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# LUBRICATION THEORY MEEN 626

Texas A&M

GPA 3.58

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This 41 page Class Notes was uploaded by Orrin Weissnat on Wednesday October 21, 2015. The Class Notes belongs to MEEN 626 at Texas A&M University taught by Luis Sanandres in Fall. Since its upload, it has received 16 views. For similar materials see /class/225989/meen-626-texas-a-m-university in Mechanical Engineering at Texas A&M University.

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Date Created: 10/21/15

MEEN 626N0tes 2 Continued Simple OneDimensional Fluid Film Bearings Load capacity and frictional drag analysis for A Plane Slider Bearing B Rayleigh Step Bearing C Elementary Squeeze Film Flow Plane Slider Bearing Analysis of 1Dslider bearing ME626L8A0808 Figure 1 shows the geometry and coordinate system of a onedimensional slider thrust bearing The lm thickness h has a linear taper along the direction ofthe surface velocity U The film wedg generates a hydrodynamic pressure eld that supports an applied load W Note that the exit minimum lm thickness h2 is unknown and must be determined as part of the bearing design The bearing taper h1h2 needs be designed appropriately L Width B bearing pad U Moving surface Fig 1 Geometry of taper slider bearing Nomenclature Assum tions U surface speed incompressible lubricant isoviscous L bearing length steady state operation dhdt0 width B gtgt L B bearingwidthBgtgtL H Iubricantviscosity no fluid inertia effects hxfim thickness h1 at inlet leading edge h2 at exit trailing edge X hx h1h2 h1f 111 gt 112 Reynolds eqn for generation of hydrodynamic pressure p reduces to 3 0 with p0 ambient at x0 and L inlet and dx 12 dx 2 exit of bearing De ne the following dimensionless variables P h 6 UL H h HXoc oc1 ocX 3 2 2 h2 X P L is a film thickness ratio or taper ratio hence Eq 2 becomes d H3d P H 0 dX dX A first integral ofthis equation renders a constant proportional to the ow rate Qx ie 3 d H ampP H QX 4 Integration of Eq 4 is rather simple for the lm tapered pro le After some algebraic manipulation and application ofthe pressure boundary conditions at the inlet and exit planes ofthe bearing the end result is Figure 2 depicts the pressure pro le for four lm taper ratios OL152 3 and 4 Note that as 0c increases the peak pressure increases However for ocgt22 the peak pressure levels off 004 quotI 0035 I quot 0025 1 I Dim pressure a15 u a20 quot39 a30 a40 Fig 2 Dimensionless pressure for 1Dslider bearing and increasing film thickness inletexit ratios 0 The analysis determines and flow rate qXoc h2UBQX 7 where QXx a 1oc Integration ofthe pressure eld over the pad surface renders the bearing reaction force opposing the applied load W Le L 2 1 w J PXdXB me w J PXdX 8a 0 1122 o Substitution ofthe found pressure eld above gives after considerable algebraic manipulation There is an optimum lm ratio on that determines the largest load capacity This optium ratio is determined from Note that too large taper ratios co gt Xopt act to reduce the load capacity The formula above shows that the machined taper h1h2 218 h1 for maximum load carrying It is also important to determine the shear force due to the fluid being dragged into the thin lm region This force equals L f J EWX dXB 9a 0 d V where w is the shear stress at the moving wall 1W 39 39 Ty X atyzo 108 The uid velocity field VX adds the Poiseuille and Couette contributions ie due to pressure and shear respectively ie Then U Tw 39 h P 10b Substitution of the found pressure pro le above and integration over the pad surface gives a shear force The power lost needed to drag the uid is PW fU 11 It is customary to define a coefficient of friction p relating the shear force to the applied load ie Since h2L ltlt 1 the friction coefficient wis actually much smaller than the dimensionless coef cien up displayed in the graph above Figure 3 below depicts the dimensionless peak pressure load capacity shear force and ow rate fo the 1Dslider bearing as a function ofthe lm thicknesses ratio 0c lDslider bearing I I 09 08 r quot39 I 07 39 06 quot 05 39 04 03 Dimensionless parameters 02 01 0 IIIquot I I I I 15 2 25 3 35 4 Film thickness ratio h1h2 Peakpressure I u a I u Load 39 39 39 Shear force quot 39 ow rate Fig 3 Dimensionless parameters for tapered 1Dslider bearing Increasing film ratios 0 if ll Thus far the analysis considers the lubricant viscosity to remain invariant However most lubricant mineral oils have a viscosity strongly dependent on its temperature In actuality as the lubricant ows through the lm thickness it becomes hotter beacuse it must carry away the mechanical powr dissipated