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Optimum surface profie design and performance evauation of incined sider bearings A. A. Ozap and H. Umur Uudag University, Mechanica Engineering Department, 6059, Goruke Bursa, Turkey The aim of the present
Optimum surface profie design and performance evauation of incined sider bearings A. A. Ozap and H. Umur Uudag University, Mechanica Engineering Department, 6059, Goruke Bursa, Turkey The aim of the present optimization study is to propose an innovative surface profie design by impementing a wavy form on the upper surface, without varying the physica imits of the compete sider-bearing structure. Differentia equations governing the fuid-fim mass, Reynods and energy equations, are soved simutaneousy by the iterative transfer matrix approach, which aso takes into account the streamwise decrease of ubricant viscosity. Computations indicated that friction coefficient vaues decrease with wave ampitude in pad incination ranges of , and for inet/it pressure ratios of.0, 3.0 and 5.0 respectivey. Wave number is determined to augment the compete pressure distribution inside the bearing and optimum wave number is evauated as 5 for pressure management. Keywords: waviness. For correspondence. (e-mai: 480 Performance optimization, sider-bearing, WITH tensive appications of sider-bearings in mechanica devices, investigations associated with their design and performance optimization have been given priviege. The present industria needs cover increased oad capacity, owered friction and power consumption and creative designs, which in return contribute to considerabe progress in computer-aided modeing of sider-bearing ubrication. Anaysis on siding surface definitions is important in predicting the system responses, where appication of a wavy pattern to the fow surfaces is a new approach and compicates the numerica-ubrication simuations, as the waviness is defined by two independent variabes: ampitude and waveength. A few studies have concentrated on the effects of surface waviness on the ubrication process numericay. A numerica mode that takes the sources of noninearities, such as surface waviness into account for ba-bearing appications, was deveoped by Harsha et a.. Rasheed 3 considered the infuence of waviness on cyindrica siding eement and proposed a critica wave number range of 9 for improved operating conditions. Sottomayor et a. 4 studied roer bearings for various waviness ampitude vaues and recorded augmented friction coefficients in higher ampitude cases. van Ostayen et a. 5 investigated the performance of a hydro-support with random waviness and Honchi et a. 6 appied a micro-waviness mode to an air sider-bearing. Kwan and Post 7 evauated augmented oad vaues of aerostatic bearings with higher wave ampitudes, whereas Ai et a. 8 showed that the ubricant fim thickness decreased with waviness in journa-bearings. Journa bearings were aso studied by Mehenny and Tayor 9, who found that the maximum pressure increased with wave number, whereas the numerica mode for journa-bearing systems of Liu et a. 0 did not converge efficienty for waviness ampitudes above 6 µm. Performance predictions of the ubrication process under various boundary and geometric conditions have aso been considered. Naduvinamani et a. investigated surface roughness effects on the oad carrying capacity. Lin, and Karkoub and Ekame 3 inspected the geometric infuences on the oad vaues. Effects of input pressure on the work and friction characteristics have been numericay evauated 4,5. Kumar et a. 6 performed a numerica study on the surface roughness effects on eastohydrodynamic ubrication of roing ine contacts. Watanabe et a. 7 predicted the infuence of structura design features on the frictiona characteristics of microgrooved bearings. Lubricant fow rate is a major design consideration studied by Luong et a. 8, both numericay and perimentay for thrust-bearing appications, and by Tian 9, numericay for porous bearings. Hargreaves and Egezawy 0 worked on the upper surface discontinuities and so occurring pressure variations in sider-bearings. Non-Newtonian character of the ubricant has a considerabe contribution in the ubrication systems; and specificay concentric and eccentric cases have been perimentay investigated,. Optimization of sider-bearing ubrication is aso incuded in the scope of numerica studies. Stokes and Symmons 3 performed a muti-dimensiona optimization on the pastohydrodynamic drawing of wires. Lin 4 tried to get an optimum fow cavity for one-dimensiona porous curved siderbearing. Stabiity is a chief concern for system durabiity and safety. Instabiities based on pressure perturbations and geometric definitions have been investigated 5 7. Lubricant viscosity is known to vary significanty with temperature. However, numerica studies on waviness and performance predictions presented above, disregarded CURRENT SCIENCE, VOL. 90, NO., 0 JUNE 006 this fact due to the compity of the numerica structure, and performed their computations with a fixed viscosity vaue for the compete fow domain. Indeed there are aso studies that focus on the effects of temperature and ubricant properties on the performance of bearings. A ineary narrowing sider-bearing, with heat conduction to the stationary ower surface has been investigated 8. Temperature dependency of ubricant viscosity was handed by imposing the temperature distribution of the previous soution set on the noda viscosity vaues, unti a convergence of 0.05% was achieved for each node between two successive soutions. A simiar approach was used by Pandey and Ghosh 9 on both siding and roing contacts. Their convergence criterion for temperature distribution was ess sensitive (0.