Stefan-Maxwell equations derivation

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 ... 

In gases, the binary MS diffusivities are normally assumed independent of composition. With this approximation, the effective diffusivity of component i in a multi-component gas mixture can be derived from the Stefan-Maxwell equation to give (Treybal, 1980)... 

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. 

It may be shown that this equation is equivalent to the phenomenological equations derived from irreversible thermodynamics, as weU as the multicomponent diffusion equations derived from the Stefan-Maxwell equations, which were first used to describe diffusion in multicomponent gases. 

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 a gas system composed of a mix of ideal gases, it is possible to derive the Stefan-Maxwell equation (4,5), which solves for the mole fraction of a component in terms of the diffusivities, concentration, and diffusion velocities [velocity of a given species U in equation (10-3)] a one-dimensional form of the equation is... 

We consider a nonreacting mixture such as iso-propanol (1) and water (2), evaporating into ambient air at constant temperature. Assuming the physical equilibrium condition (VLE) at the vapor-liquid interface and applying the Stefan-Maxwell-flux equations to the liquid phase and linear flux equations to the air-diluted gas phase, we derive the following expression for the relative flux XT... 

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 ... 

For multicomponent gas mixtures the generalised Maxwell-Stefan (GMS) equations should be used. Krishna [87b] derived an expression for the flux of specimen / ... 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

Multicomponent diffusion in the films can be rigorously described by the Maxwell-Stefan equations derived from the kinetic theory of gases [69]. The... 

Diffusion Coefficients in Multicomponent Systems. The value of the diffusion coefficient of a species in a binary system is often not the same as the value in a multicomponent system. The diffusion coefficients can be modified in multicomponent systems as a result of added frictional forces at the atomistic scale. The multiple diffusing species interact in various complex ways that can be described using equation 9, which is derived from the so-called Stefan-Maxwell relations (4) ... 

It may be noted that equation 10.86 is identical to equation 10.30. (Stefan s Law) and. Stefan s law can therefore also be derived from Maxwell s Law of Diffusion. 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

The mass diffusive flux m, of Equation (3.2) generally depends on the operating conditions, such as reactant concentration, temperature and pressure and on the microstructure of material (porosity, tortuosity and pore size). Well established ways of describing the diffusion phenomenon in the SOFC electrodes are through either Fick s first law [21, 34. 48, 50, 51], or the Maxwell-Stefan equation [52-55], Some authors use more complex models, like for example the dusty-gas model [56] or other models derived from this [57, 58], A comparison between the three approaches is reported by Suwanwarangkul et al. [59], who concluded that the choice of the most appropriate model is very case-sensitive, and should be selected, according to the specific case under study. 

Multicomponent diffusion in the films is described by the Maxwell-Stefan equations, which can be derived from the kinetic theory of gases (89). The Maxwell-Stefan equations connect diffusion fluxes of the components with the gradients of their chemical potential. With some modification these equations take a generalized form in which they can be used for the description of real gases and liquids (57) ... 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

Derivation of the Maxwell-Stefan Equation for Binary Diffusion... 

In Section 8.3 we presented a derivation of an exact matrix solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures. Although the final expression for the composition profiles (Eq. 8.3.12), is valid whatever relationship exists between the fluxes (i.e., bootstrap condition), the derivation given in Section... 

Toor (1957) derived a solution of the Maxwell-Stefan equations for ternary systems when the total molar flux is zero, = 0. Write down expressions for [P], (y), and show that, for = 0, the eigenvalue solutions are equivalent to the expressions given by Toor. 

The explicit method of Taylor and Smith (1982) for mass transfer in ideal gas mixtures is an exact solution of the Maxwell-Stefan equations for two component systems where all matrices are of order 1. Does the generalized explicit method derived in Exercise 8.40 reduce to the expressions given in Section 8.2 for a film model of mass transfer in binary systems ... 

A more rigorous derivation of these relations were given by Curtiss and Hirschfelder [16] extending the Enskog theory to multicomponent systems. FYom the Curtiss and Hirschfelder theory of dilute mono-atomic gas mixtures the Maxwell-Stefan diffusivities are in a first approximation equal to the binary diffusivities, Dgr Dsr- On the other hand, Curtiss and Bird [18] [19] did show that for dense gases and liquids the Maxwell-Stefan equations are still valid, but the strongly concentration dependent diffusivities appearing therein are not the binary diffusivities but merely empirical parameters. 

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 rigorous derivation of the Maxwell-Stefan equations by use of the kinetic theory of dilute gases has already been explained in connection with... 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [77] or the average species velocity is used [96]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. 

By solving (2.403) for VuJs and equating the result to the same gradient as derived from the simplified Maxwell-Stefan equations (2.407), we obtain an expression for the effective mass based diffusivity ... 

---