## 3 Expander Design
### Meanline Design  
Meanline design, which is also called one-dimensional design, was conducted based on the results of the preliminary design. Meanline design is a flow path design including \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Expander 1**} & **Expander 2** \\  & & (3)–(4) & (5)–(6) \\ \hline Pt\_in & bar & 9.76 & 5.98 \\ \hline T\_in & K & 49.1 & 24.8 \\ \hline Pt\_out & bar & 6.1 & 1.62 \\ \hline T\_out & K & 42.8 & 17 \\ \hline Pr & - & 1.6 & 3.69 \\ \hline Efficiency & \% & 74.7 & 76.3 \\ \hline Power & kW & 5.32 & 6.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Operation conditions of Expanders 1 and 2. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Ns** & **Unit** & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** & **1.0** & **0.392** \\ \hline Efficiency & - & 0.76 & 0.81 & 0.85 & 0.87 & 0.87 & 0.84 & 0.8 & 0.72 & 0.61 & 0.85 \\ \hline RPM & - & 38,191 & 57,286 & 76,382 & 95,477 & 114,573 & 133,668 & 152,764 & 171,859 & 190,955 & 75,000 \\ \hline Power & kW & 5.4 & 5.7 & 6.0 & 6.1 & 6.1 & 6.0 & 5.6 & 5.1 & 4.3 & 6.0 \\ \hline Ds & - & 10.0 & 6.67 & 5.0 & 4.0 & 3.33 & 2.86 & 2.5 & 2.22 & 2.0 & 5.09 \\ \hline D\_tip & mm & 105.59 & 70.39 & 52.8 & 42.24 & 35.2 & 30.17 & 26.4 & 23.46 & 21.12 & 53.77 \\ \hline U\_tip & m/s & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 & 211.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Preliminary design results for Expander 1.  
the blade shape and flow angle. In this study, meanline design was conducted using ConceptsNREC Inc's RITAL program. The Rodgers loss model was applied for the nozzle design, and the CETI passage loss model was applied for the rotor design. Figure 2 shows the meanline shape of the expander. The designed expander consists of a volute, nozzle, and rotor. Generally, a volute is symmetric. However, in this research, an overhang-shaped volute was selected to shorten the shaft length and increase the assembly. [PERSON] et al. compared losses according to the shape of volute [12]. They conducted a CFD analysis of the loss in a circular volute and a rounded square volute. In this study, the volute shape was determined based on the fact that a circular volute has less loss than that of a rounded square volute. The major dimensions of the expander are shown in Table 4. The designed tip clearance between the rotor and shroud was 0.3 mm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Dimension** \\ \hline RPM & - & 75,000 \\ \hline Number of nozzles & - & 17 \\ \hline Nozzle inlet radius (N1) & mm & 36.5 \\ \hline Nozzle outlet radius (N2) & mm & 29.5 \\ \hline Nozzle height & mm & 3.3 \\ \hline Nozzle outlet blade angle & \({}^{\circ}\) & 76.21 \\ \hline Number of rotors & - & 14 \\ \hline Rotor inlet radius (R1) & mm & 26 \\ \hline Rotor outlet shroud radius (R2) & mm & 13.2 + 0.3 \\ \hline Rotor outlet hub radius (R3) & mm & 8.4 \\ \hline Rotor outlet blade angle & \({}^{\circ}\) & \(-\)60 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Meanline design results. Figure 2: Meridional view of expander. The velocity triangles at the inlet and outlet of the rotor are shown in Figure 3. In this study, the incidence angle was \(-26.74^{\circ}\), and the deviation angle was \(22.8^{\circ}\). Figure 4 shows the performance curves derived from the meanline design. They were plotted at 80% (60,000 rpm), 90% (67,500 rpm), and 100% (75,000 rpm) of the design speed. Figure 4a shows the relation between the expansion ratio and mass flow rate. Figure 4b shows the relation between the expansion ratio and efficiency. The efficiency from the meanline design was recalculated by reflecting the loss in the expander. It is defined as the ratio of the enthalpy difference and isentropic enthalpy difference between the inlet and outlet of the expander. It was verified that the target efficiency was satisfied at the design expansion ratio.