Minimum error Fickian diffusion coefficients for mass diffusion in multicomponent gas mixtures

Mass diffusion in multicomponent gas mixtures is governed by a coupled syst em of linear equations for the diffusive mass fluxes in terms of thermodyna mic driving forces, known as the generalized Stefan-Maxwell equation. In co mputations of mass diffusion in multicomponent gas mixtures, this coupling between the different components results in considerable computational over head. Consequently, simplified diffusion models for the diffusive mass flux es as explicit functions of the driving forces are an attractive alternativ e. These models can be interpreted as an approximate solution to the Stefan -Maxwell equation. Simplified diffusion models require the specification of "effective" diffusion coefficients which are usually expressed as function s of the binary diffusion coefficients of each species pair in the mixture. Current models for the effective diffusion coefficients are incapable of p roviding a priori control over the error incurred in the approximate soluti on. In this paper a general form for diagonal approximations is derived, which accounts for the requirement imposed by the special structure of the Stefan -Maxwell equation that such approximations be constructed in a reduced-dime nsional subspace. In addition, it is shown that current models can be expre ssed as particular cases of two general forms, but not all these models cor respond to the general form for diagonal approximations. A new minimum erro r diagonal approximation (MEDA) model is proposed, based on the criterion t hat the diagonal approximation minimize the error in the species velocities . Analytic expressions are derived for the MEDA model's effective diffusion coefficients based on this criterion. These effective diffusion coefficien ts automatically give the correct solution in two important limiting cases: for that of a binary mixture, and for the case of arbitrary number of comp onents with identical binary diffusivities, Although these minimum error ef fective diffusion coefficients are more expensive to compute than existing ones, the approximation will still be cheaper than computing the exact Stef an-Maxwell solution, while at the same time being more accurate than any ot her diagonal approximation, Furthermore, while the minimum error effective diffusion coefficients in this work are derived for bulk diffusion in homog eneous media, the minimization procedure can in principle be used to derive similar coefficients for diffusion problems in heterogeneous media which c an be represented by similar forms of the Stefan-Maxwell equation. These pr oblems include diffusion in macro- and microporous catalysts, adsorbents, a nd membranes.