Performance of Tilting Pad Journal Bearings with the Same Sommerfeld Number

Abstract

This paper presents the change of non-dimensional characteristics and thermal behavior of different sized tilting pad journal bearings (TPJBs) with the same Sommerfeld number. A three-dimensional (3D) TPJB numerical model is provided considering the thermo-elastic hydro-dynamic (TEHD) lubrication model with pad thermal-elastic deformation. The pivot stiffness is assumed to be the combination of linear and cubic stiffness based on the Hertzian contact theory. The TPJBs in a configuration of load between pad (LBP) with the same Sommerfeld number having seven different sizes are simulated, and their non-dimensional dynamic and static characteristics and thermal behavior are compared. Pad thermal and elastic deformation are both taken into account. If the changes in lubricant viscosity, thermal deformation, and elastic deformation of journal/pads due to viscous shearing are ignored, the bearings with identical Sommerfeld numbers should show the same performance characteristics. However, the heat generation at the bearing clearance during operation (a) induces a decrease in viscosity and heat transfer to journal/pads and (b) results in a thermal deformation. Furthermore, the elastic deformation of the tilting pads and pivots also affects the bearing dynamic performance. For the same Sommerfeld number, the numerical analyses provide how the viscous shearing and elastic deformation lead to a change in bearing performance. For the small bearings with the same Sommerfeld number, the non-dimensional characteristics did not change significantly, where the heat generation was small being compared to the large sized bearing. The largest change in non-dimensional characteristics occurred when the maximum temperature of the oil film increased by 30 °C or more compared to the lubricant supply temperature. The root cause of the change in the non-dimensional characteristics is the viscous shearing in the oil film, and the thermal deformation of the structures surrounding the oil film acts in combination. These results provide insight into the Sommerfeld number, which can be used for the early stage of bearing design.

1. Introduction

In an initial stage of bearing design, Sommerfeld number is a good approximation to size the bearing and anticipate its performance. Identical Sommerfeld numbers for different sizes of bearings indicates the identical nondimensionalized static and dynamic force performance [1]. As Sommerfeld number is simple in nature, the assumptions for deriving Sommerfeld number neglect the effect of change in temperature influencing the decrease in viscosity and heat transfer to adjacent structures (bearing, shaft). The purpose of this paper is to give light to design engineers of the limitation of Sommerfeld number by providing how the tilting pad journal bearing (TPJB) performance differs with the thermal deformation, viscous shearing, elastic deformation of the tilting pads and pivots. The following literature review presents how the TPJB characteristics are affected by the various factors.

In 1960s, Lund [2] introduced a method for estimating the TPJB force coefficients assuming that the journal is subjected to harmonic motion. Dowson et al., 1966 [3] experimentally showed that there is a large temperature variation near the surface of the bearing and maximum temperature occurs at the minimum film thickness. This paper emphasizes the importance of estimating the effective lubricant viscosity in bearing to accurately predict the bearing performance. Khonsari et al. (1996) [4] state that a proper thermo-hydro-dynamic (THD) analysis can give a better solution to bearing analysis. They compare their prediction of temperature variation to experimental results in reference [3] and shown a good correlation. Brockett et al. [5] emphasize the importance of accounting the thermal deformation in the prediction which can be as large as 45% of minimum film thickness and can lead to high temperature increase up to ~20 °C.

Kim and Palazzolo [6] present thermo-elastic-hydro-dynamic (TEHD) model with thermal cavitation by defining thermal boundary conditions in cavitated regions. The paper well summarizes the theory behind their theoretical model: (1) energy equation in the fluid film, (2) flexibility of the pivot, (3) thermal expansion of the shaft, and (4) upwinding technique. They assumed the heat flow from the lubricant to the bearing is negligible. Nonetheless, the prediction of temperature distribution closely matches with reference [7]. Monmousseau and Fillon [8] present nonlinear analysis accounting TEHD of a TPJB. The authors showed the increase of bearing temperature leads to change in dynamic response, in particular at the critical speed. All the listed papers [2,3,4,5,6,7,8] emphasize that, without considering thermo-hydro-dynamic analysis in prediction, the outcome will likely result in unrealistic bearing performance.

