## 4 Optimization Design Scheme and Results Analysis  
### Optimization Design Scheme  
The thickness variation rule of the 791 airfoil was applied to the blade thickening of the multiphase pump; the thickening law is shown in Table 7. The 791 airfoil is shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline _x/l_ & 0 & 0.05 & 0.075 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 0.95 & 1 \\ \hline \(\delta\delta_{max}\) & 0 & 0.296 & 0.405 & 0.489 & 0.778 & 0.92 & 0.978 & 1 & 0.883 & 0.756 & 0.544 & 0.356 & 0.2 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The thickening law of a 791 airfoil; \(x\) is the length from the inlet of the airfoil, \(l\) is the chord length of the airfoil, _δ_ denotes the thickness corresponding to the \(x\) position and _δmax_ is the maximum thickness of the airfoil. Figure 7: Distribution of gas at 10% of the blade height with different inlet gas volume fractions (IGVFs) under the design working conditions. The maximum thickness of the airfoil (\(\delta_{\text{max}}\)) at the hub was determined according to the strength condition [44], usually estimated roughly according to  
\[\delta_{\text{max}}\ =\ (0.012{-}0.015)\text{D}\sqrt{\text{H}}, \tag{13}\]  
where D is the diameter of the impeller shroud of the multiphase pump; its unit is m.  
Practice has proven that the thickness of the blade calculated according to the above equation is often too thick, which is not conducive to improving the performance of the pump [45]. The thickness is related to the chord length, and the relative thickness of the hub section is generally calculated as  
\[\overline{\delta}_{\text{h}ub}=\frac{\delta}{l}(10\%{-}15\%), \tag{14}\]  
and the relative thickness of the shroud section is calculated as  
\[\overline{\delta}_{\text{shroud}}\ =\ \frac{\delta}{l}(2\%{-}5\%). \tag{15}\]  
The hub is the thickest part generally, and the thickness varies linearly from the hub to the shroud. The maximum thicknesses of the hub and shroud for the basic model were selected as 9 and 5 mm, respectively. The airfoil from the hub to the shroud of the blade is shown in Figure 9.  
To investigate the effects of the thickness variation in the airfoil from the hub to the shroud of the blade on the hydraulic performance of the multiphase pump, the thickness ratio coefficient \(\xi\) is defined as  
\[\xi=\frac{\delta_{h\text{ max}}}{\delta_{\text{s max}}}, \tag{16}\]  
where \(\delta_{h\text{ max}}\) and \(\delta_{s\text{ max}}\) denote the maximum thicknesses in the airfoil at the side of the hub and shroud, respectively, and \(\xi\) indicates the change degree of the blade thickness from hub to shroud. The thickness ratio coefficients \(\xi\) were all greater than 1 according to the usual regular thick airfoil. To investigate the influence of the multiphase pump when \(\xi\leq 1\), different thickening schemes were selected on the basis of ensuring the strength of the blade, as shown in Figure 10. The thickness ratio  
Figure 8: The 791 airfoil. Figure 9: The frame of the helical–axial blade. coefficient was selected to range from 0.8 to 1.8; \(\delta_{\text{hmax}}\) was set to 7.5, 8, 8.5, and 9 mm, representing schemes 1 to 4, respectively; and \(\xi=1.8\) was the thickening scheme of the basic model. As shown in Figure 10, the maximum thickness of the hub on the same curve was the same; the \(x\) coordinates from left to right represent the increase in the maximum thickness at the side of shroud and the vertical axis represents different thickness ratio coefficients. The main step of the optimization design of the impeller is: in order to investigate the influence of the thickness variation from hub to shroud on the characteristics of multiphase pump, the thickness ratio coefficient was defined. The variables that affect the thickness ratio coefficient were the maximum thickness of the impeller hub and the maximum thickness of the shroud.