Transcript A Novel Computational Model for Tilting Pad Journal

32nd Turbomachinery Research Consortium Meeting
A Novel Computational Model
for Tilting Pad Journal Bearings
with Soft Pivot Stiffness
May 2012
Yujiao Tao
Research Assistant
Dr. Luis San Andrés
Mast-Childs Professor
TRC 32514/15196B
Year II
1
Justification
Pivot flexibility reduces the force
coefficients in heavily loaded tilting
pad journal bearings (TPJBs).
Y
Pivot
Fluid film
W , Journal speed
XLTRC2 TFPBRG code shows poor
predictions for TPJB force
coefficients.
Housing
X
W, static load
Journal
Pad
d, Pad tilt angle
x
h
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.
2
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
XLTRC2 standards
• Compared predictions from the TPJB code
to test data
3
Tasks completed
• 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
XLTRC2 standards
• Compared predictions from the enhanced
TPJB code to test data
4
Film thickness in a pad
Cp : Pad radial clearance
CB = Cp-rp Bearing assembled clearance
Y
Rd= Rp+t : Pad radius and thickness
rp : Pad dimensional preload
dp : Pad tilt angle
xpiv, hpiv : Pivot radial and transverse
Journal
W
WY
RB
Pad Center
OP
OB
e
p
Fluid film
Bearing Center
X
RJ
h
RP
h

xpiv
dp
t
P’
P
L
Unloaded Pad
deflections
WX
Loaded Pad
Qp
x
hpiv
Film thickness:
h  C p  eX cos   eY sin  
x
piv
 rp  cos    p   h piv  Rd d p  sin    p 
5
Reynolds equation for thin film bearing
• Laminar flow
• Includes temporal fluid inertia
effects
• Average viscosity across film
Y
Housing
Pivot
Fluid film
W, static load
X
Journal
Pad
W , Journal speed
On kth pad
1   h3 P    h3 P   h W  h  h 2  2 h



 

2
RJ  12   z 12 z  t 2  12 t 2
h : fluid film thickness
μ : lubricant viscosity
RJ : journal radius
P : hydrodynamic pressure
W : journal speed
6
Thermal energy transport in thin film flows
T: film temperature
h : film thickness
U,W: circ. & axial flow velocities
, , Cv : viscosity & density, specific heat
hB, hJ : heat convection coefficients
TB, TJ : bearing and journal temperatures
W : journal speed
Neglects
temperature
variations acrossfilm. Use bulk-flow
velocities and
temperature
 


 Cv 
 U h T    W h T   hB T  TB   hJ T  TJ 
z
 R Q

2
2 2

12 
WR 
WR   CONVECTION + DIFFUSION= DISSIPATION
2

 U 
 W 


h 
12
2   (Energy Disposed) = (Energy Generated)

7
Pad inlet thermal mixing coefficient
Inlet thermal mixing coefficient l (0< l <1) is empirical parameter.
Fin  Fs  l Fh
FinTin  FsTs  l FhTh
Journal
Hot oil
Fs , Fh , Fin Volumetric flow rates
Ts , Th , Tin Fluid flow temperatures
WR
Mixing oil
Fin
Tin
Fh
Th
Upstream pad
l ~0.6-0.9 for conventional lubricant feed
arrangements with deep grooves and wide holes.
Downstream pad
Fs
Ts
Cold oil
l small (<< 1) for TPJBs with LEG feed arrangements
and scrappers.
Is l constant for all conditions?
8
Nonlinear pivot deflection & stiffness
Pivot deformation is typically nonlinear depending on the load (Fpiv),
area of contact, hardness of the materials, surface conditions.
Pivot and housing:
Ep , Eh Elastic modulus
Dp , Dh Diameter of the curvature
Contact length
2
2
 1  P2 1  H2  DH  DP



EH  DP DH
 EP
x piv
1

K piv Fpiv
• Sphere on a cylinder
x piv
2 Fpiv (1   2 )  2
4 LEDP DH ( DH  DP ) 

  ln

2
 LE
3
2.15
F
piv


x piv

D  DP
 0.52 1  3 H
DH

•Cylinder on a cylinder
*Kirk,
L
x piv  1.04 3  Fpiv 
• Sphere on a sphere
• Load-deflection function
(empirical)
np ,nh Poisson’s ratios
Fpiv  a0  a1x piv  a2 x piv   a3 x piv   a4 x piv 
2
3
4
  Fpiv   1  P2 1  H2 

