Diffusion multicomponent approaches

Yu. F. Zuev, A. B. Mirgorodskaya, B. Z. Idiatullin, Structural properties of microheterogeneous surfactant-based catalytic system. Multicomponent self-diffusion NMR approach, Appl. magnetic resonance, 2004, 27, 489-500. 

The diffusion velocities Ffe in Eqs. (3.25) and (3.26) are generally computed using mixture average diffusion, including thermal dif sion for the light species (Kee et ah, 1996), rather than from the full multicomponent approach ofEq. (3.21) ... 

Most distillation systems ia commercial columns have Murphree plate efficiencies of 70% or higher. Lower efficiencies are found under system conditions of a high slope of the equiHbrium curve (Fig. lb), of high Hquid viscosity, and of large molecules having characteristically low diffusion coefficients. FiaaHy, most experimental efficiencies have been for biaary systems where by definition the efficiency of one component is equal to that of the other component. For multicomponent systems it is possible for each component to have a different efficiency. Practice has been to use a pseudo-biaary approach involving the two key components. However, a theory for multicomponent efficiency prediction has been developed (66,67) and is amenable to computational analysis. 

There are several approaches to the preparation of multicomponent materials, and the method utilized depends largely on the nature of the conductor used. In the case of polyacetylene blends, in situ polymerization of acetylene into a polymeric matrix has been a successful technique. A film of the matrix polymer is initially swelled in a solution of a typical Ziegler-Natta type initiator and, after washing, the impregnated swollen matrix is exposed to acetylene gas. Polymerization occurs as acetylene diffuses into the membrane. The composite material is then oxidatively doped to form a conductor. Low density polyethylene (136,137) and polybutadiene (138) have both been used in this manner. 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

Despite the various drawbacks, the effective binary approach is still widely used and will be widely applied to natural systems in the near future because of the difficulties of better approaches. For major components in a silicate melt, it is possible that multicomponent diffusivity matrices will be obtained as a function of temperature and melt composition in the not too distant future. For trace components, the effective binary approach (or the modified effective binary approach in the next section) will likely continue for a long time. The effective binary diffusion approach may be used under the following conditions (but is not limited to these conditions) with consistent and reliable results (Cooper, 1968) ... 

To quantify the diffusion profiles is a difficult multicomponent problem. The activity-based effective binary diffusion approach (i.e. modified effective binary approach) has been adopted to roughly treat the problem. In this approach. 

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. 

Section 12.7.4 discusses mixture-averaged species transport. Although this approach is not rigorous, computationally it can be much faster than the multicomponent or Stefan-Maxwell formulations. In this problem we evaluate species diffusive fluxes using two different approaches to mixture-averaged transport discussed in the text. For these... 

In practice, it is often feasible to reduce the multicomponent crystal in respect of its transport behavior to a quasi-binary system. Let us assume that the diffusion coefficients are DA>DB>DC, Dd, etc. The quasi-binary approach considers C, D, etc. as practically immobile, which means that A and B are interdiffusing in the im-... 

As discussed in Section IV, Agrawal and Wei (1984) and Ware and Wei (1985b) have successfully modeled experimental deposit profiles by using the theory of coupled, multicomponent first-order reaction and diffusion. Wei and Wei (1982) employed this theory to evaluate the influence of catalyst properties on the shape of the deposit profile. Agrawal (1980) developed a model for the deactivation of unimodal and bimodal catalysts based on the consecutive reaction path. These approaches represent a more realistic consideration of the HDM reaction mechanism than first-order kinetics and will, accordingly, be discussed in more detail. 

Many practical adsorption processes involve multicomponent systems, so the problem of micropore diffusion in a mixed adsorbed phase is both practically and theoretically important. Major progress in understanding the interaction effects has been achieved by Krishna and his coworkers through the application of the Stefan-Maxwell approach. The diverse patterns of concentration dependence of diffusivity that have been observed for many systems can, in most cases, be understood on this basis. The reader is referred, for details, to the review articles cited in the bibliography. 

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]. 

Reactive absorption occurs in multiphase multicomponent fluid systems, and a single modeling approach for all - in part very different - processes, is desirable. Such an approach is suggested here, whereby an application of a reactive rate-based model as a suitable and accurate method is recommended. This method employs a kinetic description of diffusion and reaction steps. 

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. 

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. 

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... 

For liquids, there is no complete theory of multicomponent diffusion yet available. For this reason only rough theoretical approaches, as used for the description of mass transport in the porous particles filled with a liquid are discussed. The effective diffusivity concept just described is the only known approach and... 

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. 

Before closing this section we should mention several relatively recent papers that provide additional details relevant to the overview given above. The effects of evaporation, back-reaction, diffusion, and dust enrichment on isotopic fractionation in forsterite have been discussed in great detail by Tsuchiyama et al. (1999) and Nagahara and Ozawa (2000) and extended to multicomponent systems in Ozawa and Nagahara (2001). Richter et al. (2002) combined theoretical and experimental approaches to study elemental and isotopic fractionation effects due to evaporation from CMAS liquids and included consideration of the effects of temperature, gas composition, and diffusion in both the residue and in the surrounding gas. 

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