Thermoelastic Effect in a Large Lubricated Thrust Bearing

1. Introduction

Large hydrodynamic lubricated thrust bearings are widely applied in large mechanical equipment for supporting large axial loads and reducing friction and wear[1-9]. Good lubrication is critical for maintaining the performance of these bearings. However, in practice these bearings face the risk of lubricant film breakdown and pad seizure[10-12]. Such phenomena are difficult to explain using conventional hydrodynamic lubrication theory[13], which predicts that there are much thicker lubricate films in these bearings. Furthermore, the effects of fluid non-Newtonian shear thinning, surface roughness and lubricant viscosity reduction due to film viscous heating are not responsible for the film breakdown[14-16]. This behavior has been attributed to thermoelastic deformation of the thermoelastic deformation of the bearing[10-12].

Before pad seizure, there is a lubrication stage in the bearing during which the fluid film thickness is very low, so that the influence of the fluid molecule layer physically adsorbed onto the bearing surface should be considered. To date, few studies have investigated bearing performance at this stage, except for the work of Ye and Zhang, who presented the results for the step bearing[17].

This paper investigates this previously unexplored lubrication stage before film breakdown in a large-sized inclined fixed pad thrust bearing by considering both surface thermoelastic deformation and the physically adsorbed molecule layer. unlike the study of Ye and Zhang[17], the coupled shaft and bush surfaces of the bearing are made of steel and bronze, respectively. Different fluid-surface interactions are considered, and calculation results are presented for various parameter values. These results provide a new understanding of the lubrication behavior of this large thrust bearing.

2. Large hydrodynamic lubricated inclined FIXED pad thrust bearing with thermal distortion

The large hydrodynamic lubricated inclined fixed pad thrust bearing investigated in this study is shown in Figure 1. Because of surface thermal distortion, the bearing clearance profile is significantly different from that predicted by conventional theory. Under condition of very small film thicknesses, the ultrathin adsorbed molecule layer becomes significant. Although the lower shaft surface is moving and made of steel and the upper bush surface is stationary and made of bronze, the two adsorbed layers do not differ because of the same coatings applied to both surfaces. Moreover, by using different coatings, the fluid-surface interaction can be altered.

Here, u is the sliding speed, h_tot,i and h_tot,o are the surface separations at the bearing inlet and outlet, respectively; h_bf is the thickness of the adsorbed molecular layer; h is the thickness of the continuum fluid film; h_o is the value of h on the bearing outlet; l is the bearing width; and the coordinate system is also illustrated.

Fig. 1 Large hydrodynamic lubricated inclined fixed-pad thrust bearing under thermal distortion investigated in this study

3. Theoretical analysis and numerical calculation

To perform the numerical calculations for the multiscale flow problem in the bearing, Zhang’s multiscale flow model[18] was used, rather than the classical hybrid schemes[19-21]. The following assumptions were adopted:

At high sliding speeds, the fluid in the bearing may be non-Newtonian, and the interfacial slippage may occur. In addition, for very small film thicknesses, the surface roughness effect may become significant. Assumptions (1)-(3) allow this study to focus on the combined effects of the physically adsorbed layer and surface thermal distortion. The effects of lubricant non-Newtonian shear thinning, surface roughness and lubricant side leakage are not expected to alter the conclusions presented here. A new model would be required to simulatenon-Newtonian fluid behavior and side - flow effects.

Based on the above assumptions, the total mass flow rate per unit contact length through the bearing is given by[17, 18]:

