# Gas-Liquid Mixability Study in a Jet-Stirred Tank for Mineral Flotation
## 2 Methodology
### Multiphase and Turbulence Modelling
\begin{array}{l}\frac{\partial\left(\alpha_{q}\rho_{\mathrm{u}}u_{ \mathrm{q}x}\right)}{\partial t}+\frac{\partial\left(\alpha_{q}\rho_{\mathrm{ q}}u_{\mathrm{q}x}u_{\mathrm{q}x}\right)}{\partial x}+\frac{\partial\left( \alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}x}u_{\mathrm{q}x}\right)}{\partial y}+ \frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}x}u_{\mathrm{q}x} \right)}{\partial z}=\\ -\alpha_{q}\frac{\partial\alpha_{q}}{\partial x}+\frac{\partial}{\partial x} \left(\alpha_{q}\tau_{\mathrm{x}x}\right)+\frac{\partial}{\partial y}\left( \alpha_{q}\tau_{\mathrm{x}y}\right)+\frac{\partial}{\partial z}\left(\alpha_ {q}\tau_{\mathrm{x}z}\right)+\alpha_{q}\rho_{\mathrm{q}}g_{\mathrm{x}}+M_{ \mathrm{Pq}x}\\ \\ \frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}y}\right)}{\partial t }+\frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}y}u_{\mathrm{q}x }\right)}{\partial x}+\frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{ \mathrm{q}y}u_{\mathrm{q}y}\right)}{\partial y}+\frac{\partial\left(\alpha_{q }\rho_{\mathrm{q}}u_{\mathrm{q}y}u_{\mathrm{q}x}\right)}{\partial z}=\\ -\alpha_{q}\frac{\partial\alpha_{q}}{\partial y}+\frac{\partial}{\partial x} \left(\alpha_{q}\tau_{\mathrm{y}x}\right)+\frac{\partial}{\partial y}\left( \alpha_{q}\tau_{\mathrm{y}y}\right)+\frac{\partial}{\partial z}\left(\alpha_ {q}\tau_{\mathrm{y}x}\right)+\alpha_{q}\rho_{\mathrm{q}}g_{\mathrm{y}}+M_{ \mathrm{Pq}y}\\ \\ \frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}x}\right)}{\partial t }+\frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{\mathrm{q}x}u_{\mathrm{q}x }\right)}{\partial x}+\frac{\partial\left(\alpha_{q}\rho_{\mathrm{q}}u_{ \mathrm{q}x}u_{\mathrm{q}y}\right)}{\partial y}+\frac{\partial\left(\alpha_{q }\rho_{\mathrm{q}}u_{\mathrm{q}x}u_{\mathrm{q}x}\right)}{\partial z}=\\ -\alpha_{q}\frac{\partial\alpha_{q}}{\partial z}+\frac{\partial}{\partial x} \left(\alpha_{q}\tau_{\mathrm{x}x}\right)+\frac{\partial}{\partial y}\left( \alpha_{q}\tau_{\mathrm{x}y}\right)+\frac{\partial}{\partial z}\left(\alpha_{q }\tau_{\mathrm{x}z}\right)+\alpha_{q}\rho_{\mathrm{q}}g_{\mathrm{z}}+M_{ \mathrm{Pq}z}\end{array} \right. \tag{2}\]  
The equations describe the momentum conservation for each phase in the respective directions. The terms on the left side of the equations represent the changes in mass and momentum for each phase. On the right side, the terms include the pressure gradient, viscous stresses (including Reynolds stresses due to turbulence effects), volumetric forces (such as gravity), and interphase momentum coupling terms. \(\alpha_{\mathrm{q}}\) represents the volume fraction of phase q; \(\rho_{\mathrm{q}}\) denotes the density of phase q; \(\mathrm{u}_{\mathrm{q}x}\), \(\mathrm{u}_{\mathrm{q}y}\), \(\mathrm{u}_{\mathrm{q}z}\) are the velocity components of phase q in the three directions of x, y, z, respectively; \(P_{\mathrm{q}}\) represents the pressure of phase q; \(\tau_{\mathrm{x}x}\), \(\tau_{\mathrm{y}y}\), \(\tau_{\mathrm{zz}}\) are the normal viscous stress components in the x, y and z directions, respectively; \(\tau_{\mathrm{x}y}\), \(\tau_{\mathrm{x}z}\), and \(\tau_{\mathrm{y}z}\) denote the tangential viscous stress components on the x-y, x-z, y-z planes, respectively; \(g_{\mathrm{x}}\), \(g_{\mathrm{y}}\), and \(g_{\mathrm{z}}\) represent the components of gravitational acceleration in the directions of x, y, z; in this study, \(g_{\mathrm{x}}\), \(g_{\mathrm{y}}\) are zero; \(g_{\mathrm{z}}\) represents the gravitational acceleration in the z direction; \(M_{\mathrm{Pq}x}\), \(M_{\mathrm{Pq}y}\), \(M_{\mathrm{Pq}z}\) are the interphase momentum coupling terms between phases q and p in the three directions of x, y, z, respectively. In large-scale simulations, the k \(-\)\(\varepsilon\) turbulence model strikes a good balance between computational cost and accuracy when predicting shear flows.