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TRANSCRIPT
A Comparison Of Experimental And TheoreticalResults For Labyrinth Gas Seals
by fV/? _ _ - / _/
Joseph Kirk Scharrer
February 1987
TRC-SEAL-3-87
I
https://ntrs.nasa.gov/search.jsp?R=19870008663 2020-06-23T05:22:50+00:00Z
A COMPARISON OF EXPERIMENTAL AND THEORETICAL
RESULTS FOR LABYRINTH GAS SEALS
by
JOSEPH KIRK SCflARRER
Texas AZHUnlversity
Turbomachinery Laboratories
Hechanical Engineering Department
College Station, Texas 778q3
February 1987
TRC-SEAL-_-87
iii
0
ABSTRACT
A Comparison of Experimental and Theoretical Results
for Labyrinth Gas Seals. (May 1987)
Joseph K. Scharrer, B.S., Northern Arizona University;
M.S., Texas A&M University
Chairman of Advisory Committee: Dr. Dara Chllds
The basic equations are derived
for compressible flow In a labyrinth seal.
completely turbulent and Isoenergetlc. The
for a two-control-volume model
The flow is assumed to be
wall friction factors are
determined using the Blaslus formula. Jet flow theory Is used for the
calculation of the recirculation velocity In the cavity. Linearized
zeroth and flrst-order perturbation equations are developed for small
motion about a centered positlon by an expansion In the eccentricity
ratio. The zeroth-order pressure distribution ls found by satisfying
the leakage equation. The circumferential velocity distribution Is
determined by satisfying the momentum equations. The first order
equations are solved by a separation of varlable solution.
Integration of the resultant pressure distribution along and around
the seal defines the reaction force developed by the seal and the
corresponding dynamic coefficients. The results of thle analysis are
compared to experimental test results presented In thls report. The
results presented are for three teeth-on-rotor and three teeth-on-
atator labyrinth seals with different radial clearances. The theory
PEECEDING PAGE BLANK- NOT FILMED
_V
compares well wlth the cross-coupled stiffness data For both seal
types and with the direct damplng data for a teeth-on-rotor labyrlnth
seal. For a teeth-on-stator labyrlnth seal, the test resttlts show a
decrease In dlreet damping for an Increase In radial seal clearance,
while the theory shows the opposlte.
v
ACKNOWLEDGEMENT
Thls project .would not have been posslble wlthout the hard work
and dedication of Dr. Dara Chllds, Kelth Hale, Davld Elrod, Anne
Owens, and Dean Nunez. Thls project was supported In part by NASA
Grant NAS8-33716 from NASA Lewis Research Center and AFOSR Contract
F_9620-B2-K-O033.
vl
TABLE OF CONTENTS
ABSTRACT ...........................
ACKNOWLEDGEMENT .......................
TABLE OF CONTENIS .......................
LIST OF TABLES ........................
LIST OF FIGURES .......................
NOMENCLATURE .....................
CHAPTER I. INTRODUCTION ..................
CHAPTEH II. THEORETICAL DEVELOPMENT .............
Seal Analysis Overview ..................Dividing Streamline Approach ...............
Geometric Boundary Approach ...............
Perturbation Analysis ...................
CHAPTER IIl. TEST APPARATUS AND FACILITY ...........
Testing Approach .....................
Apparatus Overview ....................
Test Hardware .......................
Ins trumen tation ......................
Data Acquisition and Reduction ..............
Procedure ........................
CHAPTER IV. TEST RESULTS: INTRUDUCTION ...........
Normalized Parameters ...................
Relative Uncertainity ...................
Selection of Report Data .................
CHAPTER V. TEST RESULTS: RELATIVE PERFORMANCE OF SEALS . . .
Leakage ..........................
Direct Stiffness ......................
Cross-coupled Stiffness ..................
Direct Damping .......................
Stabillty Analysis .....................
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95
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100109118
127
vii
CHAPTER VI. TEST RESULTS: COtIPARISON TO THEORETICAL
PREDICTIONS ............
Static Results ......................
Dynamic Results .....................
CHAPTER VII. CONCLUSIONS ..................
REFERENCES .........................
APPENDIX A: GOVERNING EQUATIONS FOR TEETH-ON-STATOR SEAL . . .
APPENDIX B: DEFINITION OF THE FIRST ORDER CONTINUITY
AND MOMENTUM EQUATION COEFFICIENTS ........
APPENDIX C: SEPARATION OF THE CUNTINUITY AND MOMENTUM
EQUATIONS AND DEFINITION OF THE SYSTEM
MATRIX ELEMENTS ..................
APPENDIX D: THEORY VS. EXPERIMENT ...............
VITA ..............................
133
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LIST OF TABLES
Table 1. Seal geometries calculated by Rhode .........
Table 2. Tabulated solution of equation (56) .........
Table 3. Test stator specifications .............
Table 4. Test rotor specifications ..............
Table 5. Test seal specifications ..............
Table 6. Definitions of symbols used in figures .......
Table 7. Growth of rotor with rotational speed ........
Table 8. Normalized coefficients ...............
Table 9. Input parameters for seal program ..........
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Fig. I
Fig. 2
Fig. 5
Fig. 6
Fig. 7
Fig• 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
LIST OF FIGUHES
Small motion of a seal rotor about an eccentric
position .......................
Small motion of a seal rotor about a centered
position ................. ". ....
Flow pattern in a labyrinth seal cavity ........
Two control volume model with recirculation velocity,
U2 • • • • • • • • • • • • • • • • • • • • • • • • •
The "box-ln-a-box" control volume model of ref. [21]
A typical cavity ...................
Control volumes separated by dividing streamline.
