ARTICLE Year : 2011  Volume : 1  Issue : 2  Page : 7482 Thermal Analysis of OrificeCompensated Symmetric HoleEntry Hybrid Journal Bearings HC Garg Department of Mechanical Engineering, Guru Jambheshwar University of Science and Technology, Hisar  125 001, Haryana, India Correspondence Address: The present work describes the fluidfilm pressure (p) distributions, fluidfilm thickness (h) profiles and fluidfilm temperature (T_{f}) distributions of orificecompensated symmetric holeentry hybrid journal bearing considering the combined influence of rise in temperature and nonNewtonian behavior of the lubricant. The required governing equations have been solved using the finite element method and a suitable iterative technique. The nonNewtonian lubricant has been assumed to follow the cubic shear stress law. The computed results illustrate that variation of viscosity due to rise in temperature and nonNewtonian behavior of the lubricant affects the performance of symmetric holeentry hybrid journal bearing system quite significantly.
1. Introduction The changing technological scenario necessitated hybrid journal bearings to operate under severe conditions of heavy load and high speed resulting in a rise in temperature of the lubricant fluidfilm and bearing surface. Also, the lubricants used to lubricate the bearings are mineral in nature and additives are added to them to enhance their performance during lubrication. The behavior of polymeradded mineral oil is no longer Newtonian and there is a nonlinear relationship between shear stress and shear strain rate, called nonNewtonian behavior. It reveals that the isoviscous analysis of bearing operating with Newtonian lubricant, does not give true performance of the bearing. Hence, to predict the performance of a bearing realistically, a theoretical model must consider the combined influence of the rise of temperature and nonNewtonian behavior of the lubricant. Ferron et al., [1] performed both the theoretical as well as the experimental studies concerning the thermal effects in finite length journal bearing. In their theoretical work, they simultaneously solved the governing Reynolds equation in lubrication, 3D energy equation in the fluidfilm and 3D heat conduction equation in the bush. Heat transfer to bush and shaft was also considered. They found good agreement with measured results for pressure and temperature. Boncompain et al., [2] used an approach similar to the one employed by Ferron et al., [1] however they included the reverse flow condition in their analysis, which generally occurs in the maximum fluidfilm thickness region at large eccentricity ratios. They concluded that most of the heat generated is evacuated by the fluid flow and that the temperature gradients across and along fluidfilm are important. An efficient THD numerical procedure was developed by Vijayaraghavan [3] to analyze journal bearing performance. The conduction through the bearing and journal has been considered to accurately predict the temperature and pressure distribution. Kumar et al., [4] also described the static and dynamic performance of holeentry hybrid journal bearing operating with different flow control devices by considering the variation of viscosity due to temperature rise of the lubricant. It has been concluded that in general, a constant flow valvecompensated holeentry hybrid journal bearing gives superior performance from the point of view of minimum fluidfilm thickness and threshold speed. Kumar et al., [5] studied the effect of the variation of viscosity due to temperature on the bearing stability margin of holeentry journal bearings. The authors found that the temperatureviscosity dependence of the lubricant reduces the bearing stability. The effect of viscosity variation on the stability of the bearing system can be minimized if the value of bearing flow through holes is judiciously selected for constant flow valvecompensated bearing. Kumar et al., [6] investigated that the temperature rise of lubricant fluidfilm and/or bearing flexibility deteriorates the stability of a constant flow valvecompensated nonrecessed holeentry journal bearing system. The loss in stability margin can be regained by selecting proper value of restrictor design parameter. Garg et al., [7] presented a comprehensive review of the developments in the design and application of hydrostatic and hybrid journal bearing systems and concluded that more extensive research is needed, both analytical as well as experimental, to consider the extension of these bearings into highspeed applications. Garg et al., [8] found that change in viscosity of lubricant due to nonNewtonian behavior and rise of temperature affects the performance of the constant flow valvecompensated holeentry hybrid journal bearing system quite significantly. Garg et al., [9] studied the performance of a capillarycompensated holeentry journal bearing system and found that the value of minimum fluidfilm thickness [INSIDE:1] reduces due to the decrease in viscosity because of consideration of thermal effects and the nonNewtonian behavior of the lubricant. This reduction in the value of [INSIDE:2] may be compensated to maintain the designed value of [INSIDE:3] by taking suitable values of restrictor design parameter. Garg et al., [10] theoretically investigated the thermal and rheological effects of the lubricant on the performance of symmetric and asymmetric slotentry hybrid journal bearing system. It was found that the stability of slotentry hybrid journal bearing (symmetric/asymmetric configuration) increases with viscosity variation due to rise in temperature and nonNewtonian behavior of the lubricant, while operating at higher load [INSIDE:4] and low speed parameter (Ω). The available literature concerning the hybrid/hydrodynamic journal bearings indicates that the thermal effects together with the nonNewtonian behavior of lubricant due to additives mixed in the lubricants have been ignored in the analysis so as to obviate the mathematical complexity. The fluidfilm pressure [INSIDE:5] distributions, fluidfilm thickness [INSIDE:6] profiles and fluidfilm temperature [INSIDE:7] distributions of orificecompensated symmetric holeentry hybrid journal bearing have been presented considering the combined influence of rise in temperature and nonNewtonian behavior of the lubricant for symmetric bearing configurations shown in [Figure 1]. The results presented in this paper are expected to be quite useful to the bearing designers.{Figure 1} 2. Analysis The analysis presented in the following subsection uses finite element method to model the complete journal bearing system operating with nonNewtonian lubricants. The mathematical model, which includes the viscosity variation due to temperature rise and nonNewtonian behavior of the lubricant, involves simultaneous solution of Reynolds equation, Energy equation and Heat conduction equation. The finite element formulation of these governing equations is described below with their boundary conditions. 2.1 Reynolds Equation for Fluid Domain The generalized Reynolds equation governing the laminar flow of incompressible lubricant between the clearance space of journal and bearing considering variable viscosity and usual assumptions in the nondimensional form is written as [11] : [INLINE:1] Where [INSIDE:8] are the cross film viscosity integrals and given by the following relations: [INLINE:2] 2.2 Restrictor Flow Equations The flow of lubricant through orifice restrictor in nondimensional form is given by Sinhasan and Sah [12] : [INLINE:3] 2.3 NonNewtonian Model Most of the nonNewtonian oils follow the behavior, which is represented by cubic shear law Sinhasan and Sah [12] , The constitutive equation for cubic shear law is described in nondimensional form as: [INLINE:4] Here, [INSIDE:9] is known as nonlinearity factor. The viscosity of nonNewtonian lubricant is described by the apparent viscosity (μα) and is defined as the function of shear strain [INSIDE:10]. [INLINE:5] In the nondimensional form, the shear strain rate [INSIDE:11] at a point in the fluidfilm is function of velocity gradients [INSIDE:12], and is expressed as: [INLINE:6] 2.4 Boundary Conditions The boundary conditions used for the lubricant flow field are described as: Nodes situated on the external boundary of the bearing have zero relative pressure with respect to atmospheric pressure. [INLINE:7] The nodal flows are zero at internal nodes except those situated on holes/slots and external boundaries.Flow of lubricant through the restrictor is equal to the bearing input flow at hole/slot.At the trailing edge of the positive region [INSIDE:13] according to SwiftStieber cavitation condition. 2.5 TemperatureViscosity Relation The viscosity [INSIDE:14] is assumed to be dependent on temperature and is defined by the relation given below [1],[13] : [INLINE:8] Where, K0 , K1 and K2 are nondimensional coefficients having values 3.287, 3.064 and 0.777 respectively. For isothermal (IHS) case, the viscosity does not vary due to temperature rise. 2.6 Heat Transfer Equations The 3D energy equation and 3D conduction equation govern the heat transfer in the fluidfilm and bush respectively. The 3D energy equation is expressed in nondimensional form as given below [1],[13] : [INLINE:9] The three dimensional Fourier heat conduction equation is used to determine the temperature distribution in the bush. The heat conduction equation in cylindrical coordinates is given below [1],[13] : [INLINE:10] The journal temperature is obtained by considering the journal in thermal equilibrium, which requires that the total heat flows to and from the journal are equal as expressed by Sinhasan and Chandrawat [14] . 