# Instability Study of Magnetic Journal Bearing under S-CO\({}_{2}\) Condition
## 4 Summary and Discussion  
In this paper, a lubrication model under S-CO\({}_{2}\) conditions was developed and compared with air conditions to investigate the instability of bearings under S-CO\({}_{2}\). From the S-CO\({}_{2}\) to air comparison, it was shown that the bearing rotating under S-CO\({}_{2}\) conditions can have more issues, which is caused by the property change of S-CO\({}_{2}\) near the critical point and pseudo-critical line. From the model results, the AMB operating under S-CO\({}_{2}\) conditions was tested while thermal properties were being controlled. In these experiments, the lubrication model revealed that thermal properties of S-CO\({}_{2}\) can affect the AMB performance. The model predicted that high density and large density gradient changes can affect the shaft motion by affecting the magnitude and direction of the \(F_{LUB}\) value. The obtained \(F_{LUB}\) from tests 1 and 2 showed substantial scattering with respect to the eccentricity due to unstable rotation. The observed phenomenon seemed to be due to the high sensitivity of the density of S-CO\({}_{2}\) with respect to the pressure and temperature changes.  
The presented experimental data show the decomposition of forces acting on the rotor under the S-CO\({}_{2}\) conditions with magnetic bearings supporting the rotor. To the best of the authors' knowledge, this is the first study to actually present how various forces act on a rotor under S-CO\({}_{2}\) conditions. However, because this study is still at the early stage, further improvements of the experiment are necessary to better understand the phenomenon as well as to accumulate more data under different operating conditions to expand the database. Moreover, the lubrication analysis model has ideal assumptions, and each assumption has to be carefully re-evaluated to improve the model prediction capability. For instance, assumptions such as a single revolving center, uniform temperature distribution, and smooth geometry have to be scrutinized and improved where necessary.  
In the future, to overcome those shortcomings, the thermal properties inside the bearing should be measured to reveal clearer relationships between fluid properties and the destabilizing forces applied to the shaft. In addition, the non-ideal geometry of the electromagnets, which is considered to be another disturbance source, has to be modified in further study to match the model more closely to the experiment. By implementing the suggested improvements, it is expected that the unstable region can be clearly defined and the required performance or design of the AMB can be more precisely determined.  
The dynamics of the shaft must also be investigated. Moreover, the relationship between \(F_{LUB}\) and shaft motion can be better described as a second-order system. The trajectory of the shaft can be predicted from the data and compared to the experimental results. After validation, this model can be utilized in various S-CO\({}_{2}\) cycle systems. Then, required performance and control logic of the AMB can be suggested for each S-CO\({}_{2}\) cycle system.  
**Author Contributions:** Conceptualization, D.K., S.B. and J.L.; Data curation, D.K.; Formal analysis, D.K. and J.L.; Investigation, D.K. and J.L.; Methodology, D.K., S.B. and J.L.; Software, D.K.; Validation, D.K. and J.L.; Writing--original draft preparation, D.K. and J.L.; Writing--review and editing, D.K. and J.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was supported by the Defense Challengeable Future Technology Program of Agency for Defense Development, Republic of Korea, grant number UC190029 RD.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data sharing not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.