withn the lm Analysis of fluid lm bearings with thermal effects can be extremely complicated since the film temperature changes even across the lm thickness Such analysis is presently out of scope ie within the framework of the notes hereby presented Nonetheless a simple method follows to estimate in a global form as in a lumped system the overall temperature raise of the lubricant and its effective lubricant viscosity to use in the analysis a design of a bearing The mechanical power is not only carried away by the lubricant flow but also conducted to and through the bearing bounding solid surfaces bearing and moving collar Recall that the mechanical power dissipated equals PwfU and converted into heat that is carried away by the lubricant A balance of mechanical power and heat ow gives 12 K39PW p39Cp39qX39AT AT TeXit Tinlet where p and Cp are the lubricant density and heat capacity respectively qx is the flow rate and AT is the temperature raise Above K is an empirical coef cient denoting the fraction of mechanical power converted into heat Typically K08 Substitution ofthe shear force f and flow rate qx into the equation above gives AT K39amp395TQ 5Toc Fa 13 2 pCph2 In the expression above the viscosity is evaluated at an effective temperature Teff which is taken as weghted average between the inlet or supply temperature and the calculated exit temperature Typic AT Te 39 Tinlet 7 14 2 I E In general for applications not generating very high hydrodyamic pressures GPa the lubricant g 1395 viscosity is an exponential decaying function of 3 temperature g 1 2 15 05 I I 2 3 4 Film thickness ratio h1h2 where Tref and ref are reference lubricant temperature and viscosity expectively Xv is a viscosity temperature coef cient Lubricant technical specification charts provide the lubricant viscosity at two temperatures 400 and 1000 Thus the viscositytemperature coef cient follows from for example T1 40 T2 100 p1 1n H1 1 32 H2 6 or H2 V T241 ocVTiTl lubT 1 6 Viscosity 0 40 60 80 100 Temp erature The bearing engineering design procedure follows an iterative procedure given the taper for bearing h2h1 surface velocity and applied load a assume exit lm thickness and effective temperature set effective viscosity and b calculate bearing reaction load ow rate shear force and temperature raise if bearing load gt applied load lm thickness h2 is too small increase h2 if bearing load lt applied load lm thickness h2 is too large reduce h2 c once b is satis ed check the effective temperature if same as prior calculted then process has converged OthenNisereset effective temperature calculate new viscosity and return to A MATHCAD worksheet is provided for you to perform the analysis ME626 LSA 0808 Figure 1 shows the geometry and coordinate system of a onedimensional Rayleigh step bearing The lm thickness h is a constant over each ow region namely the ridge or step and the lm land The bottom surface moves with velocity U The sudden change in film thickness generates a hydrodynamic pressure eld that supports an applied load w Note that the minimum lm thickness h2 is unknown and must be determined as part of the bearing design The bearing step height h1h2 needs be designed appropriately LL1 L2 L1 Step or ridge U Moving surface Fig 1 Geometry of Rayleigh step bearing Nomenclature Assum tions incompressible lubricant isoviscous L bearing length steady state operation dhdt0 width B gtgt L B bear39ng W39dth39 BgtgtL no fluid inertia effects H Iubricant viscosity rigid surfaces U surface speed hxfim thickness hl at inlet leading edge h2 at exit trailing edge Two coordinate systems X1 and X2 aid to formulate the analysis 0 S X S L h X h 1 1 1 1 h1 gt h2 Over each region Reynolds eqn for generation of hydrodynamic pressure p reduces to 3 h Uh d dp 0 2 dx 12p dx 2 Integration of Reynolds Eqn over each ow region step and land is straighfonNard since the lm thicknes is constant h Uh 3 L d 1 q h2 d U hz 3 2 X P2 qx where qx is the flow rate per unit width B This ow rate is contant and equal for the two zones Not that Eq 3 shows the pressure gradient to be constant over each region step and land thus the pressure varies linearly within each region as shown in Figure 2 for the step region the boundary conditions are A L 4 at X1 1 0 1310 0 L1 L2 at L 0 43 e X1 1 p1 pstep 39 39 while forthe land region the boundary conditions are pstep at X2 1 0 1320 pstep 4b at X2 L2 p10 0 where pstep is the pressure at the stepland interface 7 Note tha this pressure is also the highest within the lm X1 X2 flow region The step pressure is determined by equating the ow rates in Eqs 3 Fig 2 pressure profile for Rayleigh step bearing De ne the following dimensionless