A Linear Fluid Inertia Model for Improved Prediction of Force

Modern Lubrication – Appendix to Notes 11 & 13

A Linear Fluid Inertia Model for Improved Prediction of Force Coefficients in Grooved Squeeze Film Dampers and Grooved Oil Seal Rings

Luis San Andres Adolfo Delgado Texas A&M University

© 2009

Bearing & seal dynamic reaction forces FX  Z  FY



rotor

  K XX  K   YX

K XY  K YY  B

 X   C XX     Y   C YX

C XY  C YY  B

 X    Y 

X

Stiffness coefficients



 Y

X

Damping coefficients

Typically: Small clearances (c/R < 0.001) Re* mass of solid steel journal [1]

Squeeze film damper oil inlet lubricant film

nti-rotation pin

Feed groove

shaft

ournal



ball bearing housing

Discharge groove

seal

In aircraft gas turbines and compressors, squeeze film dampers aid to attenuate rotor vibrations and provide mechanical isolation.

Lubricant film

Common configurations include grooves and recesses to supply oil and to prevent oil starvation in squeeze film lands.

Known issue: Poor predictions of fluid film added masses for grooved SFDs

Added mass coefficient - Literature review

Grooved SFDs

San Andrés, 1992 (Predictions) - SFD with a shallow groove behaves at low frequencies as a single land damper - Groove influences dynamic force coefficients

Arauz and San Andrés, 1994 (Prediction and experiments) - Dynamic

pressure levels at the groove (cg/c < 10) and film land are of the same order of magnitude - Model underestimates the radial force and overestimates the tangential (damping) force at the damper film land.

Qingchang et al., 1998 (Predictions and experiments) - Experimental results show that radial (inertial) force is underpredicted by a

factor of three.

Added mass coefficient - Literature review (cont.)

Grooved SFDs

Lund et al., 2003 (Predictions and experiments) Fluid inertia force coefficient is overpredicted (up to 70 %) for increasing groove volumes

Kim and Lee, 2005 (Predictions and experiments) Predictions of the inertia coefficient correlate well experimental data while damping coefficients are underestimated (two-stage liquid seal)

San Andres and Delgado, 2006 (experiments) Added mass coefficients are underpredicted by classical model (5 times smaller)

Current models do not properly predict both damping and added mass coefficients in grooved SFDs

Oil seal rings

In centrifugal compressors oil seals are commonly used to prevent leakage of process gas into ambient.

- Locked oil seal rings can induce instability in compressors.

- Seals are grooved to reduce cross-coupled stiffness and lower lock-up forces.

Oil supply (PS+P)

Anti-rotation pin Seal loading spring Inner seal

Outer seal Process Gas (PS)

Pa

Outer seal land

Inner seal land Shaft

Typical oil seal multi-ring assembly

Smooth oil seals- Experimental results Parallel oil seal configuration [1]

Childs et al., (2006, 2007)

Parallel seal configuration (balance thrust force due

to pressure drop across the seals)

Includes ‘deep’ inlet (central) groove to feed seals

Parameter identification: FSEAL=

1/2 FTest conf.

Predictions do not consider groove or fluid inertia effects (Zirkelback and San Andrés 1996)

Results

Oil supply

17 mm

Seal lengt

Force coefficients are well predicted (C,K) except added mass coefficients

Large added mass coefficients (~15 kg)

25 mm

136c 76c

Added mass predictions using Classical model (Reinhardt & Lund – c Journal [1] Graviss, M., 2005, “The Influence of a Central 1975) (single land- i.e. not including inlet groove) (2.84 kg) Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., College Station, TX.

Compare smooth and grooved seals of short length Damping, cross-coupled stiffness & added mass 2L

C XX  8

  D L3

c

flow

4c

3

; K XY

1   D L3   C XX ; M XX  8 2 20 c

Journal Constant pressure

C XX 

  D L3 4c

3

; K XY

1   D L3   C XX ; M XX  2 20 c

2-land seal: (deep groove divides lands) L

30c 5c

Coeffs are ¼ of original seal

c

C XX  2

  D L3 4 c3

; K XY

1   C XX ; M 2

XX

2

  D L3 20 c

Grooved oil seals- Literature review

Predictive models

Semanate and San Andrés, (1993)

- Bulk flow equation model

- Grooves should reduce force coefficients by a factor of four, i.e.

