Stefan-Maxwell theory

The theory of diffusion in the gas phase is well developed. In general the diffusion flux of a component i (N,) depends on all of the components. According to Stefan-Maxwell theory, the diffusion flux and the concentration gradient are governed by the matrix relation... 

Wilke has developed an approximate method to calculate the molecular diffusion coefficients starting from the Stefan-Maxwell theory... 

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 ... 

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. 

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... 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

A theory of gas diffusion and permeation has recently been proposed [56] for the interpretation of experimental data concerning molecular-sieve porous glass membranes. Other researchers [57,58], on the basis of experimental evidences, pointed out that a Stefan-Maxwell approach has to be preferred over a simple Pick one for the modeling of mass transfer through zeolite membranes. 

Concentration-dependent activity coefficients can be accommodated with relative ease by an added term (e.g., [see Helfferich, 1962a Brooke and Rees, 1968] and variations in diffusivities are easily included in numerical calculations (Helfferich and Petruzzelli, 1985 Hwang and Helfferich, 1986). In both instances, however, a fair amount of additional experimental information is required to establish the dependence on composition. Electro-osmotic solvent transfer and particle-size variations are more difficult to deal with, and no readily manageable models have been developed to date. A subtle difficulty here is that, as a rule, there is not only a variation in equilibrium solvent content with conversion to another ionic form, but that the transient local solvent content is a result of dynamics (electro-osmosis) and so not accessible by thermodynamic considerations (Helfferich, 1962b). Theories based on the Stefan-Maxwell equations or other forms of (hcrniodyiiainics of ir-... 

Early Iheoties for multicomponent diffusion in gases ware obtained from kinetic theory approaches and culminated in the Stefan-Maxwell eqmitions7 for dilute gas mixtures of constant molar density. In one dimension,... 

The last term accounts for bulk flow of the mixture. The exact form of the depends on the system under study. For ideal gases, the kinetic theory leads to the Stefan-Maxwell equations, which can be rearranged into the form of Eq. 

This form is the form suggested by the dusty gas theory, and will be formally proved in the context of Stefan-Maxwell approach in Chapter 8. 

We have presented in this chapter an account on the development of diffusion theory. Various modes of flow are identified Knudsen, viscous, continuum diffusion and surface diffusion. Constitutive flux equations are presented for all these flow mechanisms, and they can be readily used in any mass balance equations for the solution of concentration distribution and fluxes. Treatment of systems containing more than two species will be considered in a more systematic approach of Stefan-Maxwell in the next chapter. 

Adopting the dusty gas model(DGM) for the description of gas phase mass transfer and a Generalized Stefan-Maxwell(GSM) theory to quantify surface diffosim, a combined transport model has been applied. The tubular geometry membrane mass balance is givoi in equation (1). 

Nevertheless, the most rigorous, thermodynamically complete understanding of ion mobility or diffusivity in concentrated electrolytic solutions is provided by the extended Stefan-Maxwell transport theory, which can be applied to electrolytic solutions [13, 17-22], ionic melts or ionic liquids [23, 24], and ion-exchange membranes [25-28]. The diffusion driving force in any system involving ion transport is taken to be a gradient of electrochemical potential /i, typically expressed in terms of a chemical part and an electrical part as... 

The Nemst-Planck theory (under the Nemst-Einstein Eq. 4) can be derived from the extended Stefan-Maxwell equation by taking O to be a quasi-electrostatic potential referred to one ion m and taking the limit of extreme dilution. Thus it can be seen formally that Nernst-Planck theory neglects solute-solute interactions, and applies strictly only in the limit of infinite dilution. In an n-component electrolytic phase, transport can be quantified using n(n — 1) independent species mobilities, which quantify the binary interactions between each pair of species. 

In addition to the equivalent circuit method, the impedance results can also be analyzed using mathematical models based on physicochemical theories. Guo and White developed a steady-state impedance model for the ORR at the PEM fuel cell cathode [15]. They assumed that the electrode consists of flooded ionomer-coated spherical agglomerates surrounded by gas pores. Stefan-Maxwell equations were used to describe the multiphase transport occurring in both the GDL and the catalyst layer. The model predicted a high-frequency loop due to the charge transfer process and a low-frequency loop due to the combined effect of the gas-phase transport resistance and the charge transfer resistance when the cathode is at high current densities. 

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 Chapman-Enskog kinetic theory cf gases (Hirschfelder et al., 1964) is used to describe the multicomponent diffusion flux of species i in a mixture of n gas species and expressed as the Stefan-Maxwell equation (Bird et al, 2002). The diffusion flux of species i is given as... 

The foundation of concentrated solution theory is the Stefan-Maxwell multicomponent diffusion equation [16,17],... 

This equation has two major advantages over Eq. 7.1-2. First, these diffusion coefficients are the binary values found from binary experiments or calculated from the Chapman Enskog theory given in Section 5.1. Second, the Stefan-Maxwell equations do not require designating one speeies as solvent, which is sometimes an inconvenience when using Eq. 7.1-2. 

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