GT2009 59256 Childs Thorat Final
-
Upload
andrew-cao-tri-nguyen -
Category
Documents
-
view
216 -
download
0
Transcript of GT2009 59256 Childs Thorat Final
-
8/7/2019 GT2009 59256 Childs Thorat Final
1/11
Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and AirGT2009
June 8-12, 2009, Orlando, Florida, USA
GT2009-59256
PREDICTED ROTORDYNAMIC BEHAVIOR OF A LABYRINTH SEAL AS ROTORSURFACE SPEED APPROACHES MACH 1
Manish R. ThoratResearch Assistant
Turbomachinery Laboratory, Texas A&MUniversity
Dara W. ChildsLeland T. Jordan Professor of MechanicalEngineering, Turbomachinery Laboratory,
Texas A&M University, College Station,TX-77843
ABSTRACTPrior one-control-volume (1CV) models for rotor-fluid
interaction in labyrinth seals produce synchronously-reduced
(at running-speed), frequency-independent stiffness and
damping coefficients. The 1CV model, consisting of a leakage
equation, a continuity equation, and a circumferential-momentum equation (for each cavity) was stated to be invalid
for rotor surface speeds approaching the speed of sound.
However, the present results show that, while the 1CV fluid-
mechanic model continues to be valid, the calculated
rotordynamic coefficients become strongly frequencydependent.
A solution is developed for the reaction-force components
for a range of precession frequencies, producing frequency-dependent stiffness and damping coefficients. They can be
used to define a Laplace-domain transfer-function model for the
reaction-force/rotor-motion components. Calculated
rotordynamic results are presented for a simple Jeffcott rotor
acted on by a labyrinth seal. The seal radius Rs and runningspeed cause the rotor surface velocity Rs to equal the speed
of sound c0 at =58 krpm.
Calculated synchronous-response results due to imbalance
coincide for the synchronously-reduced and the frequency-
dependent models.
For an inlet preswirl ratio of 0.5, both models predict thesame log decs out to 14.5 krpm. The synchronously-reduced
model predicts an onset speed of instability (OSI) at 15 krpm,but a return to stability at 45 krpm, with subsequent increases inlog dec out to 65 krpm. The frequency-dependent model
predicts an OSI of 65 krpm. The frequency-dependent models
predict small changes in the rotors damped natural frequencies.
The synchronously-reduced model predicts large changes.
The stability-analysis results show that a frequency-dependent labyrinth seal model should be used if the rotor
surface speed approaches a significant fraction of the speed of
sound. For the present example, observable discrepancies arose
when Rs = 0.26 c0.
INTRODUCTIONLabyrinth seals are used widely in turbomachinery to
reduce leakage. However, they are known to cause latera
rotordynamic instabilities.
A tooth-on-stator/smooth-rotor labyrinth is considered hereSeveral models have been developed to predict the
rotordynamic coefficients for this seal. Kurohashi et al. [1] and
Iwatsubo [2] developed a 1CV analysis. Childs and Scharrer [3
modified the 1CV model to include angular area derivative inthe continuity and momentum equations. Wyssman et al. [4]
developed the first two-control-volume (2CV) model using abox-in-box model to account for the through flow and vortex
flow in the labyrinth cavity. An alternative 2CV model wasdeveloped by Scharrer [5]. However, experimental results by
Picardo [6] showed that the 1CV model by Childs and Scharre
yielded better predictions than Scharrers 2CV approach
Analysis using these models produced synchronously-reducedfrequency-independent results based on the assumption that the
lowest acoustic (circumferential direction) natural frequency in
the seal cavity is much higher than the rotor speed. Picardos
measured results produced frequency-independent coefficients.If the Rs is comparable to c0, frequency dependency of the
coefficients is expected. The present analysis uses the model o
Childs-Scharrer but adopts the analysis procedure by Thielekeand Stetter [7].
