## 5 Optimal Design of Four Kinds of Axial Compressors
### Results and Discussion
When the number of stages increases, the pressure ratio and density ratio of all stages decrease, as does the axial velocity. Hence a larger flow area is required to maintain the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{\
\begin{tabular}{c} **Design** \\ **Intention** \\ \end{tabular} } & \multirow{2}{*}{**4-Stage**} & \multirow{2}{*}{**5-Stage**} & \multirow{2}{*}{**6-Stage**} & \multirow{2}{*}{**7-Stage**} \\ \hline \multirow{2}{*}{Average aspect ratio} & Initial & 1.43 & 1.48 & 1.53 & 1.58 \\  & Optimal & 1.44 & 1.49 & 1.54 & 1.58 \\ \hline \multirow{2}{*}{Average solidity} & Initial & 1.35 & 1.35 & 1.35 & 1.35 \\  & Optimal & 1.33 & 1.28 & 1.30 & 1.31 \\ \hline \multirow{2}{*}{Average reaction} & Initial & 0.50 & 0.50 & 0.50 & 0.50 \\  & Optimal & 0.44 & 0.46 & 0.48 & 0.49 \\ \hline \multirow{2}{*}{Rotation speed} & Initial & 6000 & 5500 & 5000 & 4500 \\  & Optimal & 5800 & 5113 & 4551 & 4053 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Outlet axial Mach \\ number \\ \end{tabular} } & Initial & 0.34 & 0.34 & 0.34 & 0.34 \\  & Optimal & 0.34 & 0.35 & 0.34 & 0.35 \\ \hline \multirow{2}{*}{Inlet radius ratio} & Initial & 0.38 & 0.38 & 0.38 & 0.38 \\  & Optimal & 0.32 & 0.32 & 0.30 & 0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial and optimal design results of six design parameters. Figure 8: Stagewise optimal solidity distribution. same flow rate. This could be achieved with a higher aspect ratio. Figure 9 illustrate the optimal aspect ratio along the stagewise direction. As can be seen, its nearly linear descent gradually approaches the same level of around 1.25 at the final outlet. Meanwhile, the optimal efficiency corresponds to a lower value of the average reaction relative to the initial reactions. The optimal average reaction increases from 0.44 to 0.49 as the number of stages increases. Reaction reflects the compression capacity of the rotor with respect to that of the stator. A higher reaction level means higher compression capacity as well as extremely high inlet Mach number of rotors, which will deteriorate the efficiency of rotors. Similarly, a lower reaction will deteriorate the efficiency of the stators. Hence a moderate level of the reaction would be preferred for better efficiency of a stage. Table 4 also shows that optimal efficiency can be obtained with a slight decline of rotation speed. At optimal operating conditions, relative velocity or equivalently the relative Mach number is slightly reduced, and thereby the shock intensity becomes moderate, with a little less shock loss. As seen, the optimal rotation speed decreases from 6125 rpm to 4053 rpm as the number of stages increases. It can be deduced from the Euler equation that a positive correlation between the rotation speed and the theoretical work exists. Under the same total pressure ratio, the average stage theoretical work decreases as the number of stages increases, resulting in a decrease of the rotation speed. As shown in Table 4, an increase of outlet axial Mach number can give rise to improved efficiency. The initial and optimal outlet axial Mach number are basically at the same level (around 0.34). An acceleration treatment for axial velocity at the inlet and outlet of the IGV is applied to get a meridional contraction, and axial Mach number decreases almost linearly along the stagewise direction from the inlet of the first rotor to the outlet of the exit stator. Efficiency increases as the inlet radius ratio decreases. The optimal inlet radius ratio of the four types of compressors is basically at the same level (around 0.3). The lower inlet radius ratio means larger blade height and fewer blade numbers, which is beneficial to a decrease of weight and cost. Moreover, a lower inlet radius ratio can decrease the inlet axial Mach number and thus the average axial Mach number, which can weaken the intensity of shock structure. Because of the above variations of six design parameters, the flow characteristics inside the compressors have been affected.