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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.