## 2 Methodology
### Loss Models  
The selection of loss models is a significant process in the 1D method because it determines the accuracy of analysis results. [PERSON] and [PERSON] et al. adopted loss models through validation with experimental results and CFD results for an axial turbine and a centrifugal compressor, respectively [22; 23]. KAIST-TMD adopts empirical loss models and slip factor to estimate the S-CO\({}_{2}\) compressor performance. The loss models are empirical correlations for irreversibility estimation of turbomachinery. Since KAIST-TMD is validated with the S-CO\({}_{2}\) compressor experimental data, the design results can be believed to have high fidelity under various different design conditions [24]. Table 1 summarizes the selected loss models and Figures 5 and 6 illustrate the mechanisms of each loss.  
The internal losses consist of incidence loss, blade loading loss, skin friction loss, mixing loss and clearance loss for the unshrouded impeller. All the internal losses are supposed to contribute to the efficiency as well as the pressure ratio. Meanwhile, the external losses are made up of disk friction loss, recirculation loss and leakage loss. All the external losses are associated with minor flows leaking from the main flow and assumed as the energy loss outside the main flow path. Thus, it only affects the efficiency of the compressor.  
Each loss mechanism of internal loss is as follows: (1) Incidence loss is derived from non-uniform incidence along the leading edge. It increases the relative velocity in the tangential component at the entrance and has the minimum loss at the optimum incidence angle. (2) Blade loading loss is the fluid momentum loss that is associated with blade surface boundary layer growth in decelerating flows. It is also related to flow separation in the impeller. (3) Skin friction loss is similar to enthalpy loss due to the wall friction of turbulent flow in a duct. However, it does not include the effect of blade surface boundary layers. (4) Clearance loss is defined as the loss due to the flow of fluid from the pressure surface of the blade to the suction surface. It is proportional to the ratio of tip clearance to blade height. Tip clearance is the gap between the rotating impeller and the stationary shroud. (5) Mixing loss is also known as wake-jet loss. It is derived from mixing of the high momentum fluid on the pressure side, namely jet, and the low momentum fluid on the suction side, namely wake. (6) Slip loss assumes that the flow cannot be perfectly guided by impeller blades. It causes the flow angle at the impeller  
Figure 4: KAIST-TMD main algorithm.  
exit leans to the opposite direction of the rotating direction. As a result, the tangential component of absolute velocity at impeller exit decreases as shown in Figure 7.  
Each loss mechanism of external loss is as follows: (1) Leakage loss occurs when the leakage flow of tip clearance is re-entered into the blade passage. (2) The recirculation loss is the enthalpy loss brought about the recirculation of low momentum fluid from the vaneless space back into the impeller flow path. (3) Disk friction loss is due to the wall friction between the back surface of the rotating impeller and the stationary surface.