## 4 Optimization Design Scheme and Results Analysis
### Results Analysis
Conversely, when the thickness ratio coefficient \(\xi<1\), the direction of the force acts on the gas-liquid two-phase as the suction side of the blade points to the hub along the shroud. The direction of the force is opposite to the centrifugal force, which reduces part of the separation inductive force. Therefore, on the premise that the hub thickness meets the strength requirement and the shroud thickness meets the process conditions, reducing the thickness ratio coefficient \(\xi\) can improve the two-phase transportation performance of the multiphase pump. Figure 14 shows the schematic diagram of force on the fluid particle in the axial section of the impeller with the thickness ratio coefficient \(\xi>1\) and \(\xi\)\(<1\). In the figure, \(F_{c}\) is the centrifugal force, \(F_{p}\) represents the pressure gradient force, which is caused by the pressure gradient at the side of shroud and hub, \(F_{t}\) denotes the force applied by the suction side of the impeller--the direction of \(F_{t}\) is perpendicular to the suction side--and \(F_{tx}\) is the radial component force of \(F_{t}\). The separation inductive force was caused mainly because \(F_{c}>F_{p}\). The direction of \(F_{tx}\) changed due to the reduction of thickness ratio coefficient; when the thickness ratio coefficient \(\xi<1\), the direction of \(F_{tx}\) pointed to the hub along the shroud, which reduced the difference between the \(F_{c}\) and \(F_{p}\)--i.e., it reduced part of the separation inductive force. Figure 13: The gas distribution at 10% of the blade height under different thickness schemes with an inlet gas volume fraction of 0.5. (**a**) Scheme 1 with \(\delta_{hmax}=7.5\) mm and (**b**) scheme 4 with \(\delta_{hmax}=9\) mm. According to the gas distribution at 10% of the blade height, we concluded that the aggregation of gas mainly occurs at the blade outlet of the suction side near the hub. To express the accumulation phenomenon of gas in the flow passage with different thickening schemes more intuitively, the area starting from half the axial length (\(e\)) of the impeller in the axial direction to the outlet in each thickening scheme was selected as the analysis area, where the accumulation phenomenon was the most serious in the impeller and represented the accumulation degree of the gas in the flow passage, as shown in Figure 15.  
The governing equations were discretized using a finite volume method (FVM), which first involves discretizing the spatial domain using a mesh. The mesh was used to construct finite volumes, which were used to conserve relevant quantities [46]; the conceptual diagram of the finite volume method is shown in Figure 16. The scalar variables such as mass, momentum and phase-volume fraction are taken at the cell center [47]. The gas distribution and grid area date of all grids at 10% of the blade height can be derived from the center of the cell using the internal point method. Firstly, the gas volume fraction and area data whose axial coordinates were greater than half the axial length of the impeller were filtered. Then, the grid area data of the region with a volume fraction greater than 0.7 were filtered and summed. Figure 16: Conceptual diagram of finite volume method. Figure 14: Schematic diagram of force on fluid particles in the axial section of the impeller. Figure 15: Analysis area of the gas distribution at 10% of the blade height. In order to represent the degree of gas accumulation in the analysis area more intuitively, the aggregation degree \(\lambda\) was introduced in Equation (17). The larger the value, the more serious the gas accumulation is in this area:  
Figure 11:0  
where \(S_{x\,>\,0.5e}\) represents the total grid area of the analysis area, whose axial coordinate is greater than one-half axial length of the impeller; \(e\) is the axial length of impeller and \(S_{\text{{vof}}\,>\,0.7}\) is the total grid area of the region with a gas volume fraction greater than 0.7 in the analysis region.