Nilsson [9] accounted for the pad flexibility of TBJB to predict the realistic curvature of the pad by using a curved beam model. The flexible pad and pivot lead to change in the bearing film clearance due to the dynamic load on the bearing, which results in altering the bearing performance. He has shown numerically that the bearing damping can be diminished due to pad flexibility. Later, Kirk and Reedy [10] utilized the Hertzian-stress model to predict the contact stiffness of the TPJB.

Later, Wilkes and Childs [11] show experimentally that prediction not accounting for the pad and pivot flexibility at large loaded condition results in a huge deviation (~3×–9×) from the test results. While prediction considering for the pad/pivot flexibility and hot clearance provides a close match with experimental results. Wilkes emphasizes the importance of accounting for hot-bearing clearance in the prediction. Gaines [12] tested three LBP TPJBs with different pad thicknesses to experimentally identify the effect of pad flexibility on bearing performance. For decreasing pad thickness, the TPJB direct damping decreases up to ~20%; however, the reduction of direct stiffness coefficient was minor.

San Andrés and Li [13] have numerically shown that pad flexibility affects the journal eccentricity and bearing force coefficients of TPJBs. In particular, for a thin pad TPJB, the pad flexibility tends to reduce the damping by 34% compared to the thick pad TPJB. San Andrés et al. [14] present the effect of pivot flexibility versus the Sommerfeld number. For an LBP TBJP, the bearing performance is largely affected at the lower Sommerfeld number (high bearing load W).

DeCamilo et al. [15] experimentally and numerically present the effect of the starvation of oil supply in the TPJB. The limited oil flowrate induces subsynchronous vibration (SSV) of the test rig with TPJB. Apparently, the oil supply pressure affects the bearing dynamic performance. Reference [16] suspects that the oil starvation at the TPJB induces nonlinearity due to active and deactivate of the bearing when starvation develops.

San Andrés and Koo [17] present a starved flow model to predict the lack of lubrication supply in TPJB which induces the onset of sub-synchronous vibration. The oil starvation at the TPJB leads to a swift change in bearing performance, in particular, reducing both the natural frequencies and damping ratios. The author also notes that the frequency reduced KCM model cannot capture the pads’ modes of vibration, but the full-coefficient model KC model does.

Dimond et al. [18] summarize the differences of KCM for the degree-of-freedom model versus KC full bearing coefficient model. They conclude that the full coefficient KC model tends to capture better TPJB dynamic performance. Also, most TPJB applications operate under laminar flow condition for reducing Reynolds number, i.e., the effect of fluid viscosity dominates the bearing performance over fluid density.

Suh and Palazzolo [19,20,21] present a 3D TEHD model to better predict the performance and thermal distribution of the TPJB. The 3D Finite Element Method (FEM) implements the heat transfer/thermal deformation in the shaft and bearing pad and bearing pad dynamic behavior. The papers present a full-scale temperature variation of the tilting pad bearing.

Throughout the research and development of TPJB [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21], the thermo-elastic hydro-dynamic (TEHD) model has shown a better correlation to test data and it is now widely accepted for predicting the performance of the bearing in the industry. The current work presents how the TPJBs’ static/dynamic performances are affected by the elastic and thermal deformations for seven different sizes of TPJBs, but with identical Sommerfeld number. The presented numerical analysis clearly provides insight into the limitation of Sommerfeld number, in particular for large bearing sizes. The present paper implements the 3D TEHD model presented in references [19,20,21].

2. Numerical Modeling of Tilting Pad Journal Bearing

This section presents the theory behind TPJB numerical model. The presented analysis follows the work in references [19,20,21].

