Solving the Multicomponent Equations

The general method of solution that was proposed by Toor and by Stewart and Prober exploits the properties of the modal matrix [P] whose columns are the eigenvectors of [7)] (see Appendix A.4). The matrix product 

Examination of Eqs. 5.1.7-5.1.10 shows that the similarity transformation reduces the original set of n — 1 coupled partial differential equations to a set of n — 1 uncoupled partial differential equations in the pseudocompositions. Equation (5.1.10) for the /th pseudocomponent is 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if 

In order to recover the solution to the original problem in terms of real mole fractions and fluxes we apply the inverse transformations 

The application of the transformation (Eq. 5.1.20) allows us to recover the generalized Fick s law (Eq. 3.2.5) with the composition gradients then obtained by differentiation of Eq. 5.1.22 we have 

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