## 4 Results and Discussion
### Models of Skin Friction Coefficients in Shaft-Type and Disk-Type Gaps  
Based on the previous discussion presented in Sections 4.1-4.3, it is found that the windage loss and flow characteristics in the impeller back gap and labyrinth seal cavities are relatively essential, which indicates that they cannot be neglected during the primary design and operation of sCO\({}_{2}\) compressors. However, the lack of corresponding models of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) makes it challenging to accurately predict windage loss. Therefore, it is significantly urgent to propose models of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) based on previous CFD results. Figure 17: Leakage flow \(m\) versus \(\eta\) for \(\pi\) ranging from 4.5 to 6.0. The results in Figure 9 reveal that the decreasing rates of \(C_{\rm f,d}\) and \(C_{\rm f,s}\) vary for different \(Re\) ranges; hence, it is necessary to segment the models of \(C_{\rm f,d}\) and \(C_{\rm f,s}\) with an \(Re\) of \(10^{3}\) and \(10^{5}\) as the demarcation point. According to the previous analysis, the \(C_{\rm f}\) is related to \(Re\), \(\pi\), and \(\eta\). Thus, the models can be expressed as:  
\[C_{\rm f}=af(Re,\pi,\eta) \tag{11}\]  
In addition, following the approach of [PERSON] [19] and [PERSON] [27], the \(C_{\rm f}\) is to be written as follows:  
\[C_{\rm f}=aRe^{b}\pi^{c}\eta^{d} \tag{12}\]  
Adopting the Least-Square Method, the models of \(C_{\rm f,d}\) and \(C_{\rm f,s}\) are listed in Table 6. It needs to be noted that the models are only applicable to the conditions where \(Re\) is \(10^{2}\)\(\sim\)\(10^{8}\), \(\pi\) ranges from 4.5 to 6.0, and \(\eta\) is 0.018\(\sim\)0.054. Figure 18: Vorticity contours at inlet regions with \(Re\) of \(10^{2}\)–\(10^{8}\) and \(\eta\) of 0.018, 0.036, and 0.054 under \(\pi\) = 4.5 conditions. According to the Least-Square Method, the higher fitting accuracy for the models is indicated by \(SSE\) and \(R^{2}\) values approaching 0 and 1, respectively. In this paper, the maximum \(SSE\) and minimal \(R^{2}\) are 5.62 \(\times\) 10\({}^{-7}\) and 0.940 for models of \(C_{\text{f,d}}\), and 2.43 \(\times\) 10\({}^{-6}\) and 0.917 for models of \(C_{\text{f,s}}\), demonstrating that the fitting accuracy of models is remarkably high. CFD simulations are used to calculate the values of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) for the different \(Re\) in the impeller back gap, based on the dimensions and boundary conditions presented in Table 7, which are applicable to the sCO\({}_{2}\) compressor designed by [PERSON] et al. (2010) to assess the reliability of models. It needs to be pointed out that the dimensions of the labyrinth seal are consistent with those listed in Table 1.  
To quantitatively assess the predictive accuracy of the models, the relative deviation \(\Delta\) is defined, and Figure 19 illustrates the relative deviations \(\Delta\) of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) for \(Re\) ranging from 10\({}^{2}\) to 10\({}^{8}\). \[\Delta=\frac{C_{\text{f,m}}-C_{\text{f,n}}}{C_{\text{f,n}}}\times 100\% \tag{13}\]  
It is noteworthy that the \(\Delta\) is small, with maximum values of 6.23% and 6.83% for \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\). These results demonstrate the excellent agreement between the predicted values and CFD simulations, indicating that the proposed models can predict windage loss in the impeller back gap and labyrinth seal cavities. The design of the sCO\({}_{2}\) compressor in engineering practice can refer to the windage loss models in Table 6 and Equations (2) and (7). In order to reduce windage losses and increase the compressor efficiency, the following suggestions are given. Firstly, the rotational speed and windage loss present a cubic relationship; it is very effective to reduce the rotational speed in a reasonable range. Secondly, reduce the radius ratio under the condition of axial-thrust compliance. Finally, increase the sealing outlet pressure to reduce the pressure ratio, which can effectively reduce the windage loss.