## 4 Results
### Discussion  
The fluids involved in the analysis, LOx and He, mix in the drain conduit, generating a bi-component mixture, which is ejected from the turbopump, while pure He flows in the He inlet. A first check was performed comparing the He mass fraction (\(\beta_{\rm He}\)) at the drainage outlet with the ratio of the inlet flow rates of the two fluids, \(\beta_{\rm He}=0.14\). This parameter is fundamental since it is used to determine the properties of the mixture on the basis of the ones of the pure oxygen and He. Figures 7 and 8 show the pressure predictions in the two conduits. By analysing the trends, it is possible to assess a progressive pressure drop from the inlet of the ducts to the end. This loss is relatively small and is more significant in the drain conduit because of the higher flow rate and gravitational force. The leading causes of these pressure losses are line and local losses due to junctions. Taking into account the pressure losses due to the junctions is necessary for modelling: the losses they generate, despite being relatively small, influence the redistribution of flows within radial ducts and, thus, the heat flux and fluid temperature in each duct. The flow rate governs the rate of energy transport into the control volume and, in addition, the forced convection heat transfer. The results regarding the mass flow rates for the different branches are shown in Figures 9 and 10. Figure 9 shows that the flow coming from the labyrinth seal and the segmented seal is redistributed symmetrically with respect to the outlet duct. As expected, the ducts characterized by a longer flow path present a lower mass flow rate since they are less convenient because their paths have higher hydraulic resistance. All the fluid eventually flows through Branches 27-29. The same applies to the He duct, with the difference that there are fewer radial ducts and mass flow rates. In this case, the fluid flows from the inlet to the segmented seal. Figure 14: Temperature prediction at solid nodes between the He/LOx drain and He inlet conduit. Figures 11 and 12 show the predicted fluid temperature in the two conduits analysed. As seen from the two trends, the temperature of the He inlet is higher than the He/LOx drain temperature. Indeed, the He is injected at 288.1 K, while the temperature is significantly lower in the drain due to the two-phase oxygen at cryogenic temperatures from the labyrinth seal. By analysing the He inlet trend, it is possible to assess a temperature decrease from the inlet of the ducts to the end. This effect is due to heat exchange with the wall, which draws heat from the fluid. Indeed, the wall is at a lower temperature due to its proximity to the drain. In contrast, the He/LOx drain increases in temperature from the labyrinth seal to the outlet as it draws heat from the wall. Observing Figure 11, it can be assessed that the first node of the drainage has a temperature of 97.5 K, which corresponds to the saturation temperature at the pressure of Node 1; this implies that the oxygen is still in a two-phase condition at the segmented seal. However, the He injection at a high temperature caused the oxygen to rise in quality from \(x_{\rm ox}=0.15\) to \(x_{\rm ox}=0.86\). Furthermore, the presence of He, which has the characteristics of an inert gas, causes the mixture to have an overall gas fraction of \(\psi=0.88\). From the second node onward, oxygen is wholly vapourized (superheated steam). In radial ducts, the temperature assumes different values. In particular, in the He inlet, the temperature decreases from the ducts radially opposite from the inlet. In contrast, the ducts radially farther from the outlet assume higher temperatures in the drain. The cause is attributed to the different redistributions in the different branches. Considering, for example, the drain, the long residence time of the fluid in the nodes farther from the outlet determines higher temperatures, despite the low values of the heat transfer coefficient due to the low mass flow rate.