 3


D
E
EH 
P

P

2
K piv 
R.G., and Reedy, S.W., 1988, J. Vib. Acoust. Stress. Reliab. Des., 110(2), pp. 165-171.
Fpiv
x piv
9
2
Tasks completed
• Completed derivation of reduced
frequency force coefficients for TPJBs
• Developed sound iterative search scheme to
update the pad radial and transverse
displacements
Convergence to the pad and
journal equilibrium positions
• Constructed GUI for new TPJB code as per
XLTRC2 standards
• Compared predictions from the enhanced
TPJB code to test data
10
10
Iterative scheme to find pad equilibrium position
Journal displacement, pivot radial and transverse displacements
converge to equilibrium solution for TPJB with
Set initial journal center displacements
Update e
Find tilt angle for kth pad
flexible pivots
Take TPJB with rigid pivots, find
pads tilt angles and journal
eccentricity
Check convergence on loads (W0)
Estimate the initial pivot radial
displacement for kth pad
Find pivot radial and transverse
displacements for kth pad
Update e
Find tilt angle for kth pad
Check convergence on loads (W0)
Check convergence on kth pad tilt
angle and pivot displacements
Estimate the pivot radial
displacement and journal
eccentricity
Take TPJB with flexible pivots, find
pads tilt angles, radial and
transverse displacements and
journal eccentricity
End of procedure
Flow chart of iterative scheme
11
11
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
Fortran program and
Excel GUI
• Compared predictions from the enhanced
TPJB code to test data
12
12
Excel GUI and Fortran code
• Modifications/enhancements to XLPRESSDAM® 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,n
• Sphere on a sphere
• Cylinder on a cylinder
• Sphere on a cylinder
• Rigid pivot
• Load-deflection function
Different pads:
geometry parameters
13
13
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)
14
14
Predictions for a four-pad TPJB
(Childs and Harris*) Four pad, tilting pad bearing (LBP)
Specific load, W/LD 0 kPa-1,896 kPa (275 psi)
Journal speed, W
4 krpm-12 krpm
Number of pads, Npad
Configuration
Rotor diameter, D
Pad axial length, L
Pad arc angle, QP
Pivot offset
Pad preload, rP
Nominal bearing clearance, CB
Measured bearing clearance, CB
Pad inertia, IP
Oil inlet temperature
Lubricant type
Oil supply viscosity, 0
*Childs,
4
LBP
101.6 mm (4 inch)
101.6 mm (4 inch)
73o
65%
0.37, 0.58
95.3 m (3.75 mil)
54.6 m (2.15 mil)
99.6 m (3.92 mil)
7.91kg.cm2 (2.70lb.in2)
~40 oC (104 oF)
ISO VG32, DTE 797
0.032 Pa.s
Cold conditions
D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
15
15
Predictions for a four-pad TPJB
Specific load, W/LD 1.9 MPa (275 psi)
Journal speed, W
4 krpm-12 krpm
Lubricant arrangements:
Spray bar blocker, By pass cooling
Pivot type: Ball-in-socket pivot
Measured pivot stiffness: 350 MN/m
Y
Qp=73oY
CB=95 m
CB=100 m
CB=55 m
Pad 4
Pad 3
2.45o
W
W
X
CB=55 m
Pad 1
X
Pad 2
CB=100 m
Nominal cold bearing clearance
Measured cold bearing clearance
Measured cold bearing clearances on Pads #1 and #3 are
~40% smaller than the nominal cold clearance.
*Harris,
H., 2008, Master Thesis, Texas A&M University, College Station, TX.
16
16
Journal eccentricity vs. static load
Max. 275 psi
X
W
-eY
80
60
eX
40
100
20
0
0
400
800
1200
1600
2000
Specific Load (kPa)
Rotor speed W =6 krpm
Journal Displacement
(m m)
Journal Displacement
( m)
100
Y
Symbols: test data
Lines: prediction
80
-eY
60
40
eX
20
0
Predicted journal eccentricity
correlates well with
measurements.
*Childs,
0
400
800
1200
1600
2000
Specific Load (kPa)
Rotor speed W =10 krpm
D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
17
17
Film temperature rise vs. static load
Oil Inlet temperature ~40oC Max. 275 psi
40
Trailing Edge (oC)
Temperature Rise at Pad
measured pad sub-surface temperature rise at pad trailing edge
30
Pad 1
Pad 3
20
Pad 4
Trailing
edges
Y
Pad 2
Pad 3
X
W
10
Pad 4
0
0
400
800
1200
1600
