## 4 Results and Discussion
### Distributions of Flow Parameters  
#### 4.1.1 Pressure Distribution  
The pressure distribution reveals the power capacity of the turbine. In this section, the pressure distribution of the SCO\({}_{2}\) and air turbine are presented and compared. Figure 8 shows the pressure distribution at the mid-span of the SCO\({}_{2}\) radial-inflow turbine. In a SCO\({}_{2}\) radial-inflow turbine, the rotor solidity gradually increases along the flow direction as the pitch at leading edge is much wider than that of trailing edge. As shown in Figure 8, when the rotor solidity is the minimum, the suction side of the rotor blade exists an apparent low-pressure region and the pressure distribution is non-uniform. With the rotor solidity increasing, the low-pressure region tends to decrease and the working capacity for each blade decreases. The diminishing space of rotor flow passage restricts the flow of SCO\({}_{2}\) fluid. Hence, the expansion of working fluid tends to be light in (d) and (e). Condition (c) has a most uniform pressure distribution and the working capacity of blades is satisfying. In condition (a) and (b), the large space at the inlet of the rotor causes the disorder of the pressure distribution. As is known to all, when the temperature or pressure is around critical point, the physical property of SCO\({}_{2}\) can be sharply variable, especially for density and specific heat capacity. In the turbomachinery with working fluid SCO\({}_{2}\), it is a priority problem to minimize the region where SCO\({}_{2}\) transforms from a supercritical state to a subcritical state to obtain stable operation. We term the transformation from the supercritical state to the subcritical state as a trans-critical phenomenon. Hence, Figure 9 presents the trans-critical phenomenon at the trailing edge. It is a vital position in radial-inflow turbine to analyze. As Figure 9 presents, the trans-critical phenomenon is obvious at the trailing edge of the SCO\({}_{2}\) radial-inflow turbine. In condition (a) and (b), the region area is rather small and it concentrates on the tip of the rotor blade. As the rotor solidity increases, the pitch at trailing edge decreases. Hence, the flow area of SCO\({}_{2}\) reduces, which distinctly extends the area of trans-critical region. As we can see,  
Figure 8: Pressure distribution at mid-span of SCO\({}_{2}\): (**a**) solidity 7.96; (**b**) solidity 9.95; (**c**) solidity 11.94; (**d**) solidity 13.93; (**e**) solidity 15.92.  
in condition (c), the minimum pressure along the fluid domain is below 6.5 MPa. With the further decrease of flow area, trans-critical region also appears at the root of rotor blade. In (d) and (e), there exists a local constriction at the trailing edge of rotor blade, which causes an entire trans-critical region at the pressure side of trailing edge. For comparison, Figure 10 shows the pressure distribution at the mid-span with air as working fluid. Likewise, when the rotor solidity is the minimum, the pressure gradient is not uniform in the middle of the flow passage. A low-pressure region presents at the suction side of rotor blade in Figure 10a,b. In (c) and (d), the pressure distribution is uniform while in (e), the pressure difference between suction side and pressure side of the blade decreases. It is worth mentioning that in an air turbine, the low-pressure region at the suction side of rotor trailing edge vanishes. There is no trans-critical phenomenon in the air turbine. The averaged pressure along spanwise location at turbine outlet are presented in Figure 11, where (a) stands for SCO2 and (b) is air. When the value of spanwise location is 1, it corresponds to the shroud while value zero corresponds to the hub. As Figure 11a presents, the distribution of the averaged pressure along spanwise location at turbine outlet is quite different with the change of rotor solidity in a SCO2 turbine. Especially when the solidity is 15.92, the power capacity at the different cross-section of the blade varies a great deal.