## 2 Pump Model and Numerical Setting
### Numerical Scheme
#### 2.2.1 Governing Equations  
The steady-state simulation of the solid-liquid two-phase flow in the pump considered was performed using the commercial CFD code ANSYS Fluent. With the Euler-Euler method, the continuity and momentum equations were solved for each phase separately. The granular model was incorporated into the simulation. The maximum packing limit was 0.51. Since the solid volume fractions considered in the present study were relatively high, the total stress acting on the particles was separated into normal stress and shear stress. The pressure was influenced by the degree of packing, which was related to the interparticle interactions and influenced the effective stress. The high packing degree led to high resistance to the flow, and the viscosity was increased. Whether the solid phase exhibited solid-like or fluid-like behaviors depends largely on the packing fraction, which significantly influenced the dynamic behaviors of the flow. The interaction between particles was considered through the introduction of a solid pressure term \(\
abla p_{s}\) into the momentum transport equation of the solid phase. The approach of the moving reference frame (MRF) was used to model the rotation of the double-layered impeller. The continuity and the momentum conservation equations are expressed as follows:  
Liquid phase  
\[\
abla\cdot(\alpha_{L}\rho_{L}\overset{\rightarrow}{u}_{L})=0 \tag{1}\]  
\[\
abla\cdot(\alpha_{L}\rho_{L}\overset{\rightarrow}{u}_{L}\otimes\overset{ \rightarrow}{u}_{L})=-\alpha_{L}\
abla p+\alpha_{L}\rho_{L}\overset{ \rightarrow}{g}+\overset{\rightarrow}{F}_{LS}+\
abla\cdot(\alpha_{L}(\mu_{ L}+\mu_{L})(\
abla\overset{\rightarrow}{u}_{L}+(\
abla\overset{ \rightarrow}{u}_{L})^{T})) \tag{2}\]  
Solid phase  
\[\
abla\cdot(\alpha_{S}\rho_{S}\overset{\rightarrow}{u}_{S})=0 \tag{3}\]  
\[\
abla\cdot(\alpha_{S}\rho_{S}\overset{\rightarrow}{u}_{S}\otimes\overset{ \rightarrow}{u}_{S})=-\alpha_{S}\
abla p-\
abla p_{S}+\alpha_{S}\rho_{S} \overset{\rightarrow}{g}-\overset{\rightarrow}{F}_{LS}+\overset{ \rightarrow}{F}_{SS}+\
abla\cdot(\alpha_{S}(\mu_{S}+\mu_{S})(\
abla \overset{\rightarrow}{u}_{S}+(\
abla\overset{\rightarrow}{u}_{S})^{T})) \tag{4}\]  
where \(\overset{\rightarrow}{g}\) is the gravitational acceleration; \(p\) is the pressure shared by the two phases; \(p_{S}\) represents the pressure of the solid phase; and \(\mu_{S}\) is the solid phase viscosity, which is characterized by employing a Bingham plastic model, in which the material behaves as a solid until a certain stress threshold is reached. Beyond this threshold, it flows like a viscous fluid. \(\overset{\rightarrow}{F}_{SS}\) denotes the solid-solid interaction force. \(\overset{\rightarrow}{F}_{LS}\) represent the interaction force between liquid and solid phases \(L\) and \(S\)  
\[\overset{\rightarrow}{F}_{LS}=K_{LS}(\overset{\rightarrow}{u}_{S}-\overset{ \rightarrow}{u}_{L}) \tag{5}\]  
where \(K_{LS}\) is the coefficient of momentum exchange between phases \(L\) and \(S\). \(K_{LS}\) is modeled based on the Huilin-Gidaspow model (Equation (6)), which is a combination of the Wen and Yu model (Equation (8)) for solid volume fractions lower than 0.2 and the Ergun equation (Equation (11)) for higher solid volume fractions. The model incorporates both SVF and \(Re\) to adjust the drag coefficient. This adaptability is crucial in the presence of sand-water mixtures where the solid volume fraction significantly influences the drag.