Specific Load (kPa)
Rotor speed W=6 krpm
Input: inlet thermal mixing coefficient l=0.5
*Harris,
Symbols: test data
Lines: prediction
Pad 1
Pad 2
2000
At 6 krpm, film temperature rises
at pad trailing edges are
considerable even with no load
applied
H., 2008, Master Thesis, Texas A&M University, College Station, TX.
18
18
Film temperature rise vs static load
measured pad sub-surface temperature rise at pad trailing edge
Oil Inlet temperature ~40oC Max. 275 psi
Trailing Edge (oC)
Temperature Rise at Pad
40
Pad 3
30
Pad 2
Symbols: test data
Lines: prediction
Pad 1
Pad 4
Trailing
edges
Y
20
Pad 3
X
W
10
Pad 1
Pad 4
Pad 2
0
0
400
800
1200 1600
Specific Load (kPa)
2000
Film temperatures
underpredicted at 10 krpm.
Film heats little with load
Rotor speed W=10 krpm
Input: inlet thermal mixing coefficient l=0.95
*Harris,
Effectiveness of spray
bar blocker diminishes
H., 2008, Master Thesis, Texas A&M University, College Station, TX.
19
19
Stiffness coefficients
TPJB Stiffness Coefficients
(MN/m)
1000
Prediction KXX=KYY
Y
Rotor speed W=6 krpm
KXX =KYY, TFBBRG
800
l=0.5
X
W
TFBBRG code in XLTRC2TPJB model with rigid pivot
600
KXX
Symbols: test data
Lines: prediction
KYY
400
Kpiv=350 MN/m
200
0
400
800
1200
1600
2000
Specific Load (kPa)
Max. 275 psi
Rotor speed W=10 krpm
TPJB Stiffness Coefficients
(MN/m)
1000
l=0.95
800
KXX =KYY, TFBBRG
600
KYY
KXX
400
Kpiv=350 MN/m
Very soft pivot
produces
~ constant K’s,
invariant with
load and
speed.
200
0
400
800
1200
1600
2000
Specific Load (kPa)
*Childs,
D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
20
20
Damping coefficients
TPJB Damping Coefficients
(kN.s/m)
1000
800
l=0.5
CXX =CYY, TFBBRG
Symbols: test data
Lines: prediction
Y
Rotor speed W=6 krpm
X
W
600
CYY
400
Synchronous speed
coefficients
200
CXX
0
0
400
800
1200
Specific Load (kPa)
1600
Max.2000
275 psi
800
TPJB Damping Coefficients
(kN.s/m)
Prediction CXX=CYY
Rotor speed W=10 krpm
l=0.95
CXX =CYY, TFBBRG
600
400
CXX
CYY
200
0
0
400
800
1200
1600
Soft pivot renders
nearly constant
damping coefficients.
Good correlation with
test data
2000
Specific Load (kPa)
*Childs,
D.W., and Harris, H., 2009, ASME, J. Eng. Gas Turbines Power, 131, 062502
21
21
Predictions for a five-pad TPJB
(Wilkes and Childs*) Five pad, tilting pad bearing (LOP)
Fluid film
Y
W
X
Journal
Pad
Number of pads, Npad
Configuration
Rotor diameter, D
Pad axial length, L
Pad arc angle, QP
Pivot offset
Cold pad preload, rP
Cold pad clearance, CP
Cold bearing clearance, CB
Pad inertia, IP
Pad mass, mP
Oil inlet temperature
Lubricant type
Oil supply viscosity, 0
Pivot type
Inlet thermal mixing coefficient
5
LOP
101.6 mm (4 inch)
55.9 mm (2.2 inch)
58.9o
50%
0.44
120.7m (4.75 mil)
68 m (2.67 mil)
1.81 kg.cm2 (0.85 lb.in2)
0.385 kg (0.849 lb)
~36.7 oC (98.1 oF)
ISO VG32, DTE 797
0.035 Pa.s
Rocker back
0.8, 0.9
Specific load W/LD: 3,132 kPa (454 psi)
Journal speed W: 4.4 krpm-13.1 krpm
*Wilkes,
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
22
22
Pivot stiffness & hot bearing clearance
Pivot stiffness
Rocker back pivot
Pivot stiffness-deflection function
EXPERIMENTAL
14
1000
Pivot Radial Stiffness
(MN/m)
Pivot Radial Force (kN)
Pivot load-deflection function
12
10
Pivot radial force
8
6
4
2
800
600
400
200
0
Pivot radial stiffness
0
0
5
10
15
20
Pivot Radial Deflection (m)
0
5
10
15
20
Pivot Radial Deflection (m)
Hot bearing clearance Empirical
CB,cold-CB,hot=a(Thot-Tcold)
a=0.396
m/oC
Hot bearing clearance CB: 48 m~58 m
Bearing clearance
decreases due to thermal
expansion of the rotor and
pad surfaces.
Nominal cold CB=68 m
*Wilkes,
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
23
23
Pad bending stiffness
Pad bending stiffness* Derived from experiments
OP
kpad =5.4644×104Mp+1.1559×105 (N.m/rad)
RP
MP ,Pad bending moment from fluid film
Mp  
L /2 Q P
L /2 QT
  P sin  R d  dZ    P sin  R d  dZ
2
p
 L /2 Q L
MP

