Notes on Entropy Production in Multicomponent Fluids

We calculate the entropy production in a multicomponent fluid, allowing for chemical reactions and external forces that are species specific. Equations for balance of the mass of each species, energy, momentum, and entropy production are first formulated as integral equations for an arbitrary volume V with external area A and then reduced to differential equations. The treatment is rather standard insofar as the balance equations are concerned and follows closely the developments of Bird, Stewart and Lightfoot (BSL) [1] and Fitts [2]. BSL do not, however, give explicit expressions for the entropy production, although they appeal to the literature (see footnote on page 565 of [1]) in connection with the formulation of constitutive equations for the fluxes. Our treatment of entropy production is much more explicit, and extends the work of Sekerka and Mullins [3] along lines developed by Sekerka in unpublished notes written in August 1991 at a summer workshop in Aspen. Specifically, external forces and chemical reactions are now allowed. We also offer an identification of the energy flux that seems to be more self consistent with the entropy flux and with the work done by pressure in a multicomponent system. Based on this flux identification, we are able to make direct contact with the “second-law heat flux” of Fitts [2], page 28, without defining a separate heat flux, “not associated with the flow of matter, ” for the first law of thermodynamics, as does Fitts[2], page 27. We also make contact with the total energy flux in a multicomponent solution given by BSL [1], page 566. We are led finally to results that are in agreement with expressions for the entropy production of a multicomponent system by Fitts [2], page 31, and de Groot and Mazur [4], pages 26-27, and for a binary fluid system by Landau and Lifshitz [5], page 222. Furthermore, in the case for which the external forces can be can be expressed as gradients of potential functions that are independent of time, we are able to demonstrate that total chemical potentials, which are intrinsic chemical potentials augmented by the external potentials corresponding to the external forces, appear as conjugate variables to the fluxes; however, only the intrinsic chemical potentials appear as conjugate variables to the chemical reaction rates. For the case in which the external potentials are independent of