2.1. Generalized Reynolds Equation

Equation (1) describes the generalized Reynolds equation that is widely utilized in bearing force performance predictions. Equation (1) is derived from the momentum equation (Newton’s second law) and the mass conservation equation (matter is indestructible). The Reynolds equation governs the fluid characteristic of the bearing film land in between the two moving surfaces.

$$\mathbf{\nabla }·\left(D_1\mathbf{\nabla }p\right)+\left(\mathbf{\nabla }{D}_2\right)·\left({\mathit{U}}_2-{\mathit{U}}_1\right)+\left(\mathbf{\nabla }h\right)·U_1+\frac{\partial h}{\partial t}=0$$

(1)

From Equation (1), assuming that the bearing is fixed and the squeeze effect is negligible results in a more generic form as in Equation (2).

$$\mathbf{\nabla }·\left(D_1\mathbf{\nabla }p\right)+\left(\mathbf{\nabla }{D}_2\right)·\mathit{U}+\frac{\partial h}{\partial t}=0$$

(2)

where D₁ and D₂ are defined as

$$D_1=\underset{0}{\overset{h}{{\displaystyle \int }}}\underset{0}{\overset{z}{{\displaystyle \int }}}\frac{\xi }{\mu }d\xi dz-\frac{{\int }_0^h\frac{\xi }{\mu }d\xi }{{\int }_0^h\frac{1}{\mu }d\xi }\underset{0}{\overset{h}{{\displaystyle \int }}}\underset{ 0}{\overset{z}{{\displaystyle \int }}}\frac{1}{\mu }d\xi dz$$

(3)

$$D_2=\frac{{\int }_0^h{\int }_0^z\frac{1}{\mu }d\xi dz }{{\int }_0^h\frac{1}{\mu } d\xi }$$

(4)

The relationship between the viscosity and the temperature is defined as

$$\mu ={\mu }_0{e}^{-\beta \left(T-T_0\right)}$$

(5)

where Equation (5) assumes that the viscosity is a function of temperature.

2.2. Energy Equation

The energy equation governs the fluid film temperature by accounting for fluid film pressure distribution, velocity, and viscosity as shown in Equation (6).

$$\rho c\left(u\frac{\partial T}{\partial x}+w\frac{\partial T}{\partial z}\right)=k\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\mu \left[{\left(\frac{\partial u}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2\right]$$

(6)

The heat conduction is considered along the x, y, and z directions, while the velocity profiles ($u$, $w$), due to the convection, have x and z components. The viscosity term is a function of $u$ and $w$ across the bearing film y. The finite model solves the energy equation utilizing the up-winding scheme and adopts a modified mixing temperature theory [6,21].

To solve the energy equation, the thermal boundary conditions need to be defined. Equations (7) and (8) show the thermal flux and temperature conditions at the bearing pad and lubricant.

$$k_L{\frac{\partial T_L}{\partial r}|}_{\left(r=R+H\right)}=k_B{\frac{\partial T_B}{\partial r}|}_{\left(r=R+H\right)}$$

(7)

$${T_L|}_{\left(r=R+H\right)}={T_B|}_{\left(r=R+H\right)}$$

(8)

The boundary condition between the rotating shaft and lubricant is defined by orbit averaged thermal boundary condition [16].

$$k_J{\frac{\partial T_J}{\partial r}|}_{\left(\theta =0,r=R\right)}=k_L{\frac{\partial T_L}{\partial r}|}_{\left(\theta =\omega t,r=R\right)}$$

(9)

$${T_J|}_{\left(\theta =0,r=R\right)}={T_L|}_{\left(\theta =\omega t,r=R\right)}$$

(10)

The inlet temperature ($T_{in}$) to the bearing pad largely affects the performance of the TPJB because the fluid viscosity is a function of temperature which affects the pressure generation of the bearing. Hence, it is important to account for any possible change in the inlet temperature of the bearing. Figure 1 shows how the oil is transferred from one pad to another. In most cases, not all the lubricant exiting from the previous bearing pad goes into the next pad, i.e., $Q_{in}<Q_{out}$. The mixing coefficient ($\eta$) is used to account for the change in temperature due to the occurrence of mixing in between the pads. It is important to emphasize that the mixing coefficient ($\eta$) is assumed to be 0.8 in the analysis.

