Static and dynamic analysis of liquid and gas annular seals

Mihai ARGHIRProfessor, Fellow of the ASME

Institut Pprime (P’)Université de Poitiers, France

Paris

Nantes

Bordeaux

Toulouse Marseille

Lyon

Grenoble

LilleStrasbourg

Poitiers

University of Poitiers: 25000 students on three campuses

Institute P »: 270 permanent employees and as many PhD students, National Research Council’s second largest institute

Cryogenic turbopump

Shapiro, W., Hamm, R., "Seal technology for liquid oxygen (LOX) turbopumps", NASA CR·174888, MTI85TR20, 1985.

Fr

Ft

h R

ω

Ω

u

w

L

e

p0 p z

pi

pe

+

+

=

−1Y

1X

YYYX

XYXX

1Y

1X

YYYX

XYXX

1Y

1X

YYYX

XYXX

1Y

1X

e

e

MM

MM

e

e

CC

CC

e

e

KK

KK

F

F

&&

&&

&

&

I. Static problem: )M(PP &∆∆ =

RCWM m πρ 2=&mass flow rate:

pressure difference: 00exitinlet PPP −=∆

II. Dynamic problem:

Childs, 1993, Turbomachinery Rotordynamics. Phenomena, Modeling & Analysis, John Wiley and Sons.

Fr

Ft

h R

ω

Ω

u

w

L

e

p0 p z

pi

pe

( )2

W1

2m

exit

ρξ−

( )2

W1

2m

inlet

ρξ+

P0inlet

Pinlet

pressure

inlet exit

Pexit

P0exit

( ) ( )2

1222

1222

00 mexit

mminletexitinlet

WW

C

LWPP

ρξρλρξ −−++=−

I. Static problem for straight annular seals: analytic solution

Main problem: the definition of the friction factor ( )µ

ρλλ CWr m 2

Re,,Re, == K

1E+3 1E+4 1E+5 1E+6 1E+7Re

1E-3

1E-2

1E-1

λ

relation de Colebrook

relation de Moody

exp. de Nikuradse

C0/k=20

C0/k=50

C0/k=250

C0/k=2500

µρλ C2W

Re,Re3164.0 mm

25.0mBlasius == −

µρ

λ

λ

C2VRe,

VW

Re3164.0 refref

refm

25.0ref

Yamada

Blasius

==−4484476

The definition of the friction factor

2m

2mref UWV += ΩγRUm =

Fr

Ft

h R

ω

Ω

u

w

L

e

p0 p z

pi

pe

2 3 4 5log(Reax )

2

3

4

log

(Ta)

turbulent+tourbillons de Taylor

laminaire+tourb. de Taylor

laminaire

turbulent

C0 /rR=0.31

pompes

turbopompes

pompes de haute pressionà plusieurs étages

0.001

0.01

0.1

++⋅=

=

−

3164

3

Re

1010110375.1

4

ref

Moody

C

r

fλ

r : roughness height (r<<C)

For straight annular seals the « analytic » solution is a first approximation if:1. the circumferential velocity is neglected2. the rotor is centered

Fr

Ft

h R

ω

Ω

u

w

L

e

p0 p z

pi

pe

Main results:- The rotor velocity will decrease the flow rate- The rotor eccentricity will increase the flow rate

( ) ( ) ( )RzSzz

PH

x

HWU

z

HWW

t

HW ττ∂∂ρρρ ++−=

∂∂+

∂∂+

∂∂

( ) ( ) ( )0=

∂∂+

∂∂+

∂∂

x

HW

z

HW

t

H ρρρ

Numerical solutions:

( ) ( ) ( )RxSxx

PH

x

HUU

z

HWU

t

HU ττ∂∂ρρρ ++−=

∂∂+

∂∂+

∂∂

1. Resolution of the full Navier-Stokes equations: Computational Fluid Dynamics

2. Lubrication: simplified thin film flow model dominated by inertia

The « bulk flow » equations, Re*=Re.C/R>>1

How to improve the seal efficiency of the annular seal?

Are there any analytic models available? NO

What models can be used?- Simplified models (extension of the « bulk flow » model)- Computational Fluid Dynamics

By using grooves and steps: labyrinth seals

pressure

CVIII

CVIICVI CVI

axial directionLgLl

B

C

ξ21ρWI2(0)/2

PI(Ll)

PII(0)

ξ12ρWI2(Ll)/2

PII(Lg)

PI(0)

z

Simplified models for stator-grooved seals (extension of the « bulk flow » model)

2.381 1.587 1.587

1.58

7

0.11

34.925

Φ 7

6.50

4

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035axial distance [m]

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

550000

press

ure

[Pa]

rotor surfacestator andCV2/3 interfacethin film model (1)thin film model (2)

Pressure isolines

Computational Fluid Dynamics:

initial coarse mesh intermediate mesh final (refined) mesh

Mesh generation

Obtaining convergence

Interpreting the results

Commercial codes: FLUENT, CFX, FloWorks…

• Analytic model of the flow in a straight annular seal

Superposition of three effects:

