# A New Windage Loss Model for S-CO\({}_{2}\) Turbomachinery Design
## 3 Comparison of Data to Existing Windage Loss Models  
The obtained windage loss data are compared to the existing empirical windage loss models summarized in Table 2[12; 13; 14]. The measured windage loss is compared with the windage loss value obtained from the models by using inlet conditions measured during experiments. The results are shown in Figures 4-7. The plotted points in these figures show the windage loss measured by a power analyzer at the S-CO\({}_{2}\) rotor test rig as the y value. The x value of the points is the windage loss value calculated with each empirical model. The range of the points shows its uncertainty calculated at Section 2.2. The linear fitting via regression analysis is shown by a pink line. The blue line shows when the calculated windage loss and the measured windage loss are the same (i.e., y = x). If the empirical models explain the experiments well, the pink line is closer to the blue line. The two green lines represent a 20% error band. Thus, the top green line is y = 1.2x, and the bottom green line is y = 0.8x. How well the empirical model predicts the experimental results can be evaluated with the number of the plotted points inside the region between two green lines, which denotes how many data points can be predicted using the model with a 20% error. The R\({}^{2}\) value of each model can also be compared to evaluate the fitness of the model. The R\({}^{2}\) value, which is an indicator of the regression model performance for each model, is shown in Table 3. From the figures, it is concluded that empirical models based on liquid tests all underestimate the S-CO\({}_{2}\) windage loss. [PERSON] and [PERSON]'s models, which are based on the air test, show smaller differences from the experimental measurement than other models, but they all overestimate the S-CO\({}_{2}\) windage loss. It is clear that the [PERSON], [PERSON], [PERSON] and [PERSON], and [PERSON]'s models are unsatisfactory for explaining the S-CO\({}_{2}\) windage loss test results because the R\({}^{2}\) value is negative. [PERSON] and [PERSON]'s models have positive R\({}^{2}\) values. However, an R\({}^{2}\) value close to unity cannot be found amongst the tested models, which implies that a new model development is necessary. One of the major reasons why the existing models fail to predict the data successfully is due to the negligence of property variations from inlet to outlet due to viscous heating. The viscous heating from windage loss changes the CO\({}_{2}\) properties dramatically from inlet to outlet in the secondary pass of turbomachinery. However, it was originally suggested that the existing empirical models should only use inlet conditions, and therefore, this property change effect of CO\({}_{2}\) cannot be reflected properly. Figure 4: Measured windage loss vs. calculated windage loss from [PERSON]’s model. Figure 8 shows the work loss due to windage loss estimated with inlet conditions and outlet conditions, respectively, for each windage loss model to illustrate how strong this effect can be. The inlet condition is 80 bar and 36 \({}^{\circ}\)C, and the outlet condition is 79.5 bar and 53 \({}^{\circ}\)C, which are selected conditions from the actual tests conducted in the KAIST S-CO\({}_{2}\) rotor test rig [8]. As the figure shows, the existing empirical models cannot predict the windage loss under S-CO\({}_{2}\) conditions accurately by using only inlet conditions. The inlet-to-outlet density ratio is shown in Figure 9 to explain the reason for this. As shown in  
Figure 5: Measured windage loss vs. calculated windage loss from [PERSON]’s model. Figure 6: Measured windage loss vs. calculated windage loss from [PERSON] model. the figure, the density ratio is less than unity during the test. This is because the density is decreased substantially due to viscous heating, and this in turn significantly affects windage loss.