Continuum thermodynamics of chemically reacting fluid mixtures

Abstract

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 anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to 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 ..

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent

“…For a rigorous derivation of (5.6) see Bothe and Dreyer [3]. There you also find the additional contribution ∇ · S i − y i ∇ · S in the right-hand side of (5.6).…”

“…This is a condensed form of the full entropy principle. For more details see Bothe and Dreyer [3], as well as Dreyer [7]. We consider the simplest class of isotropic fluids without mesoscopic forces.…”

“…Now we model the Helmholtz free energy ρψ, where we follow Example 2 in Bothe and Dreyer [3]. The free energy can be constructed from an equation of state for the pressure p and from relations for the "chemical part" of the chemical potential µ i .…”

“…The modeling of ψ follows the concept laid out in Section 13 of Bothe and Dreyer [3] and employs a decomposition of ψ into an "elastic" part ψ el , which takes into account the mechanical (pressure) work, and a "thermal" part ψ th which accounts for the entropy of mixing.…”

“…As an interesting alternative which leads to a divergence free velocity field, we make use of the partial momentum balance for A N . According to Bothe and Dreyer [3], the partial momentum balance for A i reads as…”