[TeX:] q_{m} = - uh_{\text{bf}}\rho_{\text{bf}}^{\text{eff}} - \frac{\text{uh}}{2}\rho - \frac{h^{3}\rho}{12\eta}\frac{\partial p}{\partial x} +$q_m=-u{h}_{\mathrm{\text{bf}}}{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}-\frac{\mathrm{\text{uh}}}{2}\rho -\frac{h^3\rho }{12\eta }\frac{\partial p}{\partial x}$[TeX:] + \frac{h_{\text{bf}}^{3}\rho_{\text{bf}}^{\text{eff}}}{\eta_{\text{bf}}^{\text{eff}}}\frac{\partial p}{\partial x}\left\lbrack \frac{F_{1}}{6} - \frac{\varepsilon}{1 + \frac{\Delta x}{D}}\left( 1 + \frac{1}{2\lambda_{\text{bf}}} - \frac{q_{o} - q_{o}^{n}}{q_{o}^{n - 1} - q_{o}^{n}}\frac{\Delta_{n - 2}}{h_{\text{bf}}} \right) \right\rbrack +$+\frac{h_{\mathrm{\text{bf}}}^3{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}{{\eta }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}\frac{\partial p}{\partial x}[\frac{F_1}{6}-\frac{\epsilon }{1+\frac{\Delta x}{D}}(1+\frac{1}{2{\lambda }_{\mathrm{\text{bf}}}}-\frac{q_o-q_o^n}{q_o^{n-1}-q_o^n}\frac{{\Delta }_{n-2}}{h_{\mathrm{\text{bf}}}})]$[TeX:] + \frac{h^{3}\rho}{\eta_{\text{bf}}^{\text{eff}}}\frac{\partial p}{\partial x}\left\lbrack \frac{F_{2}\lambda_{\text{bf}}^{2}}{6} - \frac{\lambda_{\text{bf}}}{1 + \frac{\Delta x}{D}}\left( \frac{1}{2} + \lambda_{\text{bf}} - \frac{q_{o} - q_{o}^{n}}{q_{o}^{n - 1} - q_{o}^{n}}\frac{\Delta_{n - 2}}{h} \right) \right\rbrack$+\frac{h^3\rho }{{\eta }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}\frac{\partial p}{\partial x}[\frac{F_2{\lambda }_{\mathrm{\text{bf}}}^2}{6}-\frac{{\lambda }_{\mathrm{\text{bf}}}}{1+\frac{\Delta x}{D}}(\frac{1}{2}+{\lambda }_{\mathrm{\text{bf}}}-\frac{q_o-q_o^n}{q_o^{n-1}-q_o^n}\frac{{\Delta }_{n-2}}{h})]$ (1)

where the definitions of the parameters are given by Ye and Zhang[17].

Defining [TeX:] C_{y} = \eta_{\text{bf}}^{\text{eff}}/\eta$C_y={\eta }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}/\eta$ , according to Eq. (1) the pressure gradient is:

[TeX:] \frac{\text{dp}}{\text{dx}} = \frac{\frac{1}{2}\text{uρ}h + q_{m} + uh_{\text{bf}}\rho_{\text{bf}}^{\text{eff}}}{\frac{\text{cρ}h^{3}}{\eta} + \frac{d\rho_{\text{bf}}^{\text{eff}}h_{\text{bf}}^{3}}{\eta_{\text{bf}}^{\text{eff}}}}$\frac{\mathrm{\text{dp}}}{\mathrm{\text{dx}}}=\frac{\frac{1}{2}\mathrm{\text{uρ}}h+q_m+u{h}_{\mathrm{\text{bf}}}{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}{\frac{\mathrm{\text{cρ}}{h}^3}{\eta }+\frac{d{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}{h}_{\mathrm{\text{bf}}}^3}{{\eta }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}}$ (2)

where:

[TeX:] c = \frac{1}{C_{y}}\left\lbrack \frac{F_{2}\lambda_{\text{bf}}^{2}}{6} - \frac{\lambda_{\text{bf}}}{1 + \frac{\text{Δx}}{D}}\left( \frac{1}{2} + \lambda_{\text{bf}} - \frac{q_{0} - q_{0}^{n}}{q_{0}^{n - 1} - q_{0}^{n}}\frac{\Delta_{n - 2}\lambda_{\text{bf}}}{h_{\text{bf}}} \right) \right\rbrack - \frac{1}{12}$c=\frac{1}{C_y}[\frac{F_2{\lambda }_{\mathrm{\text{bf}}}^2}{6}-\frac{{\lambda }_{\mathrm{\text{bf}}}}{1+\frac{\mathrm{\text{Δx}}}{D}}(\frac{1}{2}+{\lambda }_{\mathrm{\text{bf}}}-\frac{q_0-q_0^n}{q_0^{n-1}-q_0^n}\frac{{\Delta }_{n-2}{\lambda }_{\mathrm{\text{bf}}}}{h_{\mathrm{\text{bf}}}})]-\frac{1}{12}$ (3)

