Maxwell-Stefan diffusion binary mixtures

The Maxwell-Stefan diffusion coefficients represent binary diffusivities for ideal and many nonideal mixtures, they are independent of the concentration of the species in the multicomponent mixtures. 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

The behavior of the Fick diffusion coefficient in nonideal systems may be complicated, while the Maxwell-Stefan diffusion coefficients behave quite well, and are always positive for binary systems. In nonideal binary systems, the Fick diffusivity varies with concentration. As seen in Figure 6.1, water-acetone and water-ethanol systems exhibit a minimum diffusivity at intermediate concentrations. Table 6.1 displays the dependency of binary diffusivity coefficients on concentration for selected alkenes in chloroform at 30°C and 1 atm. As the nonideality increases, mixture may split into two liquid phases at certain composition and temperature. 

The description of diffusion may be complex in mixtures with more than two components. Diffusion coefficients in multicomponent mixtures are usually unknown, although sufficient experimental and theoretical information on binary systems is available. The Maxwell-Stefan diffusivities can be estimated for dilute monatomic gases from D k Dkl when the Fick diffusivities are available. The Maxwell diflfusivity is independent of the concentration for ideal gases, and almost independent of the concentration for ideal liquid mixtures. The Maxwell-Stefan diffusivities can be calculated from... 

The activity coefficients of nonideal mixtures can be calculated using the molecular models of NRTL, UNIQUAC, or the group contribution method of UNIFAC with temperature-dependent parameters, since nonideality may be a strong function of temperature and composition. The Maxwell-Stefan diffusivity for a binary mixture of water-ethanol can be considered independent of the concentration of the mixture at around 40°C. However, for temperatures above 60°C, deviation from the ideal behavior increases, and the Maxwell-Stefan diffusivity can no longer be approximated as concentration independent. For highly nonideal mixtures, one should consider the concentration dependence of the diffusivities. 

Consider the problem of steady-state one-dimensional diffusion in a mixture of ideal gases. At constant T and P, the total molar density, c = P/RT is constant. Also, the Maxwell-Stefan diffusion coefficients D m reduce to binary molecular diffusion Dim, which can be estimated from the kinetic theory of gases. Since Dim is composition independent for ideal gas systems, Eq. (6.61) becomes... 

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. 

When the binary Maxwell-Stefan diffusivities y are equal and the mixture is ideal m j = 1). With these assumptions, Eq. 5.44 pelds... 

The preceding examples illustrated some of the ways in which the Fick matrix [D] depends on the composition of the mixture and on the binary (Maxwell-Stefan) diffusion coefficients. One further property of [D] is that both the sign and magnitude of the elements of [ >] depends on the order in which the components are numbered. Wesselingh (1985) provided a dramatic and elegant illustration of this fact for the system H2-N2-CCI2F2. At a temperature of 298 K and a pressure of 101.3 kPa the diffusion coefficients of the three binary pairs that make up the mixture are... 

In order to use the procedure of Section 4.2.2 to predict [D] we need the Maxwell-Stefan diffusivities of each binary pair in the multicomponent mixture. 

There are few methods for predicting the Maxwell-Stefan diffusivities in multicomponent liquid mixtures. The methods that have been suggested are based on extensions of the techniques proposed for binary systems discussed in Section 4.1.5 (see, e.g., the works of Cullinan and co-workers, 1966-1975 Bandrowski and Kubaczka, 1982 Kosanovich, 1975). The Vignes equation, for example, may be generalized as follows (Wesselingh and Krishna, 1990 Kooijman and Taylor, 1991). 

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. 

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. 

For non-ideal mixtures, the Pick first law binary diffusivity, >12, can thus be expressed in terms of the Maxwell-Stefan diffusivity, D 2, by ... 

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). 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

To estimate the Maxwell-Stefan and effective diffusion coefficients, diffusion data for binary mixtures is necessary. For gas systems under low pressure, the model of Fuller et al. is used most frequently [51]. The method of Wilke and Lee [40] is also valid for low pressures. Both of these methods generally agree with experimental data with an accuracy of up to 10 %, although discrepancies of about 20 % cannot be excluded [40],... 

Example 6.1 Maxwell-Stefan equation for binary mixtures For an application of the Maxwell-Stefan description of diffusion, we consider a binary isotropic mixture with components 1 and 2. To solve the mass balance equations, the diffusion flow has to be known. The binary diffusion flow without the electromagnetic field and external forces is given by... 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Maxwell-Stefan Isothermal Diffusion for Binary mixtures... 

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... 

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... 

Diffusion coefficients in binary liquid mixtures can be strong functions of composition. To illustrate this fact we have plotted experimental data for a few systems in Figure 4.1. The Maxwell-Stefan coefficient > also is shown in Figure 4.1. To obtain the Maxwell-Stefan coefficients we have divided the Fick D by the thermodynamic factor F... 

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