NOMENCLATURE
A cross-sectional area of controlvolume
[L2]
B Tooth height [L]
C, c direct and cross coupled damping [F t/L]
1 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
2/11
c0 speed of sound [L/t]
FX, FY seal reaction forces [F]
fr ,f radial and circumferential
dynamic stiffness coefficients
[F/L]
H seal Clearance [L]
j 1 K,k direct and cross coupled stiffness [F/L]kr
Jeffcott rotor stiffness [F/L]
L pitch of seal strip [L]
m&
leakage flow rate [M/L t]
P pressure [F/L2]
R Gas constant [L2/Tt2]
Rs radius of seal [L]
s complex variable for Laplace
transform
[1/t]
T Temperature [T]
t time [t]
U circumferential flow velocity [L/t]
u0(0) U(0)/Rs
, Preswirl ratio circumferential coordinate
density [M/L3]
shear stress [F/L2]
kinematic viscosity [L2/t]
rotor speed [1/t]
Excitation frequency [1/t]
SUBSCRIPTS
0,1 zeroth and first order
i Seal cavity index
r, radial and circumferential
R,S rotor and stator
THEORETICAL MODEL
The labyrinth seal model is based on the control volumeshown in Figs. 1 and 2.
Fig. 1 Axial View of Labyrinth Seal Cavity and ControlVolume
Fig. 2 Radial View of Labyrinth Seal Cavity andControl Volume
Continuity and Circumferential momentum equations are
derived for this control volume based on the analysis by Childs
and Scharrer [3]. The gas is ideal and temperature is constant.
Continuity Equation:
1( ) ( )i i i
i i i is
AUA m m
R t
+
0+ + =
& & (1)
Circumferential Momentum Equation:
2
1 1( ) ( )
( )
i i ii i i i i i i
s
i i Ri Ri Si Si is
AUAU m U m U
R t
A Pa a LR
+
+ +
= +
& &
(2)
where aR and aS are dimensionless lengths upon which shear
stress acts.Rotor and stator shear stresses are modeled using a Blasius
shear-stress model. To account for the curvature effects, the
shear stress terms are modified according to Martinez-Sanchez
et al. [8].
0.25
0.25
| |(1.0 0.075( ) )
2
| |(1.0 0.075( ) )
2
s i i iRi Ri
s
i i iSi Si
s
UR Dh D
R
Dh Dh
R
U
+
=
=
+
h
where, Dhi is the hydraulic diameter defined by
2( )
( )
i i i
i
i i i
H B LDh
H B L
+=
+ +
Leakage Model:
2 21
1 2i i
i i i
P Pm H
RT
=& (3)
1im +&
Control VolumePi
Ui
im&
ROTOR
STATOR
Pi-1Ui-1
Pi+1
Ui+1
2 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
3/11
where, 1i is the Chaplygins flow coefficient [9], 2 is the
Neumanns kinetic energy carryover coefficient [10].
PERTURBATION ANALYSIS
A perturbation analysis is used to solve the governing
equations. The eccentricity ratio is the perturbation parameterfor small motion about a centered position. The zeroth-order
equations define the leakage flow rate, plus steady-statecircumferential velocities and cavity pressures at each cavity.
The first-order equations define the perturbations in pressure
and circumferential velocity of flow due to rotor motion.
First-Order Perturbation Equations:
Continuity Equation
1 1 0 1 1 0 11
2 014 1 1 5 1 1 1 2
16
i i i i i i ii
s s
i ii i i i i i
s
P G U P G P U G G
t
G U
3 1i iP
H H
G P G P G H
R R
RG t
+
+ + +
+ + =
(4)
Circumferential Momentum Equation
1 1 0 1 0 11
2 1 0 1 1 3 1 4 1 1 5 1
i i i i i ii
s S
i i i i i i i i
U X U U A P X
t
X
R R
U m U X P X P X H
+ +
+ + + =&
(5)
Here, , are the perturbed circumferential velocities and
pressures in the ith cavity, and
1iU 1iP
1H is the first order clearance
perturbation. Due to insufficient space, the coefficients G1i and
X1i are not given in this paper; however they can be referred to
in ref. [3].
Elliptic-Orbit Solution
For small motion about a centered position, labyrinth seal
reaction forces are typically modeled as
X
Y
F K k X C c X
F k C Y c C Y
= +
&
&
i
(6)
Starting with Iwatsubo, the solution for the four unknownsk, K, c and C, was obtained by assuming an elliptical orbit of
the form:
(7), = s= cos int Y bX a tWhere the rotor is assumed to be precessing at the running
speed . The clearance function associated with this solution is:
1 cos sin sos ncH t b t a = (8)
A separation-of-variables approach is used for Eqs.(4-5) for
this clearance excitation. Substituting Eq.(7) into Eq. (6) yields:
)cos ( )sin
) cos ( ) sin
(
(
X
Y
t kb Ca t F Ka c
F t Kb caCb t
b
ka
Because of the separate sine and cosine terms, and the two
independent parameters a and b in these two equations, two
independent equations are developed from each equation to
solve for the four unknowns K, k, C, c.
