## 5 Optimal Design of Four Kinds of Axial Compressors
### Results and Discussion  
The optimal convergence curves of the four kinds of compressors are illustrated in Figure 5. The initial and optimal design results of 4-stage, 5-stage, 6-stage, and 7-stage compressors are demonstrated in Table 3. The design adiabatic efficiency of the four kinds of compressors increases effectively while mass flow rate and total pressure ratio changes are negligible after optimization. It can also be found that the optimal design adiabatic efficiency decreases with a decrease of the number of stages. With the same total pressure ratio for those four types of compressors, the increase of the number of stages brings about the decline of the pressure ratio and the Lieblein equivalent diffusion factor [24] of each stage (Figure 6). Figure 7 illustrates the spanwise distribution of optimal Lieblein equivalent diffusion factors of rotor 1 and stator 1 for 4-stage, 5-stage, 6-stage, and 7-stage compressors. The lower optimal reaction of 4-stage and 5-stage compressors (Table 4) means the relatively higher loading of the stator. The equivalent diffusion factors of stator 1 of 4-stage and 5-stage compressors are quite higher than those of 6-stage and 7-stage compressors. On the whole, for both rotor 1 and stator 1 of these four compressors, the equivalent diffusion factors along the spanwise direction decrease among the most span range as the number of stages increases. Hence fewer stages will penalize efficiency within a reasonable loading range. Figure 4: Average total pressure ratio per stage of four compressors and jet engine HPC. Figure 5: Convergence of the proposed GA for the four compressors. Figure 6: Optimal average stage equivalent diffusion factor for four compressors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Adiabatic Efficiency**} & \multicolumn{2}{c}{**Total Pressure Ratio**} & \multicolumn{1}{c}{**Mass Flow Rate**} \\  & \multicolumn{1}{c}{} & **(kg/s)** & \multicolumn{1}{c}{} & **(kg/s)** \\ \hline \multirow{2}{*}{4-stage} & Initial & 80.31\% & 3.94 & 179.08 \\  & Optimal & 83.51\% & 3.94 & 179.08 \\ \hline \multirow{2}{*}{5-stage} & Initial & 83.95\% & 3.94 & 179.08 \\  & Optimal & 84.69\% & 3.94 & 179.08 \\ \hline \multirow{2}{*}{6-stage} & Initial & 83.10\% & 3.94 & 179.08 \\  & Optimal & 85.97\% & 3.94 & 179.08 \\ \hline \multirow{2}{*}{7-stage} & Initial & 83.14\% & 3.94 & 179.08 \\  & Optimal & 86.15\% & 3.94 & 179.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial and optimal design results of four compressors. Figure 7: Optimal equivalent diffusion factor for the first stage of four compressors: (**a**) Rotor; (**b**) Stator. To reveal the aerodynamic mechanism behind the optimization of performance, the variations of the six types of design parameters, including aspect ratio, solidity, reaction, rotation speed, outlet axial Mach number, and inlet radius ratio, are analyzed in detail below. Table 4 shows that a higher efficiency is obtained with a little decline of solidity. As solidity decreases, \(\overline{\omega}_{\text{p}}^{*}\) will decrease [28]. The optimal average solidities at midspan of the four types of compressors vary within a small range from 1.28 to 1.33. As shown in Figure 8, the stagewise optimal solidities of the four compressors are around 1.3. These optimal results reveal that for a compressor with an aforementioned stage load level, an average solidity between 1.28 and 1.33 would be beneficial to the efficiency. Notably, the IGV was denoted as the first row. As shown in Table 4, optimal efficiency was achieved after the average aspect ratio slightly increased. This is also consistent with the fact that a higher aspect ratio results in a larger blade height, which in turn reduces the axial velocity. Accordingly, the relative Mach number would be kept lower as well as the flow loss would be improved. Notably, the optimal average aspect ratio increases from 1.44 to 1.58 as the number of stages increases.