[TeX:] d = \frac{F_{1}}{6} - \frac{\varepsilon}{1 + \frac{\text{Δx}}{D}}\left( 1 + \frac{1}{2\lambda_{\text{bf}}} - \frac{q_{0} - q_{0}^{n}}{q_{0}^{n - 1} - q_{0}^{n}}\frac{\Delta_{n - 2}}{h_{\text{bf}}} \right)$d=\frac{F_1}{6}-\frac{\epsilon }{1+\frac{\mathrm{\text{Δx}}}{D}}(1+\frac{1}{2{\lambda }_{\mathrm{\text{bf}}}}-\frac{q_0-q_0^n}{q_0^{n-1}-q_0^n}\frac{{\Delta }_{n-2}}{h_{\mathrm{\text{bf}}}})$ (4)

The thickness of the adsorbed layer is given by:

[TeX:] h_{\text{bf}} = nD + \Delta_{n - 2}\frac{q_{0} - q_{0}^{n}}{q_{0}^{n - 1} - q_{0}^{n}}$h_{\mathrm{\text{bf}}}=nD+{\Delta }_{n-2}\frac{q_0-q_0^n}{q_0^{n-1}-q_0^n}$ (5)

The thickness of the continuum fluid film is expressed as:

[TeX:] h(x) = h_{00} + f(x) - \frac{2}{\pi E_{v}}\int_{0}^{l}{p(s)\ln(x - s)^{2}}ds + \frac{x^{2}}{2R_{t}} - 2h_{\text{bf}}$h(x)=h_{00}+f(x)-\frac{2}{\pi E_v}{\int }_0^{l}p(s)ln(x-s{)}^{2}ds+\frac{x^2}{2R_t}-2h_{\mathrm{\text{bf}}}$ (6)

where h_oo is constant, p is the film pressure, E_v is the equivalent of Young's modulus of elasticity of the two bearing surfaces, and:

[TeX:] f(x) = x\tan\theta$f(x)=xtan\theta$ (7)

Here, θ is the tilting angle of the original geometrical shape of the bearing, R_t is defined in[22]:

[TeX:] R_{t} = \frac{R_{t,a}R_{t,b}}{R_{t,a} + R_{t,b}}$R_t=\frac{R_{t,a}{R}_{t,b}}{R_{t,a}+R_{t,b}}$ (8)

R_t,a and R_t,b are, respectively[22]:

[TeX:] R_{t,a} = \frac{k_{a}\rho_{a}c_{a}}{u\lambda_{a}\tau_{\text{av}}\alpha_{a}(1 + \nu_{a})(1 - \chi)}$R_{t,a}=\frac{k_a{\rho }_a{c}_a}{u{\lambda }_a{\tau }_{\mathrm{\text{av}}}{\alpha }_a(1+{\nu }_a)(1-\chi )}$ (9)

[TeX:] R_{t,b} = \frac{k_{b}\rho_{b}c_{b}}{u\lambda_{b}\tau_{\text{av}}\alpha_{b}(1 + \nu_{b})(1 - \chi)}$R_{t,b}=\frac{k_b{\rho }_b{c}_b}{u{\lambda }_b{\tau }_{\mathrm{\text{av}}}{\alpha }_b(1+{\nu }_b)(1-\chi )}$ (10)

k denotes the surface thermal diffusivity, c the surface specific heat, ρ the surface density, v the surface Poisson’s ratio, α the surface linear thermal expansion coefficient, λ the frictional heat input rate into the surface, χ the rate of the frictional heating removed by the lubricant flow, the subscripts “a” and “b” denote the stationary and moving surfaces, respectively, and τ_av represents the average shear stresses on each surfaces which is calculated as:

