## 3 Results  
### Effects of Large Backsweep Angle on S-CO\({}_{2}\) Compressor Performance  
Compressor performance was examined when the angle varied from 0\({}^{\circ}\) to \(-\)77\({}^{\circ}\)while using the design parameters summarized in Table 2. As shown in Figure 14, the S-CO\({}_{2}\) centrifugal compressor showed the best efficiency at \(-\)70\({}^{\circ}\), which is larger than the typical design value for the air centrifugal compressor. Also, in all cases, the total to total efficiency tends to increase as its angle at the impeller exit increases. The S-CO\({}_{2}\) compressor has the best efficiency of 84% at the backsweep angle of \(-\)70\({}^{\circ}\). It is an increase of about 3% in comparison with the recommended design region. Despite the rapid change in properties near the critical point, the compressor efficiency is insensitive to the change in the compressibility factor. Due to the structural issue of the impeller in the low-density fluids, such as air and helium, the backsweep angle is generally limited to \(-\)50\({}^{\circ}\). Figure 15 explains why large backsweep angle involves high centrifugal stress level. The angle of absolute velocity leaving the impeller decreases as the backsweep angle increases when it operates under the same operating conditions. As a result, a higher peripheral speed is required as the backsweep angle to maintain identical outlet pressure and, consequently, it causes greater stresses. Figure 16 summarizes the preliminary structural safety margin using the tip speed which is an important criterion for the centrifugal stress level. In spite that the safety margin decreases as the compressibility factor increases, an S-CO\({}_{2}\) compressor is generally less affected by the structural issues in all cases. The structural limit of centrifugal stress is set to be the same value of the air centrifugal compressor for the same design variables summarized in Table 2  
Figure 14: Total to total efficiency sensitivity to backsweep angle. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Cycle Design Results**} \\ \hline
**Case Number** & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** \\ \hline Generating power [MWe] & 10.0 & 9.97 & 10.1 & 9.95 & 9.56 \\ Cycle efficiency [\%] & 33.4 & 33.2 & 33.7 & 33.1 & 31.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cycle design results of recuperated Brayton cycle. and \(-50^{\circ}\) of backward sweep angle. Also, Figure 17 supports why the S-CO\({}_{2}\) compressor has smaller centrifugal stress than the air compressor thermodynamically. S-CO\({}_{2}\) requires only a few hundredths of the enthalpy of air when it increases the same amount of pressure. It means that the work consumption for compression is significantly reduced when the S-CO\({}_{2}\) compressor increases the same amount of pressure compared to the air compressor. Thus, smaller tip speed is required and this results in smaller centrifugal stress in the S-CO\({}_{2}\) compressor. The adopted loss models do not cover irreversibility caused by shock wave because it assumes a centrifugal compressor is operating in the subsonic region. Since the speed of sound is substantially decreased near the critical point as shown in Figure 18, a high Mach number may occur in the low-speed region. As aforementioned, the radius of an impeller increases as the backsweep angle at impeller exit increases. Thus, the relative Mach number needs to be checked whether it exceeds unity or not. Figure 19 shows that the relative Mach numbers are under the subsonic region in all cases. It proves that the additional shock loss does not need to be included and the results of Figure 14 are still valid. Also, because of the low Mach number in the S-CO\({}_{2}\) compressor, the variation of compressibility factor does not impact on its aerodynamic design in spite of its operation near the critical point where the compressibility factor change is substantial.