Isometric view of control volumes ...........
Control volume areas .................
Forces on control volumes ...............
Pressure forces on control volume I ..........
Half-infinite Jet model ................
Control volumes with geometric boundary ........
Isometric view of control volumes with geometric
boundary .......................
Forces on control volumes with geometric boundary• . .
Pressure forces on control volume I of geometric
boundary model ....................
A comparison of Theoretical and CFD results for statorwall shear stress ...................
A comparison of Theoretical and CFD results for rotorwall shear stress .................
Model of a semi-contained turbulent jet .......
CFD calculatlon of dimenslonless recirculation
velocity .......................
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7
9
9
11
14
14
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15
16
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18
27
27
29
29
32
32
33
37
X
Fig. 21 External shaker method used for coefficientidentification ....................
Fig. 22 Components used for static and dynamicdisplacement of seal rotor ..............
Fig. 23 Test apparatus ....................
Flg. 24 Shaking motion used for rotordynamlc coefficientidentification ....................
Fig. 25 Inlet guide vane detail ...............
Fig. 26 Cross-sectional view of test section showing
rotor-shaft assembly .................
Flg. 27 Detail of smooth stator ................
Flg. 28 Smooth and labyrinth stator inserts for .4ram
(0.0!6 In.) r=_=l o=_, _ .......
Flg. 29 Detail of labyrinth rotor ...............
Flg. 30 Detail of labyrinth tooth ...............
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig• 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. I;1
Hlgh speed rotor-shaft assembly ............
Test apparatus assembly ................
Exploded view of test apparatus ...........
Signal conditioning schematic for data acquisition . .
Inlet circumferential velocities for seal I......
Inlet clrcumferentlal velocities for seal 2 ......
Inlet circumferential velocities for seal 3 ......
Itcomparison of dimensional and nondimensional
direct stiffness coefficients .............
Itcomparison of dimensional and normalized direct
damping coefficients .................
Leakage versus radial seal clearance at an inlet
pressure of 3.08 bar and rotor speed of 3000 cpm. . .
Leakage versus radial seal clearance at an inlet
pressure of 3.08 bar and rotor speed of 16000 cpm. .
Page
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Fig. 42
Fig. 43
Fig. 114
Fig. 45
Fi g. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54
Fig. 55
Leakage versus radial sea] c]ear'ancp at an inlet
pressure of 8.25 bar and rotor speed of 3000 epm.
Leakage versus radial seal clearance at an inlet
pressure of 8.25 bar and rotor speed of 16000 cpm...
Direct stiffness versus inlet clrcumferentlal velocity
ratio at an inlet pressure of 3.08 bar and rotor speedof 3000 epm ......................
Direct stlffness versus inlet circumferential velocity
ratio at an inlet pressure of 3.08 bar and rotor speedof 16000 cpm .....................
Direct stiffness versus inlet eircumlferential velocity
ratio at an inlet pressure of 8.25 bar and rotor speedof 3000 cpm
Direct stiffness versus inlet circumferential velocity
rml"In mt =n 4n!et press._e ,-,r 8 o= _.... _ -^_^-............ _, ._p u_, o,,u ,uuu, speedof 16000 cpm .....................
Direct stiffness versus rotor speed for seal 1 andinlet circumferential velocity 3 ..........
Direct stiffness versus rotor speed for seal 2 and
inlet circumferential velocity 3 ..........
Direct stiffness versus rotor speed for seal 3 and
inlet circumferential velocity 3..........
Dimensionless direct stiffness versus rotor speed for
seal I and inlet circumferential velocity 3......
Cross-coupled stiffness versus inlet circumferential
velocity ratio at an inlet pressure of 3.08 bar and
rotor speed of 3000 cpm ................
Cross-coupled stiffness versus inlet circumferential
velocity ratio at an inlet pressure of 3.08 bar and
rotor speed of 16000 cpm ...............
Cross-coupled stiffness versus inlet circumferential
velocity ratio at an inlet pressure of 8.25 bar and
rotor speed of 3000 epm ...............
Cross-coupled stiffness versus inlet circumferential
velocity ratio at an inlet pressure of 8.25 bar and
rotor speed of 16000 cpm ...............
Page
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Fig. 56 Cross-coupled stiffness versus rotor speed for seal 1
and inlet circumferential velocity 3. • ......
Fig. 57 Cross-coupled stiffness versus rotor speed for seal 2
and inlet clrcumferential velocity 3.......
Fig. 58 Cross-coupled stiffness versus rotor speed for seal 3
and inlet circumferential velocity 3.......
Fig. 59 Dimensionless cross-coupled stiffness versus rotor
speed for seal I and inlet circumferential velocity 5.
Fig. 60 Direct damping versus Inlet circumferent, lal velocity
ratio at an inlet pressure of 3.08 bar and rotor speed
of 3000 cpm....................
Fig. 61 Direct damping versus inlet circumfprentlal velocity
ratio at an inlet pressure of 3.08 bar and rotor speed
of 16000 cpm ....................
Fi g. 62 Direct damping versus inlet circumferential velocity
ratio at an inlet pressure of 8.25 bar and rotor speed
of 3000 cpm ......................
Fig. 63 Direct damping versus inlet circumferential velocity
ratio at an inlet pressure of 8.25 bar and rotor speed
of 16000 opm .....................
Fig. 64 Direct damping versus rotor speed for seal 1 and inlet
circumferentlal velocity 3.............
Fig. 65 Direct damping versus rotor' speed for seal 2 and inlet
circumferential velocity 3 .............
Fig. 66 Direct damping versus rotor speed for seal 3 and inlet
circumferential velocity 3.............
