The results of numerous investigators have shown the importance of labyrinth seal inlet swirl on the calculated dynamic stiffness of labyrinth seals. These results have not included any calculation of inlet leakage swirl as a function of geometry and sealing conditions of the given seal. This paper outlines a method of calculating the inlet swirl at a given seal by introducing a radial chamber to predict the gas swirl as it goes from the stage tip down to the seal location. For a centrifugal compressor, this amounts to including the flow path from the impeller discharge, down the back of the disk or front of the cover, then into the shaft seal or eye packing, respectively. The solution includes the friction factors of both the disk and stationary wall with account for mass flow rate and calculation of radial pressure gradients by a free vortex solution. The results of various configurations are discussed and comparisons made to other published results of disk swirl

Kirk, R. Gordon

English

NASA Technical Reports Server

N87-22210
INFLUENCE OF DISK LEAKAGE PATH ON LABYRINTH SEAL INLET SWIRL RATIO
R. Gordon Kirk
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061
The results of numerous investigators have shown the importance of labyrinth
seal inlet swirl on the calculated dynamic stiffness of labyrinth seals. These
results have not included any calculation of inlet leakage swirl as a function of
geometry and sealing conditions of the given seal. This paper outlines a method of
calculating the inlet swirl at a given seal by introducing a radial chamber to
predict the gas swirl as it goes from the stage tip down to the seal location.
For a centrifugal compressor, this amounts to including the flow path from the
_mpel!er discharge, down the back of the disk or front of the cover, then into the
shaft seal or eye packing, respectively. The solution includes the friction factors
of both the disk and stationary wall with account for mass flow rate and calculation
of radial pressure gradients by a free vortex solution.
The results of various configurations are discussed and comparisons made to
other published results of disk swirl.
INTRODUCTION
Recent reports in the literature (1,2,3,4,5) have addressed the problem of
calculating the rotordynamic coefficients for a labyrinth seal having a given inlet
gas swirl, pressure drop, and resulting mass flow. The centrifugal design engineer
has at his disposal from standard aerodynamic design codes the gas swirl and
pressures at the impeller tip. The solution of the leakage gas path swirl and
resulting pressure distribution is important not only for labyrinth seal coefficients
but also for proper thrust balance calculations. This paper presents an approximate
method of calculating these desired parameters using a modified version of the
solution technique as proposed by lwatsubo (i) and later extended by Childs and
Scharrec (2). The extensions and modifications to the theory as outlined by (2) will
be discussed in this paper.
The equations of the modified formulation have been incorporated into a single
labyrinth seal analysis computer code to permit rapid evaluation of different design
conditions. The accuracy of the reported solution technique will be compared to
experimental and analytical solution results as reported in reference (6) and to the
limit case condition of zero leakage (approximately zero for computer program
results).
F ; C ING PAGE
225
https://ntrs.nasa.gov/search.jsp?R=19870012777 2020-03-20T11:37:43+00:00Z
NOMENCLATURE
Values are given in both SI and U.S. Customary Units.
made in U.S. Customary Units.