The solution from this approach is assumed to be valid ifthe lowest acoustic natural frequency in the cavity is much
higher than the rotor speed. However if the rotor speed
approaches the lowest cavity natural frequency, the frequency-independent solution breaks down. The natural frequency
considered here is that of a standing wave in the circular
annulus of a labyrinth seal. To determine the acoustic
frequency, the circular annulus can be considered to be an
open-ended pipe. If the flow velocity within the pipe isnegligible compared to the sonic velocity, and continuity is
enforced at the two ends of the pipe, the natural frequencies are
0 0 02( ) ( ) ; 1,2,.2
ni nis s
i i icpipe cavit
Ry i
L
c
R
c
= = = = ..
Hence, the running speed that equals the 1st acousticresonance is:
01n s
s
c RR
c = 0= = (10)
Note that the limiting condition 1n = is the same as the
rotor surface velocity approaching the acoustic velocity
i.e., 0sR c = . Isothermal sonic velocity, 0c R= T
in
is
considered in the present analysis, as the 1CV model is
isothermal.
Circular Orbit Solution
Using an approach developed by Childs and Kim [11] forliquid annular seals, Thieleke and Stetter [7] developed a
solution for Eqs.(4-5) assuming a circular orbit solution of the
form:(11),= = scost Y aX a t
Substituting this solution into Eq.(6) gives
(12)cos sin
ns so ic
X
Y
t ca t F Ka
k t taF Ca
+
=
=
Resolving these components into radial term that is parallel to
the rotating vector a and to a circumferential term that isperpendicular to a nets:
( )r rF
f K ca
f k Ca
F
= = +
= =
(13)
Equation (13) provides two equations for the fourunknowns; however, using two precession frequencies provides
enough equations. Providing more than two frequencies creates
a circumstance where ( ), )(rf f
1n
are curve fitted to solve for
the unknowns. Thieleke and Stetter did not consider
circumstances where ; i.e. ,where the frequency
independent model of Eq.(6) would become invalid. The
= +
= +
(9)
3 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
4/11
example in the following section, explains the inadequacy of
using the frequency independent model when 1n
Test Case for a Long Labyrinth Seal
The test conditions, seal geometry and operating conditions
used in this analysis are obtained from the tests conducted by
Picardo [6]. The input data are shown in Table 1. Air is used inPicardos tests. The mass flow rate for the given operating
conditions at rotor speed of 20200 RPM and preswirl ratio of0.578 is 0.373 kg/sec [6].The predicted exit axial flow Mach
number is 0.29
Table 1 Geometric, Operating, and Input DataReservoir Pressure 70.149 bar
Sump Pressure 36.694 bar
Temperature 288.559 K
Radial Clearance 0.198 mm
Seal Radius 57.340 mm
Tooth Pitch 4.293 mm
Tooth Height 4.293 mmRotor Friction Constant 0.079
Rotor Friction Exponent -0.250
Stator Friction Constant 0.079
Stator Friction Exponent -0.250
Compressibility Factor 1.000
Ratio of Specific Heats 1.400
Kinematic Viscosity 0.00001510 m2/s
Gas Constant 286.900 J/kg K
Number of Teeth 20
Tooth Location Stator
Speed Influence on Acoustic Resonance Location
Figures 3a and 3b show fr and f versus non-dimensionalexcitation frequencies for =15.2 krpm. The excitation
frequency is normalized with respect to the rotor speed .The plots show the resonant peaks of the 1st acoustic mode of
the labyrinth cavity. As increases, the dynamic-stiffness
coefficients at the rotor speed are strongly influenced by
resonance. Thus, the fr and f coefficients can no longer be
defined by linear expressions of Eq.(13).