%) and a unique viscosity vaue, which corresponds to the average ubricant temperature, was used for the compete fow voume. On the other hand, Yoo and Kim 30 took temperature-dependent viscosity into consideration more precisey with a convergence criterion of 0.00%. However, in their study to decrease the computation run time, convergence was not appied to each individua temperature vaue in the fow direction, but for the sum of the compete temperature set. Athough there have been considerabe efforts to determine the boundary condition cavity structure interactions and their contributions on the performance vaues of siderbearings, there is sti a gap in this subject matter from the point of upper surface design. The present optimization study aims to define an innovative design, with the impementation of a wavy form on the upper surface, by keeping the voume of the fow cavity fixed, without varying the physica imits of the compete ubrication structure. To produce a compete overview, computations are performed with three inet/it pressure vaues (β), for six ampitudes (ϕ) and wave numbers (λ) in the pad incination (θ) range of 3 5. Performance optimization outputs are discussed through streamwise pressure (P) variations and with ubricant fow rate (m ), friction coefficient ( f ) and oad capacity (W) data for various β, ϕ, λ and θ cases. Governing equations Pane sider-bearing performance anaysis covers the investigation of both momentum and energy transfer in the fow voume. Thus veocity (u), pressure (P) and temperature (T) distributions are the primary concern of the fundamenta theory. The outputs of the continuity, momentum and energy equations can be the focused items of the work, but generay the resuts of the former in the cacuation order generate the input set for the foowing, which puts forth the simutaneous handing of the three equations. As the momentum equation in the x direction (eq. ()) interprets the reation of viscous shear stress and thermodynamic pressure, the Reynods equation for one-dimensiona ubricant fows of sider-bearings is given by eq. (), where V u, V and h stand for the upper and ower surface veocities and pad height respectivey 3. d u d P, dy = µ dx () 3 d h dp Vu + V d h =. dx µ dx dx The amount of energy transferred within the ubricant fow is mainy designated by the streamwise temperature variation, but the voumetric fow rate (q x ), the terms reated to the momentum oss and the friction oss, which are interconnected with the shear stress (τ), ubricant density (ρ) and specific heat (C p ) are aso encountered in eq. (3). Lubricant viscosity (µ) appears in either of the threemain fow and energy equations. Thus the Newtonian viscosity temperature reation is characterized by Voge s rue 3 of eq. (4), where b, k and ζ are the viscosity param e- ters. () dt d P ρcpqx = Vuτu + Vτ qx, (3) dx dx b T+ ζ µ = ρke. (4) Athough the genera form of the momentum, Reynods and energy equations are as given above, cacuations on sider-bearing ubrication are frequenty performed in nondimensiona form 4,8,33. According to Hwang et a. 34, eqs () (3) can be converted into eqs (5) (7) by using the non-dimensiona parameters given in Appendix (A), with the reevant boundary conditions. d u d P =, y = 0 u = and d y d x h V y = h u V, h = = V = (5) u u d 3 d P d h 6( h = V + u ), dx d x d x in x = 0 P = P and x = P = P, (6) d T d P q x = Vuτ v + τ q x, in d x d x x = 0 T = T. (7) The non-dimensiona veocity profie (eq. (8)) can be obtained by imposing the boundary conditions in eq. (5). CURRENT SCIENCE, VOL. 90, NO., 0 JUNE 48 d P y y u = ( y h) ( V u ). d x + h + (8) Integration of the veocity profie gives the ubricant fow rate per unit width (eq. (9)). h. d P h m = ud y = h + ( Vu + ). (9) d x 0 The streamwise pressure distribution can be evauated by integrating the Reynods equation twice: (0) 6( V + ) d h d x P = d x c + P, x x u 3 3 in 0 h 0 h where the integration constant c is given in Appendix (B). The oad-carrying capacity (W) and friction force (F f ) vaues are obtained by streamwise integration of the fim pressure and shear stress distributions respectivey, and the ratio gives the friction coefficient ( f ). Expressing in terms of dimensioness quantities yieds eqs () (3). W = Pd x, () 0 F = τd x, () f 0 F f f =. (3) W Theoretica mode To generate a reaistic overview and to invove