$$T_{in}\left(z\right)=\{\begin{array}{c}\frac{Q_{out}^{i-1}\left(z\right)·T_{out}^{i-1}\left(z\right)+\left(Q_{in}^i\left(z\right)-Q_{out}^{i-1}\left(z\right)\right)·T_{supply}}{Q_{in}^i\left(z\right)} \mathrm{i}\mathrm{f} \eta ·Q_{in}^i>Q_{out}^{i-1} \\ \frac{\eta ·Q_{in}^i\left(z\right)·T_{out}^{i-1}\left(z\right)+\left(Q_{in}^i\left(z\right)-\eta ·Q_{in}^i\left(z\right)\right)·T_{supply}}{Q_{in}^i\left(z\right)} \mathrm{i}\mathrm{f} \eta ·Q_{in}^i\le Q_{out}^{i-1}\end{array}$$

(11)

2.3. Flexible Pad Model and Nonlinear Pivot Model

References [11,12,13] show that accounting for flexible pad/pivot in the bearing model provides more closely matching results compared to the experiments. For example, the flexible pivot tends to reduce the rotor critical speed while the flexible pad tends to reduce damping. Hence, the current study includes the flexible pad/pivot for the bearing predictions and compare them to TPJB with rigid pad assumption. References [19,20,21] describe the flexible pad model, but here is the summary of the model description.

Flexible Pad Model: To solve the flexible pad model, it is important to reduce the number of degree-of-freedom of a pad to lower the computational expense. In the current study, the modal reduction technique is adopted to produce the pad elastic deformation. The pad’s translation and tilting motions are expressed in terms of the rigid body mode of the pad [19].

Nonlinear pivot model: The Hertzian contact theory is used to estimate the nonlinear pivot forces for transient analysis. Reference [22] presents the pivot contact area versus the pivot stiffness based on the fact that the pivot has an oval shape. The current pivot model disregards the resistance to tilting motion and the pad translation motion along the circumferential direction. The nonlinear pivot model is well described in reference [19].

2.4. Bearing-Journal Model

Figure 2 shows the schematic of the workflow of TEHD lubrication model. The fluid film thickness (h) is fed into the Reynolds equation to calculate the fluid film force (Fn) and fluid velocity (u). Then, the fluid velocity (u) is used as an input for the Energy equation to estimate the lubricant temperature and viscosity which are fed back to the Reynolds equation. Hence, it is important to first define the fluid film thickness of the TPJB. The fluid film thickness is defined in Equation (12). The thermal expansions of the journal and bearing pad are considered as $h_{TEJ}$ and $h_{TEP}$, respectively. Figure 3 shows a schematic of the flexible pad film thickness model. The bearing film thickness ($h_n$) is estimated with respect to the nodal point of the tilting pad surface ($p_n$).

$$h\left(\theta ,z\right)=C{L}_P-\left(e_x+z{\theta }_x-p_{pvt}cos\left({\theta }_p\right)\right)cos\left(\theta \right)-\left(e_y+z{\theta }_y-p_{pvt}sin\left({\theta }_p\right) \right)sin\left(\theta \right)-\left(C{L}_P-C{L}_B\right)cos\left(\theta -{\theta }_p\right)-{\delta }_{tilt}{R}_{s}sin\left(\theta -{\theta }_p\right)-h_{TEJ}\left(\theta ,z\right)-h_{TEP}\left(\theta ,z\right)$$

(12)

The fluid film thickness account for pad thermal/elastic deformation and pivot deformation at each time step during the numerical analysis. At each time step, the nodal film thickness ($h_n$) is estimated at n-th nodal point on the pad ($p_n$) as

$$h_n=\sqrt{{\left(x_n-x_s-z{\theta }_x\right)}^2+{\left(y_n-y_s-z{\theta }_y\right)}^2}-R_s-h_{TEJ}\left(\theta ,z\right)-h_{TEP}\left(\theta ,z\right)$$

(13)

The misalignment of the bearing is not considered in the current analysis. See reference [23] for the effect of misalignment in the TPJB performance.