6.0E+07
5.0E+07
4.0E+07
3.0E+07
2.0E+07
1.0E+07
0.0E+00
1.0E+07
2.0E+07
3.0E+07
0.0 1.0 2.0 3.0 4.0 5.
fr
,RadialDyna
micStiffness
Coefficie
nt(N/m)
Non-Dimensional Frequency
Fig. 3a Radial Dynamic Stiffness Coefficient VsExcitation frequency, (Preswirl ratio=0)
8.0E+07
7.0E+07
6.0E+07
5.0E+07
4.0E+07
3.0E+07
2.0E+07
1.0E+07
0.0E+00
0.0 1.0 2.0 3.0 4.0 5
f
,CircumferentialDynamic
StiffnessCoefficient(N/m)
Non-Dimensional Frequency
Fig. 3b Circumferential Dynamic Stiffness CoefficientVs Excitation frequency, (Preswirl ratio=0)
Figure 4, shows the predicted change in the acoustic
resonant frequency versus . For lower rotor speeds, the
resonant frequency closely matches the predicted isothermaacoustic frequency of 5022 rad/sec from Eq. (10). As the 1CV
model is isothermal, acoustic frequency is evaluated using
isothermal sonic speed, 0c R= T . For higher rotor speeds
the resonant frequency increases with increasing . This
dependency can be attributed to (i) the circumferential flowvelocity in the annulus becoming an appreciable fraction of c0and (ii) convective acceleration terms that are retained here but
neglected in conventional acoustics.
4 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
5/11
0
1000
2000
3000
4000
5000
6000
7000
8000
0 10000 20000 30000 40000 50000 60000 70000
AcousticFrequency(rad/sec)
Rotor S eed RPM
Fig.4 Predicted 1st
acoustic damped naturalfrequency versus running speed. (Preswirl ratio=0)
Transfer-Function Models for Reaction Force Components
If Rs approaches c0, fr and f cannot be modeled by the
frequency-dependent model of Eq. (6). A similar situation
arises in honeycomb/hole-pattern stator seals where the
apparent acoustic velocity for flow within the seal can be
reduced due to the effect of gas compressibility within theholes/cells, dropping the lowest acoustic frequency within the
operating region. Kleynhans and Childs [12] present solutions
that produce frequency-dependent rotordynamic coefficients forthese types of seals, following the approach of Bolleter et al.
[13]. Their transfer-function model is
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
X
Y
F s s s x s
F s s s y
=
D E
E D s
]
(14)
with D and Edefined as follows:
) Re[ ( )] Im[ ( )]
) Re[ ( )] Im[ ( )]
) Re[ ( )] Im[ ( )
) Re[ ( )] Im[ ( )
(
(
]
(
(
r
rf
f
f
f
+
+
=
= +
= +
= + +
D j E j
D j E j
E j D j
E j D j
(15)
Here, the + power indicates positive excitation frequenciesand - power indicates negative excitation frequencies. The
complex functions D(j) and E(j) are obtained by adding and
subtracting terms in Eq.(15).Analytical expressions are obtained forD and Eby curve-
fitting to standard polynomial forms. Figures 5 and 6 providerepresentative results for = 15200 rpm and zero-preswirl.
101
102
103
104100
120
140
160
GaindB
Freq. [rad/sec]
Data and Curve Fit; 5 poles, 4 zeros D(s)
Curve Fit
Data
101
102
103
104
-400
-300
-200
-100
Phase[Deg]
Freq. [rad/sec]
Fig. 5 Magnitude and Phase Plot of Calculated andCurve-fitted D transfer function (Zero preswirl, RotorSpeed 15200 RPM)
101
102
103
104
110
120
130
140
150
GaindB
Freq. [rad/sec]
Data and Curve Fit; 5 poles, 4 zeros E(s)
Curve Fit
Data
101
102
103
104
-400
-300
-200
-100
0
Phase[Deg]
Freq. [rad/sec]
Fig. 6 Magnitude and Phase Plot of Calculated andCurve-fitted E transfer function (zero pre-swirl, RotorSpeed 15200 RPM)
The following D(s) and E(s) transfer functions were obtained
by curve fitting frand f for 0-8000 rad/sec:
4 3 2
5 4 3 2
8 4 13 3 17 2 19 2
12 16 20 23
5 8 13 16
3
3
5 4 7 3 11 2
4.7 10 1.1 10 1.3 10 1.3 10 2.8 10( )
2.7 10 6.5 10 1.4 10 1.7 10 1.7 10
1.24 10 2.4 10 3.6 10 3.6 10 2.3 10
3.2 10 5.4 10 1.0 1( )
0
s s s sD s
s s s s s
s s s s
s s sE s
s
+ =
+ +
+ + + =
+
25
20
14 176.8 10 4.8 10s +
(16)
In Eq. (16), D(s) has unstable poles that are discounted in the
stability analysis.