[TeX:] \tau_{\text{av}} = \frac{w(f_{a} + f_{b})}{2l}${\tau }_{\mathrm{\text{av}}}=\frac{w(f_a+f_b)}{2l}$ (11)

w denotes the bearing load per unit contact length, f_a, f_b are the friction coefficients on the stationary and moving surfaces, respectively. The effect of surface roughness can be studied using the present model only if a term accounting for surface roughness is added to Eq. (6).

The surface shear stresses at the j^th discretized point are given by[17]:

[TeX:] \tau_{a,j} = \eta\frac{{\bar{u}}_{a} - {\bar{u}}_{b}}{\frac{2\Delta_{n - 2}\left\lbrack q_{0}^{(1 + \gamma)} - q_{0}^{- (n - 2)(1 + \gamma)} \right\rbrack}{q_{0}^{(1 + \gamma)} - 1} + h_{j}} + \left. \ \frac{\text{dp}}{\text{dx}} \right|_{j}\left( \frac{h_{j}}{2} + Dn - D \right)${\tau }_{a,j}=\eta \frac{{\bar{u}}_a-{\bar{u}}_b}{\frac{2{\Delta }_{n-2}[q_0^{(1+\gamma )}-q_0^{-(n-2)(1+\gamma )}]}{q_0^{(1+\gamma )}-1}+h_j}+{\frac{\mathrm{\text{dp}}}{\mathrm{\text{dx}}}|}_j(\frac{h_j}{2}+Dn-D)$ (12)

[TeX:] \tau_{b,j} = \eta\frac{{\bar{u}}_{a} - {\bar{u}}_{b}}{\frac{2\Delta_{n - 2}\left\lbrack q_{0}^{(1 + \gamma)} - q_{0}^{- (n - 2)(1 + \gamma)} \right\rbrack}{q_{0}^{(1 + \gamma)} - 1} + h_{j}} - \left. \ \frac{\text{dp}}{\text{dx}} \right|_{j}\left( \frac{h_{j}}{2} + Dn - D \right)${\tau }_{b,j}=\eta \frac{{\bar{u}}_a-{\bar{u}}_b}{\frac{2{\Delta }_{n-2}[q_0^{(1+\gamma )}-q_0^{-(n-2)(1+\gamma )}]}{q_0^{(1+\gamma )}-1}+h_j}-{\frac{\mathrm{\text{dp}}}{\mathrm{\text{dx}}}|}_j(\frac{h_j}{2}+Dn-D)$ (13)

The pressure gradient is:

[TeX:] \left. \ \frac{\text{dp}}{\text{dx}} \right|_{j} = \frac{p_{j} - p_{j - 1}}{\delta_{x}}\text{\ \ \ \ for\ }j = 1,2,\ldots,N${\frac{\mathrm{\text{dp}}}{\mathrm{\text{dx}}}|}_j=\frac{p_j-p_{j-1}}{{\delta }_x}\mathrm{\text{for}}j=1,2,\dots ,N$ (14)

where δ_x is the distance between the neighboring discretized points.

According to Eqs. (2) and (14), the film pressure on the j^th discretized point is:

[TeX:] p_{j} = p_{0} + \delta_{x}\sum_{i = 1}^{j}\frac{\frac{1}{2}\text{uρ}h_{i} + q_{m} + uh_{\text{bf}}\rho_{\text{bf}}^{\text{eff}}}{\frac{\text{cρ}h_{i}^{3}}{\eta} + \frac{d\rho_{\text{bf}}^{\text{eff}}h_{\text{bf}}^{3}}{\eta_{\text{bf}}^{\text{eff}}}}\text{\ \ \ \ for\ }j = 1,\ 2,\ldots,\ N$p_j=p_0+{\delta }_x{\sum }_{i=1}^j\frac{\frac{1}{2}\mathrm{\text{uρ}}{h}_i+q_m+u{h}_{\mathrm{\text{bf}}}{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}{\frac{\mathrm{\text{cρ}}{h}_i^3}{\eta }+\frac{d{\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}{h}_{\mathrm{\text{bf}}}^3}{{\eta }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}}}\mathrm{\text{for}}j=1,2,\dots ,N$ (15)