3. Simulation Results and Discussions

3.1. 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.

$$S=\frac{\mu NLD}{W}{\left(\frac{R}{C_p}\right)}^2:\mathrm{Sommerfeld}\mathrm{number}$$

(14)

$$\epsilon =\frac{e}{C_b}:\mathrm{Eccentricity}\mathrm{ratio}$$

(15)

$$K_{ij}=\frac{C_p{k}_{ij}}{W}:\mathrm{Nondimensional}\mathrm{stiffness}\mathrm{coefficient}$$

(16)

$$C_{ij}=\frac{C_p\omega c_{ij}}{W}:\mathrm{Nondimensional}\mathrm{damping}\mathrm{coefficient}$$

(17)

Table 1. Lubricant properties.

Lubricant Properties	Value
Viscosity at 40 °C (N·s/m2)	0.0365
Viscosity coefficient (Pa·s)	0.0297
Heat capacity (J/kg °C)	1886
Heat conductivity (W/(mK))	0.136
Lubricant supply temperature (°C)	50
Density (kg/m3)	877

lists lubricant properties, and

Table 2. Bearing system parameters.

Bearing
Load type	LBP
Number of pads	5
Pad arc length (°)	60
Offset	0.5
Preload	0.4
Load direction (deg.)	270 (−y)
Journal Material
Young’s Modulus (Pa)	2.05 × 1011
Density (kg/m3)	7850
Poison’s Ratio	0.3
Heat Capacity (J/(kg °C))	453.6
Heat Conductivity (W/(m °C))	42.6
Therm. Exp. Coeff (1/ °C)	0.0000122
Ref. Temp. for T.exp (°C)	25
Pad Material
Young’s Modulus (Pa)	2.00 × 1011
Density (kg/m3)	7858
Poison’s Ratio	0.3
Heat Capacity (J/(kg °C)	453.6
Heat Conductivity (W/(m °C))	51.9
Therm. Exp. Coeff (1/°C)	0.0000121
Ref. Temp. for Texp (°C)	25
Babbit Material
Young’s Modulus (Pa)	5.3 × 1010
Density (kg/m3)	7390
Poison’s Ratio	0.3
Heat Capacity (J/(kg °C)	230
Heat Conductivity (W/(m °C))	55
Therm. Exp. Coeff (1/°C)	0.000021
Ref. Temp. for Texp (°C)	25
Housing Material
Young’s Modulus (Pa)	1.86 × 1011
Poison’s Ratio	0.3

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. Bearing configurations with the same Sommerfeld number.

Scale No.	Journal Diameter (mm)	Pad Thickness (mm)	Babbit Thickness (mm)	Dp (mm)	Dh (mm)	Pad Clearance (mm)	Pad Length (mm)	Bearing Load (N)
1	35	5	0.5	38	45	0.032	25	1230
2	70	10	1	76	90	0.064	50	4920
3	105	15	1.5	114	135	0.096	75	11,070
4	140	20	2	152	180	0.128	100	19,680
5	175	25	2.5	190	225	0.16	125	30,750
6	210	30	3	228	270	0.192	150	44,280
7	245	35	3.5	266	315	0.224	175	60,270

lists detailed geometry, size, and bearing load of the small to large TPJBs (35 mm → 245 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 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].

Table 4. Running conditions.

Running Condition	1	2	3
Rotor spin speed (rpm)	500	1000	3600
Sommerfeld Number	0.065	0.129	0.466
Unit Load	1.41	1.41	1.41

lists three sets of running conditions with increasing rotating speed (500 rpm, 1000 rpm, 3600 rpm), but for fixed unit load; hence, the Sommerfeld number increases with rotating speed. The current analysis assumes laminar flow condition to reduce the variables in the analysis and thus focusing on the geometrical difference of bearing affecting the bearing performance.