5 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
6/11
JEFFCOTT ROTOR MODEL-IMBALANCE RESPONSE
AND STABILITY ANALYSIS
Fig. 7 Simple Jeffcott Rotor
The Jeffcott rotor model of fig. 7, acted on by labyrinth seal
forces is used to consider the effect of frequency dependency
on synchronous response and stability analysis.
The rotor parameters are:
Mass (m) 100 kg
Natural frequency , rk
m 795.87 rad/sec = 7.6 krpm
This natural frequency is half the rotor speed of 15.2 krpm.
This choice was made to amplify any possible impact of the
labyrinth on rotor stability. The labyrinth seal parameters usedhere are shown in Table 1. Labyrinth seal forces are the only
source of viscous damping in the model.
Imbalance-Response calculations
The governing equation for the Jeffcott rotor model of figure 7
with conventional labyrinth seal forces is:
2
2
0
0
cos
sin
r
r
K k km x C c x
k K km y c C y y
me t
m te
+ + + +
=
&& &
&& &
x
(17)
where kr is the rotor stiffness and e is the rotor imbalance.
Comparisons were made for the speed-dependent
(frequency-independent) model and the frequency and speeddependent model for a range of inlet swirl ratios and rotor
speeds. Figure 8 shows the (same) calculated amplitude results
for both models. Although not shown, the phase plots also
coincide. This outcome applies because, for response toimbalance, the rotordynamic coefficients are calculated for
forward precession at for both model types.
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
10 100 1000 10000 100000
Magnitu
deRatio(X/e)
Rotor Running Speed (RPM)
transfer function model
synchronously reducedcoefficient model
Labyrinth seal
Bearing
Fig. 8 Magnitude Plot for frequency-independenmodel (synchronously reduced coefficients) andfrequency dependent model (Transfer FunctionModel) foru0(0) = 0.5
STABILITY CALCULATIONS
Speed-Dependent, Frequency-Independent Model
Approach
The frequency-independent model is implemented
considering synchronously-reduced rotordynamic coefficients
evaluated using elliptical precessional orbit at the rotor runningspeed. The homogeneous version of Eq.(6) applies.
Frequency-Dependent Models
Frequency dependency can be accounted by using Eq.(6)
with frequency dependent rotordynamic coefficients, with the
corresponding precession frequencies equal to the rotorsdamped natural frequencies. This approach is regularly used to
account for calculated frequency dependency of tilting-padbearings. Thus, the rotordynamic coefficients for forward and
backward damped natural frequencies can be extracted from theD and Etransfer functions via:
Re[ ( )]
Re[ ( )]
Im[ ( )]
Im[ ( )]
nr
nr
r
n
nr
n
r
r
n
K
k
C
c
+
+
+
+
+
+
+
+
++
=
=
=
=
D j
E j
D j
E j
Re[ ( )]
Re[ ( )]
Im[ ( )]
Im[ ( )]
nr
nr
r
n
nr
n
r
r
n
K
k
C
c
=
=
=
=
D j
E j
D j
E j
(18)
Where, nr is the damped natural frequency of the rotor, and
superscripts + and - indicate rotors forward and backward
modes.
For hole-pattern stator seals, seal forces can significantlychange the damped natural frequencies of the rotor. In such
cases, the rotordynamic coefficients are re-evaluated at the
calculated damped natural frequencies, and the procedure is
6 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
7/11
repeated until there is convergence between the assumed and
calculated natural frequency. Thus, the stability analysis
becomes iterative in nature. However, Labyrinth seal forces do
not significantly change the rotors damped natural frequencies,
and in the present example the rotors critical speed closelyapproximates the first forward and backward damped natural
frequencies. The stability analysis in this example is non-
iterative with the evaluation of rotordynamic coefficients at therotors critical speed.
