## 3 Expander Design
### Numerical Analysis  
Numerical analysis was conducted using NUMECA's FINE/Turbo program. As a turbulence model, the Spalart-Allmaras model, a one-equation model with excellent calculation efficiency, was applied to determine the exact properties of the boundary layer in the pressure gradient. Figure 3: Velocity triangles at design points: (**a**) rotor inlet (incidence angle \(-26.74^{\circ}\)) and (**b**) rotor outlet (deviation angle \(22.8^{\circ}\)). In order to increase the convergence of the calculation, the choice of boundary condition is important. In general, pressure, temperature, and flow rate are applied as boundary conditions in turbomachinery numerical analysis. Applicable pressure and temperature values are static and stagnation value. In addition, the applicable flow information is the mass flow rate and the velocity vector. In this study, numerical analysis was performed by  
Figure 4: Performance prediction curves from meanline analysis: (**a**) expansion ratio vs. mass flow rate; (**b**) expansion ratio vs. efficiency; (**c**) expansion ratio vs. power; (**d**) expansion ratio vs. expander outlet temperature. Figure 5: Generated 3D geometry of nozzle and rotor. analyzing the calculated result using the pressure condition as the main boundary condition. The applied boundary conditions were the inlet total pressure, inlet total temperature, velocity direction, turbulent viscosity, and outlet static pressure. To obtain the off-design point performance, the outlet static pressure was varied. The meanline design results were used for the flow angle information flowing into the nozzle inlet. That is, the absolute flow angle of the volute outlet was used as the nozzle inlet flow angle in the numerical analysis. A mixing plane condition was applied between the nozzle outlet and the rotor inlet. Therefore, the average physical property value in the pitch direction was applied as the rotor inlet condition. In real operation conditions, the shape of the nozzle wake changes along with the rotor revolution, and to analyze it, unsteady calculations should be conducted. In this study, the focus was on efficiency improvement, and a steady calculation was conducted. The calculation was conducted for a single periodic passage. To determine the appropriate number of grids, the efficiency according to the number of grids was analyzed. Efficiency was compared for the following number of grids: 199,079, 239,403, 345,415, 534,415, 1,088,147, and 1,332,851. When the number of grids was greater than 239,403, there was no significant change in efficiency. In this study, the number of grid cells in the calculation domain was 1,088,147. Figure 6 shows the relationship of efficiency according to grid number to check grid independence. A hexa mesh was applied to generate a dense grid near the blade wall, and Y+ was set to 2. The cryogenic expander design was conducted using the real gas properties of helium. To reduce the computation time during the numerical analysis, a property table of the working fluid was applied to the calculation. NUMCA's TabGen was used to generate the data. TabGen creates a fluid property table in connection with NIST's REFPROP, a program that provides fluid properties. Interpolation was performed based on the generated property table so that it could be used as a property value at a location necessary for numerical analysis. Figure 7 shows the performance curves of the cryogenic expander derived from the numerical analysis. In the same way as for the meanline design results, the mass flow rate, efficiency, power, and outlet temperature were plotted against the expansion ratio at 80%, 90%, and 100% of the design speed. The efficiency and power from the numerical analysis were recalculated by taking into account the extra mechanical losses in the expander. The extra mechanical losses were the loss caused by the volute and the loss caused by the windage in the rotor back disk, and in this study, the losses were assumed to be 2% and 4%, respectively.