Stefan-Maxwell formulation

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. 

Section 12.7.4 discusses mixture-averaged species transport. Although this approach is not rigorous, computationally it can be much faster than the multicomponent or Stefan-Maxwell formulations. In this problem we evaluate species diffusive fluxes using two different approaches to mixture-averaged transport discussed in the text. For these... 

Another attempt to correlate transport and self-diffusivities has been based on a generalization of the Stefan-Maxwell formulation of irreversible thermodynamics [111-113]. By introducing various sets of parameters describing the facility of exchange between two molecules of the same and of different species, the resulting equations are more complex than eqs 27 and 28 They may be shown, however, to include these relations as special cases... 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

Since the control volume in Figure 15-8 was set to move at the average molar velocity v j.ef,moi the sum of is equal to zero for any system Thus, for a binary system, only one is independent. However, there are two independent fluxes In order to use the Stefan-Maxwell formulation in practical problems, we need another relationship (called a bootstrap equation) that allows us to determine the additional independent flux In other words, we need to tie the moving to the stationary. The form of this additional equation depends on the situation. In general, = Cj yiVi, and if one of the Vj is known,... 

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 ... 

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. 

A momentum balance for multicomponent mixtures can be formulated in a manner analogous to that used to derive Equation (C.2.4) using molecule-wall and molecule-molecule (Stefan-Maxwell) relationships to give ... 

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]. 

Within micropores, surface forces are dominant and an adsorbed molecule never escapes completely from the force field of the surface. Diffusion within this regime has been called configurational diffusion, intra-crystalline diffusion, micropore diffusion, or simply surface diffusion. The Maxwell-Stefan formulation, which is generally accepted for diffusion in the bulk fluid phase, can be extended to describe surface diffusion by considering the vacant sites to be a (n + l)-th pseudospecies on the surface [38,47,49-52]. Using the Maxwell-Stefan diffusion formulation, the following relationship was obtained for surface diffusion. 

The j° term denotes the ordinary concentration diffusion (i.e., multi-component mass diffusion). In general, the concentration diffusion contribution to the mass flux depends on the concentration gradients of all the substances present. However, in most reactor systems, containing a solvent and one or only a few solutes having relatively low concentrations, the binary form of Pick s law is considered a sufficient approximation of the diffusive fluxes. Nevertheless, for many reactive systems of interest there are situations where a multi-component closure (e.g., a Stefan-Maxwell equation formulated in terms... 

On the other hand, the more rigorous Maxwell-Stefan equations and the dusty gas model are seldom used in industrial reaction engineering applications. Nevertheless, the dusty gas model [64] represents a modern attempt to provide a more realistic description of the combined bulk and Knudsen diffusion mechanisms based on the multicomponent Maxwell-Stefan model formulation. Similar extensions of the Maxwell-Stefan model have also been suggested for the surface diffusion of adsorbed molecular pseudo-species, as well as the combined bulk, Knudsen and surface diffusion apparently with limited success [48] [49]. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

A more involved procedure would allow to couple the Stefan Maxwell equations with the pressure-saturation formulation. Here, we would use the molar fractions y, as basic variables and express... 

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 ... 

Extended Stefan-Maxwell constitutive laws for diffusion Eq. 4 resolve a number of fundamental problems presented by the Nemst-Planck transport formulation Eq. 1. A thermodynamically proper pair of fluxes and driving forces is used, guaranteeing that all the entropy generated by transport is taken into account. The symmetric formulation of Eq. 4 makes it unnecessary to identify a particular species as a solvent - every species in a solution is a solute on equal footing. Use of velocity differences reflects the physical criterion that the forces driving diffusion of species i relative to species j be invariant with respect to the convective velocity. Finally, all possible binary solute/solute interactions are quantified by distinct transport coefficients each species i in the solution has a diffusivity or mobility relative to every other species j, Djj or up, respectively. 

The exact formulations of the fluxes N ) depend on the particular model being used for mass transfer principally, the whole scope is feasible, from Pick s law to the complete set of Stefan-Maxwell equations. Since the only component of importance for the gas-liquid mass transfer is hydrogen, which has limited solubility in the liquid phase, the simple two-film model along with Pick s law was used, yielding the flux expression... 

The theory of molecules diffusing in liquids is not very well developed. A rigorous formulation of multicomponent diffusion, such as the Stefan-Maxwell equation for the gas phase, is not successful in describing diffusion in a Uquid phase, because a general theory for calculating binary diffusion coefficients is lacking. However, semiempirical correlations that describe the diffusion of a dissolved component (solute) in a solvent can be used. The concentration of the dissolved component is of course assumed to be low compared with that of the solvent. The diffusion in liquids is very much dependent on whether the molecules are neutral species or ions. 

The approximations to the diffusive fluxes involve a slightly different formulation from Eq. (3.32) for the diffusion velocities. Equation (3.32) itself, together with the solutions of Eqs. (3.25c), can be rearranged to give the Stefan-Maxwell diffusion equations (Muckenfuss and Curtiss, 1958 Monchick, Munn, and Mason, 1966 Dixon-Lewis, 1968). If the thermal diffusion terms are neglected these equations become, in the first approximation. 

Many industrial processes involve mass transfer processes between a gas/vapour and a liquid. Usually, these transfer processes are described on the basis of Pick s law, but the Maxwell-Stefan theory finds increasing application. Especially for reactive distillation it can be anticipated that the Maxwell-Stefan theory should be used for describing the mass transfer processes. Moreover, with reactive distillation there is a need to take heat transfer and chemical reaction into account. The model developed in this study will be formulated on a generalized basis and as a consequence it can be used for many other gas-liquid and vapour-liquid transfer processes. However, reactive distillation has recently received considerable attention in literature. With reactive distillation reaction and separation are carried out simultaneously in one apparatus, usually a distillation column. This kind of processing can be advantageous for equilibrium reactions. By removing one of the products from the reactive zone by evaporation, the equilibrium is shifted to the product side and consequently higher conversions can be obtained. Commercial applications of reactive distillation are the production of methyl-... 

There are n -1 independent relations in the Maxwell-Stefan formulation. For a one-dimensional diffusion in direction z, Eq. (6.65) becomes... 

The coefficients of the matrix [B] have no clear physical interpretation. The advantage of the Maxwell-Stefan formulation is ftat it decouples the drag effect ([B]) from the thermodynamic effects ([T]). 

The GRM Formulated with the Maxwell-Stefan Surface Diffusion Model. 765... 

So the real driving force is the sum of the gradients of the chemical potentials as is also implicit in the general Maxwell-Stefan formulation [87-89]. 

The results of the final integration are plotted in Figure 2.5 along with the data from Carty and Schrodt (1975). The agreement between theory and experiment is quite good and support the Maxwell-Stefan formulation of diffusion in multicomponent ideal gas mixtures. This conclusion was also reached by Bres and Hatzfeld (1977) and by Hesse and Hugo (1972). For further analysis of the Stefan diffusion tube see Whitaker (1991). ... 

Matrix Formulation of the Maxwell-Stefan Equations for Nonideal Fluids... 

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