Re[ ( )]
Re[ ( )]
Im[ ( )]
Im[ ( )]
cr
cr
cr
cr
cr
cr
K
k
C
c
=
=
=
=
D j
E j
D j
E j
(19)
Where, cris the rotor critical speed.The second frequency-dependent approach involves directly
implementing the D and E transfer-function results into the
rotor model using a state-space format.Figures 9-11 present calculated log-dec results for the
model versus the inlet preswirl ratio u0(0), which is varied from0 to 0.8. Zero corresponds to a highly-effective swirl brake,
and 0.8 corresponds to a high preswirl value as might be
expected for a balance-piston seal with no swirl brake. Resultswere obtained using the following three approaches:
1. Speed dependent, frequency-independent model.2. Frequency-dependent stiffness and damping
coefficients
3. D and Etransfer-function model.Results are presented for both forward and backwards-
precessing roots.
TEST CASE 1, = 15.2 krpm; Rs = 0.26 c0
Figure 9 shows the following outcomes for forward precession:
The synchronously reduced model predicts instability forforward precession at a pre-swirl ratio of ~0.57.
The frequency-dependent model predicts instability at apre-swirl ratio of ~0.74
Transfer Function Model predicts instability at a preswirlratio of ~0.76.
Although not presented, the improved models predict that the
backward-precessing mode is more stable than predicted by the
synchronously-reduced model.
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0 0.2 0.4 0.6 0.8
LogarithmicDecrement
Preswirl Ratio
synchronously reduced coefficients
non-synchronously reduced coefficients
transfer function model
Fig. 9 Log-Dec versus u0(0) for Forward Critical Speed
TEST CASE 2, = 40 krpm; Rs = 0.7c0Figure 10 shows the following outcomes:
The synchronously-reduced model predicts instability forforward precession at all preswirl ratios up to 0.8.
The frequency-dependent and transfer-function modelspredicts instability at a preswirl ratio greater than ~ 0.66
Although not illustrated, the improved models predict reduced
stability for the backward-precessing mode as compared to
predictions from the synchronously-reduced model.
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8
LogarithmicDecrement
Preswirl Ratio
synchronously reduced coefficients
non-synchronously reduced coefficients
transfer function model
Fig. 10 Log-Dec Vs Pre-swirl Ratio for ForwardCritical Speed
TEST CASE 3 = 70 krpm; Rs = 1.2c0Figures 11a and 11b illustrate the following outcomes:
For the forward precessing mode, the synchronously-reduced model predicts stability for 0 u0(0) 0.8
7 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
8/11
The frequency-dependent model predicts instability at u0(0) 0.43
The transfer-function model predicts instability at u0(0) 0.51
For the backwards precessing mode, the improved modelspredict stability for 0 u0(0) 0.8; the synchronously
reduced model predict instability for 0 u0(0) 0.8.
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6
L
ogarithmicDecrement
Pr
0.8
eswirl Ratio
synchronously reduced coefficients
non-synchronously reduced coeffic ients
transfer function model
Fig. 11a Log-dec versus u0(0) for forward whirlingmode
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8
LogarithmicDec
rement
Preswirl Ratio
synchronously reduced coefficients
non-synchronously reduced coefficients
transfer function model
Fig. 11b Log-Dec versus u0(0) for backward whirling
mode
Evaluation of onset speed of Instability:
The onset speeds of instability for forward and backward
precessing modes are evaluated for u0(0)=0.5. Figure 12aillustrates the predicted log dec for the forward precessional
mode, showing that stability calculations are about the same for
synchronously-reduced and frequency-dependent solution up to
~14 krpm, but diverge from this speed onwards. The frequency-
dependent model predicts an onset speed of instability of 65krpm.
Fig. 12a Log-dec Vs rotor speed for forward whirling
modeFor the backwards-precessing mode, figure 12b also shows
about the same calculated log dec for speeds out to ~14 krpmThe synchronously-reduced model predicts an onset speed of
instability for the backward mode at ~50 krpm. The frequency
dependent model predicts a continuing increase in the log dec
as the running speed increases. Hence, the rotor backward
precessional mode is predicted to remain stable for the
complete speed range.
Fig. 12b Log-dec Vs rotor speed for backwardwhirling mode
The historical experience with labyrinth seals is that they
produce low values of direct stiffness and have a minimalimpact on the rotor natural frequencies. Figure 13a shows the
calculated damped natural frequency for the rotors forward
precession mode. The synchronously-reduced model shows ifirst dropping and then increasing sharply as the increases
The frequency-dependent results show a modest drop in the
8 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
9/11
natural frequency with increasing . A comparison of figures
13a and 12a suggests that differences between the log-dec
predictions for the models arise mainly due to erroneous
predictions of the damped natural frequency for the
synchronously-reduced model.