MP
2
p
 L /2 Q P
Equivalent pad-pivot stiffness: series pivot + pad bending
2
1
1
l


keq k piv k pad
Pad ½ length
*Wilkes,
Used in code TPJB
l  12 QP RP
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
24
24
Journal eccentricity vs. static load
Journal Displacement (m m)
cold CB=68 m
80
Rotor speed W =4.4 krpm
W
cB= 57m<-eY l=0.8
60
40
X
-eY
20
0
0
Journal Displacement (m m)
Y
500
Symbols: test data
Lines: prediction
1000 1500 2000 2500 3000 3500
Max. 454 psi
80
Rotor speed W =13.1 krpm
60
-eY
40
20
0
0
500 1000 1500 2000 2500 3000 3500
cB= 49m~-eY
l=0.9
Journal eccentricity slightly
under/over predicted at the
low/high rotor speeds.
Specific Load (kPa)
*Wilkes,
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
25
25
Static stiffness coefficients
Symbols: test data
Lines: prediction TPJB
Dashed: Wilkes preds.
TPJB Stiffness Coefficients
(MN/m)
K-C-M model
Y
800
Rotor speed W =4.4 krpm
600
W
l=0.8
KYY
X
400
200
KXX
0
0
500 1000 1500 2000 2500 3000 3500
Max. 454 psi
TPJB Stiffness Coefficients
(MN/m)
800
Rotor speed W =13.1 krpm
KYY
l=0.9
600
400
200
KXX
0
0
*Wilkes,
500
1000 1500 2000 2500 3000 3500
Specific Load (kPa)
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Static stiffness
coefficients over
predicted at large
specific loads
26
26
Symbols: test data
Lines: prediction TPJB
Dashed: Wilkes preds.
Damping coefficients
Y
400
Rotor speed W =4.4 krpm
300
CXX
W
X
l=0.8
200
Synchronous speed
coefficients
100
CYY
0
0
500
Rotor speed W =13.1 krpm l=0.9
1000 1500 2000 2500 3000 3500
Specific Load (kPa)
Damping coefficients
vary little with static
load.
TPJB Damping Coefficients
(kN.s/m)
TPJB Damping Coefficients
(kN.s/m)
Max. 454 psi
400
300
CXX
200
100
CYY
0
0
*Wilkes,
500
1000 1500 2000 2500 3000 3500
Specific Load (kPa)
27
27
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
Virtual Mass Coefficients (kg)
Virtual mass coefficients
100
Rotor speed W =4.4 krpm
50
Y
l=0.8
MYY
0
W
X
-50
-100
MXX
-150
0
Virtual Mass Coefficients (kg)
Symbols: test data
Lines: prediction TPJB
Dashed: Wilkes pred.
500
1000 1500 2000 2500 3000 3500
100
MYY
50
Rotor speed W =13.1 krpm
l=0.9
0
-50
MXX
-100
-150
0
500
1000 1500 2000 2500 3000 3500
Large negative virtual
masses at 4.4 krpm.
Dynamic stiffness increases
with frequency.
Specific Load (kPa)
*Wilkes,
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
28
28
Impedances
Real part
Re(Z)=K-Mw2
Re( ZYY)
800
Real part of impedance
coefficients Re(Z ) (MN/m)
Symbols: test data
Lines: prediction TPJB
Dashed: Wilkes pred.
Specific load = 227psi
700
Rotor speed W =4.4 krpm
Y
Synchronous frequency
600
X
W
500
400
300
200
100
Re( ZXX)
50
100 150 200 250 300
Excitation Frequency (Hz)
At 4.4 krpm, dynamic
stiffness HARDENs at
frequencies w > 2W.
350
Real part of impedance
coefficients Re(Z ) (MN/m)
0
0
Rotor speed W =13.1 krpm
800