Figure 4a shows the cross-section view of the 3D TPJB along the axial direction used for FE analysis. The FE analysis considers the thermal variation along the axial/circumferential directions. The estimation of the film thickness is a critical step to solve the Reynolds equation (see Figure 2). There are four elements radially and 18 elements circumferentially for each pad.

Figure 4b depicts the five-pad (LBP) TPJB and its X and Y coordinates. The rotor spins counterclockwise and the bearing load is applied at 270° away from X-axis.

Figure 4c shows the pivot diameter ($D_p$) and housing diameter ($D_h$) which are considered for the evaluation of the pivot stiffness and the resultant elastic pivot deformation. The pivot stiffness is calculated based on the Hertzian contact theory. Figure 4d shows the side view of the pivot contact area. In this research, the pivot axial length ($L_p$) is identical to the pad axial length.

3.2. Bearing Static and Dynamic Characteristics

This section presents the static and dynamic characteristic of TPJBs for Scale No. 1 (ϕ = 35 mm) through No.7 (ϕ = 245 mm) with three sets of Sommerfeld numbers: S = 0.065 (500 rpm), 0.129 (1000 rpm) and 0.466 (3600 rpm).

Figure 5 presents (a) eccentricity ratio, (b) attitude angle, (c) tilt angle of Pad No.5. and (d) non-dimensional pivot deformation ($p_{pvt}/Cb$) for both the rigid pad TPJB and the flexible pad TPJB. Tilting angles for the flexible pad model is not provided for the difficulties measuring the tilt angle of a flexible FE pad model. It is important to emphasize that at low Sommerfeld number S = 0.065, the difference of static characteristic between small (Scale No. 1, ϕ = 35 mm) to large (Scale No. 7, ϕ = 245 mm) TPJBs is minute. However, for high Sommerfeld number S = 0.466, the difference of static performance of TPJB is noticeable, in particular, from Scale No. 1 (ϕ = 35 mm) to Scale No. 4 (ϕ = 140 mm). The reason behind this change can be explained by Figure 5c,d, i.e., the pivot deforms more at higher Sommerfeld number. The nondimensional pivot deformation is defined as a change in pivot deformation in radial direction divide by the bearing clearance ($C_b$). Also, there are other thermal characteristics that affect the bearing performance which will be discussed in the later section. Both the rigid pad TPJB and the flexible pad TPJB show similar static characteristic trends. The current numerical cases present LBP (load-between-pad) condition; the name indicates that two bearing pads support the vertical bearing load. The LBP bearing configuration generally has less pivot deformation than the LOP (load-on-pad) configuration.

Figure 6 presents the dynamic performance of TPJB in terms of nondimensional stiffness and damping coefficients that are explained in Equations (16) and (17), respectively. Similar to static characteristics of TPJB, the dynamic performance of TPJB shows large variation from Scale No. 1 (ϕ = 35 mm) through Scale No. 7 (ϕ = 245 mm) for high Sommerfeld number S = 0.466. This is due to pivot deformation, change in viscosity, and thermal deformation (see later in Figure 7, Figure 8 and Figure 9). Interestingly enough, the numerical analysis for TPJB with a flexible pad tends to show a lower damping coefficient than the TPJB with a rigid pad assumption. This analysis again confirms the past findings [10,11] that assuming the pad is rigid tends to over-predict the damping coefficients compared to test data. It is obvious that, in reality, the pad has a certain degree of flexibility. The pad flexibility affects less on TPJB stiffness coefficients.

3.3. Bearing Thermal Behavior

The Sommerfeld number is a convenient method to quantify the static and dynamic performance of bearing with various geometry, size, and operating condition. However, a great deal of discretion is needed to utilize the Sommerfeld number for TPJB because TPJB has a flexible pad and undergoes pivot deformation and thermal deformation at large load conditions. These factors make it hard to pinpoint the bearing performance covering various geometry using the Sommerfeld number.