Fig. 13a Damped natural frequencies Vs rotor speedfor forward whirling mode
Fig. 13b Damped natural frequencies Vs rotor speedfor backward whirling mode
Figure 13b presents calculated results for backward-precessing natural frequency. The frequency-dependent model
shows a slow drop for out to about 48 krpm, followed by a
gradual increase. The synchronously-reduced model shows aninitial sharp drop ending at about 27 krpm, followed by a sharprise as increases further.
SUMMARY, DISCUSSION, AND CONCLUSIONS
The 1CV model for a see-through labyrinth seal continuesto be used; however, the solution procedure is modified to
account for frequency-dependency as the rotors surface
velocity approaches the speed of sound. Calculated results are
presented for a Jeffcott model acted on by a labyrinth seal
using: (i) A synchronously-reduced (traditional) model, and (ii)frequency-dependent models. Frequency dependency can be
accounted for by considering frequency dependent stiffness and
damping coefficients or by using transfer-function models. Thetransfer-function models can be integrated into a conventiona
rotordynamic code by converting them into a state-space form
The traditional speed-dependent (but frequency independent)
stiffness and damping coefficients continue to be valid for
synchronous imbalance response.
Rotordynamic calculations were performed for a Jeffcottmodel acted on by a labyrinth seal. The rotor diameter at the
seal produces a predicted surface velocity equal to the speed osound at =58 krpm. Log-dec predictions from the two mode
types coincide for speeds out to about 15k rpm. The frequency
dependent models predict an onset speed of instability (OSI)
around 65 krpm. The frequency-independent model predicts an
initial OSI at ~15k rpm followed by a return to predicted
stability around 45 krpm. The stability results show that thetraditional model is not valid for rotor surface velocities
approaching a significant fraction of the speed of sound.The new frequency dependent model needs to be validated
by experiments. Test data by Picardo [5], provides dynamic
stiffness coefficients for a frequency range of excitation of 10
to 150 Hz. However, to verify the predictions in this analysis
test data are needed from rotors running at high rotor surface
velocities (approaching the speed of sound), or the excitation
frequency range must be expanded significantly beyond therotor running speed.
ACKNOWLEDGMENTSThe work reported here was supported by the
Turbomachinery Research Consortium of the Texas A&MUniversity Turbomachinery Laboratory.
REFERENCES[1] M. Kurohashi, et al., 1980, Spring and Damping
Coefficients of the Labyrinth Seals, Paper No. C283/80
Proceedings of the 2nd International Conference on Vibrations
in Rotating Machinery (Institution of Mechanical Engineers)held at Churchill College, Cambridge University, pp. 215-222.
[2] Iwatsubo, T., 1980, Evaluation of Instability Forces of
Labyrinth Seals in Turbines or Compressors, NASA CP 2133Proceedings of a workshop at Texas A&M University 12-14
May 1980, entitled Rotordynamic Instability Problems in High
Performance Turbomachinery, pp.205-222.[3] Childs, D., Scharrer, J., 1986, An Iwatsubo-Based Solution
for Labyrinth Seals: Comparison to Experimental ResultsJournal of Engineering for Gas Turbines and Power, Vol.108
pp.325-331.
[4] Wyssman, H., Pham, T., and Jenny, R. (1984), "Prediction
of Stiffness and Coefficients for Centrifugal CompressorLabyrinth Seals," ASME J. ofEngineering for Gas Turbines andPower, 106, 920-926.
9 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
10/11
[5] Scharrer, J., 1987, A Comparison of Experimental and
Theoretical Results for Labyrinth Gas Seals, Ph.D.
dissertation, Texas A&M University.
