An analytical method to calculate the rotor excitation forces arising from labyrinth seals is presented. The objective is to model the gas flow through the seal clearance passages and cavities when the rotor is positioned eccentricly relative to the stator center. The seal flow model used in the analysis yields solutions which validate the experimentally observed influence of seal parameters on seal forces reported in the literature. The analytically predicted seal pressure distributions and forces were compared with published experimental results

Rajakumar, C.

Sisto, E.

English

NASA Technical Reports Server

ANALYTICAL PREDICTION OF LABYRINTH-SEAL-FLOW-INDUCED ROTOR EXCITATION FORCES 
C. Rajakumar and E. Sisto 
Stevens Institute o f  Technology 
Hoboken, New Jersey 07030 
An analytical method to calculate the rotor excitation forces arising from 
labyrinth seals is presented. The objective is to model the gas flow through the 
seal clearance passages and cavities when the rotor is positioned eccentric relative 
to the stator center. The seal flow model used in the analysis yields solutions 
which validate the experimentally observed influence of seal parameters on seal 
forces reported in the I i terature. The analytical ly predicted seal pressure 
distributions and forces have been compared with published experimental results. 
I NTROWCTI ON 
While performing the function of sealing, labyrinth seals have been identified 
as a potential source of destabilizing forces which cause self excited rotor whirl 
of turborotors. Typically these seals are employed to effect sealing of the working 
fluid between two stages of a compressor or turbine and in adjacent cooling air 
compartments which are maintained at different pressures. 
have a configuration as shown in figure 1 are designed to minimize the leakage flow 
by successively expanding the gas past the orifice-like clearance passages at the 
seal fin locations. 
labyrinth seals which 
An analytical model describing the excitation of turborotors by labyrinth seals 
was first published by Alford (ref. 1 ) .  In the recent literature (ref. 2 to 7) 
experimental and analytical methods of predicting labyrinth seal forces have been 
reported. Experimental results presented by Benckert and Wachter (ref. 2) show that 
a restoring force along the line of centers and a lateral force normal to the line 
act on the rotor when the rotor center i s  displaced to an eccentric position with 
respect to the seal center as shown in figure 2. The present work is aimed at 
analytically predicting these seal forces which result from the circurnferentially 
varying pressure distributions within the seal cavities. 
SYMBOLS 
C 
cc 
Ce 
co 
e 
er. 
f 
Fr. Ft 
h 
seal radial clearance 
orifice flow coefficent 
energy recovery factor 
nominal radial clearance 
rotor eccentricity 
eccentricity rat io (e/Co) 
circumferential flow area 
radial and tangential seal forces 
seal fin height 
195 
https://ntrs.nasa.gov/search.jsp?R=19860020703 2020-03-20T14:40:30+00:00Z
I 
m 
N 
n 
P 
4 
R 
Re 
Rs 
t 
T 
U 
a 
x 
; 
P 
8 
Subscripts 
7 
0 
I 
J 
r 
S 
cav i t y  width 
number of circumferential nodes 
rotor rpm 
number of  seal f i n s  
pressure (Po- seal i n l e t .  Pe- seal e x i t )  
ax ia l  mass f low rate per u n i t  circumferential length 
gas constant 
reynolds number 
ro tor  radius 
ax ia l  p i t c h  
gas temperature 
circumferential f low ve loc i t y  
i n l e t  swi r l  ve loc i t y  
length of wetted surface 
f low coef f ic ient  
f r i c t i o n  coef f ic ient  a t  the walls 
coe f f i c i en t  of  v iscos i ty  
reference angle shown i n  f igure 2 
gas density 
shear stress 
