# Effect of Thermal Boundary Condition on Tilting Pad Journal Bearing Behavior
## 2 Numerical Methods
### Three-Dimensional Energy Equation  
\[\rho c\left(u\frac{\partial T}{\partial x}+w\frac{\partial T}{\partial z} \right)=k\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{ \partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right)+\mu\left[\left( \frac{\partial u}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial y} \right)^{2}\right] \tag{6}\]  
In the above Equation (6) \(x\), \(y\), and \(z\) are circumferential, radial, and axial direction, respectively. Heat convection is considered in \(x\) and \(z\) directions, where \(u\) and \(w\) are lubricant velocity profiles which are calculated by the Reynolds Equation (2), whereas the heat conduction is taken into account along all directions. Since the purpose of this study is to analyze the steady state of the bearing, the temperature change over time was not considered. The FE model is utilized to solve the energy equation using the up-winding scheme to prevent numerical error arising from the convection terms. Thermal boundary conditions should be defined to solve the energy equation. Equations (7) and (8) are heat flux and temperature conditions at the interface between the fluid film and bearing pads, respectively (boundary B, Figure 1a). \(r\) represents the bearing radial direction. \(R\) is a journal radius, and \(H\) indicates the film thickness.  
\[k_{L}\left.\frac{\partial T_{L}}{\partial r}\right|_{(r=R+H)}=k_{B}\left. \frac{\partial T_{B}}{\partial r}\right|_{(r=R+H)} \tag{7}\]  
\[T_{L}|_{(r=R+H)}=T_{B}|_{(r=R+H)} \tag{8}\]  
Equations (9) and (10) are heat flux and temperature conditions at the interface between the spinning journal and lubricant, respectively (boundary A, Figure 1a). \(\theta\) represents the bearing circumferential direction. Unlike the thermal boundary condition between the oil film and the bearing pad, the boundary condition between the journal and the oil film is complicated due to the spinning journal. In this study, orbit averaged heat flux boundary condition suggested by [PERSON] and [PERSON] [38] is adopted. Their research was not about the bearing performance, but only to calculate the surface temperature of a spinning journal. On the other hand, temperature change of the journal, the resulting thermal expansion, and the corresponding change in the oil film thickness were implemented in MATLAB-based in-house code.  
\[k_{I}\left.\frac{\partial T_{J}}{\partial r}\right|_{(\theta=0,r=R)}=k_{L}\left. \frac{\partial T_{L}}{\partial r}\right|_{(\theta=out,r=R)} \tag{9}\]  
\[T_{J}\big{|}_{(\theta=0,r=R)}=T_{L}\big{|}_{(\theta=out,r=R)} \tag{10}\]  
The temperature at the inlet end of the bearing pad (\(T_{in}\)) greatly affects bearing performance, where \(i\) denotes the pad number. This is because the temperature of the lubricant changes the viscosity and the change in viscosity affects the bearing oil film pressure. Therefore, it is very important for bearing performance analysis to consider the temperature change of the bearing inlet. Figure 2 explains whether the lubricant from the pad and the lubricant supplied from the outside mix enter the next pad. In most cases, not all lubricants from previous bearing pads enter the next pad. A portion of the hot lubricant flowing out of the bearing pads slips out between the pads, and some is mixed with cold oil supplied from the outside to enter the next pad. In this study, the mixing temperature theory is used to account for the analysis of the inlet oil temperature as described in Equation (11). Mixing coefficient (\(\eta\)) limits the amount of oil flowing from the previous pad to the next pad. It is impossible to obtain the mixing coefficient experimentally or numerically, and 0.8, which is used in most previous studies, was also adopted in this study.  
\[T_{in}^{i}(z)=\frac{\eta\cdot Q_{in}^{i}(z)\cdot T_{out}^{i-1}(z)+\left(Q_{in} ^{i}(z)-\eta\cdot Q_{in}^{i}(z)\right)T_{supply}}{Q_{in}^{i}(z)} \tag{11}\]