The Maxwell-Stefan equations

For multicomponent diffusion in gases at low density, the Maxwell-Stefan equations provide satisfactory approximations when species / diffuses in a homogeneous mixture 

Ina simple limiting case, a dilute species i diffuses in a homogeneous mixture, 0, andEq. (2.91) int-direction becomes 

If we define a diffusivity of species i in a mixture by Dim = -.JJidxJdy)., we have 

These simple relations are applied to some ternary systems (see Chapter 6). 

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In this paper a transfer model will be presented, which can predict mass and energy transport through a gas/vapour-liquid interface where a chemical reaction occurs simultaneously in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. On the basis of this model a numerical study will be made to investigate the consequences of using the Maxwell-Stefan equation for describing mass transfer in case of physical absorption and in case of absorption with chemical reaction. Despite the fact that the Maxwell-Stefan theory has received significant attention, the incorporation of chemical reactions with associated... 

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) ... 

Single-stage simulations reveal that intermolecular friction forces do not lead to reverse diffusion effects, and thus the molar fluxes calculated with the effective diffusion approach differ only slightly from those obtained via the Maxwell-Stefan equations without the consideration of generalized driving forces. This result is as expected for dilute solutions and allows one to reduce model complexity for the process studied (143). 

The modeling of RD processes is illustrated with the heterogenously catalyzed synthesis of methyl acetate and MTBE. The complex character of reactive distillation processes requires a detailed mathematical description of the interaction of mass transfer and chemical reaction and the dynamic column behavior. The most detailed model is based on a rigorous dynamic rate-based approach that takes into account diffusional interactions via the Maxwell-Stefan equations and overall reaction kinetics for the determination of the total conversion. All major influences of the column internals and the periphery can be considered by this approach. 

Kenig EY, Wiesner U, Gorak A. Modeling of reactive absorption using the Maxwell-Stefan equations. Ind Eng Chem Res 1997 36 4425-4434. 

Matrix [D] results from the transformation of the Maxwell-Stefan equations (1) to the form of the generalized Fick s law [23]. This matrix is generally a function of the mixture composition and is assumed constant along the diffusion path [23]. The direct expressions for the elements of the diffusion matrix [D] can be found, for example, in Ref. [16]. 

Tables 2.12-2.14 show some values of diffusion coefficients in solids and polymers. In a flow of dilute solution of polymers, the diffusivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers.

We may describe multicomponent diffusion by (1) the Maxwell-Stefan equation where flows and forces are mixed, (2) the Chapman-Cowling and Hirschfelder-Curtiss-Bird approaches where the diffusion of all the components are treated in a similar way, and (3) a reference to a particular component, for example, the solvent or mass average (baiycentric) definition. Frames of reference in multicomponent system must be clearly defined. Binary diffusion coefficients are often composition dependent in liquids, while they are assumed independent of composition for gases. 

Maxwell-Stefan equations describe steady diffusion flows, assuming that shearing forces for each species are negligible. As there are no velocity gradients assumed, the Maxwell-Stefan equations can be written in the forms of fluxes. For a ternary mixture of components 1, 2, and 3, the flow of component 1 in the z direction is... 

The Maxwell-Stefan equations do not depend on choice of the reference velocity, and therefore they are a proper starting point for other descriptions of multicomponent diffusion. For ideal gas mixtures, diffusivities /, and D u are... 

Assume that a simple film model exists for the mass transfer, equilibrium is established at the gas-liquid interface, and the diffusion occurs at isobaric and isothermal conditions. Also assume that neither helium nor argon is absorbed so that N2=N3 = 0. Then, the Maxwell-Stefan equations for the diffusion of argon and helium are... 

The diffusion process in each phase may be described by a film model. By applying the Maxwell-Stefan equations for each phase, the interfacial compositions and the rates of interface transport at the bottom of the column can be estimated using the following steps ... 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

For ideal fluid mixtures, the Maxwell-Stefan equation yields... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the Maxwell-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair- directly based on the Maxwell-Stefan diffusivities. 

The principle of the Maxwell-Stefen diffusion equations is that the force acting on a species is balanced by the ffiction that is exerted on that species. The driving force for diffusion is the chemical potential gradient. The Maxwell-Stefan equations were applied to surface diffusion in microporous media by Krishna [77]. During surface diffusion, a molecule experiences friction from other molecules and from the surface, which is included in de model as a pseudo-species, n+1 (Dusty-gas model). The balance between force and friction in a multi-component system can thus be written as [77] ... 

In fact, X is a correction parameter for the Pick diffusion coefficient. This correction has a similar effect on the apparent diffusivity as the correction given in Eq. (14). When X is less then 1, the diffusivity increases with occupancy. This correction can also be applied to the Maxwell-Stefan diffusivity, which results in an even larger effect of concentration on the flux. The concentration dependence of the flux in the Maxwell-Stefan equations depends largely on the adsorption isotherm chosen, since this isotherm determines the thermodynamic factor. For Langmuir adsorption the concentration dependence of the flux increases in the following order using different models ... 

Application of the Maxwell-Stefan equations to permeation through zeolitic membranes was done by Kapteijn et al. [50,56] and Krishna and van den Broeke [57]. Kapteijn showed that both the temperature and occupancy dependence of the steady-state /i-butane flux can accurately be described by Eqs. (20), (24), and (25) [56]. The advantage of using the Maxwell-Stefan description is that it is able to describe both occupancy and temperature... 

A description consistent with NET is the Maxwell-Stefan equations. In the application of Maxwell-Stefans equations, one frequently neglects the heat of transfer, q This assumption may be good for gases. It is not so good for liquid mixtures. According to Section 3, it is likely that the heat of transfer plays an important role in flux equations of interface transport. Olivier showed that failure to include the heat of transfer leads to an error of up to 20% in the heat flux calculated for some typical phase transition conditions. 

In the DGM model, porous media are considered as arrays of heavy molecules i.e., dust) that are motionless and uniformly distributed in space. By treating the dust particles as giant molecules it is possible to use the Chapman-Enskog kinetic theory. The dust molecules are treated as an (ji - - l)th pseudo-species added to the n-component gas mixture. The dust particles are kept fixed in space (i.e., motionless) and are considered like another gas component in the Maxwell-Stefan equations. 

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

Equation 2.1.8 is the Maxwell-Stefan equation for the diffusion of species 1 in a two-component ideal gas mixture. The symbol D 2 is the Maxwell-Stefan (MS) diffusivity. 

In an attempt to check the validity of the Maxwell-Stefan equations Carty and Schrodt (1975) evaporated a binary liquid mixture of acetone(l) and methanol(2) in a Stefan tube. Air(3) was used as the carrier gas. In one of their experiments the composition of the vapor at the liquid surface was x = 0.319, x = 0.528. The pressure and temperature in the vapor phase were 99.4 kPa and 328.5 K, respectively. The length of the diffusion path was... 

Calculate the composition profiles predicted by the Maxwell-Stefan equations and compare the results with the experimental data. 

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

The Maxwell-Stefan equation for diffusion in a two component system is Eq. 2.2.11. 

Ideally one would like to be able to predict the elements of [D] from a knowledge of the infinite dilution diffusion coefficients f)°y. A comparison of the generalized Fick s law (Eq. 3.2.5), with the Maxwell-Stefan equations (Eq. 2.2.10) shows that, for a nonideal system, the... 

Both formulations of the constitutive equations for multicomponent diffusion, the Maxwell-Stefan equations and the generalized Fick s law, are most compactly written in matrix form. It might, therefore, be as well to begin by writing the continuity equations (Eq. 1.3.9) in n - 1 dimensional matrix form as well... 

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