## 2 Methodology
### Selection of Analysis Conditions  
Typically, a centrifugal compressor is known to show better efficiency for higher specific speed and larger scale. The specific speed is defined in Equation (9) and uses to estimate an isentropic efficiency, rotational speed and the number of stages of a compressor. The increase in specific speed reduces the flow resistance in an impeller because the flow passage tends to be shorter and wider as specific speed increases. Figure 10 supports this physical insight. Since this study only covers the centrifugal compressor, the specific speed of 0.64, which is the highest in the radial type of [PERSON]'s diagram, was chosen [30]. The scale effect can explain through the ratio of tip clearance to blade height. The tip clearance loss and the leakage loss decrease as the system scale increases since the ratio of tip clearance to blade height decreases as the system power capacity increases. Figure 11 represents the best achievable compressor efficiency for different system power class. The minimum tip clearance was selected considering the manufacturing tolerance. Although compressor efficiency increases as the system scale grow, in large systems, an axial type is typically recommended rather than a radial type. Therefore, in this study, a compressor for 10 MWe class power conversion system was selected to demonstrate the feasibility of the current study [31]. \[n_{s}=\frac{NQ^{0.5}}{\left(gH_{ad}\right)^{0.75}} \tag{9}\]  
Blade backsweep angle at impeller exit and static conditions at compressor inlet were chosen as the design parameters for the sensitivity analysis because those are important variables that determine the S-CO\({}_{2}\) compressor outlet conditions. Compared with the conventional compressor, the inlet operating condition is regarded as a very important design factor in the S-CO\({}_{2}\) compressor due to the steep change of thermodynamic properties near the critical point. Figure 12 shows the compressibility factor variation in the S-CO\({}_{2}\) compressor operating range. At CO\({}_{2}\) critical point, compressibility factor has the lowest value of 0.23 and it varies up to 0.65 with increasing temperature and pressure from the critical point. The inlet conditions of the analyzed compressor were selected while considering the compressibility factor. Table 2 summarizes the operating conditions and main design parameters. The design conditions represent the inlet and outlet conditions for the compressor design derived from five different simple recuperated cycle designs. Since the variation of thermodynamic properties near the critical point depends on the compressibility factor, the compressibility factor was used as the representative value for the inlet condition. It is noted that the outlet pressure of all five compressors is  
Figure 9: Predicted total to static efficiency results with SNL test data [24]. all fixed at 20 MPa when designing five different cycles studied in this paper. In order to minimize the kinetic energy loss at the compressor inlet, the inlet absolute velocity angle was chosen to be \(0^{\circ}\). To observe the effect of the backsweep angle change on the compressor performance, the range of \(0^{\circ}\) to \(-77^{\circ}\) was selected as the analysis range, which encompasses the conventional \(-50^{\circ}\). Figure 13 is an example of an S-CO2 compressor impeller after applying the design conditions of case 1 and the design parameters are summarized in Tables 2 and 3 summarizes the results of cycle analysis with recuperated Brayton cycle layout in each case. Figure 11: Peak total to total efficiency with respect to various power class. Figure 10: Peak total to total efficiency with respect to a specific speed. Figure 12: Compressibility factor contour under the S-CO\({}_{2}\) compressor operating range.