1 A Novel Computational Model for Tilting Pad Journal Bearings with Soft Pivot Stiffness Yujiao Tao Research Assistant Dr. Luis San Andrés Mast-Childs Professor 32 nd Turbomachinery Research Consortium Meeting TRC 32514/15196B Year II May 2012

2 2 Justification Pivot flexibility reduces the force coefficients in heavily loaded tilting pad journal bearings (TPJBs). XLTRC 2 T FPBRG code shows poor predictions for TPJB force coefficients. W, static load Y X Housing Pad Pivot Fluid film Journal , Journal speed , Pad tilt angle   Research objective: To develop a code, benchmarked by test data, to predict the K-C-M coefficients of TPJBs. Code accounts for thermal energy transport and the (nonlinear) effects of pivot flexibility.

3 3 Tasks completed Completed derivation of reduced frequency force coefficients for TPJBs Developed iterative search scheme to update pad radial and transverse displacements Constructed GUI for new TPJB code as per XLTRC 2 standards Compared predictions from the TPJB code to test data

4 4 Completed derivation of reduced frequency force coefficients for TPJBs Includes pivot NL deformations Developed sound iterative search scheme to update the pad radial and transverse displacements Constructed GUI for new TPJB code as per XLTRC 2 standards Compared predictions from the enhanced TPJB code to test data Tasks completed

5 5 Film thickness in a pad C p : Pad radial clearance C B = C p -r p Bearing assembled clearance R d = R p +t : Pad radius and thickness r p : Pad dimensional preload  p : Pad tilt angle  piv  piv  Pivot radial and transverse deflections Unloaded Pad Journal P  X Y pp   pp  h e RJRJ WXWX RBRB  piv OBOB RPRP WYWY OPOP  piv P’ pp t LL Bearing Center Pad Center Fluid film Loaded Pad Film thickness:

6 6 Laminar flow Includes temporal fluid inertia effects Average viscosity across film On k th pad h : fluid film thickness P : hydrodynamic pressure μ : lubricant viscosity  : journal speed R J : journal radius Reynolds equation for thin film bearing Y X Housing Pad Pivot Fluid film Journal  , Journal speed W, static load

7 7 Thermal energy transport in thin film flows T : film temperature h : film thickness U,W : circ. & axial flow velocities  C v  : viscosity & density, specific heat h B, h J : heat convection coefficients T B, T J : bearing and journal temperatures  : journal speed Neglects temperature variations across- film. Use bulk-flow velocities and temperature CONVECTION + DIFFUSION= DISSIPATION (Energy Disposed) = (Energy Generated)

8 RR Upstream pad Journal 8 Pad inlet thermal mixing coefficient Inlet thermal mixing coefficient (0<  <1) is empirical parameter. Downstream pad FhThFhTh FsTsFsTs F in T in F s, F h, F in Volumetric flow rates T s, T h, T in Fluid flow temperatures ~ for conventional lubricant feed arrangements with deep grooves and wide holes. small (<< 1) for TPJBs with LEG feed arrangements and scrappers. Hot oil Mixing oil Cold oil Is  constant for all conditions?

9 9 Nonlinear pivot deflection & stiffness Pivot deformation is typically nonlinear depending on the load (F piv ), area of contact, hardness of the materials, surface conditions. Sphere on a sphere Pivot and housing: E p, E h Elastic modulus p, h Poisson’s ratios D p, D h Diameter of the curvature L Contact length Sphere on a cylinder Cylinder on a cylinder Load-deflection function (empirical) * Kirk, R.G., and Reedy, S.W., 1988, J. Vib. Acoust. Stress. Reliab. Des., 110(2), pp

10 Tasks completed Completed derivation of reduced frequency force coefficients for TPJBs Developed sound iterative search scheme to update the pad radial and transverse displacements Constructed GUI for new TPJB code as per XLTRC 2 standards Compared predictions from the enhanced TPJB code to test data Convergence to the pad and journal equilibrium positions

