## 4 Results and Discussion
### Distributions of Flow Parameters
#### 4.1.2 Entropy Distribution  
The static entropy distribution can reflect the aerodynamic loss in the flow passage. For different rotor solidity, the difference of aerodynamic loss mainly concentrates on the outlet. In this section, the static entropy at the outlet of SCO\({}_{2}\) and air are presented and compared.  
The averaged static entropy along spanwise location at turbine outlet are presented in Figure 12. Spanwise location 1 and 0 correspond to the shroud and hub, respectively. The green plane is the contour location. It is acknowledged that higher entropy corresponds to higher aerodynamic loss. For both SCO\({}_{2}\) and air, the aerodynamic loss mainly concentrates on the hub and shroud area due to the boundary layer effect of the impeller. The distribution is quite different with the change of rotor solidity. At the shroud, the highest static entropy appears due to the relatively vacant flow passage when solidity is 7.96 in the SCO\({}_{2}\) turbine. For air, it locates at solidity 9.95, which means the decrease of solidity less affects the aerodynamic loss at the shroud. While at the hub, the highest static entropy presents when solidity is 15.92 on account of the limited flow area for both two working fluids.  
Figure 13 shows the entropy distribution at outlet with surface streamlines of the SCO\({}_{2}\) radial-inflow turbine. When the rotor solidity is the minimum, the local entropy production in the flow passage near  
Figure 11: Averaged pressure along spanwise location at turbine outlet: (**a**) SCO\({}_{2}\); (**b**) air.  
Figure 12: Averaged static entropy along spanwise location at turbine outlet: (**a**) location of graphical results; (**b**) SCO\({}_{2}\); (**c**) air.  
the shroud is apparent. Due to the low solidity, the disordered flow near the shroud results in several vortexes, thus causing local entropy concentration. With the rotor solidity increases, the vortexes gradually disappear and the increase of local entropy near the shroud completely vanishes in condition (d), i.e., solidity 13.93. The location of entropy concentration changes to the flow passage near the hub. When rotor solidity is the minimum, the flow near the shroud is disordered while solidity is the highest, the diminishing space of the rotor flow passage restricts the flow of SCO\({}_{2}\) fluid. These reasons cause the local enlargement of entropy as well as aerodynamic loss.  
Comparing to Figures 13 and 14 shows the entropy distribution at outlet with surface streamlines of the radial-inflow turbine working with air. The static entropy of the air turbine is apparently lower. In Figure 14a,b, the local vortex at the shroud causes local entropy production while in the SCO\({}_{2}\) turbine, the vortex nearly disappears when solidity is 9.95. With the solidity further increasing, the entropy production still concentrates on the shroud side even when the solidity becomes the highest. Hence, in the air turbine, the diminishing space of rotor flow passage does not cause the local enlargement of aerodynamic loss at the hub.  
Figure 14: Entropy distribution at outlet with surface streamlines of air: (**a**) solidity 7.96; (**b**) solidity 9.95; (**c**) solidity 11.94; (**d**) solidity 13.93; (**e**) solidity 15.92.  
Figure 13: Entropy distribution at outlet with surface streamlines of SCO\({}_{2}\): (**a**) solidity 7.96; (**b**) solidity 9.95; (**c**) solidity 11.94; (**d**) solidity 13.93; (**e**) solidity 15.92.