## 4 Results and Discussion
### Overview of Flow Characteristics
#### 4.1.2 Cross-Sectional Distribution of the Solid Volume Fraction  
For the two cross-sections shown in Figure 6, the distributions of the solid volume fraction are shown in Figure 7a,b. In the flow passage of the lower part of the impeller, especially near the impeller eye, few solid particles are observed. Near the blade outlet, some solid particles accumulate, and the distribution of the solid volume fraction in the circumferential direction is uneven. These particles are discharged from the upper part of the primary impeller. Meanwhile, at the inlet of the upper part of the primary impeller, a high solid volume fraction is evidenced. The discrepancy between such a pattern and those reported solid volume distributions in the centrifugal pump is noticeable [44]. Furthermore, the distributions of the solid volume fraction in each blade channel are inconsistent, implying the hydraulic loads acting on the impeller blades are different. Figure 5: Comparison between numerically and experimentally obtained pump heads at different flow rates and solid volume fractions. From another view, as observed in Figure 7c, a high solid volume fraction arises near the shaft, which is induced by the rotation of the impeller. When sand particles are well dispersed in water, the effective solid volume fraction is apparently high because the calculation does not take into account the whole volume occupied by water. Moreover, sand particles are not uniformly distributed, as shown in Figure 7, and this also causes the local solid volume fraction to exceed the packing fraction. When the water velocity is low, sand particles cannot be transported efficiently, and those subsequent particles are  
Figure 6: Distributions of pressure in the planes overlapping the mid-span planes of the (**a**) lower and (**b**) upper parts of the primary impeller. Figure 7: Cross-sectional distributions of the solid volume fraction: (**a**) lower part of the primary impeller; (**b**) upper part of the primary impeller; (**c**) lateral view; and (**d**) front view. prone to accumulating above the secondary impeller. This increases the risk of blockage for the secondary impeller as well as the inlet segment. From another perspective, the risk of the blockage of the inlet segment can be evaluated through local solid volume fractions. When the secondary impeller and the upper part of the primary impeller cannot work properly, blockage may first occur in the inlet segment. In this circumstance, sand particles accumulate in the inlet pipe, and further feeding sand particles will aggravate the accumulation. The cross-sectional velocity distributions in the flow passages of the pump are shown in Figure 8. As shown in Figure 8a, the velocity distribution varies considerably with the shift between blade channels. In accordance with the two-fluid model, the addition of sand particles induces a change in the velocity distribution. Therefore, in some blade channels of the lower part of the primary impeller, the involvement of sand particles, even with a limited number, will result in apparently low velocity. Furthermore, in the blade channels dominated by low velocity, the lowest velocity tends to arise near the outlet of the suction side of the blade. At the impeller outlet, the velocity magnitudes are moderate, which is partially related to the resistance from the mixture discharged from the upper part of the primary impeller. The upper part of the primary impeller receives the sand particles discharged from the secondary impeller. In consideration of the consistency in blade profiles between the two impellers and relative blade positions, the sharp turning of the motion of sand particles at the connection between the two impellers is avoided, so local erosive wear near the primary impeller inlet is expected to be suppressed [45]. In the flow passage of the upper part of the primary impeller, low velocity is prominent relative to that associated with the lower part of the primary impeller.