-The Lomakin force (restoring effect)-The inertia force (Bernoulli effect)-The viscous force (« oil wedge» effect)

II. Dynamic problem for annular seals

+

+

=

−1Y

1X

YYYX

XYXX

1Y

1X

YYYX

XYXX

1Y

1X

YYYX

XYXX

1Y

1X

e

e

MM

MM

e

e

CC

CC

e

e

KK

KK

F

F

&&

&&

&

&

Pinlet

Pexit

(1+ξinlet)ρWm2/2(1+ξinlet)ρWm

2/2

ρWm2/2*L/2C*λ

ρWm2/2*L/2C*λ

Wm

Wm

Pinlet

Pinlet

C

C

eccentricity

The Lomakin force, FL(restoring effect)

+

-

X

Y

FL

eX

The inertia force FI

Bernoulli equation:P+ρUm

2/2=const.

continuity equation:ρUmH=const.

Ω

+

-

H↑ → Um↓ → P↑

H↓ → Um↑ → P↓

FIeX

The viscous force(« oil wedge » effect )

Ω

+ -

eXFV

+

-

Ω

+

-

The Lomakin force(restoring effect)

The inertia forcesBernoulli effect

The viscous force(« wedge effect »)

48476

876

inertia

2

U

2

Lomakin

2m1X

m

RkWkF

+−= Ωγ

48476

viscous

U

m3Y

m

RWkF Ωγ=

( )ερπRLT

C

Rf

2

3k

3

L1 0= ( )επρ RLLRTk2 = ( )ερπ

RLTRf4

k 2V3 0

= ( ) ( )R2L

R2Ltanh1RLT −=

X

Y

FX=FL-FI

eX FY=FVΩ

FL

FI

Ω

+ -

FV

Ω

ω

X

YFYFX

Ω- ω

-ω X

Y

Ft

Fr

Fixed reference system Whirling reference system

( )4434421

mU~

22

2m1r RRkWkF ωΩγ −+−=

( )43421

mU~

m3t RRWkF ωΩγ −=

+

−+

−=

−

Y

X

Y

X

Y

X

Y

X

e

e

M0

0M

e

e

Cc

cC

e

e

Kk

kK

F

F

&&

&&

&

&

ω: whirl velocity around the centered position

2r McKC

F ωωε

+−−=

ωε

CkC

Ft −=

)Bernoulli(inertiaLomakin,C

UkWkK

2m2

2m1 −−=ε

)"wedgeoil("viscous,C

UWkk mm3

ε=

)"(",3 squeezeviscousC

RWkC m

ε=

)Bernoulli(inertia,C

Rk2c

22

εΩγ=

)Bernoulli(inertia,C

RkM

22

ε=

1. Can we accurately and efficiently calculate straight annular seals?

YES, if:- The inlet pressure drop effect,- The exit recovery effect,- The prerotation velocity,

are correctly estimated. How can these effects be estimated?

-Informed guess or parametric study,-Measurements,-Computational Fluid Dynamics (CFD).

0.0 0.1 0.2 0.3

1.52

1.56

1.60

1.64

1.68

nos résultats

Childs (1993)

ξe

K⋅107

[N/m]

0.0 0.1 0.2 0.3 0.4

1.6

2.0

2.4

2.83.2

3.6

4.0

nos résultats

Childs (1993)

1-ξs

K⋅107

[N/m]

Discussion (1/2)

2. Can grooved or stepped annular seals be calculated accurately and efficiently?

-The same problems as for straight annular seals (inlet - exit pressure drop and prerotation)

-Grooved seals can be calculated efficiently by using simplified methods (extended « bulk flow » methods). These methods are efficient but not always accurate.

-There are no simplified methods for stepped seals.

-CFD is the only accurate approach but it is difficult to handle and time consuming.

3. How accurate should be the calculation model?

-As accurate as possible?

-If inlet and exit conditions canot be accurately estimated then simplified methods are good enough and full CFD approaches are hardly justified.

1E+2 1E+3 1E+4 1E+5nombre total des volumes de contrôle

1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

seco

ndes

CP

U

solution à l'ordre zéro

solution à l'ordre un (une seule fréq. préc.)

