## 2 Computational Model  
### Governing Equations  
As explained in the introduction section, a numerical study has been conducted to assess the influence of various factors, i.e., particle's volume concentration (P\({}_{\text{vc}}\)), particle's size (P\({}_{\text{s}}\)) and pump's exit pressure (_P\({}_{exit}\)_), on the erosion rate characteristics of the pump blade and its casing based on the Reynolds-averaged Navier-Stokes equations (RANS) calculations. The steady form of the employed transport equations, i.e., continuity, and momentum, was solved for both phases (water and sand). The two-phase numerical model was built using the Eulerian approach that is provided below using Equations (1)-(4) [13]. Continuity Equation for solid and fluid phase  
\[\
abla.\left(C_{f}\rho_{f}\,\mathbf{V_{f}}\right)=0 \tag{1}\]  
\[\
abla.\left(C_{\text{s}}\rho_{\text{s}}\,\mathbf{V_{s}}\right)=0 \tag{2}\]  
Transport equations for solid and fluid phase  
\[\
abla.\left(C_{f}\,\rho_{f}\mathbf{V_{f}}\,\mathbf{V_{f}}\,\right)=-C_{f}\, P+\
abla.\left(C_{f}\tau_{\mathbf{f}}\right)+C_{f}\rho_{f}\,\mathbf{g}+\mathbf{R}+C_{f} \rho_{f}\,\left(\mathbf{F_{f}}+\mathbf{F_{1,f}}+\mathbf{F_{vm_{f}}}\right) \tag{3}\]  
\[\
abla.\left(C_{\text{s}}\,\rho_{\text{s}}\,\mathbf{V_{s}}\,\right)=-C_{\text {s}}\,P+\
abla.\left(C_{\text{s}}\tau_{\text{s}}\right)+C_{\text{s}}\rho_{ \text{s}}\,\mathbf{g}+\mathbf{R}+C_{f}\rho_{f}\,\left(\mathbf{F_{f}}+\mathbf{ F_{1,s}}+\mathbf{F_{vm,s}}\right) \tag{4}\]  
The terms, \(\mathbf{V_{f}},\mathbf{\tau_{f}}\) and \(\mathbf{V_{s}},\mathbf{\tau_{s}}\) denote flow velocity vector and shear stress tensor for both fluid and solid phases, respectively. On the other hand, the scalar quantities \(C_{f},C_{\text{s}}\) and \(\rho_{f},\rho_{s}\) are volume concentrations and densities of the fluid and solid phases, respectively. Force vector \(\mathbf{R}\) represents the interaction between the solid and liquid phases. While the term \(P\) appearing in both Equations (3) and (4) represents the average pressure at the interfacial region of the two phases [14], and it is common for both fields. The lift force is represented by \(\mathbf{F_{l}}\) and \(\mathbf{F_{vm,s}}\) denotes the virtual mass force per unit mass. The lift force for solid phase \(\mathbf{F_{l,S}}\) can be calculated utilizing Equation (5) since: \(\mathbf{F_{l,S}}=-\mathbf{F_{l,f}}\). \[\mathbf{F_{l,S}}=-0.5\,C_{s}\big{(}\mathbf{V_{f}}-\mathbf{V_{s}}\big{)}\, \times\,\big{(}\
abla.\mathbf{V_{f}}\big{)} \tag{5}\]Virtual mass force \(\mathbf{F_{vm,s}}\) is triggered by the acceleration of the solid phase through the fluid phase, and it was computed employing Equation (6). \[\mathbf{F_{vm,s}}=0.5\,C_{f}\rho_{f}\bigg{(}\frac{d_{f}}{dt}\big{(}V_{f}\big{)}- \frac{d_{s}}{dt}(V_{s})\bigg{)} \tag{6}\]  
For coupling of both fields, i.e., water and solid particles, two procedures have been implemented in literature [15], namely one-way coupled and two-way coupled, based on the value of \(\beta\) provided in Equation (7). If \(\beta\) is less than 0.14, one-way coupling is recommended, otherwise, two-way coupling techniques should be utilized. \[\beta=\frac{C_{s}\rho_{s}}{C_{f}\rho_{f}} \tag{7}\]  
The prediction of wear rate that occurs largely on the casing walls was the major objective of this study. Thus to compute the accurate values of wear rate density, predicting the boundary layer and capturing the recirculation regions near the wall region is crucial. Along with the available Reynolds-averaged Navier-Stokes (RANS) turbulence models, there are two classifications of turbulence models that have been employed in the literature depending on how the boundary layer is considered in those models [16; 17; 18]. In the first category of the RANS model, the flow in the boundary layer is not computed. Nonetheless, a wall function is utilized to approximate the velocity profile, e.g., \(k-\epsilon\) turbulence model. The \(k-\epsilon\) turbulence model manages a set of two transport equations to solve the turbulence kinetic intensity and dissipation.