Maxwell-Stefan

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

There are n Stefan-Maxwell relations in an n-component mixture, but they are not independent since each side of (2.16) yields zero on summing over r from 1 to n. Physically this is not surprising, since they describe only momentum exchange between pairs of species, and say nothing about the total momentum of the mixture. In order to complete the determination of the fluxes N.... N the Stefan-Maxwell relations must be supple-I n... 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appeahng flux equation for apphcations involving N components is... 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Graham-Uranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. 

Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M. 

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. 

Where, the diffusivity D for the transfer of one gas in another is not known and experimental determination is not practicable, it is necessary to use one of the many predictive procedures. A commonly used method due to Gilliland 6 is based on the Stefan-Maxwell hard sphere model and this takes the form ... 

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... 

Steeping parameters, 15 528t Steep tanks, 15 527-528 Stefan-Boltzmann law, 19 131 Stefan-Maxwell equations, 1 43-46, 598 Stefan s law, 7 327 Steinhart-Hart equation, 24 451 Stellite 1... 

A dusty-fluid modeF has also been used to combine the effects, which adds convection to the Stefan— Maxwell framework, as discussed in a later section.This approach is akin to eq 37. As discussed above, the concentration of water may be replaced by a function of A. 

Almost every model treats gas-phase transport in the fuel-cell sandwich identically. The Stefan— Maxwell equations are used (one of which is depend-... 

The gases in a fuel cell are typically hydrogen and water on the fuel side, and air and water on the oxidant side. Since there are not many components to the gases and one of the equations in eq 40 can be replaced by the summation of mole fractions equals 1, many models simplify the Stefan—Maxwell equations. In fact, eq 40 reduces to Tick s law for a two-component system. Such simplifications are trivial and are not discussed here. 

As the pore size decreases, molecules collide more often with the pore walls than with each other. This movement, intermediated by these molecule—pore-wall interactions, is known as Knudsen diffusion. Some models have begun to take this form of diffusion into account. In this type of diffusion, the diffusion coefficient is a direct function of the pore radius. In the models, Knudsen diffusion and Stefan—Maxwell diffusion are treated as mass-transport resistances in seriesand are combined to yield... 

One way to include the effect of gas-phase pressure-driven flow is to use eq 44 as a separate momentum equation.The models that do this are primarily CFD ones. Another way to include pressure-driven flow is to incorporate eq 44 into the Stefan—Maxwell equations, as per tihe dusty-gas... 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

For the diffusion flux (N] various approaches are possible, ranging from the complete Stefan-Maxwell set of equations to the simple law of Pick (7). The symbols of eqs. (l)-(2) are defined in Notation. 

Another frictional coefficient formalism is that represented by the generalised Stefan-Maxwell equation s>... 

Calculation of a Theoretical Rate of Uptake Where Diffusion Through the Air is Rate Determining. Assuming that diffusion of the oxide vapors through the air is rate controlling, it should be possible to use Maxwell s equation to calculate the rate of uptake of the oxide vapors under varying conditions of pressure and temperature. The use of Maxwell s equation requires knowledge of the interdiffusion constant ( Di.2 ) of the oxide vapor in air. The interdiffusion constants of the vapor species studied here are not known, but they may be estimated by the Stefan-Maxwell equation (14) ... 

In the Stefan-Maxwell setting [35,178,435], the diffusion velocities are related implicitly to the field gradients as follows ... 

Note that the Stefan-Maxwell equations involve the binary diffusion coefficients, and not the ordinary multicomponent diffusion coefficients. 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

These constraints must be satisfied in the solution of the Stefan-Maxwell equations. At a point within a chemically reacting flow simulation, the usual situation is that the diffusion velocities must be evaluated in terms of the diffusion coefficients and the local concentration, temperature, and pressure fields. One straightforward approach is to solve only K — 1 of Eqs. 3.105, with the X4h equation being replaced with a statement of the constraint. For... 

Note that the Stefan-Maxwell equations involve the binary diffusion coefficients T>kj, not the ordinary multicomponent diffusion coefficients Dy. In this context, the are also sometimes refered to as the multicomponent Stefan-Maxwell diffusivities. 

---