parameters film thickness step to land ratio step to land length ratio 5 The analysis determines the step pressure to equal 6 H U L2 oc 1 pstep 239Pstep0quot Pstep0quot 3 6 h 1 at 2 lt gt 2 1 and ow rate U Q a g 2 CPL 7 x X qx 2 2Qx l5 1a3 Note the peak pressure is largest at oc2 E Since the pressure is linear over each region integration of the pressure eld over the bearing surface is straightfonNard and renders the bearing reaction force opposing the applied load W Le 39 83 1 2 Pstep W Q 1300 dX L w J PX dXB 0 It is more meaningful to de ne the load in terms ofthe following land to total length ratio L Hence L 2 NY by and Y L W 6HU BL2Wyx Wyx h2 1a3i Y Figure 3 depicts the dimensionless load factor versus step to land ratio for four land to bearing length ratios 7 Clearly the maximum load occurs over a narrow range of film thickness ratios 15 to 225 and for land lengths around 30 of the total bearing length ie steps extending to 70 of the bearin length lDRayleigh step bearing 004 35 0 03 r s 39 tn 39 39I V 5 i s quot 4 39 g I s 5 3 002 i cc 0 0 t 39 quotu q 5 I quot 001 0101 138 176 213 25 288 325 363 4 Film thickness ratio h1h2 L L2L02 Y 2 L2L03 L quot39 L2L04 quot39 L2L05 Fig 3 Load factor for Rayleigh step bearing for three land lengths Using MATHCAD one can determine easily the optimum lm thickness ratio for a range of land lengtl v Gt1 Let Wyx 1oc3L lv Define g dw d 0 1 doc doc 1 x3 Y Y Y 3W2 on 1 a2 E guess oci Solve for g 0 ocopt rootgxyx123 Zgg 5 Comgare with sliderbearing a n 39l39 r A 4 ME626 LSA 0808 Figure 1 shows the simplest squeeze lm ow Considertwo circular rigid plates fully immersed ir a lubricant pool The plates are perfectly smooth and aligned with each other at all times The lm thickness separating the plates is a function oftime only The top circular plate radius R moves towards or away from the bottom plate with velocity Vdhc rate of change of lm thickness None ofthe plates rotates There is no mechanical deformation c the plates V Pambient Fig 1 Geometry of two circular plates for squeeze film flow Plates submerged in a lubricant pool Nomenclature Assumptions incompressible lubricant isoviscous unsteady operation dhdt ltgt 0 no fluid inertia effects no air entrainment rigid plates plates are submerged in a lubricant bath ht lm thickness only a function of time V ih top surface speed dt R plate outer radius H Iubricant viscosity The film thickness is NOT a function ofthe radial r or angular 9 coordinates Hence Reynolds equation in polar coordinate reduces to 1 A rst integral of Reynolds Eqn is straightfonNard qr 39 A 3 Note that qr O at r0 because there cannot be any ow in ot out of the center of the plates a uniqueness condition In addition the radial ow increases linearly with the radial coordinate being a maximum at rR qr gt0 flow leaves the plates if Vdhdtlt0 that is when the lm thickness is decreasing while there is lubricant in ow qrlt0 if Vgt0 when the film thickness increasing This last condition occurs if and only if the plates are submerged in a pool oflubricant OthenNise air entrains into the lm thus invalidating the major assumption for the analysis Substitution of 2 into 3 and integration leads to the pressure eld where Pa is the ambient pressure at the plate boundary rR Eq 4 shows that the pressure has parabollic shape with a peak value at the plate center r0 The peak pressure above ambient condition is Note that the peak pressure gt0 if Vlt0 lm thickness V 2 5 Ppeak 3 H 113 R decreasing Figure 2 and 3 show details of the pressure pro le and exit flow out or into the gap between the plates for the conditions of positive squeeze Vlt0 and negative squeze Vgt0 respectively outflow Vd hdt lt0 Pressure Pambient t r Fig 2 Positive squeeze film flow dhdt lt0 PgtPambient flow leaving gap Vd hdt gt0 inflow Pambient Pressure Fig 3 Negative squeeze film flow dhdt gt0 PltPambient inflow into gap The pressure acting on the plates generates a dynamic force F given by R F J P Pardr2n 6a 0 Substitution of4 into eq 6a renders where d 6b V 3th and ht a If V 0 then F 0 a squezee film flow cannot generateaforce unless V 2 0 b If V lt 0 then F gt 0 a support load opposite to the velocity of approach of both plates postive squeeze action 2 If V gt 0 then F lt 0 a load opposing the velocity of separation of both plates negative squeeze action Clearly c happens provided there is no lubricant cavitation since Ppeaklt0 This condition will not happen for sufficiently large ambient pressures In practice the needed pressure is too high to be of practical use Plate periodic motions ht ho Ahsinnt With frequency 03 are ofimportance Ah lt h0 in this case Vt Ahoacosltoatgt and the squeeze lm reaction force equals 4 D f39 th fll 39 Ah 3 R difngnesio lecssovgr gbles AH h Fo 339 H m39 2 T 03 0 ho then 131 F0 AH 0051 F gAHT 1AHsin c3 define gAHT AH 0051 1 AHsin c3 and graph the dynamic pressure eld for one period ofdynamic motion Zno Note how quicklyt squeeze llm pressure increass as the amplitude AH grows and approaches H1 Furthermore ti lm pressure albeit periodic has multiple super frequency components as AHgtgtO below note Chane in scale vertical t ONE period of motion 0 105 209 314 419 524 628 Notes 10 Thermohydrodynamic BulkeFlow Model in Thin 39ra dl UUH Hm mil Hm 39 a Incompressible liquids of large Viscosity mineral oils b Dominance of shear Couette ow hydrodynamic type c Fluid inertia and ow turbulence are usually NOT important low circumferential ow Reynolds numbers 1 Heat transfer to bushing statorand to shaft is important along with mechanical deformations induced by temperature e Fluid temperature gradient along aXial plane is negligible f Thermal effects are significant on bearing static load performance Thermohydrodynamic analyses are important for heaVily loaded hydrodynamic bearings such as tilting pad bearings a Process liquids have low Viscosity water R134 LH2 LOX b Material compressibility important LH2 c Large pressure drops along aXial direction with significant mass ow rates annular seals amp hydrostatic bearings up to 6000 psig in cryogenic turbopumps 1 Large heat capacity for transport of energy along aXial direction e Large rotor speeds well up to 100 krpm and higher induce large shear ow energy dissipation f Typically use macrotextured surfaces roughened stator to avoid generation of crosscoupled stiffness and to promote dynamic stability g Inlet uid ow circumferential swirl is important stability These operation characteristics determine the need to account for a Flow turbulence induced by shaft rotation and pressure driven ows b Fluid inertia effects temporal and advective types c Fluid properties from thermophysical equations of state 1 Accurate modeling of surface roughness effects e Two phase ow conditions at certain operating regimes Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 1 The uid flow within a thin film region see Fig1 is governed by the continuity mass conservation momentum and energy transport equations The smallness of the clearance to radius ratio cR allows for the reduction of the general governing equations to the simplified forms applicable in a thin film flow region x e 0 7239Dy e 0Hxztz e 0L Let ltUIVI15T be the uid velocity field components along the x y 2 directions and the uid pressure and temperature respectively xR6 circumferential z aXial y crossfilm coordinates UQR surface velocity L H ltlt LXI VY Vx V x 75D Figure 1 Geometry of ow region in a thin uid lm bearing Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 2 The governing equations are 6papUapV6pW0 1 6t 6x 6y 62 Mass conservation D6 615 61W C ircumferentialMomentum Transport p 2 Dt 6x 6y quot 6 AxialMomentum Transport p E 6 13 i 3 Dt 62 6y 613 CrosslemMomentum Equatzon 0 4 y EnergyTransport Equation Bird et Al 1960 pc E 2f w2i Ka T mE f i Jam my 5 p Dt 2Dt 6y 6y Dt 6x 62 6y W y where 23 i173wg 6 Dt 6t 6x 6y 62 is the material derivative The uid properties p a CPk t represent density Viscosity specific heat thermal conductivity and volumetric expansion coefficient respectively Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 3 Bulk ow primitive variables ie velocities and temperature represent average quantities across the lm thickness ie 7 Integration of Eqs 15 across the film thickness renders the bulk ow equations Continuity altpHgtMMO 6t 6x 62 6 HU2 Circumfer mp1quot 6pHU p 6pHUW Ha PT 9 6t 6x 62 6x W 6 HW2 Axial Momentum Mw Ha Pr 10 6t 6x 62 62 y Energy Transport C 6pHT I 6pHUT I 6pHWT 1 6pHVf I 6pHUVf I 6pHWVf I F ar39 6x 39 62 6t 39 6x 39 62 S 6P 6P 6P TH T lHU W RQ H 11 ar 6x 62 T yl Where K le W2 is the bulk ow speed and QShBT TBhJT TJ 12 is the heat flux from the film to the bounding bearing and journal surfaces at temperatures TB and TJ with hB and h as heat transfer convection coefficients to the bearing and journal surfaces The uid properties density viscosity and specific heat are functions of pressure and temperature ppPT uuPT CFCFPTetc 13 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 From the bulk ow theory for turbulence in thin film ows the wall shear stress differences are Hirs 1973 Launder and Leschziner 1978 r f ka kJE H 2 rzy f gkZW H 6P u xy where the turbulent shear parameters k k and kaB are local functions of the Reynolds numbers and friction factors based on Moody s friction factor formulae See Lecture Notes 8 Note that for the volumetric expansion