# Instability Study of Magnetic Journal Bearing under S-CO\({}_{2}\) Condition
## 2 Magnetic Bearing Analysis for Instability Study  
### Magnetic Bearing Lubrication Model Development  
The AMB in this paper was a hetero-polar 8-pole type. Eight electromagnets were located symmetrically with 400 \(\upmu\)m clearance with respect to the air gap. The electromagnets were placed to exert the magnetic field perpendicularly to the shaft surface, as shown in Figure 3. The fluid flows through a thin film and gaps between electromagnets. Figure 3: Cross section of a radial active-control magnetic bearing (AMB): (**a**) electromagnets in the magnetic bearing [17]; (**b**) possible secondary flow between the electromagnets. To understand the phenomenon which causes unstable shaft levitation, the research on how the working fluid affects the rotating shaft becomes important because this phenomenon seems to depend on the operating conditions and the working fluid. Fluid influences the rotating shaft primarily with pressure distribution around the shaft. It is noted that a real magnetic bearing has a gap between electromagnets, as shown in Figure 3. Due to this gap, there can be a secondary flow occurring within the gap that can affect the pressure distribution of the shaft. However, to simplify the model development process, this gap is not modeled in this paper. In the future, the secondary flow effect can be studied by changing the magnetic bearing configuration. To calculate the lubrication force exerted to the shaft, the pressure distribution around the shaft should be obtained. The method to describe the pressure distribution of the thin film fluid is suggested from many references by using the Reynolds equation [18]. This equation is obtained by substituting the velocity distribution from the Navier-Stokes equation into the continuity equation. By assuming uniform condition in axial direction due to fast rotation, the Reynolds equation with short bearing theory can be used as shown in Equations (1) and (2). It is noted that the adopted model is just the first step to model the magnetic bearing operating under the S-CO\({}_{2}\) conditions, and in the future more sophisticated models will be used to better understand and interpret the data. The turbulent flow in bearings is studied based on the Taylor vortices, which become unstable to non-axisymmetric disturbance and distorted when the Taylor number, _Ta_, increases above its critical value, _Ta\({}_{C}\)_. With the rotor and bearing design, if _Ta_ reaches its critical value before the _Re_ reaches 2000, the transition to vortex flow occurs. However, if the _Re_ exceeds 2000 while _Ta_ is still less than _Ta\({}_{C}\)_, then the transition directly to turbulent flow occurs [19]. The Reynolds number in the bearing was higher than 50,000; therefore, the bearings were considered to be operating under fully turbulent flow. The turbulence effect was considered with the Ng-Pan model as described in Equation (2) [20]. The variables in the equations are fluid density \(\rho\), pressure \(p\), shaft surface's rotating direction velocity \(u\), dynamic viscosity \(\mu\), time t, and clearance \(h\). \[\frac{\partial}{\partial X}\left(\frac{\rho h^{3}}{k_{x}\mu}\frac{\partial p} {\partial X}\right)=\frac{\partial(\rho h)}{\partial t}+\frac{1}{2}\frac{ \partial(\rho hu)}{\partial X}, \tag{1}\]  
\[k_{x}=12+0.0388(Re)^{0.8} \tag{2}\]  
To utilize Reynolds equation, the coordinate system should be determined as shown in Figure 4a. Furthermore, the shaft position and motion with this coordinate system should be prescribed. The shaft motion is assumed to be revolving with the same frequency of rotation. With this assumption, remaining information needed for the shaft position is only the biased length, \(e\). The clearance \(h\) is described with \(R_{2}\), \(e\) and \(\theta\), as shown in Equation (3). \[h=R_{2}-e*cos\theta-\sqrt{R_{1}^{2}-e^{2}sin^{2}\theta}, \tag{3}\]  
The governing equation (Equation (1)) was numerically solved with the finite difference method (FDM). The thin film geometry for the FDM is shown Figure 4b.