Continuum thermodynamics of chemically reacting fluid mixtures

We consider viscous, heat-conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, we derive a closed system of partial mass and partial momentum balances plus a mixture balance of internal energy. This is achieved by careful exploitation of the entropy principle and requires appropriate definitions of absolute temperature and chemical potentials, based on an adequate definition of thermal energy excluding diffusive contributions. The resulting interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric in case of binary interactions. For chemically reacting systems and as a new result, the chemical interaction force is a contribution being non-symmetric outside of chemical equilibrium. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations. Moreover, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations become a strict consequence. With a classification of the factors in the binary products of the entropy production according to their parity—instead of the classical partition into so-called fluxes and driving forces—the apparent antisymmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared with the speed of sound, the Maxwell–Stefan equations follow in the case without chemistry, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model with only mass being balanced individually. In the reactive case, this approximation via a scale separation argument is no longer possible. We introduce the new concept of entropy invariant model reduction, leaving the entropy production unchanged under the reduction from partial momentum balances to a single mixture momentum balance. This results in an extension of the Maxwell–Stefan equations to chemically active mixtures with an additional contribution to the transport coefficients due to the chemical interactions.