b
c i
Hi
L i
The calculations were
- tangential velocity ratio, U/(r_)
-wall separation at ith radial chamber and radial clearance for seal tooth, m
(in)
- height of ith seal element tooth, m (in)
- axial length of ith seal chamber, m (in)
-mass flow rate, Kg/sec (ibm/sec)
gas pressure, N/m 2 (ibf/in 2)
- radial position on disk, m (in)
R,R o - outer radius of disk, m (in)
R -gas constant, (ibf • ft/Ibm/°R)
RiM - mean chamber radius, m (in)
RR - average radius of rotor seal chamber surface area, m (in)
RS - average radius of stator seal chamber surface area, m (in)
Re - disk Reynolds number, = RZ_/v
s -wall separation, m (in)
S - separation ratio, = s/R o
SJ - leakage parameter, = Vs/(R2_)
O
T - gas absolute temperature, OK (OR)
U t - tangential velocity, m/sec (in/sec)
V r - radial velocity, m/sec (in/sec)
Z - gas compressibility
p - gas density, Kg/m 2 (Ibm/in 2)
T - radial wall shear stress, N/m 2 (ibf/in 2)
rw
Trd -radial disk shear stress, N/m 2 (ibf/in 2)
- gas kinematic viscosity, m2/sec (in2/sec)
X - position factor, = (Ro - r)/R o
- rotor (disk) angular spin velocity, sec -I
-I
mB - swirl velocity for free vortex, sec
226
METHOD OF SOLUTION
The solution technique proposed was developed to permit the same basic theory
and computer code to calculate both the circumferential swirl and pressure
distribution down the disk and across the labyrinth as one coupled system. This
technique was initially intended as a quick first pass method but the results have
proven to be very close to the more exact theories such that the added complication
of coupling different theories and matching boundary conditions of pressures,
temperatures, flows, and swirls may not be justified for rotordynamic evaluations. A
typical configuration is shown in Figure I for a centrifugal compressor stage disk
cover leakage flow path.
For the radial direction down the disk leakage path the equilibrium equation is
given by the following equation (6):
dV U2
-V r t I dP + rw rrd
r dr r = - _ d'_ p-_" ± p'_- [I]
For a free vortex flow neglecting the radial shear force which will be accounted for
by the crossflow factor in the circumferential equations, the pressure distribution
equation becomes
de P(r_6)2
d--{= r [2]
or dP 2
d'_ = Pr_6 [3]
whe re
_6 = gas swirl angular velocity at the radius, r.
Therefore, the pressure at any radius r is given by
2
___8(R2 2P(r) = P(R) - p - r ) [4]
This equation predicts the pressure along the disk if the gas swirl is known. The
gas swirl can be calculated from the circumferential momentum equation as outlined in
(2) and further modified to the following equation to account for variation in rotor,
stator and mean flow chamber radius. In addition, the crossflow turbulence
correction factors may be included in the solution to account for the inward flow
resistance.
1/2Po i
i i-I (R0_- Vol) 2 * YNR * ( IR_°- V°ilDHY )YMR [5]
* ARL * sign(R_ - Voi_. , wR___,RMC3
* YNS * I
l
where
RM = mean chamber radius
IVoll * DHY
)YMS , ASL * SIGN(Voi) * RMRS* C4
RR = mean rotor surface radius
22/
RS = mean stator surface radius
C3, C4 = crossflow turbulence factors
V°i = RMi _8i = average chamber swirl velocity
DHY = hydraulic diameter of chamber
YNR, YNS, YMR, YMS = turbulence factors per reference (2)
ARL, ASL = shear area for rotor and stator
Poi pressure and temperature dependent gas density in i th
ZRT i
chamber
u = gas kinematic viscosity
The leakage flow, m, can be calculated as outlined in (2) or by other suitable
calculations with the modification to radial chamber pressures given by equation (4).
The solution process requires that an initial swirl be selected to calculate the
pressure field and leakage. A swirl of 50% is suggested for starting the solution.
With this pressure field and flow, the momentum equation given by Eq. (5) is solved
for the first pass swirl values. These swirl values are then used to recalculate the
pressure drops down the disk and through the labyrinth and the resulting flow.
Another pass through Eq. (5) solution yields the second pass swirl values.
Typically, three passes give the desired convergence and the inlet swirl to the laby-
rinth is then taken from the chamber ahead of the first sealing tooth.
A general geometry input is used such that for the radial chambers a very small
tooth height and length with a tooth clearance equal to the disk to wall spacing can
be used to model the flow path. The radial surface area is calculated using the
indicated radius of each tooth location.