Kxy (1 land)= 4 Kxy(2 lands) Cxx (1 land)= 4 Cxx(2 lands)

-Fluid inertia effects not predicted (considered negligible)

Baheti and Kirk, (1995) - Reynolds and energy equation (Finite element solution) - Grooves should effectively isolate seal lands - Cross-coupled stiffness and damping coefficients are reduced by ~60 % for grooved configurations

Grooved oil seals- Experimental results

Childs et al., (2006)

Parallel oil seal configuration [1]

Single inner land groove and multiple groove oil seal (single clearance)

Childs et al., (2007)

One inner land groove with groove depths (5c,10c,15c)

Results Oil supply

17 mm

25 mm

Force coefficients are underpredicted (grooved seal)

Seal length

Inner land groove does not effectively separate seal lands [1] Graviss, M., 2005, “The Influence of a Central Groove on Static and Dynamic Characteristics of an Annular Liquid Seal with Laminar Flow,” M.S. Thesis, Texas A&M Univ., 136c College Station, TX.

Large added mass coefficients (~30 kg)-

Higher than for ‘smooth’ seal

76c

Journal

c Inner land groove

Test oil seal Predictions Inlet groove not considered (null dynamic pressure)- Null added mass coefficients

Inner land groove should reduce crossed-coupled stiffness and direct damping coefficients by a factor of four

Experiments

≠

Large added mass coefficients

≠

Groove does not effectively separate seal lands At most:

Kxy (1 land)= 2 Kxy(2 lands) Cxx (1 land)= 2 Cxx(2 lands)

Kxy (1 land)= 4 Kxy(2 lands) Cxx (1 land)= 4 Cxx(2 lands)

Null (neglected) added mass coefficients

≠

Large added mass coefficients, increasing with increasing groove depth

Need for better predictive models

Fluid flow predictive model

• Bulk flow, centered operation, incompressible fluid

• Qualitative observations of laminar flow field • Boundary Conditions • Characteristic groove depth

oil supply, Ps feed plenum groove

mid-land groove

Ps- Pd >0 Pd :discharge pressure

Pd z y

Streamlines in axially symmetric grooved annular cavity.

Pd

Groove effective depth

Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.

Ps= supply pressure Pa= ambient pressure

Ps

CFD simulations show: streamline separating flow regions IS a physical boundary delimiting the domain for squeeze film flow due to journal radial motions.

Pa Test seal Laminar flow

10c 15c Inner land groove close up (CFD -Pressure driven flow)

Linear fluid inertia model

No fluid inertia advection Oil supply

1

zI

I

2

zII

II

3

III

zIII

4

IV

zIV

n

N

n+1

zN

In each flow region:

Reynolds equation   3  P  h x x

   3  P  h   z   z 

with temporal fluid inertia       12  h 6  R    h  t x 

   h 2 

2  h  2  t   I , II ,...N

Perturbation flow analysis x= R

Centered operation eX , eY < < c

Film thickness h = c + e

i t

 eX

cos (  ) + eY sin (  )

h



R

e

Pressure field P = P0 + e i t

 e

Flow field q x = q x0 + e i t  q z = q z0 + e i t 

PX + eY PY

X



 X

 e  e

X

q xX + eY q xY

X

q z X + eY q zY









 Circumferential  Axial 

View of rotating and whirling journal

For each individual flow region

 I , II ,...N

Y

Boundary conditions

Journal centered operation

Ps= supply pressure Pa= ambient pressure Oil supply

Laminar flow

Ps

Pa

1

2

I

zI

Null axial flow rate (geometrical symmetry)

zII

4

3

IV

III

II

zIII

First-order pressures and axial flow rates must be equal

zIV

n

N

n+1

Pa

zN

No generation of dynamic pressure

Pressure field (PX) Modified local squeeze film Reynolds number

Zeroth order (Static Pressure)

P0 a s z

 c2 Re*  12 

First order (Dynamic Pressure) PX ( z )  f X ( z ) cos( )  g X ( z ) sin( )  R z z f X ( z )  c f cosh      s f sinh     12 i 3 1  i Re* R  c   R 2

;

First order solution

6 R z z g X ( z )  cg cosh      sg sinh     R  c3   R

Force Coefficients K  K

XX YX

-  2M -  2M

XX YX

(c f , cg , s f , s g )

