Solving the Multicomponent Flux Equations

In general, solving the multicomponent diffusion problems is not necessary if the analogous binary problem has already been solved (Toor, 1964 Stewart and Prober, 1964). We can mathematically convert the multicomponent problem into a binary problem, look up the binary solution, and then convert this solution back into the multicomponent one. In other words, multicomponent problems usually can be solved using a cookbook approach little additional work is needed. Some use this cookbook to convert fairly comprehensible binary problems into multicomponent goulash that is harder to understand than necessary. 

In this section, we first give the results for ternary diffusion and then for the general approach. By starting with the ternary results, we hope to help those who need to solve simple problems. They should not have to dig through the matrix algebra unless they decide to do so. 

A binary diffusion problem has a solution that can be written as 

In this, Aci is a concentration difference that generally varies with position and time, Acio is some reference concentration difference containing initial and boundary conditions, and F D) is the expUcit function of position and time. For example, for the diaphragm cell, the binary solution is (see Example 2.2-4) 

Every binary diffusion problem has an analogous ternary diffusion problem that is described by similar differential equations and similar initial and boundary conditions. The differential equations differ only in the form of Pick s law that is used. The conditions are also parallel. For example, in a binary problem the solute concentration may be fixed at a particular boundary, so in the corresponding ternary problem, solute concentrations will also be fixed at the corresponding boundary. When this is true, the ternary diffusion problems have the solutions 

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