## 3 Numerical Model  
The flow phenomena inside the pump were modeled by solving 3D mass and momentum equations in a commercial CFD code. The analysis was based on the steady, incompressible Reynolds-averaged Navier-Stokes (RANS) equations along with the \(k\)-\(\omega\) based shear stress transport (SST) turbulence model, using a finite-volume solver provided in the commercial code ANSYS CFX 19.1. The SST model uses an integrated function to switch from the standard \(k\)-\(\omega\) model near the wall region to a high Reynolds number form of the \(k\)-\(\epsilon\) model away from the boundary layer [24]. The turbulence model was chosen based on similar studies by the authors for a three-stage centrifugal pump [25; 26]. The mass conservation equation and the momentum conservation equation of the RANS can be written as  
\[\frac{\partial\overline{v_{i}}}{\partial x_{i}}=0 \tag{5}\]  
\[\rho\frac{\partial\overline{v_{i}}}{\partial t}+\rho\frac{(\overline{v_{i} \overline{v_{j}}})}{\partial x_{j}}=-\frac{\partial\overline{P}}{\partial x_{ i}}+\frac{\partial}{\partial x_{j}}[\mu(\frac{\partial\overline{v_{i}}}{ \partial x_{j}}+\frac{\partial\overline{v_{j}}}{\partial x_{j}})-\rho( \overline{v_{i}\overline{v_{j}}})] \tag{6}\]  
where, \(\rho\), \(P\), \(\mu\) and \(v_{ij}\) represent the density, pressure, dynamic viscosity and velocity components, respectively. The overbar (\(-\)) denotes the average quantity. The entire fluid domain in the computational model included the inlet domain, double-suction impeller, twin solute, single-suction second stage impeller, diffuser vanes, return channel vanes, Stage 3 and 4 domains, fifth stage impeller, diffuser and outlet domain. The impeller domains were rotating domains, while all other domains remained stationary. The impeller, diffuser and return channel domains were chosen as a single passage because of the periodically repeating nature of the domains. The periodic boundary condition saved the computational time required for a converged solution without compromising the accuracy of the results [27]. The outlet domain was extended to avoid any possible backflow phenomena at the pump outlet. The ambient pressure inlet and mass flow outlet were chosen for the boundary conditions. The interfaces between the rotating and stationary domains were taken as a stage (mixing-plane) model for the steady-state calculations due to the high pitch ratio between the domains. The interfaces between the stationary domains were chosen as a general connection with the general grid interface (GGI) mesh connection. The numerical meshes for the fluid domains were generated using ANSYS meshing module for the stationary domains. The rotating impeller domain meshes are created using  
Figure 3: The theoretical performance curve of the pump design. Turbogrid software which provides high-quality hexagonal meshes with boundary layer control. To resolve the boundary layers, multiple layers of boundary meshes are installed along the blade suction and pressure side surfaces to keep the y\({}^{+}\) value below 30. Similar boundary layer meshes were also given for diffuser and return channel vanes. The generated meshes were tested for grid independency to obtain the optimum mesh and reduce the influence of the grid density on the flow field. The tests were carried out for the design point using water as the initial fluid. The grid independence study was carried out for Stages 1 and 2 separately to find the optimum grid for the first and second stage impellers. Determining the optimum grid for the first and second stage was deemed adequate for the entire pump model, since the third, fourth and fifth stage geometries were a repetition of the second stage. The mesh of the outlet domain in the fifth stage was generated in a similar way to the inlet domain in the first stage. Figure 4 shows the grid independency test results for Stages 1 and 2 plotted against efficiency.