the combined definitiona necessities of both the physica and the thermo-fuid structura information of the compete ubrication environment, necessary compounds are chosen from the avaiabe recent numerica studies. The siderbearing in concern here is narrowing in inear stye (Figure ), simiar to that of van Ostayen et a. 5 and Kwan and Post 7. As in the study of Honchi et a. 6, the upper bearing surface is kept stationary (V u = 0 m/s). On the other hand, the ower surface veocity is chosen as V = 5 m/s, which is the mean vaue of the corresponding data of Liu et a. 0 ( m/s) and of Ai et a. 8 (8.79 m/s). Bearing ength (L) is seected as 0 mm, ying between the choices of Honchi et a. 6 (.5 mm) and Ai et a. 8 (4.5 mm). As the inet height (h in ) is fixed as mm, the it height (h ) range is seected as mm, where these vaues are within the most frequenty appied bearing pad height and journa-bearing cearance data range of mm 0,. The empoyed inet and it heights resut in the h in /h ratio range of. 8 and the mean pad incination range (θ) of 3 5 (Figure ). As the resuting h in /h ratio range is compatibe with those of Das 35 and Kumar et a. 8, where the upper imit of their corresponding h in /h ratio ranges was 6 and 0 respectivey, the consequentia pad incination range aso suits the appied vaues of Watanabe et a. 7 (θ =.78 ), Honchi et a. 6 (θ = 6.8 ) and Dadouche et a. 36 (θ = ). Investigations have been carried out with SAE 0-type ubricant that has comparabe viscosity vaues with the appication of Mehenny and Tayor 9 and with the inet temperature (T in ) of 0 C, which is cose to that of Sottomayor et a. 4 (4 C). The anaysis is based on the fact that unused ubricant is pumped in and emerges to the atmosphere. Therefore inet (P in ) and it (P ) oi pressure vaues are fixed at 0 50 and 00 kpa respectivey, with β = P in /P of The upper surface of the bearing is defined in a convenient way to fit the main aim of the work, by eq. (4), where the wave ampitude and the number of waves are indicated by ϕ and λ respectivey. The cosine curve is impemented to the streamwise-structure, where the reative positions of nodes (i) are integrated, in comparison to the tota meshing scae (n + ). ϕ i hx ( ) = hin xtanθ + cos π λ. n+ (4) To visuaize the effects of ϕ and λ on the bearing performance, wave ampitude data are rated in the range 0 00 µm, covering those of Harsha et a., Rasheed 3, and Liu et a. 0. The imposed number of waves is kept within the imits of 5 05, incuding the most recurrent vaues in the iterature 4,6,7. Computationa procedure To sove the continuity, momentum and energy equations in harmony, the geometric domain of Figure is divided into 000 sequentia ces, where higher numbers, in the eary Figure. Schematic ayout of sider-bearing with/without wavy structure at the upper surface. CURRENT SCIENCE, VOL. 90, NO., 0 JUNE 006 stages of code deveopment, appeared to increase ony the run times and not the accuracy of the streamwise convergence. Second-order finite difference marching procedure 37 in the streamwise direction with a constant ce width ( x) of L/n, where for the sake of generaity the number of ces is denoted by n, is appied for the simuation of one-dimensiona incompressibe ubricant fow. Since the ubricant fow rate (m ) is constant in the fow direction, equating the derivative of eq. (9) to zero forms a system of n inear equations, which competey represent the reation of geometric structure, static pressure and veocity distributions. The new impementation (eq. (5)), which can be designated in sigma notation, consists of two coefficient matrices (γ, ϖ) whose eements are mainy defined by the groove geometry and the upper and ower surface veocities of the bearing and can be cacuated as given in Appendix (B). As the nodes i = and i = n + represent the inet and it panes, the picit form of the n equations constitutes the transfer matrix (eq. (6)) of the system. Since the eft-hand side of eq. (6) is a banded matrix with a bandwidth of three, Thomas agorithm 37 is used in the evauation procedure, where the outputs are the scope of continuity and Reynods equations for nodes i = to n. γ i P i = ϖ i, (5) i = γi+ γi i = γi γi γ i+ i = 3 0 γi γi+ γi i = n γi γi γi i = n γi γi+ γi+ i = n γi γ i+ P i+ ϖ i γip in P i+ ϖi P i ϖ + i =. P ϖ i i+ ϖ i Pi+ ϖ i γi+ P Pi+ (6) In addition to the inet conditions and surface veocities, resuts of the transfer matrix, especiay the ubricant fow rate and streamwise pressure gradient aso participate as inputs when the temperature variation is under inspection. Superimposing the finite difference sense into the energy equation and rearranging the terms brings up a therma reation (eq. (7)) within two consecutive nodes in the mesh, which in return dispays the ubricant temperature distribution in the fow direction. V τ + τ d P T x T u u i+ = + i. q d x x i (7) Evauation of the temperature-dependent nature of viscosity covers both the traditiona isotropic 4,6 0 method and the present iterative transfer matrix approach, where the cassica soution