## 4 Results and Discussion
### Effect of Rotational Speed on the Flow Pattern and Pump Performance  
At \(q_{w}=8.03\) m\({}^{3}\)/min and \(n=1050\) rpm, the cross-sectional distributions of pressure in the mid-span plane of the upper part of the primary impeller are shown in Figure 15. Relative to those obtained at \(n=850\) rpm, the patterns of the pressure distribution are nearly invariant when \(n\) varies. At each feed rate of sand, the overall magnitude of pressure decreases with increasing rotational speed. Especially in the flow passage of the secondary impeller and at the inlet of the upper part of the primary impeller, low pressure prevails. Figure 14: Cross-sectional distributions of particle velocity at \(n=850\) rpm. Furthermore, the low-pressure zone in the flow passage of the secondary impeller extends with an increasing feed rate of sand. At \(n=1050\) rpm, the cross-sectional distributions of velocity in the _xoy_ plane at different feed rates of sand are shown in Figure 16. It is seen that relative to that obtained at \(n=850\) rpm, the particle velocity increases when \(n\) increases, and the uniformity in the overall velocity distribution is improved, especially at \(q_{s}=3.12\) m\({}^{3}\)/min, as indicated in Figures 14 and 16. In comparison, at \(q_{s}=0.89\) m\({}^{3}\)/min, low velocity is predominant in the flow passage of the primary impeller. In this circumstance, the rotational speed of the impellers should be controlled since high rotational speed induces a high linear velocity of sand particles, which will aggravate the erosive wear of both the primary and secondary impellers. At \(n=1050\) rpm, the distributions of the solid volume fraction in the _xoy_ and _yoz_ planes are shown in Figures 17 and 18, respectively. Compared to those obtained at \(n=850\) rpm, as shown in Figure 13, the distributions of SVF in the _xoy_ plane obtained at different rotational speeds are similar. Non-uniform distributions of SVF in the circumferential direction are evidenced. However, at \(q_{s}=3.12\) m\({}^{3}\)/min, the overall SVF in the flow passage of the upper part of the primary impeller decreases with increasing rotational speed. This implies that the increase in the rotational speed facilitates the migration of sand particles in the flow passage of the pump, especially at high feed rates of sand. Figure 15: Cross-sectional distributions of pressure at \(n=1050\) rpm. Figure 16: Cross-sectional distributions of velocity at \(n\) = 1050 rpm. Figure 17: Cross-sectional distributions of solid volume fractions in the _xoy_ plane in the flow passage of the upper part of the primary impeller at \(n\) = 1050 rpm. As shown in Figure 18, with increasing \(q_{s}\), the accumulation of sand particles in the sand inlet segment is intensified, resulting in an increase in the risk of the blockage of the sand inlet segment. Nevertheless, in the impeller passage and the volute chamber, elements with high SVF are dispersedly distributed, and no severe accumulation of sand particles is identified. Based on the velocity distribution shown in Figure 16, the velocity distribution differs with the blade passage. High-velocity elements are not concentrated in the blade channels. In comparison, a high velocity is more remarkable in the primary impeller relative to that in the secondary impeller, which is prominent even at \(q_{s}=3.12\) m\({}^{3}\)/min. This demonstrates that the pump can effectively deliver the mixture at such a high solid volume fraction. Based on the aforementioned flow parameter distributions, high rotational speeds of the impellers are beneficial for the transport of the sand-water mixture of high SVFs. Nevertheless, in this case, the linear velocity of the solid particles in the pump chamber can be very high. Since sand particles interact directly with the impeller blades and the inner wall of the volute, erosive wear may be incurred. In contrast, at low rotational speeds of the impellers, sand particles cannot be promptly discharged from the secondary impeller.