Since p₀=0, the bearing load is:

[TeX:] w = \int_{0}^{l}\text{pdx}$w={\int }_0^l\mathrm{\text{pdx}}$ (16)

The surface frictional forces per unit contact length are, respectively:

[TeX:] F_{a} = \int_{0}^{l}\tau_{a,j}\text{dx}$F_a={\int }_0^l{\tau }_{a,j}\mathrm{\text{dx}}$ (17)

[TeX:] F_{b} = \int_{0}^{l}\tau_{b,j}\text{dx}$F_b={\int }_0^l{\tau }_{b,j}\mathrm{\text{dx}}$ (18)

The surface friction coefficients are respectively:

[TeX:] f_{a} = \frac{\left| F_{a} \right|}{w}$f_a=\frac{\left|F_a\right|}{w}$, [TeX:] f_{b} = \frac{\left| F_{b} \right|}{w}$f_b=\frac{\left|F_b\right|}{w}$(19)

Here, the solution can only be obtained numerically. The numerical procedure follows that presented by Ye and Zhang[17] and, for conciseness, is not repeated here.

It was defined that [TeX:] C_{q} = \rho_{\text{bf}}^{\text{eff}}/\rho$C_q={\rho }_{\mathrm{\text{bf}}}^{\mathrm{\text{eff}}}/\rho$[22]. C_q and C_y were formulated in Ref.[22]. F₁, F₂ and ε were regressed out in Ref.[18]. The values of the parameters for different fluid-surface interactions are provided in Ref.[22]. The type of fluid-surface interaction is determined by strength of the interaction between the fluid molecules and the coating molecule on the solid surface. For example, a hydrophobic surface coating produces a weak fluid-surface interaction, a strongly hydrophilic coating produces a strong fluid-surface interaction, and a normally hydrophilic coating may produce a medium fluid-surface interaction. Different fluid-surface interactions lead to variations in local viscosity and density across the adsorbed molecular layer thickness, as well as different discontinuities and non-continuum effects within the adsorbed molecule layer. Definitions of weak, medium, and strong fluid-surface interactions used in this study are given in Ref.[23]. Table 1 shows the input operational parameter values, and Table 2 provides the material property data of the bearing.

Table 1 Operational parameter values

Table 2 Material properties of the bearing

4. Results and discussion

4.1 Minimum film thickness

Figure 2 shows that, when the surface thermoelastic effect is incorporated, for w=5000 kN/m, the minimum bearing clearance h_tot,min increases rapidly with the sliding speed u when is below 40 m/s. With a further increase in speed, the slope of the variation slope of h_tot,min with u is significantly reduced. When u exceeds approximately 70 m/s, h_tot,min slightly decreases with increasing u. Stronger fluid-surface interactions result in a slightly higher values of h_tot,min, even when h_tot,min=0.5 µm. Figure 3 shows that, for w=5000 kN/m and weak fluid-surface interaction (i.e. the nearly negligible adsorbed layer effect), for the elastic surface without thermal deformation, the calculated h_tot,min is slightly smaller than that predicted by classical lubrication theory calculation[13]; however, its variation with u follows the classical hydrodynamic theory. For the elastic surface with thermal deformation, the calculated h_tot,min is much lower than the classical theory prediction, and its variation with u does not follow conventional expectations; that is, very high sliding speeds are determinal to bearing lubrication.