generates the initia set of guesses for the first iteration step. As shown in Figure, ubricant viscosity is kept constant in the compete fow voume, being equa to the inet vaue, for the isotropic approach. In the first step of the iterative method, the temperaturedependent noda viscosity variation is cacuated by using the temperature distribution of the isotropic approach, together with the viscosity parameters of k, b and ζ. Soving the transfer matrix of eq. (6), with the obtained viscosity distribution, gives the initia temperature set of the iterative method. The computations continue unti two consecutive temperature distributions are not more than 0.0% distant at each node within the mesh. The appied convergence criterion (0.0%) is more sensitive than that of Kumar et a. 8, and Pandey and Ghosh 9. On the other hand, athough the criterion (0.00%) of Yoo and Kim 30 appears to be more precise, the present method differs from the former by imposing the convergence on each noda temperature, not on the sum of the compete set. This appication makes the streamwise temperature determinations more reiabe, since n + times more contro oops for every iteration step ist in the current approach. Resuts and discussion Numerica investigations are performed with wave numbers (λ) of 5 05, wave ampitude (ϕ) vaues are in the range of 0 00 µm, sider-bearing pad incination (θ) is varied between 3 and 5, and the inet pressures (β) are times the it vaue. Resuts of various structura-design and boundary condition cases are discussed through ubricant fow rate (m ), friction factor ( f ), bearing oad (W) and streamwise variations of ubricant pressure (P), and dispayed in non-dimensiona form () for the sake of generaity. Variations in m are given in terms of θ, ϕ and λ β in Figure 3 a b respectivey. It can be seen from Figure 3 a that, for β =.0, ubrication systems with higher ϕ resut in ower ubricant fow rate for a sider pad incinations (θ = 3 5 ). Moreover the infuence of ϕ on m becomes more apparent in cavities with θ cr 3.77 for the λ range of 5 5. Mass fow rate appeared not to be infuenced by wave number for sma θ; however, as in the case of wave ampitude, above the pad incination of 3.77 ubricant CURRENT SCIENCE, VOL. 90, NO., 0 JUNE Figure. Fow chart of appied computationa procedure. fow rate decreases with higher wave numbers, regardess of the eve of ϕ. Figure 3 a additionay indicates that ubricant fow rate increases ineary with higher θ in the ϕ range of 0 80 µm (ϕ ϕ 3 ) for 5 λ 5. On the other hand, as the wave ampitude is further increased to ϕ 4 ϕ 6 (0 00 µm), simiar trends of the ϕ ϕ 3 cases are dispayed for the θ range of Whereas the curves of Figure 3 a impy that beyond θ cr = 3.77, surface designs with ϕ 0 µm create a counter effect on ubricant fow rate and the outcomes become more remarkabe in cases with higher ϕ, λ and θ. Infuence of inet pressure (β), with the combined handing of various pad styes for the compete set of ϕ and θ vaues on ubricant fow rate for the maximum wave number of λ = 05 is dispayed in Figure 3 b. The m curves for β =.0 and λ = 05 are quite simiar from the point of both the vaues attained and the variation styes, to those for β =.0 and λ = 5 (Figure 3 a). Thus increasing wave number from 5 to 05 does not create any noticeabe change in the ubricant fow rate. On the other hand, increasing β augmented the fow rate vaues consideraby for the compete pad incinations. However, the increase is more remarkabe in the pad structures, particuary with ower incinations (θ = ), which can be attributed to the resistive character of higher pad incinations (θ = ) on ubricant fow rate. The maximum ϕ of 00 µm resuts in a decrease in m vaues of the θ = 5 case by 6, 68 and 7% for β of.0, 3.0 and 5.0 respectivey. This denotes that the infuence of ϕ on m increases with β. Moreover, the curves for ϕ = 0 80 µm of the β = 3.0 and 5.0 cases differ from those of β =.0 by showing continuous decreasing trends with higher θ, demonstrating increasing sopes with the ascend of inet pressure. Watanabe et a. 7 (for journa-bearing ubrication), Tian 9 (for inear porous thrust pad appication) and van Ostayen et a. 5 (for hydro-support design) aso determined the fow rate vaues to decrease with higher cavity cearance, height and pad incination. Since the power demand of the system is reated with ubricant fow rate, it can directy be controed/owered by θ and ϕ, especiay in cases with higher β. Athough the compete curves of β = 3.0 and 5.0 hibit decreasing character with θ, the decine rates become sharper at θ 3.77 for the compete ϕ set, putting forth that the pad structure hibits a critica character at θ = 3.77, aso at β = and λ = 05 cases, simiar to the findings with β =.0 and λ = 5 5. Combined effects of θ, λ, ϕ and β on friction factor ( f ) of sider-bearings are presented in Figure 4. Up to θ cr = 3.77, impementation of higher wave number (λ = 5 5) or ampitude (ϕ = 0 00 µm) did not have any contribution on f for the specific
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