## 4 Results and Discussion
### Prediction Method of Optimum Gap  
In the radial impeller machinery, the axial gap loss behind the impeller is similar to the above research object. However, it is only a unilateral gap. When the flow in the gap is laminar, the skin-friction coefficient increases with the decrease of the gap size in Region I. The experimental and numerical results in Ref. [3] show that when the Reynolds number is constant, the skin-friction coefficient decreases first and then increases with the decrease of the gap. Therefore, based on the skin-friction coefficient model of Region III and Region IV, this paper proposes a method to predict the optimal gap of the radial impeller to minimize the windage loss [25]. The calculation formula is as follows:  
\[\underset{C_{m}=\min}{Opt(S/R_{0})}=\frac{0.403}{R^{o.187}} \tag{5}\]  
The impeller parameters of the supercritical carbon dioxide (sCO\({}_{2}\)) unit designed by Sandia National Laboratory [26] are shown in Table 5. It should be noted that the parameters such as the impeller inlet pressure and temperature in the table are obtained according to the thermodynamic design and analysis program of the centripetal turbine of the research group. The accuracy of the program is verified and reliable and is used as the known flow parameters in the gap behind the impellers. Figure 17 shows the relationship between the skin-friction coefficient and the relative gap in Region III and Region IV under the turbine parameters in Table 5 and the working medium is CO\({}_{2}\). For the turbine parameters in Table 5, the curve is obtained by a fixed Reynolds number of 1.86 \(\times\) 10\({}^{7}\). It can be seen that when the Reynolds number is constant, according to the theoretical model, the skin-friction coefficient decreases with the increase of the relative gap in Region III. When the flow falls into Region IV, the skin-friction coefficient increases with the increase of the relative gap. The optimal relative gap covering the two flow regions is 0.0176 (the corresponding axial gap behind the impeller is 0.6 mm). Since the back of the impeller has only a single surface, the corresponding minimum skin-friction coefficient is 0.0012 and the windage loss at this time is 0.897 kW. Similarly, the optimum relative gap of the compressor is 0.0156 (the corresponding axial gap behind the impeller is 0.3 mm), the minimum skin-friction coefficient is 0.00104 and the windage loss is 0.307 kW. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Units** & **Turbine** & **Compressor** \\ \hline Speed & r/min & 75,000 & 75,000 \\ Power rate & kW & 178 & 51 \\ Impeller outer radius & mm & 68.1 & 37.3 \\ Impeller inner radius & mm & 39.2 & 12.0 \\ Inlet temperature & K & 780.57 & 320 \\ Inlet pressure & kPa & 10,764.89 & 10,814 \\ Inlet density & kg/m\({}^{3}\) & 72.13 & 543.02 \\ Kinetic viscosity & Pa-s & \(3.54\times 10^{-5}\) & \(4.16\times 10^{-5}\) \\ _Re_ of gap & - & \(1.86\times 10^{7}\) & \(3.57\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of sCO\({}_{2}\) turbine and compressor from Sandia, Albuquerque, NM, USA [26]. Since the gap size of the turbine and the compressor designed by the Sandia laboratory is unknown, it is assumed that the optimal relative gap is adopted. Figure 17 shows the optimal gap by using turbine parameters; in order to verify whether the windage loss is the smallest under the optimal gap condition, two cases are taken before and after the optimal gap value. The boundary conditions are consistent with the parameters in Table 5. The optimal gap verification results are shown in Table 6. The windage loss obtained by the \(S/R_{0}=0.0172\) and \(0.0180\) is greater than the optimal gap. The windage loss of the gap behind the impeller between the sCO\({}_{2}\) turbine and the compressor is numerically verified. The boundary conditions are from Table 5. The final results are shown in Table 7. It can be seen that the numerical calculation results are greater than the model prediction results and the deviation is less than 5%.