Figure 4 shows that for u=40 m/s and the strong fluid-surface interaction, for the elastic surface without thermal deformation, the sensitivity of h_tot,min to load variation w is slightly higher than predicted by conventional theory. However, for the elastic surface with thermal deformation, the sensitivity of h_tot,min to the variation w is much greater; this indicates that the film stiffness is considerably lower than classical predictions and the bearing load performance is actually significantly worse due to the surface thermoelastic deformation at large loads and high sliding speeds. Figure 5 shows that, when h_tot,min is below 0.1 µm, the effect of the adsorbed layer becomes pronounced, and is particularly strong for h_tot,min≤0.01 µm. Strong fluid-surface interactions generate a substantially thicker lubricating film when h_tot,min is on the 1 nm scale. To improve bearing performance, strong interfacial adsorption on the bearing surface is required, which can be achieved by applying a special coating (strongly hydrophilic or strongly oil-philic) on the bearing surface.

Fig. 2 Minimum bearing clearance (h_tot,min) versus sliding speed (u) curves for different fluid–surface interactions at w=5000 kN/m

Fig. 3 Minimum bearing clearance (h_tot,min) versus sliding speed (u) curves for different contact regimes at w=5000 kN/m with weak fluid-surface interaction

Fig. 4 Minimum bearing clearance (h_tot,min) versus load (w) curves for different surfaces at u=40 m/s with strong fluid-surface interaction

Fig. 5 Minimum bearing clearance (h_tot,min) versus load (w) curves for different fluid–surface interfaces at u=40 m/s with thermoelastic surface

4.2 Film pressure and film thickness distributions

Figures 6(a) and (b) show that surface thermal distortion significantly alters the film pressure profile at large loads and high sliding speeds. Due to surface thermal distortion, as the load increases, the film pressure is reduced across most of the lubricated area, while only in localized narrow regions does the film pressures increase sharply. This behavior deviates from predictions of classical hydrodynamic lubrication theory. Figures 7(a) and (b) show that surface thermal distortion also significantly alters the film thickness distribution at large loads and high sliding speeds. Although the surface thermoelastic effect does not change the location of the minimum bearing clearance, it shifts the location of the maximum film thickness toward the bearing entrance. Due to the surface thermoelastic effect, as the load increases, the film thickness increases across most of the lubricated area, while it decreases sharplyin localized narrow regions. Figure 7(b) corresponds closely to Figure 6(b). The bearing lubrication performance, therefore, deviates from classical hydrodynamic lubrication theory under severe surface thermal distortion. Moreover, Figure 7(b) highlights the practical necessity of cooling large hydrodynamic lubricated thrust bearing, operating under heavy loads and high sliding speeds in order to mitigate the significant effects of surface thermoelastic deformation.

Fig. 6 Film pressure distributions for different loads and contact regimes at u=40 m/s with medium fluid-surface interaction

Fig. 7 Film thickness distributions for different loads and contact regimes at u=40 m/s with medium the fluid-surface interaction

4.3 Friction coefficient

Figures 8(a) and (b) show that surface thermal distortion to strongly affects the bearing friction coefficient. Due to surface thermal distortion, the friction coefficient on the stationary bush surface slightly decreases with increasing sliding speed when u exceeds 15 m/s, while the friction coefficient on the shaft surface increases monotonously with increasing speed. Overall, surface thermal distortion significantly reduces the bearing friction coefficient particularly at high sliding speeds.

Fig. 8 Friction coefficients on the stationary and moving bearing surfaces for different contact regimes at w=5000 kN/m with weak fluid-surface interaction

5. Model validation

The multiscale flow model used for lubrication analysis, which incorporates the physically adsorbed molecule layer, has been validated by Jiang and Zhang[24]. The film thickness results obtained in the present study qualitatively agree with experimental observations of large hydrodynamic lubricated thrust bearing, showing film collapse due to surface thermoelastic deformation[4, 5].

6. Conclusions

The numerical computation was performed to evaluate the performance of a large hydrodynamic thrust bearing under high operational parameters, incorporating the effects of surface thermoelastic deformation and the adsorbed fluid layer. The moving shaft surface is made of steel, and the stationary bush surface is made of bronze.

Based on the obtained results, the main conclusions are as follows: