptions contain the same information and are interrelated as proven by Curtiss and Bird [18] [19] for dilute mono-atomic gas mixtures. 

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 ... 

For the simulation of RD columns in which the chemical reactions take place at heterogeneous catalysts, it is important to keep in mind that a macrokinetic expression (5.55) has to be applied. Therefore, the microkinetic rate has to be combined with the mass transport processes inside the catalyst particles. For this purpose a model for the multicomponent diffusive transport has to be formulated and combined with the microkinetics based on the component mass balances. This has been done by several authors [50-53] by use of the generalized Maxwell-Stefan equations. 

Krishna, R., Multicomponent surface diffusion of adsorbed species A description based on the generalized Maxwell-Stefan equations, Chem. Eng. Sci., 45(7), 1779-1792 (1990). 

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

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