## 2 Modeling
### Research Model  
The working fluid first flows into the nozzle flow passage. In the nozzle, the pressure gradually declines and accordingly, the velocity increases. Figure 2 presents the geometry model of nozzle. A straight blade with uniform geometry angles from hub to shroud direction is adopted due to the low blade height. The geometry of nozzle is identical in all conditions with different solidity structures. The red part represents high temperature and pressure while the blue part stands for lower temperature and pressure. Detailed geometric parameters, i.e., the diameter of inlet and outlet, the blade height, and the geometry angle of the inlet and outlet are given in Figure 2.  
After the process in the nozzle, the working fluid enters from a radial direction into the impeller. In this process, SCO\({}_{2}\) expands in the impeller and exits from axial direction. The backward bent vane is adopted to guarantee the designed velocity triangle and a shrouded impeller is used to promote the aerodynamic efficiency. Figure 3 shows the geometry model of impeller, taking blade number 12, as an example. After the calculation of impeller diameter, blade height of inlet and outlet, external diameter of outlet and axial length, the arc to connect each point is determined by design experience. In Figure 3, the red arrow stands for high temperature and pressure while the blue arrow represents relatively low temperature and pressure. [PERSON] (2018) gave the definition of solidity for radial pumps. Likewise, we define the solidity with cord length and pitch in a radial-inflow turbine. These parameters are obtained in Workbench BladeGen while modelling. In this research, the rotor solidity is defined by \(c\)/s, where \(c\) is the cord length of the rotor blade and \(s\) is the trailing edge pitch at the outlet of the impeller, as shown in Figure 3.  
Figure 2: Geometry model of nozzle.  
The concrete geometric parameters of impeller are shown in Table 1. Generally, the number of nozzle blades and the number of impeller blades should be relatively prime for the consideration of reducing the exciting force of turbomachinery. Hence, while the nozzle blade number is designed as 13, we choose five kinds of rotor solidity, which are realized by changing the rotor blade numbers. They are even from 8 to 16. The large range of rotor solidity can present its effect on aerodynamic performance.  
High solidity usually results in large weight and axial thrust of the wheel. In some application environments such as aerospace and warships, the weight of the wheel is limited. The overlarge axial thrust and the trans-critical problem of the high solidity case also need consideration in the turbine design. Hence, for the low solidity case of SCO\({}_{2}\) turbine mentioned above, we try to establish a new structure to improve the flow condition at the leading edge of the rotor blade. A splitter, which is normally used in centrifugal compressors, is introduced here. We establish a pressure side (ps) offset splitter and a suction side (ss) offset splitter to compare with the normal one, as shown in Figure 4. The offset value is 20% of the local pitch, that is, the distance between the splitter blade the closest main blade is 30% of the local pitch.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **A** & **B** & **C** & **D** & **E** \\ \hline Blade number & 8 & 10 & 12 & 14 & 16 \\ Impeller diameter/mm & & & 92.2 & & \\ External diameter of outlet/mm & & & 40.6 & & \\ Axial length/mm & & & 20 & & \\ Blade height of inlet/mm & & & 2.97 & & \\ Blade height of outlet/mm & & & 24.6 & & \\ Inlet geometry angle/\({}^{\circ}\) & & & 90 & & \\ Outlet geometry angle/\({}^{\circ}\) & & & 27 & & \\ Chord length (\(c\))/mm & & & 50 & & \\ Pitch at trailing edge (s)/mm & 6.28 & 5.03 & 4.19 & 3.59 & 3.14 \\ Solidity (\(c\)/\(s\)) & 7.96 & 9.95 & 11.94 & 13.93 & 15.92 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of the impeller.  
Figure 3: Geometry model of the impeller.