RESULTS OF SWIRL PREDICTED IN GAS LEAK PATH
The evaluation of the proposed swirl calculation procedure has been based upon
numerous similar conditions as reported by Jimbo (6). Initial comparisons of actual
compressor swirl results from similar geometry is overplotted in Fig. 2. The
parameter for leakage flow was noted to be similar to those given by Jimbo. The
leakage flow parameter is defined as
where
V , s
SJ = R"'_ _- [6]
o o
V = radial gas velocity
R = disk outer radius
o
= rotor speed
s = wall separation
For SJ = 0.0002 it is obvious that the compressor disk swirl does not agree with the
reported complete analytical solution. The swirl rates are greater from the
labyrinth program approximate solution. However, the other parameter, the disk
Reynolds number given by
228
Re = R 2 _/_ [7]
o
was calculated and found to be considerably different from the test results. The
analysis results by Jimbo used air and a R = 9.82 x 105 was quoted. These initial
compressor test results had an R e = I.I x _08. To match the parameters for the two
systems the gas, pressure drop, and rotor speed were changed as given in _able 1
under test rig conditions. This gave a disk Reynolds number of Re = 3 x I0_, only
off by a factor of 3 from Jimbo, compared to a factor of 112 for the compressor stage
results.
The results for the labyrinth analysis overplot to the accuracy that the curves
can be evaluated. The swirl results for the compressor stage and the test result
condition are given in Tables 2 and 3, respectively. The swirl down the disk and
through the labyrinth are given in the table with the radius X-position factor
indicated for comparison to the analysis results of Figure 2. The results are in
excellent agreement for the case of near zero flow (i.e., swirl _0.5) and for SJ =
0.0002 where the swirl at X-position factor of 0.31 is now calculated as 0.563 as
compared to 0.63 for the compressor gas. A comparison of the compressor swirl, test
case calculation, Jimbo calculation, and test results reported by Jimbo are shown in
Figure 3. The test results show a slowing of the swirl that is not predicted. The
test rig was equiped with numerous flow and pressure measuring ports in the stator
wall and it is very possible that the cause of the test rig result reduced swirl was
the increased surface roughness resulting from the measuring instruments. Complex
flow fields could also be the cause of the discrepancy and are beyond the scope of
the present analysis. Results for reduced leakage, SJ = .000052, are given in Fig. 4
and labyrinth analysis compressor results for SJ = .0000372 overplotted. This case
of reduced leakage compares closely even though the disk Reynolds numbers are not
similar. The test rig results once again show a reduced swirl ratio with great
restriction noted in the ×-position factor range of 0.4-0.5. The overall trend is
similar as concluded by Jimbo.
One additional labyrinth program result is given in Table 4 for the condition of
leakage from the final compressor stage through a balance piston full labyrinth.
These results have a wall spacing that reduces as the radius reduces. A swirl of
0.82 is predicted for this geometry and gas properties, even though the flow SJ
parameter is 0.00021 and a uniform spacing air test result would give a swirl rate
closer to 0.6 (see Fig. 2 for ×-position ~0.4).
CONCLUSIONS
(I) The proposed approximate calculation procedure produces results that are
acceptable for rotor dynamic evaluations of labyrinth seals.
(2) The flow parameter, SJ, and disk Reynolds number, Re, used by Jimbo to present
results are very useful in comparing results for different designs and give
great insight into disk swirl behavior.
(3) Non-uniform leak path geometry can be used to increase or decrease the swirl in
the gas leak path.
(4) Increased stator surface roughness will suppress the swirl due to the increased
shear drag on the swirling flow.
229
(1)
(2)
(3)
RECOMMENDATIONS
Test evaluations using current technology flow measurement capability should be
conducted on typical compressor and turbine disk design gas leak path
configurations.
The importance of gas properties and actual system configurations must be
closely evaluated.
The proposed calculation procedure can be used, with a high degree of
confidence, for entry swirl evaluation of compressor labyrinth designs.
(1)
(2)
(3)
(4)
(5)
(6)
REFERENCES
lwatsubo, T., N. Matooka, and R. Kawai, "Spring and Damping Coefficients of the
Labyrinth Seal," NASA CP 2250, 1982, pp. 205-222.