2.7 Thermal Boundary Conditions The thermal boundary conditions used for the solution of energy equation of and conduction equation are expressed as [13],[14] : [INSIDE:15] i.e., fluidfilm journal interface[INSIDE:16] i.e., fluidfilm bush interfaceOn the fluidbush interface [INSIDE:17] [INLINE:11] On the outer part of the bush [INSIDE:18], the free convection is assumed [INLINE:12] On the lateral faces of the bearing (β = ± λ) [INLINE:13] At the inlet edge of the hole [INLINE:14] Boundary conditions are modified after each iteration according to the bush temperature distribution obtained from heat conduction equation. The global system equations of governing equations (1), (6) and (7) are obtained by employing Galerkin's orthogonality criterion and then solved after applying appropriate boundary conditions. The entire lubricant flow field is discretized using 8noded hexahedral linear isoparametric elements. The three dimensional finite element grid for the thermal analysis is made compatible with the twodimensional (2D) grid used for the solution of the Reynolds equation along the remaining two directions (i.e. circumferential and axial). The bush is discretized using 8noded hexahedral isoparametric elements for the thermal analysis. The grid is made compatible with the one used for lubrication and analysis of fluid domain for energy equation. 3. Solution Procedure In the present study an iterative numerical solution scheme is used to establish pressure and temperature fields in the lubricant fluidfilm. The thermohydrostatic (THS) rheological solution of a holeentry hybrid journal bearing system requires the simultaneous solution of Reynolds equation , energy equation and conduction equation along with appropriate boundary conditions. [Figure 2] shows the overall solution scheme used in the present study. The overall solution scheme comprises three different modules, namely, Module NNWTNLUBRA, Module ENRGY, and ModuleCNDUC. In module NNWTNLUBRA, the generalized Reynolds equation governing the flow of the nonNewtonian lubricant is solved along with restrictor flow equation so as to obtain fluidfilm pressures. The solution for the Newtonian lubricant is obtained as the initial trial solution to be used for the nonNewtonian case. The values of cross viscosity integrals [INSIDE:19] are obtained at each gauss point using Numerical integration (Simpson's rule). The shear strain rate [INSIDE:20] is computed using equation (4), and the corresponding equivalent shear stress is obtained from equation (2) using NewtonRapson's method. The apparent viscosity is computed at each gauss point using equation (3).The temperatureviscosity relationship as expressed by equation (5) is used to compute the viscosity. For the continuity of the flow between restrictor and bearing, the system equation is modified accordingly. The nodal pressure obtained after achieving journal center equilibrium position is used as the input variable by the ModuleENRGY for solution of energy equation. The journal temperature is then computed using the fluidfilm temperatures obtained from the solution of the energy equation. Energy equation is repeatedly solved after modifying for the boundary conditions at the fluidfilm journal interface . The iterations are terminated when index ITJR attains a value equal to unity. ModuleCNDUC is used to compute the temperature in the bush. To establish the converged solution for the fluid bush interface and bush surface temperature, energy and conduction equations are solved simultaneously using the boundary conditions. When the value of index ITFB becomes unity it indicates the convergence of the fluidbush interface and bush surface temperature. Since the temperature distribution alters the fluidfilm viscosity field, a new fluidfilm pressure field is obtained using Module NNWTNLUBRA. The iterative procedure is repeated till the converged solution for the fluidfilm pressure field (ITPR = 1) is obtained.{Figure 2} 4. Results and Discussions The analysis and solution algorithms as described in the previous sections have been used to compute the fluidfilm pressure distribution, fluidfilm thickness profile and temperature distribution in orificecompensated symmetric holeentry hybrid journal bearing operating with nonNewtonian lubricant for representative values of the bearing geometric and operating parameters as shown in [Table 1]. The values of inverse Peclet number [INSIDE:21] and dissipation [INSIDE:22] number are computed from lubricant properties, bearing geometric and operating parameters already published in the literature [1],[14],[15],[17] .