Fig. 67 Normalized direct damping versus rotor speed for seal
I and Inlet circumferential velocity 3........
Fig. 68 Forces on a synchronously precessing seal .......
Fi g. 69 Whirl frequency ratio versus inlet circumferentialvelocity ratio at an Inlet pressure of" 3.08 bar" and
rotor speed of 16000 cpm ...............
Fig. 70 Whirl frequency ratio versus inlet circumferential
velocity ratio at an inlet pressure of 8.25 bar and
rotor speed of 16000 cpm ...............
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Fig. 71
Flg. 72
Fig. 73
Fig. 74
Fig. 75
Fig. 76
Fig. 77
Fig. 78
Fig. 79
Fig. 80
Fig. 81
Fig. 82
Whirl frequency ratio versus rotor speed for the seals
of table 5 at an inlet pressure of 8.25 bar and inlet
circumferential velocity 5 ..............
Pressure gradients of seal I for the rotor speeds oftable 6 at an inlet pressure of 8.25 bar and inlet
clrcumferential velocity I..............
A comparison of experimental and theoretical pressure
gradients of seal I for a rotor speed of 3000 epm and
inlet circumferential velocity 3...........
A comparison of experimental and theoretical pressure
gradients of seal 2 for a rotor speed of 3000 cpm andInlet circumferential velocity 3 ..........
A comparison of experimental and theoretical pressure
gradients of seal 3 for a rotor speed of 3000 ClXn andinlet circumferential velocity 3...........
A comparison of experimental and theoretical leakage
versus inlet circumferential velocity ratio for seal I
at a rotor speed of 3000 cpm .............
A comparison of experimental and theoretlcal leakage
versus inlet circumferential velocity ratio for seal 2
at a rotor speed of 3000 cpm .............
A comparison of experimental and theoretical leakage
versus inlet circumferential velocity ratio for seal 3
at a rotor speed of 3000 cpm .............
A comparison of experimental and theoretical direct
stiffness versus inlet circumferential velocity ratio
at an inlet pressure of 3.08 bar and rotor speed of
3000 cpm.......................
A comparison of experimental and theoretical direct
stiffness versus inlet circumferential velocity ratio
at an inlet pressure of 3.08 bar and rotor speed of16000 cpm .......................
A comparison of experimental and theoretical direct
stiffness versus inlet circumferential velocity ratio
at an inlet pressure of 8.25 bar and rotor speed of
3000 cpm .......................
A comparison of experimental and theoretical direct
stiffness versus inlet circumferential velocity ratio
at an Inlet pressure of 8.25 bar and rotor speed of
16000 cpm .......................
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IL17
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FIE. 83
Fig. 84
Fig. 85
Fig. 86
Fig. 87
Fig. 88
Fig. 89
Fig. 90
Fig. 91
Fig. 92
Fig. 93
A comparison of experimental and theoretical direct
stiffness versus rotor speed for seal I and inlet
clrcumferentlal veloclty 5 ..............
Itcomparison of experimental and theoretical direct
stiffness versus rotor speed for seal 2 and inlet
elrcumferentlal veloclty 5 ..............
Itcomparison of experimental and theoretlcal direct
stiffness versus rotor speed for seal 3 and inlet
circumferential velocity 5 ..............
It comparison of experlmental and theoretical cross-
coupled stiffness versus inlet clrcumferential velocity
ratio at an inlet pressure of 3,08 bar and rotor speed
of 3000 cpm ......................
It comparison of experimental and theoretical cross-
coupled stiffness versus inlet circumferential velocity
-_'^ at an Inlet pressure of _ _o_.vv b__ and rotor speed
of 16000 cpm .....................
A comparison of experimental and theoretical cross-
coupled stiffness versus Inlet circumferential velocity
ratio at an Inlet pressure of 8.25 bar and rotor speed
of 3000 cpm ......................
A comparison of experlmental and theoretical cross-
coupled stiffness versus inlet circumferential velocity
ratio at an inlet pressure of 8.25 bar and rotor speed
of 16000 cpm .....................
Itcomparison of experimental and theoretical cross-
coupled stiffness versus rotor speed for seal I andinlet circumferential veloclty 5 ...........
Itcomparison of experimental and theoretical cross-
coupled stiffness versus rotor speed for seal 2 and
Inlet clrcumferentlal velocity 5 ...........
Itcomparison of experimental and theoretlcal cross-
coupled stiffness versus rotor speed for seal 3 andinlet clrcumferentlal velocity 5...........
Itcomparison of experimental and theoretical direct
damping versus Inlet clrcumferentlal velocity ratio
at an Inlet pressure of 3.08 bar and rotor speed of
3000 cpm .......................
Page
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150
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156
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160
XV
Fig. 94 A comparison of experimental and theoretical direct
damping versus inlet circumferential velocity ratio
at an inlet pressure of 3.O8 bar- and rotor speed of
16000 cpm ....... ................
Fig. 95 A comparison of experlmental and theoretlcal direct
damping versus inlet cireumlferential velocity ratio
at an Inlet pressure of 8.25 bar and rotor speed of
3000 epm .......................
Fig. 96 A comparison of experimental and theoretical direct
damping versus inlet circumferential velocity ratio
at an inlet pressure of 8.25 bar and rotor speed of
I6000 cpm .......................
Fig. 97 A comparison of experimental and theoretical direct
damping versus rotor speed for seal I and Inlet
circumferential velocity 5 ..............
Fig. 98 8 nr_nn_r|_nn n? m,lnmrim_ntml _nrl th_nmotimml Hirpot
damping versus rotor speed for seal 2 and Inletcircumferential velocity 5 .............