coefficient i P p T 0 forincompressibleliquids t l foridealgases 15 Note that ET for LH2 is not in the range of 0 to l Substitution of the bulk ow momentum Eqs 910 into Eq 1 l and using the mass conservation principle Eq 8 renders a more suitable form of the energy transport equation Or after substitution of the wall shear stresses C 6pHT6pHUT 6pHWTQS p it 6x 62 x mpr aP 6P oRaP U W 2 6x H k U2 WZU k QR U 16b 6t 6x 62 2 4 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 5 This equation shows the energy balance in the uid film as CONVECTION DIFFUSIONCOMPRESSION WORK DISSIPATION Energy Disposed Energy Generated In annular seals and hydrostatic bearings the variation of temperature along the aXial direction and the energy needed for compression work are retained since the pressure drop across the seal or bearing can be quite large These conditions differentiate the present development from conventional THD analyses of incompressible uid fihn journal bearings for example P T p T U U B T 17 C pp i CFCF z tT A M pi 2 01R and U 18 MR is a characteristic flow speed due to pressure The subscript denotes characteristic values In dimensionless form the governing equations on the film lands become Continuity huhwaa6prh 0 19 Circumferential Mgmentum h k u k Res u Re th 6 Zhu w 20 Axial Momentum 4 fag w Res 611 w Re 51m wh wz 21 Energy Transport Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 a Re Fhf5p Re 66 mumgzhw me a 22 x E Rh 06 pu6 7w6 7 ThA6 k vfluA kJ lAZ uA 6139 6x 62 26x h 2 4 The dimensionless ow parameters are 2 012 kg U U 71C pwc2 pUc c ReS 039Rep Rep ReFRep E 23 M M The frequency 039 and speed A numbers denote the importance of squeeze fihn and shear ow effects relative to the pressure induced ow respectively The reference Reynold numbers Re p denotes the ratio of uid advection forces to Viscous ow induced forces In hydrostatic bearings and annular seals the large pressure differentials can generate ow turbulence even without journal rotation The Eckert number E denotes the ratio of kinetic energy to heat convection in the uid film The ratios Re p E or ResE5 represent the effect of heat convection relative to shear dissipation In the bulk ow model the heat transfer from the uid film to the bounding surfaces is QShBT TBhJT TJ A1 Where hB and hJ are the heat transfer convection coefficients to the bearing and journal surfaces respectively These coefficients are determined from the ReynoldsColburn analogy between uid friction and heat transfer Hohnan 1986 The average heat transfer over the entire laminarturbulent boundary is St 5033 f2 AZ where Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 7 S h Stanton number A3 Cp 2 C 50 2 Prandtl number A4 fa 1 6 Lbm 2 A5 m m H R2 39 Is the Fanning friction factor based on Moody s equation From the relationships above the heat transfer coefficient is 1 2 3 h EpC K fso A6 and by analogy 1 2 3 1 2 3 113 5pCFVBfBsox h 3pcpmso A7 Where VBJ and 1 J are the uid velocities and friction factors relative to the bearing and journal surfaces In actuality the archival literature presents many other formulas empirically based for turbulent flow heat transfer coefficients Holman 1986 The above expressions are used because of their simplicity and ability to include surface roughness effects Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 8 Consider the unsteady flow in the thin film lands of a uid film bearing or seal The governing bulk ow equations are Continuity pHU1JHO i12 1 I kaxkJi Momentum H6 P 2 pHUipHUU 2 6x H 6t 1 6x1 1 iJ391x2 y a 6 6P 6P HaP Energy C HT HUT HT U QR Fiat 6x10 j Q3 at 79ch 26x 2 2 k UUZ k U 2R 3 H y 2 4 where i j x y are the circumferential and aXial coordinates K QR Ky 0 journal surface speeds kkykj are turbulent shear parameters The uid properties density Viscosity specific heat and thermal expansion are thermodynamic variables ie PPPTPT CF CFPT 5031 with QS hB T TB hj T TJ as the heat flow convected into the bearing and journal surfaces The film thickness is typically given by HcxyXtcos Ytsin6 4 where cxy is the radial clearance distribution and X t and Y t are time dependent journal center displacements Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 9 Continuity Exhu y7huyaho 5 C ircumferenlial momentum A kxux k 7 h Res 0 huxRe hup huxuy 63 Axialmomentum apkyUy a a a 2 ha TResgphuyRep altphufuygtaphuy 6b Where xxR39 yYR39 tra39 hHc uUV uy UyV A 2RV VcZPsauR39 amRV pltP PP Zpp Eat Cpl Rep RepE EC 2 is a typical advection flow Reynolds number Res pquot mo Re 0 is a typical squeeze lm Reynolds number and a is a characteristic u whirl frequency typically equal to the rotational frequency or that of an external excitation The flow