Stefan diffusion

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 MaxweU-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 directiy based on the MaxweU-Stefan diffusivities. 

Whitaker, S, Role of the Species Momentum Equation in the Analysis of the Stefan Diffusion Tube, Industrial and Engineering Chemistry Research 29, 978, 1991. 

Thus the gas/vapor/liquid-liquid mass transfer is modeled as a combination of the two-film model and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium state exists only at the interface. 

Figures 23 and 24 show the liquid-phase compositions for, respectively, the reboiler and condenser as functions of time. After column startup, the concentration of methanol decreases continuously whereas the distillate mole fraction of methyl acetate reaches about 90%. A comparison of the rate-based simulation (with the Maxwell-Stefan diffusion equations) and experimental results for the liquid-phase composition at the column top and in the column reboiler demonstrates their satisfactory agreement (Figures 23 and 24). Figure 25 shows the simulation...

The mass transfer within a rigid droplet is determined by the Maxwell-Stefan diffusion. The appropriate diffusion coefficients experimentally determined... 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

Bosse [48] proposed a new model to predict binary Maxwell-Stefan diffusion coefficients Dij, based on Eyrings absolute reaction rate theory [49]. A correlation from Vignes [50] which was shown to be valid only for ideal systems of similar-sized molecules without energy interactions [51] was extended with a Gibbs-excess energy term... 

Diffuslonal interaction methods. These calculate component efficiencies, but account for diffusional interactions. The calculation procedure is based on the Maxwell-Stefan diffusion equations, as developed by Krishna et al. (200,201). The equations are complex and are presented in the original reference. Lockett (12) has an excellent summary. For a ternary system, the steps below are followed (12) ... 

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... 

Negative Maxwell-Stefan diffusivities are allowed if they satisfy... 

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... 

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... 

The principle of the Maxwell-Stefan diffusion equations is that the force acting on a species is balanced by the friction that is exerted on that species. The driving force for... 

The first term on the right-hand side of the equation denotes the friction between species the second term represents friction between a species and the surface. and )Ji+i are the Maxwell-Stefan diffusivities. The first term on the right side is often referred to as an exchange term that represents the probability of molecules exchanging places on the surface. Since this exchange is not likely to occur in narrow zeolite channels, it is commonly neglected. This is called single-file diffusion. 

On comparing Eqs. (20) and (5) and (8), it can be seen that for Langmuir adsorption, the Maxwell-Stefan diffusivity in a one-component system is identical to corrected diffu-sivity in the Darken equation. 

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 ... 

The same model was applied to permeation of lighter hydrocarbons (C1-C3) through the silicalite-1 membrane [50]. In the case of methane, ethane, and ethene, some concentration dependence of the Maxwell-Stefan diffusivity was observed. This can be caused either by the importance of interfacial effects, which are not taken into account, or by the contribution of activated-gas translational diffusion to the net flux. The diffusivities calculated from these permeation experiments were, however, in rather good agreement with diffusivity values from the literature, which implies that these zeolitic membranes could also be a valuable tool for the determination of diffusion coefficients in zeolites. 

The Maxwell-Stefan diffusivities defined in Eqs. (20) and (21) for one- and two-component systems should be identical if they are independent of occupancy. Diffusivities... 

Figure 16b Fickian and Maxwell-Stefan diffusivities of n-butanc as a function of 1/7, calculated from the w-butane permeation results in Fig. 3. (Adapted from Ref. 56.)...

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 more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann s equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Pick s formulation with the aid of several examples. The MaxweU-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 

Por a binary fluid mixture, the force balance on component 1 expresses the equality between the gradient of its chemical potential and the drag force, so MaxweU-Stefan diffusion is given by [38]... 

Comparing Eqs. 5.28 and 5.32 with those stating Pick s law (Eqs. 5.27a to 5.27c) for a binary system yields the following relationship between Pick the diffusivity, D and the Maxwell-Stefan diffusivity V. 

If we introduce the (n — l)-dimensional matrix [T] of thermodynamic factors, we can recast the LHS of Eq. 5.35 in terms of the mole fraction gradients. The Maxwell-Stefan diffusion for multicomponent systems is thus [38]... 

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

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