Minimum Error Fickian Diffusion Coefficients for Mass Diffusion in Multicomponent Gas Mixtures

Abstract

Mass diffusion in multicomponent gas mixtures is governed by a coupled system of linear equations for the diffusive mass fluxes in terms of thermodynamic driving forces, known as the generalized Stefan‐Maxwell equation. In computations of mass diffusion in multicomponent gas mixtures, this coupling between the different components results in considerable computational overhead. Consequently, simplified diffusion models for the diffusive mass fluxes as explicit functions of the driving forces are an attractive alternative. 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 functions of the binary diffusion coefficients of each species pair in the mixture. Current models for the effective diffusion coefficients are incapable of providing a priori control over the error incurred in the approximate solution. 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-dimensional subspace. In addition, it is shown that current models can be expressed as particular cases of two general forms, but not all these models correspond to the general form for diagonal approximations. A new minimum error diagonal approximation (MEDA) model is proposed, based on the criterion that 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 coefficients automatically give the correct solution in two important limiting cases: for that of a binary mixture, and for the case of arbitrary number of components with identical binary diffusivities. Although these minimum error effective diffusion coefficients are more expensive to compute than existing ones, the approximation will still be cheaper than computing the exact Stefan‐Maxwell solution, while at the same time being more accurate than any other diagonal approximation. Furthermore, while the minimum error effective diffusion coefficients in this work are derived for bulk