Abstract
This work is devoted to the consistent modeling of a three-phase mixture of a gas, a liquid and its vapor. Since the gas and the vapor are miscible, the mixture is subjected to a non-symmetric constraint on the volume. Adopting the Gibbs formalism, the study of the extensive equilibrium entropy of the system allows to recover the Dalton’s law between the two gaseous phases. In addition, we distinguish whether phase transition occurs or not between the liquid and its vapor. The thermodynamical equilibria are described both in extensive and intensive variables. In the latter case, we focus on the geometrical properties of equilibrium entropy. The consistent characterization of the thermodynamics of the three-phase mixture is used to introduce two Homogeneous Equilibrium Models (HEM) depending on mass transfer is taking into account or not. Hyperbolicity is investigated while analyzing the entropy structure of the systems. Finally we propose two Homogeneous Relaxation Models (HRM) for the three-phase mixtures with and without phase transition. Supplementary equations on mass, volume and energy fractions are considered with appropriate source terms which model the relaxation towards the thermodynamical equilibrium, in agreement with entropy growth criterion.
Highlights
The modelling of compressible multiphase flows is crucial for a wide range of applications, notably in the nuclear framework, for instance in loss of coolant accident in pressurized water reactors or in vapor explosions in steel industry [5, 45]
The equations of state obtained are introduced in two Homogeneous Equilibrium Models (HEM) models depending on whether phase transition occurs or not between the liquid and its vapor
Hyperbolicity of the systems is investigated, taking advantage of the entropy structure studied in the previous section
Summary
This constraint makes the whole modeling difficult since it prevents from using convenient tools of convex analysis such as sub-convolution and Legendre transform, see relative works in [23, 36] At this stage, the maximization of the mixture entropy allows to recover a consistent characterization of the thermodynamical equilibrium: the Dalton’s law for the gaseous phases and the equality of the temperatures apply. We have used the same thermodynamical model in [2] to model the rebound and collapse of a bubble of air and vapor surrounded by liquid In this preliminary work, each phase follows a stiffened gas law.
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