# A New Windage Loss Model for S-CO\({}_{2}\) Turbomachinery Design
## 4 New S-CO\({}_{2}\) Windage Loss Model Development  
A new windage loss model is developed by modifying the existing model and the validity of the newly proposed model is presented in this section. The Taylor number is first examined to determine whether it alone is enough to describe the windage loss test data. The measured windage loss is plotted with respect to the Taylor number, as shown in Figure 10. Since the data have wide dispersion without a clear pattern with respect to the Taylor number, it is concluded that other variables are required to describe the windage loss data better. The heat capacity ratio \(\gamma\) is therefore considered to be used next in conjunction with the Taylor number to better explain S-CO\({}_{2}\) windage loss test data. The measured windage loss data are now plotted with respect to \(Ta_{intel}\) and \(\gamma_{intel}\) in Figure 11. By correlating the data with the Taylor number and \(\gamma\), the data can now have  
Figure 8: Difference between \(W_{intel}\) and \(W_{outlet}\). Figure 9: Inlet-to-outlet density ratio of the S-CO\({}_{2}\) windage loss test. \begin{table}
\begin{tabular}{c c} \hline \hline
**Empirical Models** & **R2 Value** \\ \hline [PERSON] & 0.8604 \\ [PERSON] & 0.7133 \\ [PERSON] & \(-\)6.2276 \\ [PERSON] & \(-\)31.8137 \\ [PERSON] and Boulos & \(-\)0.4311 \\ [PERSON] & \(-\)0.7444 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The R2 value of existing models for measured S-CO\({}_{2}\) windage loss. one-to-one correspondence. Therefore, these two variables are used to correlate S-CO\({}_{2}\) windage loss data. Since [PERSON]'s model has the best regression result as shown in Table 3, [PERSON]'s model is selected to correct with \(Ta_{inlet}\) and \(\gamma_{inlet}\). The modified model should produce identical results with the original model when the \(\gamma\) value becomes equal to air, since [PERSON]'s model is developed from data obtained from air tests. The assumed skin friction coefficient with this constraint can thus have the following relations shown in Equations (15) and (16). The least square method is used to find the value for exponent \(a\). The newly proposed model is shown in Equation (17) for the supercritical phase and in Equation (18) for the gas phase. \[C_{f,new}(Ta_{inlet},\gamma_{inlet})=\frac{K(\gamma_{inlet})}{Ta_{inlet}^{0.1}} \tag{15}\]  
Figure 11: Measured windage loss vs. Taylor number and \(\gamma\). Figure 10: Measured windage loss vs. Taylor number. \[K(\gamma_{inlet})=\frac{1.8}{Ta_{crit}^{0.4}c_{r}^{-1.25}}\frac{(1+c_{r})^{2}}{(1+c_ {r})^{2}-1}\bigg{(}\frac{\gamma_{inlet}}{\gamma_{air}}\bigg{)}^{a} \tag{16}\]  
\[C_{f,new,s}= \frac{1.8}{Ta_{crit}^{0.4}c_{r}^{-1.25}}\frac{(1+c_{r})^{2}}{(1+c_{r })^{2}-1}Ta_{inlet}^{-0.1}\bigg{(}\frac{\gamma_{inlet}}{\gamma_{air}}\bigg{)}^{ -0.4051} \tag{17}\] \[C_{f,new,g}= \frac{1.8}{Ta_{crit}^{0.4}c_{r}^{-1.25}}\frac{(1+c_{r})^{2}}{(1+c_ {r})^{2}-1}Ta_{inlet}^{-0.1}\bigg{(}\frac{\gamma_{inlet}}{\gamma_{air}}\bigg{)}^ {-0.6254} \tag{18}\]  
Even though the developed model is based on one S-CO\({}_{2}\) test rig, it is expected that this model could be applied to the design of other turbomachinery. This is because the developed model is based on non-dimensional numbers, and as long as the similitude holds, the model can be applied to the design of other equipment operating under S-CO\({}_{2}\) conditions. The newly developed model is a model modifying the base model (i.e., [PERSON]'s model) by multiplying the term \(\Big{(}\frac{\gamma_{inlet}}{\gamma_{air}}\Big{)}^{a_{1}}\) to reflect the change in properties of the working fluid due to viscous heating. Therefore, this model can reflect not only S-CO\({}_{2}\) conditions but also other fluids with similar characteristics if the fluid is in a superheated gas or supercritical phase. Other conditions such as geometry or rotation speed are different in the designed machinery, the developed model will follow the trend of [PERSON]'s model, which is based on a larger test data set.