Fig. 99 A comparison of experimental and theoretical direct
damping versus rotor speed for seal 3 and inlet
elrcumferentlal velocity 5 ...............
Fig. |00 A comparison of experimental and theoretical results
of this report with those of [18] for cross-coupled
stiffness
Fig. 101 A comparison of experimental and theoretical results
of thls report wlth those of [18] for direct damping.
Fig. DI Direct stiffness versus rotor speed for seal I and
inlet circumferential velocity 5...........
Fig. D2 Direct stiffness versus rotor speed for seal 2 and
inlet circumferential velocity 5...........
Fig. D3 Direct stiffness versus rotor speed for seal 3 and
inlet circumferential velocity 5...........
Fig. D4 Cross-coupled stiffness versus rotor speed for seal ]
and inlet circumferential velocity 5.........
Fig. D5 Cross-coupled stiffness versus rotor speed for seal 2
and inlet circumferential velocity 5........
Fig. D6 Cross-coupled stiffness versus rotor speed for seal 3and inlet circumferential velocity 5.......
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192
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Fig. D7 Direct damping versus rotor speed for meal 1 and Inletcircumferential velocity 5 ..............
Fig. D8 Direct damping versus rotor speed for seal 2 and inletcircumferential velocity 5 ..............
Flg, D9 Direct damping versus rotor speed for seal 3 and inlet
clrcumferentlal velocity 5..............
Fig. DIO A comparlmon of experimental and theoretical leakage
versus inlet circumferential velocity ratio for seal I
at a rotor speed of 16000 cpm .............
Fig. D11A comparison of experimental and theoretical leakage
versus inlet circumferential velocity ratio for seal 2at a rotor speed of 16000 cpm ............
Flg. D12 A comparlmon of experimental and theoretical leakage
versus inlet clrcumferentlal velocity ratio for seal 3at a rotor speed of 16000 cpm .............
Fig. D13 A comparison of experimental and theoretical pressure
gradients of seal I for a rotor speed of 3000 cpm and
inlet circumferential velocity I..........
Fig. D14 A comparison of experimental and theoretical pressure
gradients of seal I for a rotor speed of 3000 cpm andinlet circumferential velocity 2..........
Fig. D15 A comparison of experimental and theoretical pressure
gradients of seal I for a rotor speed of 3000 cpm andinlet circumferential velocity t ...........
Fig. D16 A comparison of experimental and theoretical pressure
gradients of seal I for a rotor speed of 3000 cpm and
inlet circumferential velocity 5 ...........
Fig. D17 A comparison of experlmental and theoretical pressure
gradients of seal 2 for a rotor speed of 3000 cpm and
inlet circumferential velocity I...........
Flg. D18 A comparison of experimental and theoretlcal pressure
gradients of seal 2 for a rotor speed of 3000 cpm andinlet clrcumferential velocity 2 ...........
Fig. D19 A comparison of experimental and theoretical pressure
gradients of seal 2 for a rotor speed of 3000 cpm and
inlet circumferential velocity 4...........
Page
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197
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199
200
201
202
203
204
205
206
207
@ xvll
Fig. D20 A comparison of experimental and theoretical pressure
gradients or seal 2 for a rotor speed of 3000 cpm and
inlet clrcumferentlal veloclty 5 ...........
Page
208
Fig. D21A comparison of experimental and theoretical pressure
gradients of seal 3 for a rotor speed of 3000 cpm and
• inlet circumferential velocity 1 ........... 209
Fig. D22 A comparison of experimental and theoretical pressure
gradients of seal 3 for a rotor speed of 3000 cpm and
inlet circumferential velocity 2 ...........
Fig. D23 A comparison of experimental and theoretlcal pressure
gradients of seal 3 for a rotor speed of 3000 cpm and
inlet circumferential velocity 4 ...........
;)10
211
Fig. D24 A comparison of experimental and theoretical pressure
gradients of seal 3 for a rotor speed of 3000 cpm and
• inlet circumferential velocity 5 ........... 212
xvttt
NOHENCLATURE
A1 Cross sectional area of control volume (La); illustrated in
figures (g) and (1_)
B1 Helght o1" labyrinth seal strlp (I.); Illustrated In flgure (6)
C Direct damplng eoe1"flclent (Ft/L)
Cr Nomlnal radial clearance (L); illustrated In rlgure (6)
Dh Hydraulic diameter or cavity (L); introduced In equation (_4)
H Local radial clearance (L)
K Direct stiffness eoe1"1"Iclent (F/L)
L Pitch of seal strips "');t,- iilu_trated in figure (6)
NT Number of seal strips
NC=NT-I Number of cavities
P Pressure (F/L 2)
R Gas constant (L21Tt 2)
Rs, Radius of control volume I (L); illustrated In figure (6)
Rs 2 Radius of control volume II (L); illustrated In figure (6)
Rsm Surface velocity of rotor (L/t)
T Temperature (T)
Tp Tooth tlp width (L); illustrated In figure (6)
U I Average axial velocity for control volume I (L/t); illustrated In
figure (7)
U2 Average axial velocity for control volume II (L/t); illustrated In
figure (7)
N, i Average clrcum1"erentlal velocity for control volume I (L/t);
illustrated in figure (7)
@
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xix
circumferential velocity for control volume II (L/t);W2i Average
illustrated in figure (7)
Wo I Average circumferential velocity in the interface between control
volumes I and II (L/t); introduced in equation (40)
a,b Radial seal dieplacement components due to elliptical whirl (L);
introduced in equation (73)
ar Dimensionless length upon which shear stress acts on rotor;
introduced in equation (42)
as Dimensionless length upon which shear stress acts on stator;
introduced in equation (42)
c Cross coupled damping coefficient (Ft/L); in equation (41)
eo Displacement of the seal rotor from centered position (L)
k Cross coupled stiffness coefficient (F/L); in equation (41)
