# Instability Study of Magnetic Journal Bearing under S-CO\({}_{2}\) Condition
## 2 Magnetic Bearing Analysis for Instability Study
### Magnetic Bearing Lubrication Model Analysis  
To find the instability source, results from stable (air) and unstable (S-CO\({}_{2}\)) operating conditions were compared. The inlet pressure range was selected to cover from 1 bar to 100 bar for the comparison. The temperature range was from room temperature (25 \({}^{\circ}\)C) to 70 \({}^{\circ}\)C. The rotational speed range was from 10,000 rpm to 30,000 rpm. The eccentricity ratio, \(\varepsilon\), is the ratio between the eccentricity, \(e\), and the averaged gap distance, c = \(R_{2}-R_{1}\). The range for \(\varepsilon\) was between 0.01 and 0.1. From the lubrication model, the pressure distributions for several conditions were obtained. Figure 6 shows part of the calculation results. Figure 6 shows that larger differences in peak and valley values occur for pressure distribution when the eccentricity ratio is higher. This is because the skewed shaft forms larger density and physical wedges, contributing to larger pressure differences with respect to the angle [18]. From the obtained pressure distribution for various pressure and temperature conditions, forces were calculated. \(F_{r}\) and \(F_{t}\) of CO\({}_{2}\) and air for 30,000 rpm are shown in Figures 7 and 8, respectively. Figure 5: Flow chart of the lubrication model. Figure 6: Pressure distribution around a rotating shaft: (**a**) 7.5 MPa, 30 \({}^{\circ}\)C CO\({}_{2}\) inlet and 30,000 rpm conditions with various eccentricity ratio. (**b**) description of fluid force exerted on the shaft caused by pressure distribution  
The calculated \(F_{r}\) and \(F_{t}\) values under various conditions for CO2 show that operating conditions can potentially affect the stability due to the magnitude and directional change of forces under different fluid conditions. From Figure 7, it is shown that \(F_{r}\) acts in the opposite direction of the shaft position, and the \(F_{t}\) consistently acts in the \(\overset{\rightarrow}{\omega}\times\overset{\rightarrow}{e}\) direction; thus, the attitude angle which is the shaft position when the net force is in the opposite direction of gravity will have a value between \(1.5\pi\) and \(2\pi\). From the results, it is concluded that \(F_{r}\) in the S-CO2 condition has a peak value at the pseudo-critical line, so \(F_{r}\) will dramatically change around it. In the case of \(F_{t}\), it seems to be primarily proportional to the density. In contrast, rapid change of \(F_{r}\) and \(F_{t}\) near the pseudo-critical line is not shown for an air-lubricated condition. From the comparison, the reason seems to be a nearly constant and low density of air, which is different from CO2. Further calculations with high-density air (case 1) and S-CO2 conditions (case 2) were performed using the developed lubrication model to analyze the influence of the dramatic change in \(F_{r}\) near the pseudocritical line. The obtained results imply that forces acting on the shaft are influenced either by the fluid's high density or density gradient. To identify the major factor for the result, high-density air conditions under high pressure as well as low-density air under atmospheric conditions were evaluated with the developed model to contrast with CO2 cases. To  
Figure 8: (**a**) \(F_{r}\) and (**b**) \(F_{t}\) of air for various conditions (30,000 rpm, \(\varepsilon=0.05\)). explicitly show the influence of density and density gradient terms on the forces, the right-hand side (RHS) of Equation (3) is further separated, as shown in Equation (13).