Two-Phase Thermodynamic Equilibrium

The two-phase thermodynamic equilibrium model is a multiphase mixture approach that is restricted to modeling two phases. The two phases—liquid and vapour—must be of the same substance and single-component, for example, water and steam. The two phases are assumed to be in thermodynamic equilibrium.

Phase changes that can occur are boiling and condensation. The suspension of water and steam is assumed to be a homogeneous single-phase system. The transport equations Eqn. (2937) to Eqn. (2939) for mass, momentum, and energy are solved. The distribution of phases, that is, the volume fraction $\alpha$ , is not obtained by solving a transport equation, but from the distribution of static enthalpy under the assumption of thermodynamic equilibrium.

Assuming that the liquid and vapor phase are in thermodynamic equilibrium, steam mass quality (vapor mass fraction) $Y$ is a function of the static thermal enthalpy $h$ ([634], [640]):

where $h_{ls}$ and $h_{vs}$ are the enthalpies of liquid and vapor at saturation temperature $T_{\mathrm{sat}}$ .

The vapor volume fraction ${\alpha }_v$ is calculated using the expression:

where ${\rho }_{vs}$ and ${\rho }_{ls}$ are the densities of vapor and liquid at saturation temperature.

Continuity Equation

The conservation of mass for the two-phase mixture is given by:

$$\frac{\partial }{\partial t}{\int }_V^{}{\rho }_m\mathrm{dV}+{\int }_A^{}{\rho }_m{\mathbf{\text{v}}}_m\cdot da={\int }_V^{}{S}_u\mathrm{dV}$$

(2937)

where:

• ${\rho }_m$ is the mixture density
• $v_m$ is the mass averaged velocity
• $S_u$ is a user-defined mass source term

Momentum Equation

The momentum balance for the two-phase mixture is given by:

$$\frac{\partial }{\partial t}{\int }_V^{}{\rho }_m{\mathbf{\text{v}}}_m\mathrm{dV}+{\int }_A^{}{\rho }_m{\mathbf{\text{v}}}_m\otimes {\mathbf{\text{v}}}_m\cdot da=-{\int }_A^{}p\mathbf{\text{I}}\cdot da+{\int }_A^{}{T}_m\cdot da+{\int }_V^{}{\mathbf{\text{f}}}_b\mathrm{dV}+{\int }_V^{}{s}_u\mathrm{dV}-\underset{\mathrm{Drift}\mathrm{Flux}\mathrm{Term}}{\underbrace{{\int }_A^{}\frac{{\alpha }_v}{1-{\alpha }_v}\frac{{\rho }_v{\rho }_l}{{\rho }_m}{\overline{v}}_{\mathrm{dr}}\otimes {\overline{v}}_{\mathrm{dr}}\cdot da}}$$

(2938)

where:

• $I$ is the unity tensor
• $p$ is pressure
• $T_m$ is the viscous stress tensor
• $f_b$ is the body force vector
• $s_u$ is a user-defined momentum source term
• ${\alpha }_v$ is the volume fraction of the vapor phase
• ${\rho }_v$ , ${\rho }_l$ are the density of vapour and of liquid, respectively
• ${\overline{v}}_{\mathrm{dr}}$ is the mean drift flux velocity of the vapor phase

The last term of Eqn. (2938) appears only when the drift flux model is used to account for the relative motion between the phases.

Energy Equation

For the two-phase equilibrium model, the energy equation is given by:

$$\frac{\partial }{\partial t}{\int }_V^{}{\rho }_m{E}_m\mathrm{dV}+{\int }_A^{}({\rho }_m{H}_m{\mathbf{\text{v}}}_m+p)\cdot da=-{\int }_A^{}\dot{\mathbf{\text{q}}}\cdot da+{\int }_A^{}T\cdot {\mathbf{\text{v}}}_m\cdot da-\underset{\mathrm{Drift}\mathrm{Flux}\mathrm{Term}}{\underbrace{{\int }_A^{}\alpha \frac{{\rho }_v{\rho }_l}{{\rho }_m}\left(h_{vs}-h_{ls}\right){\overline{v}}_{\mathrm{dr}}\cdot da}}+{\int }_V^{}({\mathbf{\text{f}}}_b\cdot {\mathbf{\text{v}}}_m+S_u)\mathrm{dV}$$

(2939)

where:

• $E_m$ is the total energy of the mixture
• $H_m$ is the total enthalpy of the mixture
• $\dot{q}$ is the heat flux vector
• $S_u$ is a user-defined energy source term
• $h_{vs}$ is the enthalpy of vapor at saturation temperature $T_{\mathrm{sat}}$
• $h_{ls}$ is the enthalpy of liquid at saturation temperature $T_{\mathrm{sat}}$