## 4 Results and Discussion
### Effects of Reynolds Number and Pressure Ratio  
Figure 9 illustrates the behavior of \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) versus \(Re\) for Case I, with a different \(\pi\) ranging from 4.5 to 6.0, respectively. The results reveal that both \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) decrease with increasing \(Re\), and the decreasing rate remains unchanged when \(Re\leq 10^{3}\) and gradually decreases to 0 for \(Re\) ranging from 10\({}^{4}\) to 10\({}^{8}\). Consequently, \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) become independent of \(Re\) while \(Re\) exceeds 10\({}^{5}\). Given the variability in the rates at which \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) decrease with respect to \(Re\), the models of \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) in Section 4.4 need to be segmented, with \(Re\) being 10\({}^{3}\) and 10\({}^{5}\) as the demarcation point. It is also observed that \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) increase with \(\pi\) from 4.5 to 6.0 under the conditions of the same \(Re\) since a larger \(\pi\) corresponds to a higher fluid density, and thus windage loss and \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) are larger as known from Equations (1) and (5). Moreover, the comparison between Figure 9a,b reveals that the \(C_{\mathrm{f,d}}\) is larger than \(C_{\mathrm{f,s}}\), which is because disk-type windage loss is significantly greater than shaft-type. In Figure 10, disk-type and shaft-type windage losses for \(Re\) from 10\({}^{2}\) to 10\({}^{8}\), corresponding to rotational speed \(\Omega\) being 10\({}^{-1}\)-10\({}^{5}\) r/min under \(\pi\) = 4.5 conditions are presented. It is observed that windage losses rapidly increase with \(Re\) and are much larger in disk-type gaps than in shaft-type gaps. For example, at \(Re=10^{6}\) and \(\Omega=10^{3}\) r/min, disk-type and shaft-type windage losses are 0.08 W and 0.008 W, and at \(Re=10^{8}\) and \(\Omega=10^{5}\) r/min, they are 780 W and 84.250 W, respectively, which is basically consistent with the findings reported by [PERSON] et al. [39]. This observation is because the impeller back gap is the primary region compared to the labyrinth seal, according to Figures 1 and 3, leading to much larger disk-type windage loss than shaft-type, as confirmed in Figure 10, which further supports the previous conclusions that \(C_{\mathrm{f,d}}\) is larger than \(C_{\mathrm{f,s}}\) in Figure 9.  
Figure 9: Skin friction coefficients \(C_{\mathrm{f,d}}\) and \(C_{\mathrm{f,s}}\) versus \(Re\) under for \(\pi\) ranging from 4.5 to 6.0. (**a**) Disk-type skin friction coefficient \(C_{\mathrm{f,d}}\). (**b**) Shaft-type skin friction coefficient \(C_{\mathrm{f,s}}\). Figure 11 describes the proportion of disk-type and shaft-type windage losses. The proportion of shaft-type windage loss increases with \(Re\), while it decreases with \(\pi\). On the contrary, the proportion of disk-type windage loss exhibits the opposite trend to that of shaft-type, falling as \(Re\) increases and rising with \(\pi\). Notably, the proportion of disk-type windage loss is consistently far larger than that of shaft-type loss under various \(Re\) and \(\pi\) conditions, with its minimum reaching 90%, which indicates that windage loss in the impeller back gap significantly contributes to total windage loss. Nonetheless, it is still necessary to consider the influences of shaft-type windage loss in the preliminary design and operation of sCO\({}_{2}\) compressors and TAC units. In order to evaluate the flow characteristics in the impeller back gap and labyrinth seal cavities, the leakage flow rate \(m\) is a crucial parameter. The \(m\) versus \(Re\) for \(\pi\) ranging from 4.5 to 6.0 is presented in Figure 12. As the velocity of fluid in gaps is higher as \(\pi\) increases,  
Figure 11: Proportion of disk-type and shaft-type windage loss. Figure 10: Windage losses versus \(Re\). the \(m\) gradually grows when the area at the inlet and outlet is same.