This section gathers the multiphase-flow models of TrioCFD. The multiphase CFD developments are gathered in the CMFD module, which designates all RANS approaches for multiphase flows and is based on the Pb_Multiphase problem. TrioCFD/CMFD uses the PolyMAC_MPFA numerical scheme [66], [65], initially developed for multiphase component-scale codes, with semi-implicit ICE and SETS solvers located in TRUST. CMFD regroups the interfacial terms specific to CFD applications: interfacial forces, energy transfer, wall heat-flux partitioning, bubble-diameter determination, etc.

Note: Work is still being done on the models presented in this section, and they are not fully validated.

2-fluid 6-equation framework

TrioCMFD is based on the Pb_Multiphase framework, which contains the multiphase terms common to component-scale codes and CFD. The following system presents the terms already included in TRUST and the terms added by TrioCMFD (for each phase \(k\), mass \(\mathcal{M}_k\), momentum \(\mathcal{Q}_k\) and energy \(\mathcal{E}_k\)):

\[\begin{aligned} (\mathcal{M}_k)\quad & \frac{\partial \alpha_k \rho_k}{\partial t} + \nabla \cdot (\alpha_k \rho_k \vec{u_k}) = \Gamma_k \\ (\mathcal{Q}_k)\quad & \frac{\partial \alpha_k \rho_k \vec{u_k}}{\partial t} + \underline{\nabla} \cdot (\alpha_k \rho_k \vec{u_k} \otimes \vec{u_k}) = -\alpha_k \underline{\nabla} P + \underline{\nabla} \cdot [ \alpha_k \mu_k \underline{\underline{\nabla}} \vec{u_k} - \alpha_k\rho_k \overline{u_i'u_j'}] + \vec{F}_{ki} + \vec{F}_k \\ (\mathcal{E}_k)\quad & \frac{\partial \alpha_k \rho_k e_k}{\partial t} + \nabla \cdot (\alpha_k \rho_k h_k \vec{u_k}) = \nabla \cdot [ \alpha_k\lambda_k \underline{\nabla} T_k - \alpha_k\rho_k \overline{u_i'e_k'}] -p\left[\frac{\partial \alpha_k}{\partial t} + \nabla \cdot (\alpha_k u_k)\right] + q_{ki} + q_{kp} \end{aligned} \]

In these equations, the terms \(\Gamma_k\) (mass transfer), \(\vec{F}_{ki}\) (interfacial forces), the Reynolds-stress terms \(-\alpha_k\rho_k\overline{u_i'u_j'}\) and \(-\alpha_k\rho_k\overline{u_i'e_k'}\), and the interfacial/wall heat fluxes \(q_{ki}\), \(q_{kp}\) are the contributions added by TrioCMFD; the rest is provided by TRUST. The single-phase turbulence (the Reynolds-stress terms) is treated in Multiphase Turbulence Modelling. One physical quantity necessary for the multiphase terms is the bubble diameter (see Interfacial Area & Bubble Diameter).

Notations

Variable	Expression
Vapour / liquid phase	\(v\) / \(l\)
Distance to the wall	\(y\)
Normal unit vector from the wall	\(\overrightarrow{n}\)
Velocity of phase \(k\)	\(\overrightarrow{u_k}\)
Dynamic / kinematic viscosity of phase \(k\)	\(\mu_k\) / \(\nu_k=\mu_k/\rho_k\)
Density of phase \(k\)	\(\rho_k\)
Reynolds stress tensor of the carrying phase	\(\underline{\underline{\tau_R}}=-\langle u_iu_j\rangle\)
Turbulent kinetic energy / dissipation of the carrying phase	\(k\) / \(\epsilon\)
Temperature of phase \(k\) / wall	\(T_k\) / \(T_\text{wall}\)
Thermal conductivity / heat capacity of phase \(k\)	\(\lambda_k\) / \(Cp_k\)
Liquid / void fraction	\(\alpha_l\) / \(\alpha_v\)
Internal energy of phase \(k\)	\(e_k\)
Mass transfer to phase \(k\)	\(\Gamma_k\)
Interfacial / volume forces applied to phase \(k\)	\(\vec{F}_{ki}\) / \(\vec{F}_k\)
Interfacial / wall heat flux to phase \(k\)	\(q_{ki}\) / \(q_{kp}\)
Interfacial area between \(l\) and \(v\)	\(a_i\)
Bubble diameter	\(d_b=\dfrac{6\alpha_v}{a_i}\)
Bubble departure diameter	\(d_\text{dep}\)
Saturation temperature / latent heat / surface tension	\(T_\text{sat}\) / \(L_\text{vap}\) / \(\sigma\)
Bubble Reynolds number	\(Re_b=\dfrac{\\|\overrightarrow{u_v}-\overrightarrow{u_l}\\| d_b}{\nu_l}\)
Weber number (turbulent fluctuation)	\(We=\dfrac{(\epsilon d_b)^{2/3}\rho_l d_b}{\sigma}\)
Eötvös number	\(Eo=\dfrac{g(\rho_l-\rho_v)d^2}{\sigma}\)
Superheat Jakob number	\(Ja_\text{sup}=\dfrac{\rho_l Cp_l(T_\text{wall}-T_\text{sat})}{\rho_v L_\text{vap}}\)
Subcooling Jakob number	\(Ja_\text{sub}=\dfrac{\rho_l Cp_l(T_\text{sat}-T_l)}{\rho_v L_\text{vap}}\)