Diffusion in multicomponent systems

In earlier chapters we examined systems with one or two types of diffusing chemical species. For binary solutions, a single interdiffusivity, D, suffices to describe composition evolution. In this chapter we treat diffusion in ternary and larger multicomponent systems that have two or more independent composition variables. Analysis of such diffusion is complex because multiple cross terms and particle-particle chemical interaction terms appear. The cross terms result in TV2 independent interdiffusivities for a solution with TV independent components. The increased complexity of multicomponent diffusion produces a wide variety of diffusional phenomena. 

The general treatment for multicomponent diffusion results in linear systems of diffusion equations. A linear transformation of the concentrations produces a simplified system of uncoupled linear diffusion equations for which general solutions can be obtained by methods presented in Chapter 5. 

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Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Uphill diffusion in a binary system is rare and occurs only when the phase undergoes spinodal decomposition. In multicomponent systems, uphill diffusion occurs often, even when the phase is stable. The cause for uphill diffusion in multicomponent systems is different from that in binary systems and will be discussed later. 

Unfortunately, environmental problems do not solely involve dilute binary systems. To mention just one example, the dissolution of a patch of spilled diesel fuel into the groundwater (see below Illustrative Example 19.4) involves diffusion in multicomponent systems for which adequate data are extremely rare. In many cases diffusivities of diluted compounds in air or water must serve as best estimates for more complex systems. Yet, the above remarks should remind us that things may be more complicated. 

Diffusion in multicomponent system is difficult to analyze. Transport of one component is affected by the presence of the other component due to their mumal interaction. This results in the coupling of fluxes. Thus, single-component diffusion equation cannot be used to predict diffusion in a multicomponent system. Greenlaw et al. [38] proposed a simple relationship in which the diffusion coefficients for components i and j are interdependent on both component concentrations ... 

Catalytic systems are inherently multicomponent. An attractive method for determining the difFusivities in multicomponent systems is Fourier transform (FT) PFG NMR spectroscopy, which allows the simultaneous determination of the self-difFusivities of the individual components in a. mixture. If the chemical shifts of the individual species are sufficiently different, the Fourier transform of the spin echo yields separate peaks for the various adsorbates, and then similar to normal PFG NMR the attenuation of the separate peaks with increasing applied field gradient intensity yields the self-diffusivities. Because the technique can also be applied at elevated temperatures, it provides the opportunity for in-situ diffusion measurements under reaction conditions. The experiment also yields the time dependence of the relative concentrations of the reactant and product molecules and thus the intrinsic reaction rate. 

Kirkaldy, J. S., Isothermal Diffusion in Multicomponent Systems, Advances in Materials Research, 4, 55-100 (1970). 

The 15 chapters fall into three parts. Part I (Chapters 1-6) deals with the basic equations of diffusion in multicomponent systems. Chapters 7-11 (Part II) describe various models of mass and energy transfer. Part III (Chapters 12-15) covers applications of multicomponent mass transfer models to process design. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

Obviously, diffusion in multicomponent systems is the rale rather then the exception in most systems of practical interest. Yet, unusual or significant effects occur only rarely in multicomponent systeies. Such... 

A phenomenological theory of ordinety diffusion in multicomponent systems bused on irreversible thermodyonmics1 suggests that the ordinary diffusive flux of any species is a linear function of all die independent composition gradients. In one dimension. 

Multicomponent Effects. Only limited experimental data are available for multicomponent diffusion in liquids. The binary correlations are sometimes employed for (he case of a solute diffilsing through a mixed solvent of uniform composition.3,33 It Is clanr that thermodynamic nonideslities In multicomponent systems can cause sigaiflcanl effects. The resder is referred to Cussler s book4 for a discussion of available experimental information on diffusion in multicomponent systems. 

The Maxwell-Stefan equations for diffusion in multicomponent systems can become very complicated, and they are sometimes handled by using an effective dif -fusivity or pseudobinary approach. The effective diffusivity is defined by assuming that the rate of diffusion of component i depends only on its own composition gradient that is,... 

In order to be able to analyze diffusion in multicomponent systems, we look into the phenomenological theory of irreversible processes (Fitts, 1962). At constant temperature and pressure, the diffusion current density ji relative to the center of mass is obtained as... 

Ghorayt b, K.. and Firoozabadi, A. Molecular Pressure and Thermal Diffusion in Multicomponent Systems, J. Chemical Physics to be submitted, 1998). 

The phase composition of the diffusion zone for reactive diffusion in multicomponent systems is rather difficult to predict [19-21]. The reason is that some of the phases which are present in the equihbrium phase diagram do not appear even after a long period of anneaHng [22, 23]. This effect depends on the initial composition of the samples and on the regime of diffusion interaction. Note that in an unHmited binary system, aU intermediate phases will sooner or later definitely appear. The increase of the number of components participating in diffusion results in an increase in the number of the possible regimes for... 

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