# CFD and Energy Loss Model Analysis of High-Speed Centrifugal Pump with Low Specific Speed
## 2 CFD Simulations and Energy Loss Model
### CFD Simulations  
As for the water pump studied in this paper, the fluid can be regarded as incompressible fluid, that is, the density is constant, and both the change of temperature and the energy conservation equation can be ignored. Thus, the isothermal model with 25 \({}^{\circ}\)C is adopted. The inlet condition of simulation is the total pressure, 0.1 MPa, and the outlet condition is the mass flow rate from 1 to 5 m\({}^{3}\cdot\)h\({}^{-1}\). Meanwhile, the convergence criterion in the Solver Control is set as \(10^{-5}\). As for the incompressible turbulent flow, the Reynolds time averaged method, which is usually adopted in engineering, is used to deal with the transient Navier Stocks equations. The instantaneous value of each pulse momentum in the flow field is expressed as the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline \multirow{5}{*}{Semi-open impeller} & Number of blades & \(Z_{1}\) & 6 \\  & Inlet diameter/mm & \(D_{1}\) & 27.2 \\  & Outlet diameter/mm & \(D_{2}\) & 64.0 \\  & Outlet blade width/mm & \(b_{2}\) & 3.8 \\  & Outlet angle/\({}^{\circ}\) & \(\beta_{2}\) & 20 \\ \hline \multirow{5}{*}{Vaned diffuser} & Number of blades & \(Z_{2}\) & 7 \\  & Inlet diameter/mm & \(D_{3}\) & 65.0 \\ \cline{1-1}  & Outlet diameter/mm & \(D_{4}\) & 82.0 \\ \cline{1-1}  & Outlet angle/\({}^{\circ}\) & \(\beta_{3}\) & 5.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of hydraulic components of pump. Figure 1: Image of the low specific speed high-speed centrifugal pump (**a**) and schematic of low specific speed high-speed centrifugal pump; (**b**): 1—inducer; 2—inpeller; 3—diffuser; 4—casing; 5—main axle; 6—motor stator; 7—tail end fixed bearing; 8—front fixed bearing. sum of the average value and the pulsation value, and the characteristics of turbulence are expressed by other equations. The Navier Stocks equations are shown below. \[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial\rho\overline{u_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}} \left(\rho\overline{u_{i}u_{j}}\right)=-\frac{\partial\overline{p}}{\partial x _{i}}+\frac{\partial}{\partial x_{i}}\left[\mu\left(\frac{\partial\overline{u _{i}}}{\partial x_{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}}\right)- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right]+f_{i} \tag{2}\]  
The \(k\)-\(\varepsilon\) model is the most widely used one for its good robustness and reasonable prediction of large-scale turbulence. The dissipation rate is defined as:  
\[\varepsilon=\frac{\mu}{\rho}\left(\frac{\partial u_{i}^{\prime}}{\partial x_{ k}}\right)\left(\frac{\partial u_{i}}{\partial x_{k}}\right) \tag{3}\]  
Turbulent viscosity \(\mu_{\mathrm{t}}\) is a function of \(k\) and \(\varepsilon\):  
\[\mu_{\mathrm{t}}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{4}\]  
Then, there is the standard \(k\)-\(\varepsilon\) turbulence equation for incompressible fluid:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}\left[\rho u _{j}k-\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{\partial k}{\partial x _{j}}\right]=\rho(p_{k}-\varepsilon) \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{j}} \left[\rho\mu_{j}\varepsilon-\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right) \frac{\partial k}{\partial x_{j}}\right]=\rho\frac{\varepsilon}{k}(C_{1}p_{ k}-C_{2}\varepsilon) \tag{6}\]  
where \(C_{\mu}\), \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(C_{1}\), \(C_{2}\) refer to the turbulence model coefficient and the values are 0.09, 1.00, 1.30, 1.44, 1.92, respectively. As for the impressive flow, the pressure correction algorithm SIMPLE method is used to solve the governing equations. The channels of inducer and impeller are set as a rotating state at 7500 rpm, of which the blades, shroud, and hub are set as a rotating wall. The surfaces of the internal flow channel are also set as a rotating wall and the rest of the surfaces are set as a frozen wall.