Leakage mass flow rate per circumferential length (H/Lt)
mr, nr, ms, ns Coefficients for friction factor; introduced in
equation (43)
t Time (t)
v Total velocity (L/t); introduced in equation (48)
w Shaft angular velocity (I/t)
Q Shaft preceeslonaL1 velocity (I/t)
p Density of fluid (M/L')
Kinematic viscosity (L'/t)
¢ - eo/C r Eccentricity ratio
• Turbulent viscosity (Ft/L');introduced in equation (9)
= 3.141592
Y Ratio of specific heats
xx
Subscripts
o Zeroth-order component
I First-order component, control volume I value
2 Control volume II value
t 1-th chamber value
J Value along the dlvtdlng streamline
x X-dlrectlon
y Y-dlrectlon
r Reservoir value
s Sump value
CHAPTEH I
INTRODUCTION
The problems of instability and synchronous response In
turbomachlnes have arisen recently because of the trends In design
toward greater efficiency with higher performance. To achieve these
design goals, the machines are designed for hlgher speeds, larger
loadlngs, and tighter clearances. In order to achieve the higher
speeds, rotors frequently traverse several critical speeds (speeds
whlch coincide wlth the rotor's damped natural frequency). The
characteristics of synchronous reponse, when the rotor vibrates at a
frequency coincident wlth the running speed, are such that the
vibrational amplitude reaches a maximum at each critical speed. In
order to limit the peak synchronous vibration levels, damping must be
introduced Into the rotor system. As loadlngs are increased and clear-
ances decreased, fluid forces increase and can lead to unstable or
self-exclted vibrations. This motion Is typically subsynchronous,
whlch means that the rotor whirls at a frequency less than the rotating
speed, and occurs wlth large amplitudes which grow as running speed
increases. Thls eltuatlon can also be lmproved by adding damping to
the rotor system, which would help curb the growth of the amplitudes.
One of the rotordynamlc force mechanisms which plays a role In self-
excited vlbratlon and synchronous response ls that of the forces devel-
oped by labyrinth seals.
Journal Model: ASME Journal of Trlbology
2I
A limited amount of experimental data has been published to date
on the determination of stiffness and damping coefficients for
labyrinth gas seals. The flrst publlshed results for stiffness
coefficients were those of Wachter and Benckert [I,2,3]. They
investigated the following three types of seals: a) teeth-on-stator, b)
teeth on the rotor and stator, and c) teeth on the stator and steps or
grooves on the rotor. These results were limited in that the pressure
drop was small, much of the data were for nonrotatlng seals, no data
were presented for seals wlth teeth on the rotor, the rotor speed was
llmlted, and tests where rotation and Inlet tangential velocity existed
simultaneously were very scarce.
The next Investigation was carried out by Wright [4], who measured
an equivalent radial and tangential stiffness for slngle-cavlty seals
wlth teeth on the stator. Although for a very limited and special
case, Wrlght's results do glve insight Into the effect of pressure
drop, convergence or divergence of the clearance, and forward or
backward whirl of a seal. These results could be reduced to direct and
cross-coupled stiffness and dampln E, hence, they are the first
published damping coefficients for teeth-on-stator labyrinth seals.
Brown and Leon E [5] Investigated the same seal configurations as
Wachter and Benckert, In an effort to verify and extend their work.
Their results include variations of pressure, geometry, rotor speed,
and inlet tangential velocity. Although the investigation was
extensive, the published results are 11mlted beca_aee of the lack of
information concerning operating conditions for the various tests.
Childs and Scharrer [6] investigated geometrically similar teeth-
on-rotor and teeth-on-stator labyrinth seals for stiffness and damping
coefficients up to speeds of 8000 cpm. Kanemltsu and Ohsawa [7]
Investigated multistage teeth-on-stator and interlocking labyrinth
seals up to speeds of 2400 cpm. They measured an effective radial and
tangential stiffness while varying the whirl frequency of the rotor.
These data could be reduced to stiffness and damping coefficients.
Hisa et al. [8] investigated teeth-on-stator seals wlth 2-4 teeth and a
teeth-on-stator seal wlth steps on the rotor up to speeds of 6000 epm.
These data only included statle tests for direct and cross-coupled
stiffness using steam.
In the area of theoretlcal analysis of labyrinth seals, there Is
much more published information. The first steps toward the analysis
of a labyrinth seal were taken by Alford [9_, who neglected
circumferential flow and Spark et al [I0] who neglected rotation of the
shaft. Vance and Murphy [11] extended the Alford analysis by
introducing a more realistic assumption of choked flow.
Kostyuk [12] performed the first comprehensive analysis, but
failed to include the change In area due to eccentrloity whlch is
responsible for the relationship between cross-coupled forces and
parallel rotor displacements, lwatsubo [13,14] reflned the Kostyuk
model by Includlng the time dependency of area change but neglected the
area derivative In the circumferential direction. Kurohash{ [15]
Incorporated dependency of the flow coefflelent on eccentrlclty into
hls analysis, but assumed that the circumferential velocity in each
cavity was the same. Gans [16] Improved on the Iwatsubo model by
Introducing the area derivative In the circumferential direction.
Martinez-Sanchez et al. [17] produced results similar to Gans, but used
4 •
empirical flow coefficients to improve their results. Childs and
Seharrer [18] improved the Iwatsubo solutlorl by using a modified set of
reduced governing equations and an efficient solutlon technique.
However, their solution continued to neglect axial veloclttes.