domain is divided into a number of staggered control volumes CV as shown in Figure 2 Each control volume encloses a particular flow variable circumferential and axial velocities pressure and temperature as a nodal quantity denoted by its P value The boundaries of the C V are surfaces through which flow comes in or out The control volumes are surrounded by nodal variables denoted as East West North and South The notation defines with lowercase the uxes mass energy or momentum through the surfaces of the C Vs ie east west north and south Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 10 ILI 39F39PTr 51 Slammed mid and whim whims 17 F 5 1 EIMUMBrEw al limit 1 IDEle 39Il39Oilll39ll lE Axial valncilry mmml vulumi Figure 2 Depiction of staggered control volumes for integration of flow equations Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 The axial momentum transport equation 6 b is integrated over the aXial velocity Vcontrol volume shown in Figure 2 PE 1 V igva B Atlall wallamu1 wnlml vulums 3 h idxdy kyu 1 dxdyiReS 631Fhuy Re huxuy dxdy 7 Consider the following approximations for the various terms in Eq 7 2 n a 2 n j h dxdy j h apL dx thpS pP5xV 8a ie assume the pressure is uniform over the south and north faces of the control volume with an average uniform film thickness evaluated at the center of the Vcontrol volume V HA3 dxdy if VP 5xV5yV 8b w s P ie assume an average film thickness viscosity and turbulent shear coefficient ky for the whole control volume For the momentum flux terms Assume uniform circumferential ows Fhu EyV across the east and west faces ii wuxuywxdwgwmyW am And a uniform aXial flow across the south and north faces of the Vcontrol volume 2 n a 2 n n huyuy dxdy p huyuy dep huyuy S ExV 8 1 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 For the temporal unsteady term 2 n a a 2 n 8 IIEphuydxdy Ejjphuydxdy e Since the control volume size is fixed in space Thus Eq 7 over the Vcontrol volume becomes ky V V V a 2 V n V h VP 5x 5y Resa Tjjphuydxdy Rep phuxuy W 5y phuyuys 5x P 9 th pSpP6xV Before proceeding further integration the continuity equation 1 over the Vcontrol volume gives 2 V n V a 2 quot N phuJWEy phuys 5x 0 phdxdy O 10 The flow rates across the faces ensw of the control volume are denoted by F Zhuj 5y F Zhuj 5f 11 FHV Ehuy 1 ExV F Fhuy 5 ExV With these definitions Eq 10 is rewritten as a 2 n V V V V F FW Fn FS aa TJV phdxdy O 12 which establishes a balance of ows in and out through the Vcv faces and equaling the rate of mass accumulation within the cv The momentum flux terms in equation 9 are treated using the upwind scheme of Launder and Leschziner 1978 This scheme establishes a selection of velocity based on whether the flow is 1m or out of the face in a control volume For example F V V Zhuxuyf 5 EV u e P if F gt 0 where FV 50 2 w a FZV VE if FZV lt O 2 That is if flow leaves the e face Fegt0 it carries the upstream velocity VP On the other hand if flow comes into the eface it carries the downstream velocity VE This procedure is known as UPWINDING Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 13 Define the following operator a0maxa0 b Then statement a can be conveniently written as F 0 VP 4210 V5 Hence the momentum uxes are written as Zhu uy W 5yV FWV j FWV0VW FWV0VP Zhu uy 5y Fjuj Fj 0VP EV0VE EhuyuyaxV FSVSuj 1 V0VS 1 V0VP 13 Zhuy My 6 Ffu 1V0VP EV0VN The differences in momentum uxes in Eq 9 are thus written as Zhux uy 5yV Ehuy uy ExV FZV u FWVuV FnVuy Fsyuj VP EV0 FWV0EV0 1 V0 14 FV0 V FV0VW EV0VN FSV0VS w Let a Re FV0 a ReFWV0 aRe Ff0 a Re 133 15 Using the identities l 1 610 Elaa a0 El aa The following relationship for the RHS of Eq 14 is obtained F Fj0 FWV FWV0Ff 1f0 FSV0 F FW Fn FSV 163 1 Rep a2 61 61 aJVv And using the discrete form of the continuity equation Eq 12 F1 enol lemlw1 mcolmwl anol a 2 n 1 16b o39a T p hdxdy Re gal Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 14 where nb refers to nodes e w n s on each of the surfaces bounding the control volume Substitution of Eq 16b into the axial momentum equation 14 gives k hpspp6xy 1 V JP ExVEyV Res dxdy Rev worn FactMOM Jacob 17 Re ll Frolic FV 0VW Ff0lVN lEV OlllS W 5 And substituting Eq 16b V kl a hZpspp5xyaLTnb 2 J ga iVPResngPWygtdXdy P 13 a VP Res hdxdy Hence the difference form of the axial momentum transport eguation is k V hzltpS ppgt6xVZam if ZaLlVPReSFh6xV6 6in 19 r117 n17 T P A suitable approximation for the unsteady term time derivative is needed An implicit rstorder scheme is used ie aVPNVPVP 6139 N Ar where VPis the axial velocity at time tAt and V is the axial velocity at time t respectively For the scheme to be implicit all field variables velocity and pressure in Eq 19 must be