Diffusion Stefan-Maxwell equation

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

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). 

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. 

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... 

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) ... 

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... 

An alternative is to solve the Stefan-Maxwell equations [35,178,435] in which the diffusion velocities are related implicitly to the field gradients ... 

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. 

The Stefan-Maxwell equations (12.170 and 12.171) form a system of linear equations that are solved for the K diffusion velocities V. The diffusion velocities obtained from the Stefan-Maxwell approach and by evaluation of the multicomponent Eq. 12.166 are identical. 

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

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

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] ... 

The multi-component diffusivities in the gas mixture can be approximated by the modified Stefan-Maxwell equations(8,9) i.e ... 

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. 

For ideal gases an expression for the effective binary diffusion coefficient, D m, as a function of the binary diffusion coefficient, DA], can be derived as in this case the Stefan-Maxwell equation holds ... 

Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-13. Stefan-Maxwell equations were discussed earlier. Smith and Taylor [23] compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-214) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. 

Wilke [29] obtained solutions to the Stefan-Maxwell equations. The first, Eq. (5-216), is simple and reliable under the same conditions as Blanc s law. This equation applies when component i diffuses through... 

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. 

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