## 2 Calculation Model
### Theoretical Model  
#### 2.2.1 Control Equation  
The internal flow of the vortex pump is very complex, with phenomena such as an attached boundary layer, backflow vortex, and free shear turbulence formed during the impeller rotation. In order to ensure the accuracy and reliability of numerical simulation results, the RNG k-\(\varepsilon\) turbulence model combined with the Reynolds averaged Navier-Stokes (RANS) method is used for calculation. The RNG k-\(\varepsilon\) model is particularly suitable for dealing with both high Reynolds number and low Reynolds number flows and can effectively capture the detailed characteristics of the internal flow field of the vortex pump, which provides accurate data support for pump design and performance optimization. The continuity equation plays a constraining role in ensuring mass conservation within the governing equations. When the fluid medium is an incompressible fluid, the equation is expressed as follows:  
\[\frac{\partial C_{i}}{\partial x_{i}}=0 \tag{1}\]  
The Reynolds time-averaged equation is expressed as follows:  
\[-\rho\overline{C^{\prime}_{i}C^{\prime}_{j}}=\mu_{t}\left(\frac{\partial C_{i} }{\partial x_{j}}+\frac{\partial C_{j}}{\partial x_{i}}\right)-\frac{2}{3} \delta_{ij}\rho k \tag{2}\]  
In which \(\mu_{t}\) is the turbulent eddy viscosity and \(\delta_{ij}\) is the Kronecker delta symbol. In the Reynolds time-averaging method, different turbulence models are derived using various assumptions about eddy viscosity and different modeling approaches. The continuity equation and momentum equation for turbulent mean motion are as follows:  
\[\frac{\partial p}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\rho u_{j} \right)=0 \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\left( \rho u_{i}u_{j}\right)=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\left(\mu\frac{\partial u_{i}}{\partial x_{j}}-\rho\overline{u^ {\prime}_{i}u^{\prime}_{j}}\right) \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Channel width, \(E\) (mm) & 10.32 \\ Impeller width, \(b\) (mm) & 5.15 \\ Inlet diameter, \(d_{1}\) (mm) & 15 \\ Outlet diameter, \(d_{2}\) (mm) & 15 \\ Tongue wrap angle, \(\theta\) (\({}^{\circ}\)) & 60 \\ Blade groove radius, \(r\) (mm) & 15 \\ Impeller diameter, \(D_{2}\) (mm) & 45.9 \\ Number of blades, \(Z\) & 30 \\ Distance between the impeller outer diameter and the & 2.75 \\ channel, \(a\) (mm) & 2.75 \\ Distance between the impeller end face and the channel, & 2.36 \\ \(c\) (mm) & 6.35 \\ Blade height, \(h\) (mm) & 6.35 \\ Design head of the vortex pump, \(H\) (m) & 10 \\ Rated speed, \(n\) (r/min) & 3000 \\ Rated flow rate, \(Q_{\text{d}}\) (L/min) & 10 \\ Total weight (kg) & 3 \\ Impeller materials & 2 CrB-GJB2294-1995 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main design parameters of the vortex pump. In which \(p\) is the pressure, \(\mu\) is the dynamic viscosity, and \(i\) and \(j\) are the component indices of the vector. In order to close the equation system, the transport equation can be established for the turbulent kinetic energy \(k\). This \(k\) can then be used to construct the turbulent viscosity. The transport equation is as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho eu_{i})}{\partial x_{i} }=\frac{\partial}{\partial x_{j}}\Bigg{[}\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\Bigg{]}+\mu_{t}\left(\frac{\partial u _{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\frac{ \partial u_{i}}{\partial x_{j}}-C_{D}\rho\frac{k^{\frac{3}{2}}}{L} \tag{5}\]  
\[\mu_{t}=\rho C_{\mu}\sqrt{kL}\]  
In which \(k\) is the kinetic energy (m\({}^{2}\)/\({}^{2}\)) and \(C_{\mu}\), \(\sigma_{k}\) is the empirical constant.