11 Iterative scheme to find pad equilibrium position Journal displacement, pivot radial and transverse displacements converge to equilibrium solution for TPJB with flexible pivots Flow chart of iterative scheme Set initial journal center displacements Check convergence on loads (W 0 ) Find tilt angle for k th pad Estimate the initial pivot radial displacement for k th pad Find pivot radial and transverse displacements for k th pad Find tilt angle for k th pad Update e Check convergence on k th pad tilt angle and pivot displacements End of procedure Check convergence on loads (W 0 ) Update e Take TPJB with flexible pivots, find pads tilt angles, radial and transverse displacements and journal eccentricity Estimate the pivot radial displacement and journal eccentricity Take TPJB with rigid pivots, find pads tilt angles and journal eccentricity

12 Tasks completed Completed derivation of reduced frequency force coefficients for TPJBs Developed sound iterative search scheme to update the pad radial and transverse displacements Constructed GUI for new TPJB code as per XLTRC 2 standards Compared predictions from the enhanced TPJB code to test data Fortran program and Excel GUI

13 Excel GUI and Fortran code Modifications/enhancements to XL PRESSDAM ® code FEM to solve Reynolds equation (hydrodynamic pressure) Uses control volume method to solve energy transport eqn Excel GUI Parameters of pivot: type, radii of contact & material properties (E  Different pads: geometry parameters Sphere on a sphere Cylinder on a cylinder Sphere on a cylinder Rigid pivot Load-deflection function

14 Tasks completed Completed derivation of reduced frequency force coefficients for TPJBs Developed sound iterative search scheme to update the pad radial and transverse displacements Constructed GUI for new TPJB code as per XLTRC2 standards Compared predictions from TPJB code to test data Bearings tested by Childs and students (TurboLab)

15 Predictions for a four-pad TPJB (Childs and Harris*) Four pad, tilting pad bearing (LBP) * Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, Number of pads, N pad 4 ConfigurationLBP Rotor diameter, D101.6 mm (4 inch) Pad axial length, L101.6 mm (4 inch) Pad arc angle,  P 73 o Pivot offset65% Pad preload,0.37, 0.58 Nominal bearing clearance, C B 95.3  m (3.75 mil) Measured bearing clearance, C B 54.6  m (2.15 mil) 99.6  m (3.92 mil) Pad inertia, I P 7.91kg.cm 2 (2.70lb.in 2 ) Oil inlet temperature~40 o C (104 o F) Lubricant typeISO VG32, DTE 797 Oil supply viscosity,  Pa.s Specific load, W/LD 0 kPa-1,896 kPa (275 psi) Journal speed,  4 krpm-12 krpm Cold conditions

16 C B =55  m 16 Predictions for a four-pad TPJB * Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX. Specific load, W/LD 1.9 MPa (275 psi) Journal speed,  4 krpm-12 krpm Lubricant arrangements: Spray bar blocker, By pass cooling Pivot type: Ball-in-socket pivot Measured pivot stiffness: 350 MN/m Nominal cold bearing clearance C B =100  m C B =55  m C B =100  m Measured cold bearing clearances on Pads #1 and #3 are ~40% smaller than the nominal cold clearance. X Y 2.45 o  p = 73 o W Measured cold bearing clearance C B =95  m Pad 1 Pad 4 Pad 2 Pad 3 X Y W Pad 1 Pad 4 Pad 2 Pad 3

17 Journal eccentricity vs. static load -e Y eXeX Rotor speed  =10 krpm Rotor speed  =6 krpm Predicted journal eccentricity correlates well with measurements. Symbols: test data Lines: prediction * Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, e Y eXeX Max. 275 psi

18 Film temperature rise vs. static load measured pad sub-surface temperature rise at pad trailing edge Rotor speed  =6 krpm X Y W Pad 3 Pad 2 Pad 4 Pad 1 Trailing edges Input: inlet thermal mixing coefficient = 0.5 At 6 krpm, film temperature rises at pad trailing edges are considerable even with no load applied Pad 1 Pad 2 Pad 3 Pad 4 * Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX. Symbols: test data Lines: prediction Max. 275 psiOil Inlet temperature ~40 o C