When designing turbomachinery, an engineer needs to fully account for thermal effect in their machine. This practice helps to improve long-term reliability and prevent unexpected failures. Hence, this section presents the thermal characteristic of TPJB for three sets of different Sommerfeld numbers—S = 0.065 (500 rpm), 0.129 (1000 rpm), and 0.466 (3600 rpm).

Figure 7 presents the thermal effect for different Sommerfeld number by showing (a) pad trailing edge (outlet) temperature, (b) peak lubricant temperature, (c) peak bearing temperature, and (d) peak journal temperature. As expected, for low Sommerfeld number S = 0.065 (500 rpm), the variation of the temperature for small (Scale No. 1, ϕ = 35 mm) to large (Scale No. 2, ϕ = 245 mm) bearing sizes is minute. But, for high Sommerfeld number S = 0.466 (3600 rpm), the peak temperature increases with the increasing size of the bearing. There is ~40 °C increase in temperature from Scale No. 1 to Scale No. 7. This is most likely due to the large bearing load for No. 7 bearing, but not necessarily large enough pad thickness and Babbitt thickness. In other words, the thermal energy generated by the large bearing load is larger than the thermal reservoir of the journal, the pad, and Babbitt. Hence, the thermal expansion can be more significant than the smaller bearing No. 1 which results in a large increase in temperature. In practice, to reduce the peak temperature at the bearing, in particular for larger size, an engineer could (1) introduce bleed holes in the bearing sidewall to relief the hot lubricant to the sump and/or (2) increase the lubricant flow rate.

Figure 8 shows the thermal gradient of TPJB for small (Scale No. 1, ϕ = 35 mm) to large (Scale No. 2, ϕ = 245 mm) bearing sizes. It shows the temperature distribution at the bearing pad, film thickness, and the shaft. For all cases, the loaded bearing pads (#4, #5) have the highest temperature. TPJBs with higher speed 3600 rpm, tends to have larger film thickness than the lower speed 500 rpm cases. This is due to larger bearing stiffness generation at higher operating speed.

Figure 9 presents the minimum film thickness and journal thermal expansion versus three sets of Sommerfeld number S = 0.065, 0.129 and 0.466. It is again clearly showing that for large Sommerfeld number S = 0.466, the variations of film thickness and thermal expansion of journal are significant. Figure 10 shows the film thickness distribution of five-pads (LBP) TPJB (Scale No. 2, ϕ = 245 mm) for S = 0.065 (500 rpm), and 0.466 (3600 rpm), respectively. The visual graphic shows the rotor spinning speed largely affects the film thickness distribution.

As shown in Figure 7, Figure 8, Figure 9 and Figure 10, these thermal distortions affect the static/dynamic performance of TPJB which lead to the dissimilarity of bearing performance from bearing-to-bearing for constant Sommerfeld number.

4. Conclusions

This study presents the thermo-elastic-hydro-dynamic effect on TPJB performance by comparing seven different sizes of bearings with the same Sommerfeld numbers. The 3D numerical model with TEHD lubrication includes the pad thermal-elastic deformation and pivot stiffness. The numerical results vividly show that the static and dynamic bearing characteristics vary with bearing sizes albeit the Sommerfeld number is identical. This is because the thermal condition affects the bearing geometry and viscosity, which eventually affects the bearing performance. The change of bearing performance is pronounced at higher operating speeds and for larger bearings. This is due to the thermal energy that is generated at high speed and high bearing load conditions.

If the heat source supplied from outside the bearing system is ignored (except for the lubricant), the oil film viscous shear heat changes the lubricant viscosity and is simultaneously transferred to the structure around the oil film. The rotating shaft and bearing pads cause thermal deformation, which changes the oil film thickness. The changed bearing clearance changes the viscous shear heat again to change the heat transfer conditions to the surrounding structures. These changes interact and generally converge under constant conditions.

Hence, it is important to thoroughly forecast the bearing performance accounting for lubricant supply conditions when designing a relatively large bearing with high-speed operation. At higher operating speeds, and possibly for larger bearings, bearing performance and health can be more sensitive to lubrication supply pressure and temperature.