## 2 Computational Modeling  
### Pump Model  
The object of study in this present work is one stage of a three-stage centrifugal type 11 MT32 pump with a specific speed of \(\mathrm{n_{q}}=22.6\) min\({}^{-1}\). This study is numerical and performed using ANSYS 14.0. Geometric models are established using the Design Modeler subroutine, while computational meshes are generated in the mesh subroutine of the ANSYS Workbench 14.0. The numerical simulations are accomplished by applying ANSYS CFX. A geometric model of the flow part (fluid region) of two of the pump stages is established. Each stage consists of an impeller, vane diffuser and RGV. The vane diffuser and the RGV are united in a common body, which is called Radial diffuser 1 for the first stage and Radial diffuser 2 for the second one, respectively. Both impellers are identical, have six blades each and do not have rear balance holes. The vane diffuser has eight vanes, which is the same number as the RGV. Due to limited computing resources, the impeller models represent 1/6 of a whole impeller, each containing one blade. The vane diffuser and RGV models make up 1/2 of the total elements with four vanes each. A diagram of the newly established geometric model is shown in Figure 1a,b, which displays the main sizes of a RGV. Table 1 presents the main geometric parameters of the pump stage studied, as well as the three RGVs studied.  
Figure 1: Geometry of the studied object: (**a**) geometric pattern of the two stages created in Design Modeler; (**b**) geometric sizes of RGV.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pump Element** & **Parameters** & **Symbol** & **Value** \\ \hline \multirow{6}{*}{Impeller} & Number of blades & z\({}_{\text{La}}\) & 6 \\ \cline{2-4}  & Inlet diameter (mm) & D\({}_{1}\) & 76.0 \\ \cline{2-4}  & Outlet diameter (mm) & D\({}_{2}\) & 168.0 \\ \cline{2-4}  & Outlet blade width (mm) & b\({}_{2}\) & 9.3 \\ \cline{2-4}  & Outlet angle (deg) & \(\beta_{2}\) & 30 \\ \hline \multirow{2}{*}{Radial Diffuser} & Number of blades & z\({}_{\text{Le}}\) & 8 \\ \cline{2-4}  & Inlet diameter (mm) & D\({}_{3}\) & 170.0 \\ \cline{2-4}  & Outlet diameter (mm) & D\({}_{4}\) & 234.5.0 \\ \hline \multirow{6}{*}{Original RGV} & Radius at blade inlet (mm) & R\({}_{5}\) & 98.5 \\ \cline{2-4}  & Radius at blade outlet (mm) & R\({}_{6}\) & 42.5 \\ \cline{2-4}  & Angle at blade inlet (deg) & \(\alpha_{5}\) & 24 \\ \cline{2-4}  & Angle at blade outlet (deg) & \(\alpha_{6}\) & 90 \\ \cline{2-4}  & Placement angle (deg) & \(\gamma\) & 27 \\  
The angle \(\alpha_{5}\) at the inlet of RGV1 is determined by the condition that the streamline at the inlet of the vane is tangent to the midline of the vane. The angle is calculated by applying the following equation:  
\[\alpha_{5}=\arcsin\frac{c_{r}}{c} \tag{1}\]  
Velocity values for c and c\({}_{r}\) are obtained after running a simulation using a diffuser without a RGV and a nominal pump operating mode with a flow rate of Q = 11.39 dm\({}^{3}\)/s closest to the nominal one, Q\({}_{n}\) = 11 dm\({}^{3}\)/s.  
The inlet angle \(\alpha_{5}\) of RGV2 is set to be 24 degrees--equal to that of the original RGV.  
The original vane angle \(\alpha_{6}\) is 90 degrees. [PERSON] [1], p. 510, recommends \(\alpha_{6}\) to be selected in the range of 94 deg to 96 deg. For RGV1, it is decided to be \(\alpha_{6}\) = 110 deg, while for RGV2--\(\alpha_{6}\) = 128 deg. In the authors' opinion, turning the flow against the rotation of the next impeller somewhat leads to an increase in its head.  
The second pump stage is used in validating the results, as the experimental data obtained are based on studying this current pump stage. Furthermore, the simulation calculations with the three types of RGV are accomplished by using Radial diffuser 1 and Impeller 2 to establish the effect of RGV on the flow in front of the impeller inlet and the performance parameters of the pump stage.