Hauck [19] introduced the use of multl-control-volume analysis in
the study of labyrlnth seals by applylng the equations of Impulse and
"balance of moments" to a three-control-volume model. These equations
were written In the axial dlrection only and neglected the effects of
rotor speed. PuJlkawa et al. [20] Introduced the use of two control
volumes into the analysls of labyrlnth seals, but their analysls, which
neglects the axlal velocity components, was heavlly dependent on
emplrlcal information which is not customarlly available. Flnally,
Jenny et al. [21] used the two-control-volume approach In conjunction
with a two dlmenslonal solutlon to the Navler-Stokes equations to
account for the free shear stress between the Jet flow and the cavity
flow. However, they neglected the reeirculation velocity in the
cavity, assumed the flow to be incompressible, and their free shear
stress relatlon required a correction factor to fit the experlmental
data. Further, the present author obtains different signs in the
expansion of the continuity equation and different perturbation
equations. These discrepancies are explained in detail in the
following section.
The most extensive comparison of analytical predictions and
experimental results was carried out by Seharrer [22] using the theory
of Chllds and Scharrer [18] and the results of Childs and Scharrer [6],
This comparison showed that the theory [18] predicts cross-coupled
5
stiffness reasonably wel], but underpredlets direct stlffness, direct
damping, and cross-coupled damping.
In reviewing the state of the art in labyrinth seal
experimentation and analysls, it becomes clear that there is a need for
(a) an improved theory for the prediction of damping coefficients, (b)
more extensive testing of teeth-on-rotor seals, and (c) test results
showing the effects of change of radial seal clearance and higher
speeds on stiffness and damping coefficients. This report will
describe the revised test facillty and program designed to measure the
forces developed in a gas labyrlnth seal. Some results, showing the
effect of radial clearance change and higher rotor speed on teeth-on-
rotor and teeth-on-stator labyrinth seals, will be presented. Also, a
new analysis, which more accurately describes the physics of the flow
field, will be presented and a comparison made between the theoretlcal
and experlmental mass flow rate, pressure distribution, and seal
coefficients for the new results presented here.
The _Jor contribution of this report is a new analysis which
incorporates the reclrculatlng velocity in the cavity into the shear
stress calculations. In addition, the analysis Is based on a close
comparison with the CFD results of Rhode [23,24]. The CFD results
were reinforcement for the
stresses and velocity profiles.
CFD and experimental results
seal. However, not enough data was provided in the
results for a comparison In this report.
assumptions and modelling of the shear
Stoff [25] carried out a comparison of
for incompressible flow In a labyrinth
paper to use the
6 •
CHAPTEH 11
THEORETICAL DEVELOPMENT
SEAL ANALYSIS OVERVIEW
As related to rotordynamics, seal analysis has the objective of
determining the reaction forces acting on the rotor arising from shaft
motion within the seal. There are two linearlzed seal models, expressed
in terms of dynamic coefficients, which have been suggested for the
force-motion relationship. For small motion about an eccentric
position, as shown in figure I, the relations of equation (I) have been
proposed.
Kyy (Co
$
[cxx co cx,c ICyx(¢o) Cyy(CoU(_ l
(I)
where the dynamic coefficients {Kxx,Kyy,Cxx,Cyy} and {Kxy,Kyx, Cxy,Cyx}
represent the direct stiffness and damping and the cross-coupled
stiffness and damping, respectively. These coefficients are functions
of the equilibrium eccentricity ratio, ¢o = eo/Cr, where eo is the
displacement of the rotor from the centered position and Cr is the
nominal radial clearance. The cross-coupling terms result when motion
in one plane results in a reaction in a plane orthogonal to it. These
cross-coupllng terms depend on the magnitude and direction (with
respect to the rotor's rotation) of the fluld's clrcumferentlal
veloclty. This velocity may exist at entry to the seal or may develop
as the fluid passes through the seal. The cross-coupled stiffness term
\
\
Fig. 1 Small motion of a'seal rotor about an eccentric position.
is the rotor spin speed,f_ is the precessional orbit
frequency.
\
Fig. 2 Small motion of a seal rotor about a centered position.
_Is the rotor spin speed,J'L is the precessional orbit
frequency.
8
usually produces a destabilizing
considerable interest.
much less significant
respect to stability.
force component, and Is therefore of
The cross-coupled damping term is generally
than the cross-coupled stiffness term wlth
The second Itnearized seal model Is applleable
for small motion about a centered position, as
form of the model Is
where the dynamic coefficient matrices are
Is used in the analysis which follows.
Preamble
The flow in a labyrinth
[14] and calculatlon [23] to be
shown in figure 2. The
skew-symmetrl c. This model
end, correction factors had to be
of the shear stress to improve the
damping coefficients, but, in the
incorporated into the calculatlon
correlation wlth test data.
flow region In the leakage path and a reclrculating velocity region In
the cavity Itself(see figure 3). The first attempts at analysis of
thls system neglected the axial velocity components in the flow and
concentrated on the clrcumferentlal components. Thls was the slngle
control volume approach, used in refs [9-18]. In an attempt to improve
upon the results of these analyses the two-control-volume approach was
introduced, see refs [20,21]. These analyses incorporated the axial
veloclty of the Jet flow into the solution but not the reclrculating
velocity component of the cavity flow. The results from Jenny et al.
[21] showed substantial improvement In the prediction of stiffness and
seal has been shown by experiment
comprised of two flow regimes: a Jet
\\\\\\\\_
Fig. 3 Flaw pattern in a labyrinth seal cavity.
STATP__._ r• I,.i, 'e ,
II" .... a
I .
Fig. 4 _ntml-volume model with
recirculation velocity, U2.