N + i C XX  2    + i C YX   1

L 2  R

  0

0





2

Obtained from boundary conditions

  f X ( z  ) c o s( ) 2    d x   d z 2    g X  ( z  ) sin ( )  

  R N   2   1   R 

L

 0

L

 0

 f X  ( z ) d z     g X  ( z ) d z  

Test Squeeze film damper

Oil inlet

Oil inlet plenum

Journal diameter: 127 mm Clearance= 127 m

Inlet groove

Flow in

6 mm

y Squeeze film land Journal

78 c

c

z

Inlet groove

I

II

25 mm

Film land

Shaft Discharge groove

31 c

III

Discharge groove

4 mm 2 in (50.8 mm)

Sealed-end SFD assembly cut view.

Grooved SFD

Mass coefficient [kg]

Damping coefficient [kN.s/m]

200

Good correlation with experimental results for both damping and added mass coefficients

150 100 50 0 1 30

Dynamic

Experiments 5

10

15

20 6 mm

y z

I

78 c

20

c

10

31 c

1

5

10

15

Inlet groove-to-clearance ratio (cI/c)

pressure

20

II

Inlet groove

25 mm

III

Film land

Discharge groove 4 mm

Test grooved oil seal

Inlet

z x=R x=R 

Oil supply

17 mm

25 mm 2 mm

Clearance= 86 m

Seal length Discharge plenum

136c 76c

Journal diameter: 117 mm

Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.

Journal

c (0-15) c

Parallel oil seals Configuration [Childs et al]

Buffer seal

200 Best predictions

150 100

Mass coefficient [kg]

Damping coefficient [kN.s/m]

Smooth seal (inlet groove)

Damping decreases rapidly as groove effective depth increases (~1/c

Experiments

50 0 1 60

5

10

15

20

I

136c

II

c

Best predictions

Mass coefficient less sensitive to groove effective depth (~1/c)

40

20

0 1

Experiments Classical model prediction 5

10

15

Inlet groove-to-clearance ratio (cI/c)

20

Grooved seal (mid or inner land groove)

Damping coefficient [kN.s/m]

200

c 12c I c7c

150

I

Best predictions

Experiments

100

cIIIc cIII= 16c

50 I

Mass coefficient [kg]

0 1 60

5

10

Best predictions

Experiments

I

cIII= 16c cIIIc

20

0 1

5

10

III

IV

c

15

c 12c I c7c

40

II

Mid-land groove-to-clearance ratio (cIII/c)

15

Best correlation with experimental results for groove effective depth at ~ 50% of groove physical depth

Oil seal (grooved and ungrooved) Cross-coupled stiffness [kN/m]

60 Kmodel (c Kxy(model)

I

=11c, cIII =c

(no inner land groove)

40

kxy -kyx

cIII= cII (exp.)

Kmodel (cI Kxy(model)

(16c groove)

20

0 4000

=11c, cIII =7c

kxy -kyx

6000

8000

cIII=cII +15cII (exp.)

10000

Rotor speed [RPM]

Model effectively predicts reduction of cross-coupled stiffness due to inner land groove.

Cross-coupled stiffness

I

II

III

IV

c

Finite element solution

Off-centered journal operation

Film thickness

x= R

h = h0 + e i t

 eX

cos (  ) + eY sin (  )

Reynolds eqn. with temporal fluid inertia

h R

 e

e

X

   3  P   h    z    z    22 22 12   h   6 R  hh   h   hh 22  t x tt   3  P  h x x

 Y

 

Test grooved oil seal

Clearance= 86 m

Outlet plane

Journal diameter= 117 mm

h0

4 1

Inlet

e 3

2

e

Groove s z  0 Inlet plane

z

x=R x=R 

Childs, D. W., Graviss, M., and Rodriguez, L. E., 2007, “The Influence of Groove Size on the Static and Rotordynamic Characteristics of Short, Laminar-Flow Annular Seals,” ASME J. Tribol, 129(2), 398-406.

