Modelling of thermal diffusion and thermodynamic restrictions

Upon a review of the balance equations for a mixture, diffusion fluxes are shown to satisfy an evolution equation which, with slight differences, describes diffusion relative to a constituent or relative to the barycentric reference. As a consequence, the particular case of binary mixtures is considered with the view of establishing a connection with the current models. The fact that the diffusion flux is governed by a balance (evolution) equation is quite unusual in the literature since customary models are based on constitutive equations (Fick’s law, Soret and Dufour models). This in turn motivates the analysis for a more appropriate thermodynamic setting. The second law for the whole mixture, along with some constitutive assumptions in classical (nonequilibrium) thermodynamics, is shown to provide the constitutive equations which are currently applied in the literature. Instead, the model of diffusion flux through a balance equation proves to be fully consistent with rational thermodynamics and the assumption that the functions, relative to a single constituent, depend on variables pertaining to that constituent whereas interaction terms are allowed to depend also on quantities pertaining to the other constituents.