## 2 Pump Model and Numerical Setting
### Numerical Scheme  
A brief review of the research on the numerical simulation of solid-liquid two-phase flows in the impeller pump is shown in Table 2. Since the pump structure, operating parameters, and solid phase properties are remarkably different, it is difficult to compare the accuracy of the numerical methods employed in different works. Moreover, some newly developed numerical methods are devised for specific scenarios. For the Euler-Euler approach, the solid and liquid phases are treated as interpenetrating continua. Such a two-fluid model can be applied to predict internal solid-liquid flows as well as the operating efficiency of the impeller pump. However, since the solid phase is treated as a single continuous entity, the behavior of individual particles cannot be described [21]. In comparison, an explicit advantage of the Euler-Lagrange approach is that it considers the motion of each particle [22]. Meanwhile, this approach entails high computational cost when the number of particles is great [23]. Based on the Euler-Lagrange approach, the strategy coupling CFD and DEM (discrete element method) has been developed, and this strategy is especially suitable for the study of granular dynamics. A big barrier to extending the application of the strategy is that the number of particles that can be handled is limited [24].  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Diameter of water inlet (mm) & 356 \\ Diameter of sand inlet (mm) & 328 \\ Diameter of pump outlet (mm) & 203 \\ Diameter of lower part of primary impeller (mm) & 558 \\ Diameter of upper part of primary impeller (mm) & 609 \\ Inlet diameter of upper part of primary impeller (mm) & 320 \\ Blade number of lower part of primary impeller & 6 \\ Blade number of upper part of primary impeller & 6 \\ Blade number of secondary impeller & 6 \\ Blade outlet angle of lower part of primary impeller (\({}^{\circ}\)) & 16 \\ Blade outlet angle of upper part of primary impeller (\({}^{\circ}\)) & 25 \\ Blade outlet angle of secondary impeller (\({}^{\circ}\)) & 43 \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Refs** & \(\mathbf{q}\) **(m3/h)** & \(\mathbf{H}\) **(m)** & **Methods** & \(\mathbf{\rho_{p}}\) **(kg/m3)** & \(\mathbf{d_{p}}\) **(um)** & \(\mathbf{C_{V}}\) **(\%)** \\ \hline
[25] & - & - & Euler–Euler & - & 50–1250 & 5–25 \\
[26] & 135 & 60 & DPM & - & 50–450 & 0.2–0.6 \\
[27] & 61,200 & 240 & Euler–Lagrange & 1550–3550 & 1530 & - \\
[28] & 485 & 14 & DPM & 2740 & 40–427.5 & 0.36 \\
[29] & - & - & DPM & 2300 & 20–70 & \(\approx\)1.7 \\
[30] & 420 & 100 & Euler–Lagrange & - & 6000 & 7.5 \\
[31] & - & - & Euler–Euler & - & - & 5–20 \\
[32] & 9000 & 34 & Euler–Lagrange & 1950 & 200 & 5–10 \\
[33] & 485 & 14 & DPM & - & 20 & 0.4 \\
[34] & 165 & 32 & DPM & 2650 & 25 & 0.43–4.3 \\
[35] & 1260 & 43.5 & DPM & 2650 & 4–250 & 0.5–6.1 \\
[36] & 180 & 26 & DEM & 2500 & 100 & 1 \\
[37] & 27 & 7.8 & DEM & 4290 & 1000–3000 & 5–15 \\
[38] & 400 & 100 & DEM & 2040 & 30,000 & 5 \\
[39] & 11,100 & 50 & Euler–Lagrange & - & 25 & 0.1–1 \\
[40] & 100 & 40 & Euler–Euler & 2650 & 1 & 10 \\
[41] & 5000 & 24 & DDPM & - & 300 & 4–12 \\
[42] & 690 & \(\approx\)40 & Euler–Euler & - & 500 & 5 \\
[43] & 15 & 4 & CFD-DPM & 2500 & 500 & 0.1–2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Research on the numerical simulation of solid–liquid two-phase flows in impeller pumps.