On the Solution of the Stefan-Maxwell Equations

Abstract

The well–known Hirschfelder, Curtiss, and Bird [HCB] zero–diagonal formalism for the multicomponent mass and thermal diffusion in dilute gases is reviewed. A formal expression for the thermal diffusion coefficient in terms of the thermal diffusion ratio is obtained. The theory of thermal diffusion is reviewed and practical expressions for estimating the thermal diffusion ratio are summarized. Application to the chemical oxygen–iodine laser is briefly discussed. An exact solution of the Stefan–Maxwell equations is obtained for the species diffusion velocities and mass fluxes. From this solution it is shown that the effective binary mass diffusion model is the lowest order approximation to the exact solution and, hence, the correct form of the effective binary mass diffusion coefficient is obtained. Numerical examples illustrate the pitfalls of using an incorrect effective mass diffusion coefficient. Also, from this solution for the mass fluxes an expression for the zero– diagonal multicomponent mass diffusion matrix is derived. It is shown that the linear equations satisfied by the zero–diagonal multicomponent diffusion coefficients are equivalent to a generalized Stefan–Maxwell equation whose direct solution for the diffusion matrix agrees with that obtained from the solution for the mass fluxes. A practical, exact expression for the thermal diffusion coefficient in terms of the thermal diffusion ratio and the correct form of an effective binary thermal diffusion model proposed by Paul and Warnatz is derived. Using the HCB formalism it shown how to express the Dufour flux in terms of the thermal diffusion ratio. In separate appendices the effective binary mass diffusion model is derived from its intuitive definition and a phenomological definition of the effective binary mass diffusion coefficient is given.