circumferential coordinate 
angular ve loc i t y  of ro tor  ro tat ion 
f i n  and cav i ty  number designations 
circumferential node number 
ro tor  
stator 
GOVERNING EQUATIONS 
The i n t r i c a t e  f low path of  labyr inth seals. combined wi th  the asymmetry caused 
by rotor eccentr ic i ty  complicate the analysis employing fundamental f l u i d  mechanics 
governing equations. A r e l a t i v e l y  simpler set of  labyr inth seal governing equations 
derived by Kostyuk ( r e f .  8)  are employed i n  the analysis here. For the f l u i d  
macro-element shown i n  f igure 1, the fo l lowing conservation of mass and 
circumferential momentum equations are derived, neglecting the curvature ef fect  f o r  
small clearance-to-radius rat ios.  
F r i c t i ona l  resistance at  the seal walls are calculated using the fol lowing 
relat ionships. 
196 
where: 
The Reynolds numbers are given by: 
Proper signs are attached t o  the f r i c t i o n  coef f ic ient  X based on the r e l a t i v e  
ve loc i t y  of  the f l u i d  wi th respect t o  the rotor and stator walls. 
isothermal f low process, Mart in’s ( r e f .  9) leakage f low equation f o r  small pressure 
drop accross each f i n  i s  given by: 
Assuming 
where: 
O r i f i c e  f low coefficients Cc fo r  annular o r i f i c e s  presented by Vermes ( r e f .  10) 
have been used i n  the calculat ions. The residual energy factor Ce accounts for  the 
part  of  the k i n e t i c  energy i n  the leakage f low from the ( i - l l t h  o r i f i c e  that i s  
present a t  the i n l e t  t o  the i - t h  o r i f i c e .  Representing the o r i f i c e  f low by one hal f  
of  a f l a t  symnetrical j e t ,  Vermes ( r e f .  10) derived the fol lowing expression f o r  the 
residual energy factor:  
8.52 
( l i / C i )  + 7.23 
Cei = 
Equations (1). (2). (5) and the ideal gas re la t  
P i  p i R T  
onsh 
(7) 
P 
( 8 )  
form the system o f  equations t o  be solved f o r  the circumferential pressure 
d i s t r i b u t i o n  resul t ing due t o  the rotor eccentr ic i ty.  Although f igure 1 shows f i n s  
on the rotor,  the equations derived are also v a l i d  f o r  a labyr inth wi th  f i n s  on the 
s tator .  
NUMERICAL PROCEDURE 
For the rotor  undergoing steady rotat ional  motion about a given eccentric 
posi t ion ( f i g .  2 ) .  we can drop o f f  the time der ivat ive terms i n  equations ( 1 )  and 
(2). 
( 2 1 become : 
After subst i tu t ing f o r  qi and pi from equations (5) and ( 8 ) .  equations (1) and 
197 
The seal clearance and circumferential f low area f o r  the eccentric rotor 
configuration are given by: 
C i  = COi - e COS@ 
f i  = I i  (h i  + Coi - e cos01 (12) 
(11) 
Equations (9)  and (70) are nonlinear ordinary d i f f e r e n t i a l  equations t o  be 
solved t o  get the pressure and ve loc i ty  d is t r ibut ions,  p i  (0 )  and u i  (81 i n  each 
seal cavi ty.  
A f t e r  introducing the central f i n i t e  dif ference approximations given below we 
d iv ide the circumferential coordinate length, Rs0 i n t o  m equal parts. 
The resul t ing system of nonlinear algebraic equations are represented 
symbolically as: 
' F ( P i j ,  u i j )  = 0 (15) 
G ( P i j ,  u i j )  = 0 i=l.Z..(n-l)  j=1,2..m (16) 
Equations (15) and (16) are solved employing the Newton Raphson i t e ra t i on  
procedure i n  the multidimensional space formed by the variables P i j  and u i j .  
dropping the second and higher order terms from the Taylor series expansions of 
equations (15) and (161, we a r r i ve  a t  the i t e r a t i o n  scheme given by: 
Af ter  
198 
By assuming an initial guess for the vector [Pi, ui,]k, the Jacobian matrix and 
