# Effect of Thermal Boundary Condition on Tilting Pad Journal Bearing Behavior
## 2 Numerical Methods
### Modeling for Heat Transfer and Thermal Deformation of Journal-Bearing System  
The heat generated by the viscous shear in the oil film is transferred to the bearing pads and journals bordering the oil film. The transferred energy heats the bearing and shaft, and the resulting temperature change causes thermal deformation of the structure. In this study, a 3D FE model of bearing and shaft structure was created using in-house code, and heat transfer and thermal deformation were analyzed.  
Thermal deformation of the shaft and bearing pad changes the oil film thickness. The changed oil film thickness changes the pressure, thereby changing the viscous shear and changing the temperature boundary conditions of the oil film. Such heat generation, heat transfer, and thermal deformation influence each other and reach a state of dynamic and thermal convergence.  
Figure 2: Schematic diagram of oil inlet groove and boundary conditions at pad edges.  
Figure 3 depicts the schematic of the workflow of the THD lubrication model presented in this study. Firstly, the film thickness profile of the TPJB is defined by the user input initial states which includes journal position, initial temperature guess of bearing structure, and the resulting thermal deformation. The film thickness profile is depicted in Equation (12), where each notation is described in Figure 4 schematically. The amount of fluid film thickness reduction by the thermal expansions of the journal and bearing pad are considered as \(h_{TEJ}\) and \(h_{TEP}\), respectively. Fluid film thickness (\(h\)) and its time derivative (\(\dot{h}\)) is used in the Reynolds equation for the evaluation of the fluid force (\(F_{n}\)) and velocity profile (\(u,\ w\)). The fluid velocity (\(u,\ w\)) is fed into the energy equation to produce the fluid film temperature and viscosity which are fed back into the generalized Reynolds equation. There, three closed loops in the bearing analysis algorithm as shown in Figure 3. The first one is in the dynamic model, where the fluid force calculated by the Reynolds equation changes the dynamic behavior, and the changed dynamic behavior affects the film thickness again. The second one is in the coupled model of the Reynolds equation and the energy equation. The lubricant temperature updated by the energy equation affects the Reynolds equation, where the changed viscosity affects the pressure distribution. The third closed loop is in the heat flux and thermal deformation model. The lubricant temperature changes the thermal boundary condition among the spinning journal, lubricant, and bearing pads. The changed boundary condition influences the thermal deformation and the resultant lubricant film thickness. The updated film thickness affects the lubricant temperature again. The main three closed loops are implemented in the in-house code. The dynamic model adopted in this study is based on the rigid body dynamic model where the pivot elastic deformation is considered based on the Hertzian contact theory.  
\[\begin{array}{ll}h(\theta)=CL_{P}&-\Big{(}e_{x}-p_{pvt}cos\big{(}\theta_{p} \big{)}\Big{)}cos(\theta)-\Big{(}e_{y}-p_{pvt}sin\Big{(}\theta_{p}\Big{)} \Big{)}sin(\theta)\\ &-(CL_{P}-CL_{B})cos\Big{(}\theta-\theta_{p}\Big{)}-\delta_{tilt}R_{s}sin \big{(}\theta-\theta_{p}\big{)}-h_{TEJ}(\theta,z)\\ &-h_{TEP}(\theta,z)\end{array} \tag{12}\]  
Figure 3: Thermo-hydrodynamic lubrication model.