Mixture Multiphase (MMP)

The Mixture Multiphase (MMP) model assumes that the phases are miscible. It is suitable for modeling dispersed multiphase flows such as bubbly and droplet flows. Typical applications include fuel cells, boilers, and steam turbines. When simulating liquid droplets dispersed in a gas, the Mixture Multiphase (MMP) model accounts for evaporation and condensation.

The Mixture Multiphase (MMP) model models fluid phases by solving transport equations for mass, momentum, and energy for the mixture of phases as a whole rather than for each phase separately. To calculate the distribution of phases, the volume fraction transport equation is solved for each phase. For phases that are moving at different velocities, algebraic relations are used to compute the relative velocities.

Volume Fraction Transport Equation

The volume fractions are transported according to the following conservation equation:

$$\frac{\partial }{\partial t}{\int }_V{\alpha }_{i}dV+{\int }_A{\alpha }_i{\mathbf{v}}_m\cdot da={\int }_V(S_{u,i}-\frac{{\alpha }_i}{{\rho }_i}\frac{D{\rho }_i}{Dt})dV+\underset{\mathrm{Turbulent}\mathrm{Term}}{\underbrace{{\int }_A\frac{{\mu }_t}{{\sigma }_t{\rho }_m}\nabla {\alpha }_i\cdot da}}-\underset{\mathrm{Slip}\mathrm{Velocity}\mathrm{Term}}{\underbrace{\underset{V}{\int }\frac{1}{{\rho }_i}\nabla \cdot \left({\alpha }_i{\rho }_i{\mathbf{v}}_{d,i}\right)}}dV$$

(2875)

where:

• $t$ is time
• $V$ is volume
• ${\alpha }_i$ is the volume fraction of phase $i$
• $v_m$ is the mass averaged velocity
• $a$ is the surface area vector
• $S_{u,i}$ is the user-defined source term for phase $i$
• ${\rho }_i$ is the density of phase $i$
• ${\mu }_t$ is the turbulent dynamic viscosity
• ${\sigma }_t$ is the turbulent Schmidt number
• $v_{d,i}$ is the diffusion velocity

The turbulent term is set to zero for laminar flow cases.

To clarify the turbulent diffusion term, the exact term is:

$\frac{{\mu }_t}{{\rho }_i{\sigma }_t}\nabla Y_i=\frac{{\mu }_t}{{\rho }_i{\sigma }_t}\nabla \left(\frac{{\rho }_i}{{\rho }_m}{\alpha }_i\right)=\frac{{\mu }_t}{{\rho }_i{\sigma }_t}\left(\frac{{\rho }_i}{{\rho }_m}\nabla {\alpha }_i+{\alpha }_i\nabla \left(\frac{{\rho }_i}{{\rho }_m}\right)\right)$

where $Y_i={\alpha }_i{\rho }_i/{\rho }_m$ is the mass fraction of phase $i$ .

However, the second component can be omitted, as typically:

$|\frac{{\rho }_i}{{\rho }_m}\nabla {\alpha }_i|\gg \left|{\alpha }_i\nabla \left(\frac{{\rho }_i}{{\rho }_m}\right)\right|$

Omitting the second component results in the turbulent diffusion term that is given in Eqn. (2875).

Continuity Equation

The conservation of mass for the mixture of phases is given by:

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

(2876)

where:

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

Momentum Equation

The momentum balance for the mixture of phases is given by:

$$\frac{\partial }{\partial t}{\int }_V^{}{\rho }_m{\mathbf{\text{v}}}_m\mathrm{dV}+{\int }_A^{}{\rho }_m{\mathbf{\text{v}}}_m\otimes {\mathbf{\text{v}}}_m\cdot da=-{\int }_A^{}p\mathbf{\text{I}}\cdot da+{\int }_A^{}{T}_m\cdot da+{\int }_V^{}{\mathbf{\text{f}}}_b\mathrm{dV}+{\int }_V^{}{s}_u\mathrm{dV}-\underset{\mathrm{Slip}\mathrm{Velocity}\mathrm{Term}}{\underbrace{\sum _i{\int }_A{\alpha }_i{\rho }_i{\mathbf{v}}_{d,i}\otimes {\mathbf{v}}_{d,i}\cdot da}}$$

(2877)

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

The stress tensor, $T_m$ , is estimated as:

$$T_m={\mu }_{\mathrm{eff}}\left[(\nabla {\mathbf{\text{v}}}_m+{(\nabla v_m)}^T)-\frac{2}{3}\left(\nabla \cdot {\mathbf{\text{v}}}_m\right)\mathbf{\text{I}}\right]$$

(2878)

where ${\mu }_{\mathrm{eff}}$ is the effective viscosity.

Energy Equation

The energy equation for the mixture of phases reads:

$$\frac{\partial }{\partial t}{\int }_V^{}{\rho }_m{E}_m\mathrm{dV}+{\int }_A^{}({\rho }_m{H}_m{\mathbf{\text{v}}}_m+\underset{\mathrm{Slip}\mathrm{Velocity}\mathrm{Term}}{\underbrace{\sum _i{\alpha }_i{\rho }_i{H}_i{\mathbf{v}}_{d,i}}})\cdot da=-{\int }_A^{}\dot{\mathbf{\text{q}}}\cdot da+{\int }_A^{}{T}_m\cdot {\mathbf{\text{v}}}_m\cdot da+{\int }_V^{}({\mathbf{\text{f}}}_b\cdot {\mathbf{\text{v}}}_m+S_u)\mathrm{dV}$$

(2879)

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

Total energy $E_m$ and total enthalpy $H_m$ of the mixture are defined as:

$$E_m=H_m-\frac{p}{{\rho }_m}$$

(2880)

$$H_m=h_m+\frac{{\left|{\mathbf{\text{v}}}_m\right|}^2}{2}$$

(2881)

where $h_m$ is the mixture-specific static enthalpy.

The terms that contain the diffusion velocity $v_{d,i}$ in Eqn. (2875), Eqn. (2877), and Eqn. (2879) are due to the slip between phases. See Phase Slip.