the function values are evaluated to calculate the new vector of variables 
[Pi, Ui,]k+l. The iterations are continued until the change in each component of 
the solution vector is less than a predetermined error value. The seal forces are 
obtained by integration of the pressure distribution over the rotor surface using 
the sumatlon equations given by: 
n-1 m 
Fr = RsAB I i Pi, cosej 
n-1 m 
Ft = RsA9 l i  I: Pij sin9j 
i=l j=l (19) 
RESULTS 
Seal cavity pressure distributions obtained were compared with the experimental 
results of Leong and Brown (ref. 4)'shown in figure 3. The ll-cavity seal having 
fins on the stator with inlet and exit pressure of Po-2.974 bar, Pe=l bar was 
investigated at a rotor speed of 7504 rpm and eccentricity ratio of 0.6. The 
analytically predicted pressure distribution patterns in the first two seal cavities 
are in agreement with the experimentally measured patterns, with a peak pressure 
occuring near the 180 degrees location in the first cavity while the pressure is 
minimum near this location in the second cavity. Pressure levels predicted by the 
theory beyond the first three cavities are in fair agreement with the experimental 
results. The difference in cavity pressure levels between the theory and experiment 
I S  attributable to possible difference between the orifice flow coefficients used in 
the calculations and the actual flow coefficients in the experiment. Also, possible 
random variations of the seal clearance at successive fin locations in the test seal 
could contribute to the difference between the experimental and ana'lytical results. 
The experimental results of Benckert and Wachter (ref. 2 )  were used to verify 
the seal forces calculated analytically. Figures 5 and 6 show the variation of seal 
forces with eccentricity ratio for a 17-cavity seal with fins on the stator. 
Tangential forces generated when the rotor rotates at 9482 rpm and 4773 rpm, for the 
seal inlet and exit pressures of Pot1 bar, Pee1 bar are shown in figure 4(a). The 
negative eccentricity ratios indicate that the rotor center has been displaced along 
the negative x-axis in figure 2. The analytically predicted tangential seal forces 
are in good agreement with the experimental results showing an average deviation of 
about 16 percent, excluding the zero eccentricity ratio case where the experimental 
results show a nonzero force. Radial forces predicted by the analysis are shown in 
figure 4(b); experimental results on radial forces were not reported in the 
reference for the case investigated. These results show that the seal force 
magnitudes increase with rotor rotational speed for any given eccentricity ratio. 
The tangential forces act in a direction that would initiate or promote forward . 
whirl motion of the rotor center. The radial forces are decentering where they tend 
to push the rotor away from the seal center. 
The seal forces variation with eccentricity ratio for zero rotor speed, but 
nonzero swirl velocity at the seal inlet are presented in figures 5(a) and 5(b). 
The seal pressures in this case are Po=2.041 bar, Pe-1 bar. The tangential seal 
forces have been predicted quite close to the experimental results, the maximum 
difference being 17 percent. The radial forces from the experiment, again were not 
199 
reported i n  the reference. The two different inlet swirl velocities investigated 
snon that the seal forces ~ncrease with the inlet swirl velocity and appreciable 
magnitudes of tangential forces are generated which would promote forward whirl 
instability in rotor systems. The radial forces in this case are of relatively 
