Diffusion equation multi-component systems

Figure 3-21 Calculated diffusion profiles for a diffusion couple in a ternary system. The diffusivity matrix is given in Equation 3-102a. The fraction of Si02 is calculated as 1 - MgO - AI2O3. Si02 shows clear uphill diffusion. A component with initially uniform concentration (such as Si02 in this example) almost always shows uphill diffusion in a multi-component system.

Equation (7.145) results from the rigorous dusty gas model, but unfortunately, it is not easy to implement for a multi-component system. Therefore, we will use simplified equations for the flux relations (7.145). The ordinary diffusion term in formula (7.145) can be approximated by... 

The principle of the Maxwell-Stefen diffusion equations is that the force acting on a species is balanced by the ffiction that is exerted on that species. The driving force for diffusion is the chemical potential gradient. The Maxwell-Stefan equations were applied to surface diffusion in microporous media by Krishna [77]. During surface diffusion, a molecule experiences friction from other molecules and from the surface, which is included in de model as a pseudo-species, n+1 (Dusty-gas model). The balance between force and friction in a multi-component system can thus be written as [77] ... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

Equation (5.177) resulting from the rigorous model (dusty gas model) for diffusion and flow is not easy to implement for a multi-component system. Some investigators (Soliman et al., 1988 Xu and Froment, 1989b Elnashaie and Abashar, 1992) have used simplified... 

A first attempt to consider the role of the Debye counterion atmosphere on the transport of a surfactant ion through the DL was made by Mikhailovskij (1976, 1980) (cf. Kortilm 1966, Lyklema 1991). In contrast to a macro-kinetic model, Mikhailovskij derived kinetic equations for a multi-component system under the influence of an external electric field. The basis of this derivation was the set of Bogolubow equations for the partial distribution functions. As the result of the model derivation the following set of electro-diffusion equations is obtained. 

The diffusion coefficients used to describe multi-component diffusion are mutual diffusion coefficients. In the multi-component system, mutual diffusion coefficients are defined by Equation 4-13 the matrix of diffusion coefficients depends on the concentration of individual components. The diffusion coefficients used in the earlier sections of the chapter, however, describe solute molecules diffusing in a medium at infinite dilution. The isolated molecule is called a tracer these tracer diffusion coefficients are defined by the physics of random walk processes, as described in Chapter 3. The self-diffusion coefficient, used in Equation 4-11, is a tracer diffusion coefficient in the situation where all of the molecules in the system are identical. The self-diffusion coefficient, T>aa is defined by (recall Equation 3-12) [62] ... 

As a preliminary test, the governing equation (i.e., convection-diffusion-reaction PDE) is solved using an ODE time integrator for the binary system Including only A and B components. Fig. 2 shows effects of the three kernels on liquid concentrations. Abnormal concentration profiles are found in inactive zones for the conventional rate model (Fig. 2a) and that with the sum kernel (Fig. 2b). The product kernel will play an effective role for multi-component systems. 

The basic expressions for the mass fluxes and the equations of continuity for multi-component mixtures are given in Sec. II,B. For a -component mixture of ideal gases in a system in which there is no pressure diffusion, forced diffusion, or thermal diffusion, the fluxes are given by... 

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

The j° term denotes the ordinary concentration diffusion (i.e., multi-component mass diffusion). In general, the concentration diffusion contribution to the mass flux depends on the concentration gradients of all the substances present. However, in most reactor systems, containing a solvent and one or only a few solutes having relatively low concentrations, the binary form of Pick s law is considered a sufficient approximation of the diffusive fluxes. Nevertheless, for many reactive systems of interest there are situations where a multi-component closure (e.g., a Stefan-Maxwell equation formulated in terms... 

Even the binary system diffusivities in liquid mixtures are composition-dependent. Therefore, in multi-component liquid mixtures with n components, predictions of the diffusion coefficients relating flows to concentration gradients are empirical. The diffusion coefficient of dilute species i in a multicomponent liquid mixture, Dj , may be estimated by Perkins and Geankoplis equation ... 

The analysis of macropore diffusion in binary or multicomponent systenis presents no particular problems since the transport properties of one compos nent are not directly affected by changes ini the concentration of the bther components. In an adsorbed phase the situation is more complex since ih addition to any possible direct effect on thei mobility, the driving force for each component (chemical potential gradient is modified, through the multi-component equilibrium isotherm, by the coiicentration levels of all components in the system. The diffusion equations for each component are therefore directly coupled through the equilibrium relationship. Because of the complexity of the problem, diffusion in a mixed adscjrbed phase has been studied tjs only a limited extent. 

The qualitative nature of the departure from the pseudo-stationary state can be explained on the basis of very simple considerations. Let us treat our hypothetical chain system and simplify the multi-component diffusion equations by assuming that the free radical B diffuses through the mixture of other components (regarded as a single species) with an effective binary diffusion coefficient, Then, combining the equation of continuity with the equation for diffusion of the free radical. 

Nonetheless, some concentrated systems are best described using multicomponent diffusion equations. Examples of these systems, which commonly involve unusual chemical interactions, are listed in Table 7.0-1. They are best described using the equations derived in Section 7.1. These equations can be rationalized using the theory of irreversible thermodynamics, a synopsis of which is given Section 7.2. In most cases, the solution to multi-component diffusion problems is automatically available if the binary solution is available the reasons for this are given in Section 7.3. Some values of ternary diffusion coefficients are given in Section 7.4 as an indication of the magnitude of the effects involved. Finally, tracer diffusion is detailed as an example of ternary diffusion in Section 7.5. 

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