Childs, D. W., and J. K. Scharrer, "An lwatsubo-Based Solution for Labyrinth
Seals - Comparison to Experimental Results," Rotordynamic Instability Problems
in High Performance Turbomachinery - 1984, Texas A&M University, College
Station, Texas, May 28-30, 1984.
Benckert, H., and J. Wachter, "Flow Induced Spring Coefficients of Labyrinth
Seals for Application in Rotordynamics," NASA CP 2133 Proceedings of a Workshop
held at Texas A&M University May 12-14, 1980, Rotordynamic Instability Problems
of High Performance Turbomachlnery, pp. 189-212.
Jenny, R., "Labyrinths as a Cause of Self-Exclted Rotor Oscillations in
Centrifugal Compressors," Sulzer Technical Review 4, 1980, pp. 149-156.
Kirk, R. G., "Evaluation of Aerodynamic Instability Mechanisms for Centrifugal
Compressors," ASME Paper 85-DET-147. Presented at Design Engineering Vibration
Conference, Cincinnati, Ohio, September 10-13, 1985.
Jimbo, H., "Investigation of the Interaction of Windage and Leakage Phenomena in
a Centrifugal Compressor," ASME Paper 56-A-47, presented at ASME Annual Meeting,
New York, NY, July 30, 1956.
230
PARAMETER
TABLE 1 SYSTEM PARAMETERS FOR ACTUAL COMPRESSOR STAGE
DESIGN CONDITIONS AND AN ASSUMED SYSTEM TO APPROXIMATE THE
RESULTS OF TEST CONDITIONS FROM REFERENCE 6.
COMPRESSOR STAGE TEST RIG CONDITIONS
(Ref. 6 assumed conditions)
MW 21.33 25.95
_, m2/s 2.79 x 10 -7 4.64 x 10-6
(in2/sec) (4.32 x 10-4) (7.19 x 10-3)
Z .89 .955
PS, N/m 2) 5.90 x 106 3.43 x 105
(ib/In 2) (855) (49.7)
PE, N/m 2 5.17 x 106 1.01 x 105
(ib/in 2) (750) (14.7)
T, OK 331.6 301.4
(°R) (602.9) (548.5)
N, Hz 207.4 i00.0
(RPM_ (12566) (6000)
Ro, m .154 .154
(in) (6.05) (6.05)
Re i.i x 108 3 x 106
231
TABLE 2 RESULTS FOR COMPRESSOR STAGE GIVING PREDICTED
CHAMBER _.L
DISK TIP 0
2 .08
3 .17
4 .25
5 .31
RADIAL + 6 .34
TURN 7
SEAL 8
+ 9
I0
Ii
12
CHAMBER FLOW SWIRL RATIO
SJ = 0.000023 0.00014 0.000236 0.00034
.52 .52 .52 .52
.5033 .534 .548 .557
.5053 .5496 .571 .5867
.5095 .5689 .600 .6225
.5144 .5938 .632 .6603
.5045 .5997 .6425 .6731
•452 .5605 .6111 .6472
.384 .5121 .571 .6131
.359 .475 .5377 .5834
.350 .447 .5097 .5574
.346 .425 .4862 .5347
.345 .408 .4663 .5146
R2 a_
R = o 108
--= i.i x
e
CHAMBER
Disk Tip
2
3
4
5
RADIAL t 6
TURN 7
SEAL 8
+ 9
I0
11
12
TABLE 3 RESULTS FOR TEST CASE GIVING PREDICTED CHAMBER
SWIRL RATIO FOR ASSUMED SUPPLY PRESSURE CONDITIONS
0
.08
.17
.25
.31
.34
(SEE TABLE I)
SJ = .000015 0.0001 0.00022 0.00034
.52 .52 .52 .52
.496 .509 .522 .534
.496 .513 .532 .549
.497 .519 .544 .567
.492 .529 .563 .592
.47 .524 .564 .597
.43 .473 .518 .558
.3513 .4105 .465 .511
.3445 .3798 .431 .478
.3438 .364 .408 .453
.3438 .3558 .392 .434
.3438 .3512 .382 .421
R
e
R2_
o
-- = 3 x 106
232
CHAMBER
0
1
2
3
4
5
RADIAL + 6
TURN 7
SEAL + 8
9
I0
11
12
13
14
15
16
17
18
19
20
21
TABLE 4 RESULTS OF LEAKAGE AND GAS SWIRL
FOR FLOW FROM LAST STAGE TO A BALANCE PISTON
LABYRINTH HAVING 15 TEETH
X SWIRL RADIUS WALL SPACE
(dim.) (dim.) (in.) (in.)