10 I
This report introduces the calculation of the reclrculatlon
veloclty Into the analysis. The model for the recirculation veloclty,
U:, used here Is illustrated in figure 4. Thls velocity component Is
important In the calculatlon of the eavlty shear stresses. The focus is
on the shear stresses, because experimental results [6] have shown that
the stiffness and damping eoefflelents are very sensltlve to the
circumferentlal velocity In the seal. In the control volume analysis
to be presented, the solution to the eIrcumferentlal momentum equatlon
yIelds the clroumferentlal velocity In the seal. An Improvement in the
shear stress calculation wl11 yield an Improvement In the calculation
of the stiffness and damping coefflclents.
Before proceeding wlth the solutlon development, the approach
taken In modelling the flow will be discussed. As mentioned
previously, the flow In a labyrinth seal is known to have two distinct
regions: a Jet flow region In the leakage path and a reclrculatlng flow
region In the cavity Itself(see figure 3). Therefore, a two-control-
volume model seems appropriate. The choice is between the "box-In-a-
box" model(see figure 5) of Jenny et al [21] or a more conventional
model wlth a control volume for the Jet flow and one for the
reclrculatlng flow In the cavity, as shown in figure 4. The two-
separate-control-volume model was chosen, since It Is suggested by the
known physics of the flow. The flow enters the seal and separates into
two distinct flow regions which are separated by the dividing
stream1 Inc.
The flnal question Is whether the control volumes should be
defined using a geometric boundary or using the dividing streamline as
ll
S TATO,qC.V.__7-
,P W,,U,;
..[--" '
"1I
I,I4
'LJ
II
IP,C.V X i
I
] mi,,1
,c_II
1., ," j-I'I.t
II
1
/
Fig. 5 '/he "box-in-a-box" control voltm_
O
0
Z2 •
reviewed and a method of solution discussed. The geometric
approach and solution is provided In the following section.
DIVIDING STREAMLINE APPROACH
Assumptions
The followlng assumptions are used
the boundary. The dividing streamline approach seems, at first, to be
the obvious choice. The governing equations would be simplified by the
restriction of no flow across a streamline, the free shear stress
relations are derived for flow along the dividing streamline, and the
solution for the velocity of the reclrculatlng flow may be derived for
flow along the dividing streamline. Despite these advantages, the
dividing streamline approach was not used, however, It wlll now be
boundary
equations:
I)
2)
in deriving the governing
The fluid is considered to be an ideal gas.
Pressure variations within a chamber are small compared to the
pressure difference across a seal strip.
3) The lowest frequency of acoustic resonance In the cavity ls much
higher than that of the rotor speed.
4) The eccentricity of the rotor Is small compared to the radlal seal
clearance.
5) Although the shear stress is slgnlfleant In the determlnatlon of
the flow parameters (velocity etc.), the contrlbutlon of the shear
stress to the forces on the rotor are negligible when compared to
the pressure forces.
6) The cavity flow Is turbulent and Isoenergetic.
7) The reclrculation velocity, U,, is unchanged by viscous stresses as
it swirls within a cavity.
13
ProceduF_
The followlng analysis is developed for the teeth-on-rotor "see-
through" labyrlnth seal shown In figure 6. The continuity and
circumferential momentum equations are derived for the two-control-
volume model shown in figures 7 through 11. A procedure Is discussed
for determining the approximate location of the dividing streamline and
the perturbation of the dlvldlng
centered positlon.
Continuity Equations
Figures 7 and 8 show
streamline. These control volumes
streamline for small motion about a
the control volumes deflned by the dlvldln 8
have a unity clrcumferentlal width.
Their continuity equations are:
I: + ml+l - ml - 0 (3)
II:
where the control
I)pW2A2
BpAI @pW:AI-- +
_t Rs,BB
SpA2-- ÷
Bt Rs2B8
volume areas, A, and
:, 0 (4)
are defined by
A=, are shown In figure 9 and
L L
A, " LCr + $ ydx ; A, - LB - $ ydx0 0
(5)
The followlng momentum equations for control volumes I and
derived using figures 10 and 11 which show
shear stresses acting on the control volumes.
that the reclrculatlon veloclty, U2, Is included In the shear stress
deflnltlons used In equations (6) and (7). These definitions are
developed in a subsequent sectlon of this chapter.
II are
the pressure forces and
It Is important to note
14 •
t!
Cr
li".t ¸ I
1 i
Fig. 6 A typical cavity.
Fig.
(Dividingstrea_nline)
7 Control volumes separated
by dividing streamline.
15
O
O
V_i_'_/de
+ 00 d6--,.. A,;
i
/ '__ _'W,_
Rs dO_._
F.jg. 8Isometric view of control volumes.
A2
Fig. 9 Control vo]ume areas.
16 •
_; Li Rs, de_- Ts as L, Rs,de
"Trar L,R
k_Jcu
Fig.lO Forces on control volumes
L!
RsdE)
Fig.ll Pressure forces on control volume I
17
apWIA l 2pW,A, aw, pw I aAj WIA l ap1. • + -- -- + ----- (6)
at RB, ae Rs, Be Rs, a6
• . A, aPi
÷ ml+lW,l - mlW,l-1 -Rs, aB
+ zJILl- xsiaslLl
apW2A ' 2pW2A 2 BW2 pW= aA= WzA z apII: -- + * -- -- + (7)
at Rs= ae Rs= ae Rs 2 ae
Streamline Location
A= aPI
As= aOZJlLl + xrlarlLl
The maln difficulty In obtaining a solution to the above equations
(3) through (7) is in the determination of the locatlon of the dividing
streamline. The streamllne definition, y(x), must be found to
determine the control-volumes areas, A, and A=, defined In equations 5.