19 Film temperature rise vs static load measured pad sub-surface temperature rise at pad trailing edge X Y W Pad 3 Pad 2 Pad 4 Pad 1 Trailing edges Input: inlet thermal mixing coefficient = 0.95 Film temperatures underpredicted at 10 krpm. Film heats little with load Rotor speed  =10 krpm Pad 1 Pad 2 Pad 3 Pad 4 * Harris, H., 2008, Master Thesis, Texas A&M University, College Station, TX. Symbols: test data Lines: prediction Max. 275 psi Effectiveness of spray bar blocker diminishes Oil Inlet temperature ~40 o C

20 Stiffness coefficients Rotor speed  =10 krpm Rotor speed  =6 krpm =0.5 =0.95 K YY K XX =K YY, TFB BRG K piv =350 MN/m Very soft pivot produces ~ constant K’s, invariant with load and speed. Symbols: test data Lines: prediction * Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, TFB BRG code in XLTRC 2 - TPJB model with rigid pivot K XX Prediction K XX =K YY K YY K XX =K YY, TFB BRG K XX Max. 275 psi

21 Rotor speed  =10 krpm Rotor speed  =6 krpm =0.5 =0.95 Soft pivot renders nearly constant damping coefficients. Good correlation with test data Damping coefficients Symbols: test data Lines: prediction * Childs, D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, C YY C XX =C YY, TFB BRG C XX C YY C XX =C YY, TFB BRG C XX Synchronous speed coefficients Prediction C XX =C YY Max. 275 psi

22 Predictions for a five-pad TPJB (Wilkes and Childs*) Five pad, tilting pad bearing (LOP) * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Specific load W/LD: 3,132 kPa (454 psi) Journal speed  4.4 krpm-13.1 krpm X Y Pad W Journal Fluid film

23 Pivot stiffness & hot bearing clearance Pivot load-deflection function Pivot stiffness Hot bearing clearance Rocker back pivot Pivot stiffness-deflection function C B,cold -C B,hot =  (T hot -T cold ) Bearing clearance decreases due to thermal expansion of the rotor and pad surfaces.  =  m/ o C Pivot radial force Pivot radial stiffness * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. EXPERIMENTAL Hot bearing clearance C B : 48  m~58  m Nominal cold C B =68  m Empirical

24 Pad bending stiffness Pad bending stiffness* k pad =5.4644×10 4 M p ×10 5 (N.m/rad) M P,Pad bending moment from fluid film * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Equivalent pad-pivot stiffness: series pivot + pad bending Derived from experiments Pad ½ length  RPRP MPMP MPMP OPOP Used in code TPJB

25 Journal eccentricity slightly under/over predicted at the low/high rotor speeds. * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Journal eccentricity vs. static load Symbols: test data Lines: prediction -e Y Max. 454 psi Rotor speed  =13.1 krpm Rotor speed  =4.4 krpm c B = 57  m<-e Y c B = 49  m~-e Y =0.8 =0.9 cold C B =68  m

26 Rotor speed  =13.1 krpm 26 Static stiffness coefficients over predicted at large specific loads K YY K XX K YY K XX K-C-M model * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Static stiffness coefficients Symbols: test data Lines: prediction TPJB Dashed: Wilkes preds. Max. 454 psi Rotor speed  =4.4 krpm =0.8 =0.9

27 Damping coefficients vary little with static load. C YY C XX C YY C XX * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Damping coefficients Max. 454 psi =0.8 =0.9 Rotor speed  =13.1 krpm Rotor speed  =4.4 krpm Synchronous speed coefficients Symbols: test data Lines: prediction TPJB Dashed: Wilkes preds.