evaluated also at time t At Finally the discrete form of the axial momentum transport equation is given as V V V Res Elly gxygyy V hp pspP6x Zananb A VP aPVP r117 where V ExVEyV Ar P V a ExVEyV 2 an Re5 H17 P Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 15 Integration of the circumferential momentum transport Eq 63 over the Uvelocity control volume shown in Figure 2 and using the continuity equation to simplify some terms same as for the Vequation leads to the following algebraic equation I U Clrcumlaramlal Walecm minim vudluma U k R u h3ltpP pg5yUagUEaIEVIUWagUSaZUN ExUEyUAif phpUP5xU5yUaUP P 21 Where U U at time t and U ky UExUEyU aU ay ExUEyUReS ph P g1 l P Ar a Re U0 agReFWU0 agRe FHU0 agReFSU0 EEGWXYEV Fhuxw yquotE Fhuyquot6xv FS huy56xU Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 The difference equations for momentum transport are summarized as U p 22 V V V V V V P 3 P nb nb P 4 P P P 23 hp p5xZaVSSV aV nb where aV Icy VaxVa VZaVSV39 SV Re Wyw P h y quotb r117 r a 139 s p P AT P aU 1 05an UZaUSU39 SURe WM P h y b r117 r a r s p P AT P n P U SUth aanyU 3130 P In general F Fhux5y Fw Fhux 5 r w where r U or V Integration of mass conservation Eq 5 on the pressure control volume Pcv shown in Figure 2 if Fun 7 leads to Pg 1 where the ow rates across the faces of the Pcv are Fe FWP Ff Ff maimzh xdy 0 24 TWS Pressure mnlmi wlum Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 17 Fj hu25yp hZUP6y FPhP5PhPU 5 u y I 1 w y 25 FnPphuxn 5xPph UNExP The unsteady integral term is approximated as 62quot a P P P6hp P6513 p p E phdxdy ta 1php5x 5y pPa hp 6 5x 5y which implies that the size of the Pcv is fixed in space and that the values of film thickness h and density 2 are uniform within the control volume The algebraic form of the continuity equation establishes the flow balance on a finite size control volume as M P FZP FWP P EP0395xP5yPJ h 0 26 Since h x y r cos 9 rsin t9 then gm 0 6139 a at ost9Psmt9P where the time derivatives of g and ie journal center position are given by the solution of the rotorbearing equations of motion Also a P P P M where p pt and p ptAr 6139 A1quot The continuity equation in the Pcv is thus written as FJ FJ Ff F a xP6yP Zihwhp fl 0 27 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 18 Let the ow variables be expressed as UEu Vl7v39pp39 23 where the current velocities U V satisfy the momentum equations but not the mass continuity equation u v and p are correction fields Substituting Eq 28 into the momentum equations 22 and 23 leads to 3613 EE5yU Zag l nb S j SEU r117 29 hf3l p39P p39E5yU 2057 39nb aZUp a u39P r117 Then hi5 p P p 55yquot Zalbu39nb a 391 303 rib and identically from the aXial transport equation hi pis pip6xy 26th GZV P r117 Introducing the SIMPLEC procedure Van Doormal amp Raithby 1984 U U V V Zanbu 3911 N Z anbu 39P7 2 anbv 3911 N Z anbv 39P Equations 30 become dU V WP P517 P pf dz zhggyv ap Z anb dV v v v33 PS pf dghV5xV 32 GP Z anb Where P V ExVEyV 261 STV P Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 U ExUEyUZaZ7STU 3 Res 2 h ExZ iyv Sf Res Eh LEV L U dU V dV et D P P P 613 2615 P aZ Zanyb Substitution of the correction fields Eq 28 into the continuity Eq 27 gives P P RF P 1wF WP17nF nP ISF SPo6xp ypEphphpe0 33 T P P F2phux6y ph UPEyP Where 34 F5 phuy 5x phs VP Exp etc F Fhu P 5 FhDZp Pp g5ypafp Pp g 0 Fh 5WD F 3033 Exp 3h DZ p sp p5xp a p sp P ai 3h 5WD 35 F W ap39W p P F a P39p P39N P Let S 77 75 1775 7 05xP5yP FIJI hp palm 36 Then Eq 33 becomes F Z F WF n F S 373 or apiPpiEafpiWp PapiPpiNapisp PEE 515193 Zafbp39nb EP 37c nb where aj a5 aW aN aSP 2a 38 H17 Note that if p39nb p39P 0 then S1quot 0 and mass continuity is satisfied Thus the momentum equations are also satisfied and conversion of the flow field is achieved Once the perturbed pressure field p39P is obtained is obtained correction to the circumferential u and aXial v fields are performed using Eqs 27 In the numerical procedure the pressure is typically underrelaxed as Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 20 pnew pold apv Unew Uold uv Vnew Void vv 39 with a as a relaxation parameter whose value is typically less than one The dimensionless equation for turbulent bulk ow energy transport is Yang 1992 EpRe a EpRequot a Re 5 hT F huT huT F E 61p E5 6xp x 6yp y E5 Q5 2 06 puxa puya p thThA6 p kxuujiuxlky A uXA 6139 6x 6y 2 6x h 2 4 where xXR yYR tra hHc uXUXU uyUyU AQRU amRU TTT pP PaPsa E5 UfY C is the Eckerd number Re Rep cR ptM UtRc2 is a modi ed Reynolds number and Res pt 4 wc2 is the squeeze lm Reynolds number as 28 f TBZJ f TJ is the heat ow to bearing andjournal surfaces TLcv T I 421quotquot Fun 39 i TIMMPMEIUIB mutual whmme Define the following source terms 