# A New Windage Loss Model for S-CO\({}_{2}\) Turbomachinery Design
## 2 Description of S-CO\({}_{2}\) Windage Loss Experiment  
### Experimental Design of S-CO\({}_{2}\) Windage Loss Measurement  
An S-CO\({}_{2}\) rotating machinery test section is used for measuring windage loss as shown in Figure 2[8]. The schematic diagram of the test section is shown in Figure 3. The radius of the shaft is 32 mm, and the shaft length is 348 mm as shown in Figure 3. The clearance between the shaft and the stationary wall (i.e., gap width) is 1.5 mm. The test section rotor is supported by Active Magnetic Bearing (AMB). The experimental test rig has no impeller by referring to [PERSON]'s paper [2] to make the internal loss negligible; thus, windage loss can be measured. Since the AMB has a gap ratio greater than other bearing types and the impeller is removed from the rotor, the test section is appropriate to test the shaft rotor dynamics and interaction between shaft and working fluid. The test section is designed to operate under 12 MPa and 100 \({}^{\circ}\)C conditions and can achieve a 35,000 RPM rotational speed. Figure 3: Schematic diagram for S-CO\({}_{2}\) rotor test rig. Figure 2: KAIST magnetic bearing supported S-CO\({}_{2}\) rotating machinery test section. The external loss can be obtained from the power measurement. The motor output power is measured by a power analyzer. In general, the motor power will be converted to mechanical work to overcome the external loss, enthalpy rise of S-CO\({}_{2}\) from inlet to outlet, and heat losses via cooling in motor winding and heat transfer to ambient air from the test section external wall. Therefore, by measuring motor power and other converted quantities, the windage loss can be generally obtained using Equation (1) [9]. \(L_{ext}\) is external loss. \(P_{input}\) is motor power input and \(\Delta H_{CO_{2}}\) is CO\({}_{2}\)s enthalpy change for major flow. \(Q_{water}\) is the heat transferred from the test rig to the cooling water, and \(Q_{leak}\) is the heat removed by leaking of the working fluid. \[L_{ext}=P_{input}-\Delta H_{CO_{2}}-Q_{water}-Q_{leak} \tag{1}\]  
However, only the minor flow exists in the test section used in this study since the test rotor has no impeller attached and the major flow is blocked from the main loop while the test is conducted as shown in Figure 2. Moreover, the test section has no cooling water jacket for the motor winding and the leakage test is performed for the test rig. Therefore, all heat losses can be ignored. Consequently, the measured motor output can be regarded as the external loss. The obtained external loss consists of different losses as shown in Equation (2) [9]. It is first noted that the test conducted under S-CO\({}_{2}\) conditions has shown the motor power is measured in the order of a few kW. Thus, losses smaller than a few tens of Watts can be neglected. Other external losses are predicted based on the previous research. If the Daily and Nece model as shown in Equations (3) and (4) is used to estimate the disk loss, it is estimated to be only 0.5 W [10]. Thus, the disk loss can be neglected. The small value is due to the test rotor configuration with no impeller and only the shaft end surface will contribute to the disk loss. The bearing loss can also be negligible because the AMB has a greater gap ratio than other types of bearing. Previous research [9] used the Schlichting model to calculate it as shown in Equation (5), and the maximum value is estimated to be 10 W. Therefore, the measured motor power is the external loss of the test section rotor, and the measured external loss can be viewed as equivalent to the windage loss.