# Performance of Tilting Pad Journal Bearings with the Same Sommerfeld Number
## 3 Simulation Results and Discussions  
### Simulation Model  
In this paper, the TEHD analyses of seven different size of TPJBs but with identical Sommerfeld number is performed. Equation (14) defines the Sommerfeld number that determines the static and dynamic characteristic of TPJBs. Equations (15) through (17) are nondimensional eccentricity ratio, stiffness, and damping coefficients. It is important to note the limitation of the Sommerfeld number. The Sommerfeld is a good estimation of bearing characteristics, but it neglects the effect of change in viscosity and shaft/bearing deformation. \[S=\frac{\mu NLD}{W}\bigg{(}\frac{R}{C_{p}}\bigg{)}^{2}:\text{ Sommerfeld number} \tag{14}\]  
\[\varepsilon=\frac{e}{C_{b}}:\text{ Eccentricity ratio} \tag{15}\]  
\[K_{ij}=\frac{C_{p}k_{ij}}{W}:\text{ Nondimensional stiffness coefficient} \tag{16}\]  
\[C_{ij}=\frac{C_{p}\alpha c_{ij}}{W}:\text{ Nondimensional damping coefficient} \tag{17}\]  
Table 1 lists lubricant properties, and Table 2 lists the material properties and geometry of the TPJB presented in the analysis. The current study delves into the characteristic of TPJB with identical Sommerfeld number but with different sizes of bearings. Table 3 lists detailed geometry, size, and bearing load of the small to large TPJBs (\(35\,\text{mm}\to 245\,\text{mm}\)) with the same Sommerfeld number that is numerically investigated. As shown in Equation (14), to maintain the identical Sommerfeld number for different sizes of the bearings when the rotating speed is fixed, the bearing load requires  
Figure 3: Schematic of flexible tilting pad model with nodal film thickness. to proportionally follow the change of the bearing sizes. Scale No. 1 with bearing diameter 35 mm through Scale No.7 with bearing diameter 245 mm have identical Sommerfeld number. The 245 mm bearing is denoted as a \"large\" bearing for labeling purposes only. There are even larger bearings as large as 900 mm in diameter for large turbine applications [24]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Lubricant Properties** & **Value** \\ \hline Viscosity at 40 \({}^{\circ}\)C (N-s/m\({}^{2}\)) & 0.0365 \\ Viscosity coefficient (Pa-s) & 0.0297 \\ Heat capacity (J/kg \({}^{\circ}\)C) & 1886 \\ Heat conductivity (W/(mK)) & 0.136 \\ Lubricant supply temperature (\({}^{\circ}\)C) & 50 \\ Density (kg/m\({}^{3}\)) & 877 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lubricant properties. \begin{table}
\begin{tabular}{c c} \hline \hline
**Bearing** & \\ \hline Load type & LBP \\ Number of pads & 5 \\ Pad arc length (\({}^{\circ}\)) & 60 \\ Offset & 0.5 \\ Preload & 0.4 \\ Load direction (deg.) & 270 (\(-\)y) \\ \hline \hline \multicolumn{2}{c}{**Journal Material**} \\ \hline Young’s Modulus (Pa) & \(2.05\times 10^{11}\) \\ Density (kg/m\({}^{3}\)) & 7850 \\ Poison’s Ratio & 0.3 \\ Heat Capacity (J/(kg \({}^{\circ}\)C)) & 453.6 \\ Heat Conductivity (W/(m \({}^{\circ}\)C)) & 42.6 \\ Therm. Exp. Coeff (1/ \({}^{\circ}\)C) & 0.0000122 \\ Ref. Temp. for T.exp (\({}^{\circ}\)C) & 25 \\ \hline \hline \multicolumn{2}{c}{**Pad Material**} \\ \hline Young’s Modulus (Pa) & \(2.00\times 10^{11}\) \\ Density (kg/m\({}^{3}\)) & 7858 \\ Poison’s Ratio & 0.3 \\ Heat Capacity (J/(kg \({}^{\circ}\)C)) & 453.6 \\ Heat Conductivity (W/(m \({}^{\circ}\)C)) & 51.9 \\ Therm. Exp. Coeff (1/\({}^{\circ}\)C) & 0.0000121 \\ Ref. Temp. for Texp (\({}^{\circ}\)C) & 25 \\ \hline \hline \multicolumn{2}{c}{**Babbit Material**} \\ \hline Young’s Modulus (Pa) & \(5.3\times 10^{10}\) \\ Density (kg/m\({}^{3}\)) & 7390 \\ Poison’s Ratio & 0.3 \\ Heat Capacity (J/(kg \({}^{\circ}\)C) & 230 \\ Heat Conductivity (W/(m \({}^{\circ}\)C)) & 55 \\ Therm. Exp. Coeff (1/\({}^{\circ}\)C) & 0.000021 \\ Ref. Temp. for Texp (\({}^{\circ}\)C) & 25 \\ \hline \hline \multicolumn{2}{c}{**Housing Material**} \\ \hline Young’s Modulus (Pa) & \(1.86\times 10^{11}\) \\ Poison’s Ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bearing system parameters.