700
Synchronous frequency
Re( ZYY)
600
500
400
300
200
100
Re( ZXX)
0
0
*Wilkes,
50
100 150 200 250
Excitation Frequency (Hz)
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
300
29
29
350
Impedances
Symbols: test data
Lines: prediction TPJB
Dashed: Wilkes pred.
Specific load = 227psi
Imaginary part Ima(Z)=Cw
Imaginary part of impedance
coefficients Im (ZXX ) (MN/m)
500
400
Y
Rotor speed W =4.4 krpm
Im( ZXX)
Synchronous frequency
W
300
200
Im(ZYY)
100
Rotor speed W =13.1 krpm
0
50
100 150 200 250 300
Excitation Frequency (Hz)
At 4.4 krpm, predicted
damping is frequency
dependent for w>2W
350
Imaginary part of impedance
coefficients Im(Z ) (MN/m)
0
500
Synchronous frequency
Im( ZYY)
400
300
Im( ZXX)
200
100
0
0
*Wilkes,
50
100 150
200 250
300 350
Excitation Frequency (Hz)
30
30
J.C., 2011, PhD. Thesis, Texas A&M University, College Station, TX.
X
Predictions for other test TPJBs
Kulhanek and Childs, 2012, ASME, J. Eng. Gas Turbines Power, 134,
052505 1-11.
• TPJB code delivers good predictions for stiffness and damping by
estimating the (actual) hot bearing clearances and using a constant pivot
radial stiffness.
Delgado et al., 2010, ASME Paper GT2010-23802
• 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.
In the works. Will be part of forthcoming Y. Tao M.S. Thesis
(TRC report 2012)
31
31
Conclusions
• For TPJBs with very soft pivots (Kpiv<<Kfilm), 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
due
thermal
expansion
Bearing &
& pad
pad clearances
clearances change
change a
lotto
due
to thermal
&
mechanical
deformation
of the rotorof
& the
padrotor
surfaces.
expansion
& mechanical
deformation
& padUsing
nominal
bearing
& pad
clearances
a BAD
idea. is a
surfaces.cold
Using
nominal
cold
bearing & is
pad
clearances
BAD idea.
•A-priori knowledge of pivot stiffness and bearing &
pad clearances is required to obtain accurate
predictions of TPJB performance.
32
32
2012 Proposal to TRC (2 years)
Objective: Enhance TPJB code to accurately
predict pad surface deformations
Pad surface elastic &
thermal deformations
Hot oil flow change bearing & pad
clearances
Hydrodynamic pressure P
Kd = P + C∆T
K, Pad stiffness matrix
P,
Fluid film pressure vector
C, Mechanical-thermal stiffness
Pivot constraint
FE pad structural analysis by Yingkun Li
d,
matrix
Pad displacement vector
33
33
Proposed work 2012-2013
• 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 XLTRC2
• Digest more test data and continue to update
predictions using enhanced code.
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TRC Budget
2012-2013 Year III
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)
$
1,600
2012-2013 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.
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Questions (?)
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