small magnitude and they act in a centering direction tending to restore the rotor 
center back to the seal center. 
CONCLUSIONS 
Employing the seal flow governing equations derived by Kostyuk (ref. 81. 
methods of predicting seal forces for an infinitesimal eccentricity have been 
reported in the literature where seal stiffness coefficients are presented. In this 
analysis the radial and tangential comonents of seal forces for large eccentricity 
ratios have been predicted, for the first time, employing Kostyuk's equations. The 
analysis yields results which validate the experimental ly observed variation of seal 
forces with rotor eccentricity, inlet swirl velocity and rotor rotational speed. 
The tangential force components predicted by the theory, especially the swirl 
flow-induced forces are in good quantitative agreement with the experimental 
results. The analysis also shows the nonlinear dependence of the forces with 
eccentricity beyond e,== .4. 
consideration in large amplitude rotor dynamic investigations where nonlinear 
stability problems may be encountered. 
This behavior needs to be given appropriate 
REFERENCES 
1. 
2. 
3.  
4. 
5 .  
6 .  
7 .  
8. 
Alford, J.S.: Protecting Turbomachinery From Self Excited Rotor Whirl. 
ASME Journal of Engineering for Power, Vol. 87, No. 4, Oct. 1965, 
pp. 333-344. 
Benckert, H.; and Wachtet. J.: Flow Induced Spring Coefficients o f  
Labyrinth Seals for Application in Rotor Dynamics. NASA CP 2133, 1980, 
pp. 189-212. 
Hauck, L: Exciting Forces Due to Swirl Type Flow in Labyrinth Seals. 
Proceedings of  the IFTOMM Conference on Rotordynamics. Rome, Sept. 1982. 
Leong. Y.M.M.S.; and Brown. R.D.: Circumferential Pressure Distributions 
in a Model Labyrinth Seal. NASA CP 2250, 1982, pp. 223-241. 
Leong, Y.M.M.S.; and Brown. R.D.: Experimental Investigations o f  Lateral 
Forces Induced by Flow Through Model Labyrinth Glands. NASA CP 2338. 1984, 
pp. 187-210. 
Murphy, B.T.; and Vance, J.M.: Labyrinth Seal Effects on Rotor Whirl 
Instability. Institution of Mechanical Engineers CP C306/80, f980. 
pp. 369-373. 
Wright. D.V . :  Labyrinth Seal Forces on a Whirling Rotor. ASME 
Publication on Rotor Dynamical Instability AMD. Vol 55. 1983, pp. 19-31. 
Kostyuk. A.G.: A Theoretical Analysis of the Aerodynamic Forces in the 
Labyrinth Glands of Turbomachines. Teploenergetica. lQ(11) 29-33, 1972, 
- -  
pp. 39-44. 
200 
9. Martin, H . M . :  Labyrinth Packings. Engineering. Vol. 85, Jan. 10, 1908, 
pp. 35-38. 
10. Vermes, G: A Fluid Mechanics Approach to the Labyrinth Seal Leakage 
Problem. ASME Journal of Engineering for Power, Vol. 83, April 1961. 
PP. 161-169. 
4 L e  
Figure 1. - Labyrinth seal f luid element. 
T '  
o f  centers 
Figure 2. - Seal forces. 
201 
Rs = 120 mm h = 9.525 mm 
theory t = 9.525 mm 
4 ,o  expt. Co= -635 mm 
m m 7  
c) Q c) 
Q m Q 0 9  " 
10 
CI 1 m 0 
Y - - 
c) 
11 
0 Q 0 0 
00 120.00 $40.00 
A n g l e  # ( d e g .  1 
Figure 3. - Seal pressure distribution. 
Rs = 150 mm 
0 ,  A expt. 
theory 
h = 2.5 mm 
t = 5  mm 
Co = .25 mm 
P 
n 
Z 
W 
c 
0 
LL 
8 
P 
I 
8 
ai 
-0.40 0.00 0.40 0 
E c c .  r a t i o  er 
I 
a 
c(Q 
W 
Q) 
0 
LQ 
o? 
LLQ k0 
ea a 
4 
0.00 0. a0 0. 
I -0. A0 
E c c .  r a t i o  e,.. 
(a) Tangential force Ft. 
(b) Radial force Fr. 
.9@ 
30 
Figure 4. - Seal forces with variable rotor speed, 
202 
Rs = 150 mm 
- theory 
0 , A expt. 
h = 2.5 m 
t = 5  mm 
Co = .25 m 
I 
0.00 0.40 c I '  -0.40 
E c c .  r a t i o  er 
(a) Tangential force Ft. 
(b) Radial force Fr. 
Figure 5. - Seal forces with variable inlet swirl velocity. 
.90 
203 


Analytical prediction of labyrinth-seal-flow-induced arotor excitation forces

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