Oil supply

17 mm

2

25 mm 2 mm

Seal length Discharge plenum

136c 76c

Journal

c (0-15) c

Parallel oil seals Configuration [Childs et al]

Buffer seal

Seal operating conditions

Supply pressure: 70 bar 1.0

Test data

Y

Shaft speed: 10,000 rpm

Smooth smooth seal seal

0.8

Grooved seal

Load

Static eccentricity ratio: 0, 0.3, 0.5, 0.7

Grooved seal 0.5

 y

0.3

-1.0

X

0.0 -0.8

-0.5

-0.3

0.0

0.3

-0.3

-0.5

-0.8

-1.0

x Journal locus

0.5

0.8

1.0

Journal center locus indicates seal operates with oil cavitation at the largest test eccentricities

Seal Leakage

10,000 rpm, 70 bar

0.45 0.45

Smooth Seal Grooved Seal

0.40 0.40 Leakage [kg/s]

0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20

Smooth seal- Predictions

0.15 0.15

Smooth seal- Experiments

0.10 0.10

Grooved seal- Predictions (c = 7c)

0.05 0.05

Grooved seal- Experiments (cg= 16c)

0.00 0.00 00

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

Static journal journal eccentricity eccentricityratio ratio (e/c) (e/c) Static

0.6 0.6

Predicted leakage correlates well with experiments for both smooth land and grooved seal

Direct damping Damping [kN.s/m]

300

10,000 rpm, 70 bar

Cxx

200 0

Smooth Seal

100 0

Grooved Seal

0 0

0.2

0.4

Model predicts accurately reduction in direct damping due to inner land groove.

0.6 0

Static journal eccentricity ratio (e/c)

Damping [kN.s/m]

300

Cyy

200

Smooth Seal 100

Grooved Seal

Smooth seal- Predictions Smooth seal- Experiments Grooved seal- Predictions (c = 7c) Grooved seal- Experiments (cg= 16c)

0 0

0.2

0.4

Static journal eccentricity ratio (e/c )

0.6

Cross-coupled stiffness

70 bar

60

Eccentricity ratio=0

Stiffness [MN/m]

50

Smooth Seal

40 30

Grooved Seal

20

Kxy

10 0 0

2000

4000

6000

8000

10000

12000

Rotor speed (RPM)

60

Eccentricity ratio=0.3

Stiffness [MN/m] [MN/m] Stiffness

50

Smooth Seal

40

Grooved Seal

30 20

= 0, 0.3

Model effectively predicts reduction in cross-coupled stiffness due to midland groove. Smooth seal- Predictions Smooth seal- Experiments Grooved seal- Predictions (c = 7c) Grooved seal- Experiments (cg= 16c)

Kxy

10 0 0

2000

4000

6000

8000

Rotor speed (RPM)

10000

12000

Added mass Experimental data shows relatively large added mass coefficients. Predictions correlate well with experimental results.

10,000 rpm, 70 bar 40

Mxx

Added Mass [kg]

35

Grooved Seal

30 25

Smooth Seal

20

Added mass coefficients are larger for grooved seal

15

Classical theory [1] predicts ~ 1/10 of test value

10 5 0 0

0.1

0.2

0.3

0.4

0.5

Static journal eccentricity ratio (e/c)

0.6

Smooth seal- Predictions Smooth seal- Experiments

[1] Reinhardt, F., and Lund, J. W., 1975, “The Influence of Fluid Inertia on the Dynamic Properties of Journal Bearings,” ASME J. Lubr. Technol., 97(1), pp. 154-167.

Grooved seal- Predictions (c = 7c) Grooved seal- Experiments (cg= 16c)

Conclusions:

• Damping and cross-coupled stiffness decrease rapidly as the effective groove depth increases(~1/c3).Added mass coefficients less sensitive to effective depth (~1/c).

• Predicted force coefficients (K,C,M) correlate well with experimental data for a narrow range of groove effective depths

• For shallow and short mid-land groove (oil seal) predicted (K,C,M) correlate best with test data when using 50% of actual groove depth.

• In oil seals, an inner land groove does not uncouple adjacent film lands!!

Conclusions (cont):

 The dynamic pressure field in deep grooves may be difficult to measure for most practical excitation frequency ranges. (fluid inertia pressures are proportional to 2).

 Deep grooves do generate dynamic pressures of mainly inertial nature, which lead to large added mass coefficients.

 There is a specific groove depth where the added mass coefficient peaks. This groove depth value for the studied configurations is < 10c.

 Force coefficients in test configurations, like SFDs and oil seals, are a function of the ancillary geometries like feeding/discharge arrangements.