Diffusion multicomponent systems

One other textbook deserves a special mention. The book by G. Froment and K. Bischoff, Chemical Reactor Analysis and Design, aims not to be easy but elegant, introducing the reader directly to the advanced theories of reaction engineering and to the frontiers of research by including complex reaction networks, advanced models for catalytic systems, multicomponent diffusion, and the surface renewal theory for gas-liquid contact. The book is excellent for students who wish to become scientists in chemical reaction engineering. 

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 ... 

One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B -h C). Umesi and Danner compared the three equations given below for 49 ternaiy systems. All three equations were equivalent, giving average absolute deviations of 25 percent. 

All the experimental data in Table 6.1 refer to pure gases. Separation experiments, in which surface diffusion is the separation mechanism, are scarcely reported. Feng and Stewart (1973) and Feng, Kostrov and Stewart (1974) report multicomponent diffusion experiments for the system He-Nj-CH in a y-alumina pellet over a wide range of pressures (1-70 bar), temperatures (300-390 K) and composition gradients. A small contribution of surface diffusion (5% of total flow) to total transport could be detected, although it is not clear, which of the gases exhibits surface difiusion. The data could be fitted with the mass-flux model of Mason, Malinauskas and Evans (1967), extended to include surface diffusion. 

Diffusion in a system with three or more components is called multicomponent diffusion. One example is diffusion of Ca, Fe, Mn, and Mg in a zoned garnet (Ganguly et al., 1998a). Another example is diffusion between an andesitic melt... 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

The full treatment of multicomponent diffusion requires a diffusion matrix because the diffusive flux of one component is affected by the concentration gradient of all other components. For an N-component system, there are N-1 independent components (because the concentrations of all components add up to 100% if mass fraction or molar fraction is used). Choose the Nth component as the dependent component and let n = N 1. The diffusive flux of the components can hence be written as (De Groot and Mazur, 1962)... 

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. 

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) ... 

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

Kirkaldy J.S. and Purdy G.R. (1969) Diffusion in multicomponent metallic systems, X diffusion at and near ternary critical states. Can. J. Phys. 47, 865-871. 

Mungall J.E., Romano C., and Dingwell D.B. (1998) Multicomponent diffusion in the molten system K20-Na20-Al203-Si02-H20. Am. Mineral. 83, 685-699. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

For a two-component mixture the multicomponent diffusion coefficients D, become the ordinary binary diffusion coefficients Sh,. For these quantities 2D,-, = 2D,- and 2D = 0. For a three-component system the multicomponent diffusion coefficients are not equal to the ordinary binary diffusion coefficients. For example, it has been shown by Curtiss and Hirschfelder (C12) in their development of the kinetic theory of multicomponent gas mixtures that... 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

After this formal discussion of chemical diffusion, let us now turn to some more practical aspects. In order to compare the formal theory with experiment, we have to carefully define the reference frame for the diffusion process, which is not trivial in the case of binary or multicomponent diffusion. To become acquainted with the philosophy of this problem, we deal briefly with defining a suitable reference frame in a binary system. Since only one (independent) transport coefficient is needed to describe chemical diffusion in a binary system, then according to Eqn. (4.57) we have in a one-dimensional system... 

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. 

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

For a system with no kinetic or adsorption complications, the forward transition time x decreases while xr increases until finally x = xr in the limit, at steady state. (Because the convergence rate is slow, equality of x and xr is not commonly achieved experimentally before the onset of natural convection and nonplanar diffusion effects.) Quantitative treatments for single component systems, multicomponent systems, stepwise reactions, and systems involving chemical kinetics have been derived. The technique has not been used extensively. 

This simplification is not possible for some CVD systems in which large density changes are associated with the deposition process. The growth of CdHgTe is a typical example that shows how the depletion of Hg next to the substrate creates an unstable density gradient that drives recirculations (205), as discussed earlier and illustrated in Figure 14. LPCVD processes use little or no diluent and often involve several species, and multicomponent diffusion may be an important factor (21). Fortunately, these reactors are isothermal, and the relative insensitivity of reactor performance to details of the fluid flow greatly simplifies the analysis. 

A. Pfennig, Multicomponent Diffusion, in Int. Workshop "Transport in Fluid Multiphase Systems From Experimental Data to Mechanistic Models. 

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... 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

Zielinski and Hanley [AlChE J. 45,1 (1999)] developed a model to predict multicomponent diffusivities from self-diffusion coefficients and thermodynamic information. Their model was tested by estimated experimental diffusivity values for ternary systems, predicting drying behavior of ternary systems, and reconciling ternary selfdiffusion data measured by pulsed-field gradient NMR. 

In all experiments, a constant silver layer thickness of three monolayers was initially deposited. In the multicomponent system, Cd diffuses towards S under thin layer conditions" established by a 2D CdxAgy surface alloy. In the one-component systems, Cd diffuses towards S under semi-infinite conditions. A phase transition... 

For multicomponent systems, the diffusion flux of each component depends on the concentration gradient of all the system components. In order to account for these interactions, a matrix of diffusion coefficients is considered and the two laws of diffusion are written [5] ... 

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