# Development of Hybrid Airlift-Jet Pump with Its Performance Analysis
## 3 Governing Equations and Numerical Methods  
The compressible unsteady Reynolds-averaged Navier-Stokes equations are used as the governing equations for the flow in the hybrid pump. The homogeneous mixture model, which assumes that each phase traverses at different velocities but with a local equilibrium over short spatial length scales, is used for the two-phase flow. The mixture model consists of the continuity, momentum, and energy equations for the mixture, and the volume fraction equations for the secondary phase. The continuity equation for the mixture can be described as:  
\[\frac{\partial}{\partial t}(\rho_{m})+\frac{\partial}{\partial x_{i}}(\rho_{m }u_{m,i})=0 \tag{4}\]  
where \(u_{m}\) is the mass-averaged mixture velocity vector and \(\rho_{m}\) is the mixture density. The momentum equation for the mixture can be obtained by adding the individual momentum equations for all phases and be expressed as:  
\[ \begin{array}{ll}\frac{\partial}{\partial t}(\rho_{m}u_{m,i})+&\frac{ \partial}{\partial x_{j}}\left(\rho_{m}u_{m,i}u_{m,i}\right)\\ &=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[u_{ m}\left(\frac{\partial u_{m,i}}{\partial x_{j}}+\frac{\partial u_{m,i}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{m,k}}{\partial x_{k}} \right)\right]+\rho_{m}\overrightarrow{g}+\overset{\rightarrow}{F}\\ &+\frac{\partial}{\partial x_{j}}\left[\mu_{t}\left(\frac{\partial u_{m,i}}{ \partial x_{j}}+\frac{\partial u_{m,i}}{\partial x_{i}}\right)-\frac{2}{3} \left(\rho_{m}k+\frac{\partial u_{m,k}}{\partial x_{k}}\right)\delta_{ij} \right]\end{array}  \tag{5}\]  
where \(p\) is the static pressure, \(\mu_{m}\) is the viscosity of the mixture, \(\mu_{t}\) is the turbulence viscosity, \(\overrightarrow{g}\) is the gravity, and \(\overset{\rightarrow}{F}\) is the body force. The energy equation for the mixture assumes the following form:  
\[\frac{\partial}{\partial t}\sum_{k=1}^{n}(\alpha_{k}\rho_{k}E_{k})+\
abla\cdot \sum_{k=1}^{n}\left(\alpha_{k}\overset{\rightarrow}{v}_{k}(\rho_{k}E_{k}+p) \right)=\
abla\cdot\left(k_{eff}\
abla T\right) \tag{6}\]  
where \(k_{eff}\) is the effective conductivity, and \(T\) is the temperature. From the continuity equation for the secondary phase \(p\), the volume fraction equation for the secondary phase \(p\) can be obtained:  
\[\frac{\partial}{\partial t}\big{(}\alpha_{p}\rho_{p}\big{)}+\frac{\partial}{ \partial x_{i}}\big{(}\alpha_{p}\rho_{p}u_{m,i}\big{)}=0 \tag{7}\]  
The standard \(k\)-\(\epsilon\) model is used for the turbulence model and the standard wall function was employed for the near-wall treatment. The effects of turbulence models on two phase numerical solutions were intensively investigated in the preceding study [24]. The standard \(k\)-\(\epsilon\) turbulent model is based on the transport equations for the turbulent kinetic energy \(k\) and its dissipation rate \(\epsilon\), respectively. The transport equations can be written as:  
\[\frac{\partial}{\partial t}(\rho_{m}k)+\frac{\partial}{\partial x_{i}}(\rho_{m }ku_{m,i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu_{m}+\frac{\mu_{t}}{ \sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho_{m }\epsilon-Y_{m}. \tag{8}\]  
\[\frac{\partial}{\partial t}(\rho_{m}\epsilon)+\frac{\partial}{\partial x_{i}}( \rho_{m}\epsilon u_{m,i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu_{m}+ \frac{\mu_{t}}{\sigma_{\epsilon}}\right)\frac{\partial\epsilon}{\partial x_{j }}\right]+C_{1\epsilon}\frac{\epsilon}{k}(G_{k}+C_{3\epsilon}G_{b})-C_{2 \epsilon}\rho_{m}\frac{\epsilon^{2}}{k} \tag{9}\]  
In these equations, \(G_{k}\) represents the generation of turbulence kinetic energy due to the mass-averaged mixture velocity gradients. \(G_{b}\) is the generation of turbulence kinetic energy due to buoyancy. \(Y_{m}\) represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. \(C_{1\epsilon}\), \(C_{2\epsilon}\) and \(C_{3\epsilon}\) are constants.