1 6x 2 6x h Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 40 39a zz zT r 2 S 06 pura puya p th S2hA6 p kt uu2lux kJ A uxA 41 6139 By y 2 4 21 Integration of Eq 40 over the temperature control volume Tcv shown above leads to cpElje IT Vhfdxdy CF Rep Uhuf dyi5huy dx 126 Q AxAy S1 fpszAxAy 42 Implementation of the upwind scheme for the thermal ux transport terms gives Muff AyFZ i 10 is 10TE Emir Ay FEW FW0TW FW0TP 43 Muff Ax Fnofp Fn0 Muff Ax 1133 FS0TP where E ZhuxAy Fw ZhuxAyW F ZhuyAx FS ZhuyAxs are the momentm uxes through the controlvolume faces Using these expressions the LHS of equation 42 is rewritten as 5 Res 6 MSW FEE EphTdxdyZanbTp Zaannb if M if 44 F F F Ff Rep Q AxA EC 2 w n S P E S y EpRep Ep Re Ep Re Ep Re where 50 mi moi Moles The discrete form of the continuity equation on the Tcv gives EFwEE0I J gammy 46 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 22 Substitution of Eq 46 into 44 gives LHsm ENE j j thdxdyzanbfp awn QSAxAy E Trcv E a E h 47 5 3713 Resj jw at dxdy Re aj j Zhdxdy Trcv Since Res Re 0 the last two terms on the RHS of the preVious expression add to zero ie continuity equation Then 5 Re a Requot LHSM s IT VphdedyZ awn Zaannb E QSAxAy 48 The integral form of the energy transport Eq 42 becomes a Res a Re E5 TL3 h dar xdyZanb TP Zaannb E Q AxAy 49 S1AxAyTp 32 AxAy airy F Let and Q TP 113 hJ hBTB hJTJ With Tp as the fihn temperature 6139 A1quot in the preVious time step Then the discrete form of Eq 49 becomes a Res Ephp Re 2 am E ETAxAy E m hJAxAy S1AxAy TP t 50 CResph Re Zaannb S2AxAypE E AxAyTp E hm hJTJAxAy The algebraic form of the energy transport equation is finally written as aiTpa TEaVTwaISTNa TsSf 51 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 23 a 4220 a CF Rep 420 E E where 523 Re Re a7 FE P Ff 6137 p PH Ff a a aVTVa a S S POHS 3 52b Sf S 1ng 353 S20T2 52c Requot s EEP h3hJAxAy 039 p p S zphp mywpm pwAyVPpn PAx 52d 5 Re E h S7 p 5 1 P AxA P3 EC AT y Z 1 A2 S51 kxuuj3uxlkj T uXA AxAy P A 32 3hppg pwAy 52e ST 123sz Ay C3 E 5 Res E h S54 FE AxAyTp Note the terms involving Ar correspond to unsteady ow conditions As for the source term SleAy p l S1AxAy goes into aP if S1AxAygt 0 or 2 SleAy TP into the source term ofthe RHS if S1AxAy lt 0 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andres 2007 24 POSTCRIPT 2006 The CFD numerical method detailed above was quite popular in the 1980s and throughout the mid 1990s Nowadays however CFD methods use efficiently non staggered and nonstructured grids and implement very fine meshes large number of nodal points without incurring into excessive computational costs The CFD field has rapidly evolved thanks to the ever increasing speeds and low cost of computers References Hirs GG 1973 A BulkFlow Theory for Turbulence in Lubricant Films ASME Journal of Lubrication Technology pp 137146 Launder BE and Leschzinger M 1978 Flow in Finite Width Thrust Bearings Including Inertial Effects ILaminar Flow IITurbulent Flow ASME Journal of Lubrication Technology Vol 100 pp330345 Van Doormall JP and Raithby GD 1984 Enhancements of the SIMPLE Method of Predicting Incompressible Fluid Flows Numerz39cal Heat Transfer Vol 7 pp 147163 Yang Z L San Andres and D Childs 1995 quotThermohydrodynamic Analysis of Process Liquid Hydrostatic Bearings in Turbulent Regime Part I The Model and Perturbation Analysisquot ASME Journal of Applied Mechanics Vol 62 3 pp 674679 Yang Z L San Andres and D Childs 1995 quotThermohydrodynamic Analysis of Process Liquid Hydrostatic Bearings in Turbulent Regime Part II Numerical Solution and Resultsquot ASME Journal of Applied Mechanics Vol 62 3 pp 680684 San Andres L 1995 quotThermohydrodynamic Analysis of Fluid Film Bearings for Cryogenic Applicationsquot AIAA Journal of Propulsion and Power Vol 11 5 pp 964972 San Andres L Yang Z and Childs D 1993 quotThermal Effects in Cryogenic Liquid Annular Seals I Theory and Approximate Solutionsquot ASME Journal of Tribology Vol 115 2 pp 267276 Yang Z San Andres L and Childs D 1993 quotThermal Effects in Cryogenic Liquid Annular Seals II Numerical Solution and Resultsquot ASME Journal of Tribology Vol 115 2 pp 277284 San Andres L 1992 quotAnalysis of Turbulent Hydrostatic Bearings with a Barotropic Fluidquot ASME Journal of Tribology Vol 114 4pp 755765 San Andres L Yang Z and Childs D 1993 quotImportance of Heat Transfer from Fluid Film to Stator in Turbulent Flow Annular Sealsquot WEAR Vol 160 pp 269277 San Andres L 1991 quotAnalysis of Variable Fluid Properties Turbulent Annular Sealsquot ASME Journal of Tribology Vol 113 pp 694702 Notes 10 TH BulkFlow Model Thin Film Lubrication Dr Luis San Andre39s 2007 25

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