[6] Picardo, A., 2003, High Pressure Testing of See-Through
Labyrinth Seals, M.S. thesis, Texas A&M University.[7] Thieleke, G., Stetter, H., 1990, Experimental Investigations
of Exciting Forces caused by Flow in Labyrinth Seals, TAMU
Instability Workshop.[8]Martinez-Sanchez, M., Lee, O.W.K., Czajkowski, E., 1984,
The prediction of Force Coefficients for Labyrinth Seals,
Nasa CP 2338,Rotordynamic Instability Problems in High
Performance Turbomachinery, proceedings of a work held at
Texas A&M University, pp. 235-256.[9]Childs, D., Chang-Ho Kim,1984, Analysis and testing of
Rotordynamic Coefficients of Turbulent Annular Seals with
Different, Directionally Homogeneous Surface-RoughnessTreatment for Rotor and Stator Elements, Nasa CP
2338,Rotordynamic Instability Problems in High Performance
Turbomachinery, proceedings of a work held at Texas A&M
University, pp. 313-340.
[10]Gurevich, M.I., 1966, The Theory of Jets in an IdealFluid, Pergamon Press, pp. 115-122.[11]Neumann, K., 1964, Zur Frage der Verwendung von
Durchblickdichtungen im Dampfturbinenbau,
Maschinentechnik, Vol.13, No.4, pp.188-195.[12]Kleynhans, G., and Childs, D., 1997, The Acoustic
Influence of Cell Depth on the Rotordynamic Characteristics of
Smooth-Rotor/Honeycomb-Stator Annular Gas Seals, ASME
Trans., J. of Engineering for Gas Turbines and Power, Vol.
119, No. 4, pp. 949-957.
[13] Bolleter, U., Leibundgut, E., Strchler, R., and McCloskey,T., 1989, Hydraulic Interaction and Excitation Forces of High
Head Pump Impellers, in: PumpingMachinery1989,
Proceedings of the Third Joint ASCE/ASME MechanicalConference, La Jolla, CA, pp. 187-194.
APPENDIX A
Circular Orbit Solution:
The seal clearance function can be defined as:
1 ( ) cos ( ) sinH yx t t =
The pressure and velocity perturbations can be expressed
as:
1 1 1
1 1 1
cos sin
cos sin
i ic is
i ic is
P
U
P P
UU
+
+
=
=
Substituting the above variable definitions in the first order
continuity and momentum equations eliminates the theta
dependency. The time dependency is eliminated by considering
circular whirl orbit:
$
1 0
1 1
1 1
( ) ( ) j t
j ti i
j ti i
H x t jy t r e
P p e
U u e
= + =
=
=
uur
ur
ur
Where is the radius of the circular whirl orbit. and
are complex amplitudes:
0r
1 1
1i
p
$1iu
$
1
1
i
i 1 1
ic is
ic is
p P jP
u U jU
= +
= +
The separation of variables approach yields algebraic
equations of the form:
[ ]{ } [ ]{ } { } {1 1 2 1 3 1 21 1i i i }A x A x A x x ++ + =
Where,
{ } { }{ }
[ ]
1 1 1 11
2 0 2 6 52 0 0 0
4
41
4 0
4 0
0
0 0 0
0 0 0
0 0
0 0
T
ic is ic isi
T
i i i i i
i
i
i
i
s
i
x P P U U
G U G G X x r r r
R
G
GA
X m
X m
=
0r
= +
=
&
&
[ ]
1 0 1 01 3
1 0 1 03 1
2
0 1 03 1 2
0 13 2
0
0
i i i ii i
s s
i i i ii i
s s
i i ii i is s
i ii i
s s
U G PG G
R
U GG G
RA
A UX X XR R
A UX X
G
R
G
R
X
XR R
X 01
i
P
=
+
5
53
0 0
0 0 0
0 0 0 0
0 0 0 0
i
i
G
GA
0 =
The above set of equations is for a single cavity. The cavity
equations are assembled to form a 4NC 4NC matrix equation
(NC= number of cavities). Pressure & velocity perturbations a
the entry and exit are zero. These form the boundary condition
for the problem.
10 Copyright 2009 by ASME
-
8/7/2019 GT2009 59256 Childs Thorat Final
11/11
The solution is of the form:
01 1
01 1
ic ic
is is
rP P
rP P
=
=
The radial and circumferential forces, and)(rF
( )F ,
on the seal are evaluated from the pressure perturbations in the
labyrinth seal cavities.
0 1
1
( )
NC
r s
i
F rR
=
= icp
Similarly,
0 1
1
( )
NC
s is
i
F R r
=
p=
The radial and circumferential dynamic stiffness
coefficients are obtained from these forces:
0
0
((
( )(
))
)
rrf
f
F
rF
r
=
=
11 Copyright 2009 by ASME