28 Large negative virtual masses at 4.4 krpm. Dynamic stiffness increases with frequency. M YY M XX M YY M XX * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Virtual mass coefficients Rotor speed  =13.1 krpm Rotor speed  =4.4 krpm =0.8 =0.9 Symbols: test data Lines: prediction TPJB Dashed: Wilkes pred.

29 Real part At 4.4 krpm, dynamic stiffness HARDENs at frequencies  2 . Re( Z XX ) Re( Z YY ) Synchronous frequency Re( Z XX ) Re( Z YY ) Synchronous frequency Re(Z)=K-M  2 * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Impedances Specific load = 227psi Rotor speed  =13.1 krpm Rotor speed  =4.4 krpm Symbols: test data Lines: prediction TPJB Dashed: Wilkes pred.

30 At 4.4 krpm, predicted damping is frequency dependent for  >2  Im( Z XX ) Im(Z YY ) Synchronous frequency Im( Z XX ) Im( Z YY ) Synchronous frequency Ima(Z)=C  * Wilkes, J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX. Impedances Rotor speed  =13.1 krpm Rotor speed  =4.4 krpm Imaginary part Specific load = 227psi Symbols: test data Lines: prediction TPJB Dashed: Wilkes pred.

31 Predictions for other test TPJBs Kulhanek and Childs, 2012, ASME, J. Eng. Gas Turbines Power, 134, TPJB code delivers good predictions for stiffness and damping by estimating the (actual) hot bearing clearances and using a constant pivot radial stiffness. In the works. Will be part of forthcoming Y. Tao M.S. Thesis (TRC report 2012) Delgado et al., 2010, ASME Paper GT TPJB code takes TPJB pivots as rigid and uses hot bearing clearances. Predicted stiffness and damping correlate well with test data (45 psi). Tschoepe and Childs, 2012, not yet published TPJB code uses measured pivot load-deflection function and hot bearing clearances. Predicted stiffness and damping are in agreement with test TPJB data.

32 Conclusions For TPJBs with very soft pivots (K piv <<K film ), pivot stiffness determines bearing stiffness. Film temperatures at no load condition are high. At high rotor speeds (> 10 krpm), LEG and spray bar blockers have less effectiveness in cooling a bearing. Bearing & pad clearances change due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea. A-priori knowledge of pivot stiffness and bearing & pad clearances is required to obtain accurate predictions of TPJB performance. Bearing & pad clearances change a lot due to thermal expansion & mechanical deformation of the rotor & pad surfaces. Using nominal cold bearing & pad clearances is a BAD idea.

Proposal to TRC (2 years) Hydrodynamic pressure P Pad surface elastic & thermal deformations change bearing & pad clearances K  = P + C∆T K, Pad stiffness matrix P, Fluid film pressure vector C, Mechanical-thermal stiffness matrix , Pad displacement vector Objective: Enhance TPJB code to accurately predict pad surface deformations Hot oil flow Pivot constraint FE pad structural analysis by Yingkun Li

34 Proposed work Build a 3-D FE model of commercial pads ( ANSYS® or SolidWorks® ) to obtain pad stiffness matrix. Reduce model with active DOFs, perform structural modal analysis for easy off-line evaluation of pad surface deformations and pivot deflections. Implement oil feed arrangements (LEG, spray bar blockers etc.) in the FE model Construct new Excel GUI and Fortran code for XLTRC 2 Digest more test data and continue to update predictions using enhanced code.

35 TRC Budget Year III Support for graduate student (20 h/week) x $ 2,200 x 12 months$ 26,400 Fringe benefits (0.6%) and medical insurance ($197/month)$ 2,522 Travel to (US) technical conference$ 1,200 Tuition & fees three semesters ($227/credit hour)$ 9,262 Other (PC+software+storage supplies)$ 1, Year III$ 40,984 Enhanced TPJB code will model current (commercial) TPJBs and improve predictions of force coefficients with minimum User expertise for specification of empirical parameters Year III

36 Questions (?)