## 4 Results and Discussion
### Prediction Method of Optimum Gap
The theoretical model is still reliable when CO\({}_{2}\) is used as the medium and less than the deviation when water is used. Figure 18 shows the velocity distribution in the gaps behind the turbine and compressor impellers. Due to the high Reynolds number of the two cases, the boundary layer separation of the two walls still occurs although the axial gap is small. When the medium is CO\({}_{2}\), the viscosity near the critical point is low and the influence of the boundary layer on the flow loss is reduced. In addition, the different working temperatures of the turbine \begin{table}
\begin{tabular}{c c|c|c} \hline
**Parameters** & \multicolumn{2}{c}{**Value**} \\ \hline Relative gap (\(S/R_{0}\)) & 0.0172 & 0.0176 & 0.0180 \\ Loss model/kW & 0.956 & 0.897 & 0.961 \\ Numerical simulation/kW & 0.992 & 0.908 & 0.980 \\ Relative deviations & 3.8\% & 1.2\% & 2.0\% \\ \hline \end{tabular} Gray cells: optimal gap and corresponding results.  
\end{table} 
Table 6: Optimal gap verification results. Figure 17: Relationship between \(C_{m}\) of the relative gap behind the turbine impeller. and the compressor make a difference in the viscosity coefficient. The friction torque on the rotating disk is proportional to the viscosity coefficient, so the deviation between the turbine and compressor is also different. It should be noted that there is a discrepancy in the ratio of the inner and outer diameters of the impeller between the two, with different relative areas of the impeller back surface, which is one of the reasons for the deviation. Through the verification of this paper, the prediction method of the optimal gap has good applicability for water-liquid and CO\({}_{2}\) working fluids. The optimal gap obtained by this method will provide a basis for the design of a radial turbine structure. It should be noted that once the structure is manufactured, it may not be at the optimal gap when operating under variable conditions. The windage loss of the gap is the smallest under the design condition. Since the method is based on the model of two regions of the turbulent state, it has higher applicability in the range of Reynolds numbers from \(1.58\times 10^{5}\) to \(10^{8}\). In fact, in radial turbomachinery such as the supercritical carbon dioxide cycle, the windage loss of the gap behind the impeller is greater than the current predicted value. The main reasons are as follows: (1) The size value of the gap is greater than the predicted value due to the axial expansion of the rotor and the axial movement; (2) There are generally deduplication pits on the back side of the impeller and the back side is not completely circumferentially smooth; (3) The existence of the inlet and outlet in the gap causes the flow to have a certain radial flow which is different from the hypothesis of the prediction model; (4) The temperature distribution of the impeller and stator will affect the flow and the ratio of the temperature of the gap to the temperature of the flow also has a great influence on the distribution of the heat transfer [27]. However, in the design of such radial turbomachinery, the best gap size can be obtained according to the flow parameters at the gap between the impeller and the guide blade and the prediction model of Region III and Region IV. Through certain structural design measures, the best gap can be as close as possible to minimize the windage loss of the gap behind the impeller.