16 rainures

jointlisse

4 rainures(Hr/C0=2.5)

19 rainures

11 rainures

21 rainures

Discussion (2/2)

The same problems but compressibility must be taken into account:

Gas (air) annular seals

speedsoundLocal

velocityFluidMnumberMach =,

2

2

222

12

11

2

Mdz

dHfM

MH

M

dz

dM

−

−

−+=γγ

M>1 M=1

Friction force

Um

M<1 M=1

Friction force

Um

Acceleratingflow

Deceleratingflow

M=1

UmDecreasingflow section

M<1

Um

M>1 M=1

Decreasingflow section

Non-viscous nozzle flow

Viscous flow in a parallel channel

Incompressible ρ=const, M<0.3

Compressible ρ≠const, subsonic 0.3<M<0.9transonic 0.9<M<1.1

supersonic 1.1<M<5highly supersonic M>5

Choked flow: Mach number M=1 in the exit section of the annular seal

0 0.2 0.4 0.6 0.8 1dimensionless axial length

0

0.2

0.4

0.6

0.8

1

(P-P

exit

)/(P

inle

t-P

exit

)

0.2

0.4

0.6

0.8

1

Mac

h nu

mbe

r

pressure, subsonic exitMach, subsonic exitpressure, choked exitMach, choked exit

Lomakin effect in gas (air) annular seals: subsonic flow

+

-

X

Y

FL

eX

Pinlet

Pexit

(1+ξinlet)ρWm2/2(1+ξinlet)ρWm

2/2

ρWm2/2*L/2C*λ

ρWm2/2*L/2C*λ

0 0.01 0.02 0.03axial distance [m]

1.75E+005

2.00E+005

2.25E+005

2.50E+005

2.75E+005

3.00E+005

3.25E+005

3.50E+005

pres

sure

[P

a]

Subsonic exit (Pexit=1.75 bar), isothermal flowcentered seal10% eccentricity, minimum film thickness10% eccentricity, maximum film thickness

0 0.01 0.02 0.03axial distance [m]

0.2

0.4

0.6

0.8

1

Mac

h nu

mbe

r

Pexit=1.75 bar, isothermal flowcentered seal10% rel. eccentricity, minimum film thickness10% rel. eccentricity, maximum film thickness

Lomakin effect in gas (air) annular seals: choked flow and static instability

0 0.01 0.02 0.03axial distance [m]

5.0E+004

1.0E+005

1.5E+005

2.0E+005

2.5E+005

3.0E+005

3.5E+005

pres

sure

[P

a]

Sonic exit (Pexit=0.5 bar), isothermal flowcentered seal10% eccentricity, minimum film thickness10% eccentricity, maximum film thickness

0 0.01 0.02 0.03axial distance [m]

0.2

0.4

0.6

0.8

1

Mac

h nu

mbe

r

Pexit=0.5 bar, isothermal flowcentered seal10% rel. eccentricity, minimum film thickness10% rel. eccentricity, maximum film thickness

-

+

X

Y

-FL

eX

Inversed Lomakin effectNegative direct stiffnessStatic instability

Are the analysis methods good enough?

-It depends on the accuracy needed in rotordynamic analysis.

-CFD is becoming popular (FLUENT, CFX, FloWork, …)

Further needs: improvement of simplified, efficient methods for damper seals (honeycombed, dimpled stator surface).

Conclusion

References used in this presentation:

- CRC Handbook of Lubrication, 1982.

- Childs, 1993, Turbomachinery Rotordynamics. Phenomena, Modeling & Analysis, John Wiley and Sons.

- Arghir, M., Lucas, V. and Frêne, J., “Comparisons Between Algebraic Nonlinear Turbulence Models. Application to Seals Analysis”, Tribology Series n° 30, Elsevier Science B.V., 1995, pp. 605-614.

- Arghir, M., Amoser, M., Dueymes, E. - "L'influence de joints annulaires lisses sur le comportement dynamique de ligne d'arbres de machines hydrauliques", La Houille Blanche, N° 6, pp.14-28, 1997

- Arghir, M. Frêne, J., "A Quasi 2D-Method for the Rotordynamic Analysis of Centred Labyrinth Seals", ASME Journal of Engineering for Gas Turbines and Power, 121(1), pp. 144-152, 1999.

- M. Arghir, J. Frêne – “A bulk-flow analysis of static and dynamic characteristics of eccentric circumferentially-grooved liquid annular seals” ASME Journal of Tribology, 126(2), pp 316-326, 2004.

- M. Arghir, J. Frêne - “Numerical Solution of Lubrication’s Compressible Bulk Flow Equations. Applications to Annular Gas Seals Analysis”, 2001-GT-117.

- M. Hélène, M. Arghir, J. Frêne - “Systematic investigation of a labyrinth gas seal static characteristics. An industrial case”, 17th International Conference on Fluid Sealing, York, UK, 8-10 April, Published by BHR Group Limited, pp. 487-503, 2003.

- Amoser, M., « Stromungsfelder und Radialkrafte in Labyrinthdichtungen hydraulischer Stromungsmaschinen, Dissertation ETH Zurich, nr. 11150.

- Kundig, P. , 1992, «Gestufte Labyrinthdichtungen hydraulischer Maschinen », Dissertation, ETH Zurich, 10366.

- Mihai Arghir, Cyril Defaye, Jean Frêne, “The Lomakin Effect In Annular Gas Seals Under Choked Flow Conditions”, ASME Journal of Engineering for Gas Turbines and Power, 129(4), pp. 1028-1034, 2007.