{Table 1} To check the validity of the analysis, the results of orificecompensated holeentry hybrid journal bearing have been computed and compared with the available experimental and theoretical results of Ferron et al. [1] The operating conditions in the present results are chosen identical to that of Ferron et al., [1] in their theoretical work. The results computed from the present study show a good agreement between the theoretical and experimental results of Ferron et al., [1] as depicted in [Figure 3]. The results from the present work are found to be quite close (nearly 2%) to the analytical results of Ferron et al., [1] at lower load and the results of the study at higher load are almost average of the experimental and analytical results of Ferron et al. [1] .{Figure 3} 4.1 FluidFilm Pressure [INSIDE:23] Distributions The fluidfilm pressure distribution along circumferential (β = 0.0) direction for symmetric bearing configuration is shown in [Figure 4]. It can be observed from [Figure 4] that pressure is maximum nearly at about α = 270° i.e. in the direction of external load for both Newtonian and nonNewtonian lubricants. The reduction in the pressure of the order of 20% is observed for [INSIDE:24] as compared with Newtonian case [INSIDE:25] at a particular location i.e. α = 270° . This is consistent with the earlier results of Boncompain et al., [2] and Boncompain and Frene [16] obtained for plain hydrodynamic bearing.{Figure 4} 4.2 Fluidfilm Thickness [INSIDE:26] Profiles The fluidfilm thickness profiles along circumferential direction for symmetric bearing configurations are shown in [Figure 5]. It is observed from the Figure that film thickness at the axial midplane (β = 0.0) along the circumference is minimum at nearly α = 330 °. The fluidfilm thickness along circumferential direction at any point is found smaller for the nonNewtonian lubricant between angular locations from 230 ° to 50 ° as compared to the Newtonian lubricant whereas the film thickness is found higher for the remaining angular locations i.e. from 50 ° to 230 °. The maximum percentage increase in the minimum fluidfilm thickness of the order of around 13.6% is found at angular locations 150 ° when the bearing is operating with nonNewtonian lubricant [INSIDE:27] with as compared to [INSIDE:28]. The maximum percentage decrease of the order of around 22% is observed at angular location 330 ° when the bearing is operating with nonNewtonian lubricant with [INSIDE:29] as compared to [INSIDE:30] for the symmetric bearing configuration.{Figure 5} 4.3 Fluidfilm Temperature [INSIDE:40] Distributions Plot of fluidfilm temperature distribution along circumference at axial midplane (β = 0.0) and across midfilm [INSIDE:31] for symmetric bearing configuration is shown in [Figure 6]. A temperature rise as high as up to 1.617 is noted for symmetric bearing configuration operating with nonNewtonian lubricant with [INSIDE:32]. It is found that the temperature rise is more when the bearing is operating with the nonNewtonian lubricant with [INSIDE:33]. A total of about 6.6% change in temperature is observed along circumference at [INSIDE:34] for symmetric bearing configuration. For a designer it becomes imperative to consider the thermal effects and nonNewtonian behavior of the lubricants to predict the accurate performance of the bearing system. The 2D temperature profile of symmetric bearing configuration with nonNewtonian lubricant is [INSIDE:35] shown in [Figure 7].{Figure 6}{Figure 7} 5. Conclusion The following conclusions are made from the results presented in this study. When the influence of viscosity variation due to temperature rise and nonNewtonian behavior of the lubricant is considered, it is observed that pressure distribution get affected. This alters the performance characteristics of the nonrecessed hybrid journal bearings.The viscosity of the lubricant decreases due to the rise in temperature and increase in the nonlinearity factor [INSIDE:36] of the lubricant. The effect of the decrease in the viscosity of the lubricant is to reduce the fluidfilm thickness [INSIDE:37] for the bearing with the given operating and geometric parameters. Such reduction in the value of [INSIDE:38] should be accounted while designing a bearing to maintain a safe design value of minimum fluidfilm thickness [INSIDE:39]. 6. Nomenclature [INLINE:15] [INLINE:16] [INLINE:17] References


