Transport Stefan-Maxwell equation

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... 

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

Species diffusive transport may also be considered within the Stefan-Maxwell framework, discussed in the previous section. The Stefan-Maxwell equations were written as... 

Equation (51) is based on the Stefan-Maxwell equations describing the mass transport in the gas phase. 

More complicated and realistic models which allow the prediction of transport processes in porous media have been suggested, and have been validated in recent years. For example, it was realized that there might be significant contributions to the overall flux by components which are adsorbed at pore walls but possess a certain mobility [30]. To quantify such surface diffusion processes, a Generalized Stefan-Maxwell equation has been proposed [28] ... 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

The BFM is developed in Ref. [49] by considering transport within a pore structure and applying the Stefan-Maxwell equations to the fluid mixture [48]. In Ref. [24] we showed that it can be written as... 

Another model referred to in the literature as a diffusion model [50] is similar in nature to the BFM, but is derived by assuming the membrane can be modelled as a dust component (at rest) present in the fluid mixture. The equations governing species transport are developed from the Stefan-Maxwell equations with the membrane as one of the mixture species. The resulting equation for species i is identical to Eq. (4.4) [50], thus the BFM and this diffusion model are equivalent. 

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

A model, frequently referred to as dusty-gas model [1-3], can be used to describe multi-component diffusion in porous media. This model is based on the Stefan-Maxwell approach for diluted gases which is an approximation of Boltzmann s equation. The pore walls are considered as consisting of giant molecules ( dust ) distributed in space. These dust molecules are treated as the n+l-th pseudo-species in a n-component gaseous mixture. The dust particles are kept fixed in space, and are treated like a gas component in the Stefan-Maxwell equations. This model analyzes the transport problem by distinguishing three separate components 1) diffusion, 2) viscous flow and 3) structure of the porous medium. 

One of the early mechanistic models for a PEM fuel cell was the pioneering work of Bemardi and Verbrugge [45, 46]. They developed a one-dimensional, steady state, isothermal model which described water transport, reactant species transport, as well as ohmic and activation overpotentials. Their model assumed a fully hydrated membrane at all times, and thus calculated the water input and removal requirements to maintain full hydration of the membrane. The model was based on the Stefan Maxwell equations to describe gas phase diffusion in the electrode regions, the Nemst-Planck equation to describe dissolved species fluxes in the membrane and catalyst layers, the Butler Volmer equation to describe electrode rate kinetics and Schlogl s equation for liquid water transport. 

The liquid phase eonsists of pure water, while the gas phase has multi components. The transport of each species in the gas phase is governed by a general convection-diffusion equation in conjunction which the Stefan-Maxwell equations to account for multi species diffusion, as described in section 3, with the addition of a source term accounting for phase... 

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. 

From a physical point of view, diffusion and permeation are coupled processes inside the GDL with diffusion usually being the dominant process. Due to the large average pore sizes, Knudsen diffusion does not play a role volume diffusion is the predominant process, which can be described by the Stefan-Maxwell equation. Knudsen diffusion will dominate the transport process only when the pore diameters are of the same order of magnitude as the mean free path of the gas molecules. 

Charge transport in the electrolyte and mass transport in the gas phase are the dominant transport phenomena in fuel cells. Mass transport in gas mixtures is generally described by the Stefan-Maxwell equations ... 

Mass transport Nernst equation, drag coefficient, or Stefan-Maxwell equation... 

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. 

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. 

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... 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

The Mean Transport Pore Model (MTPM) described diffusion and permeation the model (represented as a boundary value problem for a set of ordinary differential equations) are based on Maxwell-Stefan diffusion equation and Weber permeation law. Parameters of MTPM are material constants of the porous solid and, thus, do not dependent on conditions under which the transport proeesses take place. 

A similar model often used by reaction engineers is derived for the limiting case in which all the convective fluxes can be neglected. Consider a dilute component s that diffuses into a homogeneous mixture, then J 0 for r 7 s. To describe this molecular transport the Maxwell-Stefan equations given by the last line in (2.298) are adopted. With the given restrictions, the model reduces to ... 

The Stefan-Maxwell (Maxwell, 1860 Stefan, 1872) equation gives implicit relations for the fluxes when the system is isothermal and the wall effects are negligible, this means negligible viscous transport (i.e. constant pressure) and Knudsen diffusion. For multicomponent mixture the equation has the form ... 

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). 

Generally, the increased complexity of the model based on Stefan-Maxwell relations does not justily improvement in model accuracy, which in most cases is redundant. Complex equations can be tolerated in numerical modelling however, in anal3dical modelling, clarity and simplicity of the resulting expressions are of the highest priority. Below we will use a simple Pick s law of diffusion to describe species transport in GDLs and in catalyst layers. 

Two standard methods (mercury porosimetry and helium pycnometry) together with liquid expulsion permporometry (that takes into account only flow-through pores) were used for determination of textural properties. Pore structure characteristics relevant to transport processes were evaluated fiom multicomponent gas counter-current difhision and gas permeation. For data analysis the Mean Transport-Pore Model (MTPM) based on Maxwell-Stefan diffusion equation and a simplified form of the Weber permeation equation was used. 

As a model we have used the Mean Transport-Pore Model (MTPM) [6] which assumes that the decisive part of the gas transport takes place in transport-pores that are visualized as cylindrical capillaries. The transport-pore radii are distributed around the mean value (first model parameter). The width of this distribution is characterized hy the mean value of the squared transport-pore radii, (second model parameter). The third model parameter is the ratio of porosity, y/i, and tortuosity of transport-pores, qt, q/= Pore diffusion is described by the Maxwell-Stefan diffusion equation extended to account for Knudsen transport [6]. For gas permeation the simplified form of Weber equation [8-10] is used. 

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