The Maxwell-Stefan Equations for Ternary Systems

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus, 

At constant pressure the driving forces are equal to the composition gradients Of the three equations 2.1.11-2.1.13, only two are independent due to the restriction Vx + Vx2 + Vx3 = 0. It is interesting to note that for a binary system, this restriction is sufficient to prove that 12 = 21- multicomponent ideal gas mixture we need a more detailed analysis (Hirschfelder et al., 1964 Muckenfuss, 1973) to show that 

As the molecules of all three constituents are, in general, in relative motion with average velocities m, it is hard to see how any simpler formulation will suffice. Equation 2.1.11 reduces to the proper binary equation in the limits X3 0 and 0 for the 1-2 and 1-3 binaries, respectively, so that both terms are necessary. It is to be noted that Eq. 2.1.11 does not include a term (m2 — M3) for the first constituent as it is not reasonable to assume that the relative velocity of these constituents alone will produce a potential gradient of the first constituent as there would be no direct drag on the molecules of constituent 1. If an additional term of the form X]X2X3(m2 — W3)/ i23 were to be introduced into Eq. 2.1.11, it 

13 are the only consistent generalization of Eqs. 2.1.9 and 2.1.10 to a ternary mixture, assuming a linear relation between the potential gradients and the constituents relative velocities. 

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Toor (1957) derived a solution of the Maxwell-Stefan equations for ternary systems when the total molar flux is zero, = 0. Write down expressions for [P], (y), and show that, for = 0, the eigenvalue solutions are equivalent to the expressions given by Toor. 

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