0 .637 9 -
.001 .641 8.94 .15
.07 .6756 8.34 .4
.15 .705 7.65 .36
.22 .732 7.02 .32
.293 .765 6.36 .3
.375 .81 5.62 .26
.42 .823 5.18 .22
- .781 5.14 -
- .74 5.14 -
- .71 5.14 -
- .68 5.14 -
- .66 5.14 -
- .64 5.14 -
- .62 5.14 -
- .61 5.14 -
- .598 5.14 -
- .588 5.14 -
- .580 5.14 -
- .573 _ 5.14 -
- .567 5.14 -
- .563 5.14 -
N = 11097. RPM
MW = 18.3
PS = 948 PSI
PE = 253 PSI
v = 7.63 x 10-4 in2/sec
Z = 0.979
y = 1.255
leakage = 1.09 ibm/sec
SJ = 0.00021
92(1162)
= 1.23 x 108
Re = -4
7.63 x I0
Cp = 0.52
c = 0.0115 in. radial clearance
type seal = interlocking
233
/
/
FOR RADIAL CHAMBERS;
Figure I. - Typical compressor stage showing disk cover gas
leakage path with nomenclature for analysis.
234
ZI I I I I I I
.00034
.00024
S J :,00020
SJ=.O0010
S J ,,00005
NO LEAKAGE FLOW
I I I I 1
O. I 0.2 0.3 0.4 0.5 0.6 O.7 O.8 0.9
X .'POSITION FACTOR
Figuce 2. - Velocity pcofile along cadius, b = Ul(c_)
versus X = (Ro-r)/R o at Re = R_ _/v = 9.82×105 .
Overplot of typical compressor result with
Re = 1.1×108 (From Jimbo - fig. 6 ref. (6)).
I I I I l I I [ |
O--C) S= .--i_o=.0551 J
0 0--0 S =-i_-o --.0362
I.O- bo ESTIMATED =.440 //
U /Io,
-/-bo;O.52
/
_ o.5',
°t ° i,_ • PROGRAM RESULTS FOR APPROXIMATE TEST RIG CONDITIONS
0.0 1 I l I I I I I I
O.O O.I 0.2 03 0.4 0.5 O,6 0.7 0.8
X : POSITION FACTOR
Figure 3. - Tangential velocity pcofile along radius
b = U/(c_) versus X = Ro-r/R o for constant
leakase flow SJ = (S/Ro)[V/Ro_)] = 0.000104 at
Re = 9.82X105. Overplot of a typical compressor
result with SJ = 0.00014 and Re = 1.1X108; test
case points for SJ = 0.0001 and Re = 3X106 (over-
plot on fig. 18 of cef. (6)).
235
I I i f i I I I
0--0 S =_ =.0551
_ s:-_0=.o3o2"°
_- bo ESTIMATED=.415
I"- I.
..J
_bo:0"65
Z _ 0 0--_0 0 ./_
z°"'
0.0 1 I I I I I I I
0.0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8
X" POSITION FACTOR
Figure 4. - Tangential velocity profile along radius
b = U/(r_) versus X = Ro-r/R o for constant
leakage flow SJ = (S/Ro)[V/(Ro_)] = 0.000052 at
Re = 9.82×105 . Overplot of compressor having
SJ = 0.0000372 and R e = 1.3×108 (overplot on
fig. 20 of ref. (6)).
09
236


Influence of disk leakage path on labyrinth seal inlet swirl ratio

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