There Is no known solution for the location of a dividing streamllne
for the three dimensional flow fleld found In a labyrinth cavity• An
approximation for the location of the dividing streamllne can be
obtained using the theory for the flow of a two-dlmenslonal, turbulent,
lsoenergetlc, half-lnflnite Jet. Figure 12 shows the model for this
theory. The flow ls assumed to enter wlth one velocity component, in
the x-dlrectlon, and spread Int0 the cavity, developlng a F-component
of velocity. This model does not account for the clrcumferentlal velo-
city component, which ls the same order of magnitude as the axial velo-
city, in a labyrinth seal flowfield. The solution procedure Involves
solving the Inflnltesslmal form
dlmenslonless velocity profile and
the continuity and momentum equations
of the x-momentum equation for the
then solving the integral form of
for the location of the dividing
18 •
y
Control Surface
Fig. 12 Half-infinite jet model.
19
streamline. A complete
et al. [26].
The following is a
determine the location of the Jet
derlvatlon uses the assumption
discussion of this theory can be found In Korst
derivation of the equations necessary to
dividing streamline. The following
that the curvature In the dividing
where the
averaged. The lrfflnitessimal x-momentum equation,
reduced using equation (8), is:
pu _u/_x ÷ pv _u/_y - _(ep_u/_y)/_y
where e ls the apparent(turbulent) kinematic viscosity.
streamline Is small. The Inflnltesslmal form of the continuity equation
for the flow lllustrated In figure 12 Is:
_(pu)/@x + _(pv)/@y = 0 (8)
x and y velocity components, u and v, respectively, are time
whlch has been
(9)
Since the flow
111ustrated In figure 12 Is a quasl-one-dlmenslonal Jet flow where
there Is little or no Inltlal vertlcal velocity component, equation (9)
can be llnearlzed using the followlng perturbation method:
u - Ul ÷ _" ; v = v" ; P = Pl ÷ P" (10)
where _v"l<<lU:l and lu.I<<Iu,l. The resultant equatlon Is:
plum @u"/Bx + p=Vz @u"/@y = epl 8=u"/_=y (11)
where e-e(x) and the second term (V= term) ls corL_ldered small. The
final form of the equatlon Is:
U l _u"/_x- • _:u"l_y: (12)
The following dlmenslonless variables are introduced:
¢ = U/U l - I+u"/U_,- y16
- x/6 (13)
F. = I • d_/(U,6)0
where 6 ls the lnltlal boundary layer thickness shown In figure 12.
20 ill
Equation (12) becomes:a¢/aE - a=¢/aE =
with the initial conditions:
$ - ¢(0,{) = 0 for -- < { < 0
¢ - ¢(0,{) - ¢o({) for 0 < { < 1.0
¢ - ¢(0,_) - 1.0 for 1.0< { < -
and boundary conditions:
¢ - ¢(E,--) - 0 for E > 0
¢ - $(E,-) = 1.0 for E > 0
The solution to
conditions Is:
where
(14)
equation (14) for the above Inltial and boundary
q -B =
¢ " 0"5[l-err(rip-n)] + 11vr_'_I ¢[(n-B)/np]e
n-np
np . I/(2/[) ; n = _npx
err(x) - 2/,_" $ exp(-B=)dB0
err (- x) - -err (x)
dB (15)
The apparent
viscosity far from the mixing region, e_:
e = e_f($)where
f($) --> 1.0 as $ --> ,,
According to Prandtl, this can be rewritten as:
el - Kb(x)[Uma X - Umin]
wlth ae/ay - 0
For a balf-lnflnite Jet, equation (17) is:
e I - Kb(x)U I
where b(x) is the width of the mixing region.
increases linearly, i.e.
b(x) = cx = c$8
where c is a constant, equation (16) becomes:
e = c$6U_f($)
viscosity, e, can be expressed in terms of the apparent
(16)
(17)
(18)
Assuming that the mixing
(19)
(20)
Substltutlng this Into 6, from equatlon (13) yields:
$
- e f $r($) de (21)0
Looking at a limiting case of equation (21):
as x/6--) -
then $--> -
6--> "
rip- 1/2V_--> 0
This limiting case is for either no initial boundary layer, which is a
good assumption for labyrinth seals, or fully developed velocity
profiles. Since np--> O, the variable n is now undefined. Liepman
and Laufer [27] have defined n for this limiting condition using the
following development. By definition:
as $--> -then f($) --> 1.0
Inserting this into equation (21) yields:
= c$=/2
By definition:
q - Cnp - C/2_-- E/(_Z_) - y/(x_2-d-)
Letting c = I/(2a=), yields the desired result:
q = oylx (22)
where a is the Jet spreading parameter. Korst and Tripp [28] used
experimental data to find the following relation for 0:
o = 12.0 + 2.758M, (for air) (23)
Goertler [29] has shown that the dimensionless velocity, $, follows
directly from equation (15) when np --> O:
@ = 0.5(1+err(n)) (24)
Equations (24) is a solution for the dimensionless velocity profile, $,
at any dimensionless position, n. The goal of this development is to
21
determine the dimensionless dividing streamline position, nj. nJ can
be obtalned by solving the integral form of the continuity and x-
momentum equations for the system shown In figure 12.
Control Volume Anal_sls
The coordinate systems and definition of the control surface are
shown in figure 12. The (x,y) coordinate system Is the Intrinsic coor-
dinate system whlle the (X,¥) coordinate system Is the reference
system. Equations (22) and (24) are approximate relations; exact
relatlonshlps, if known, would provide for conservation of momentum for
the constant pressure mixing region. The reference coordinate system
is the coordinate system In which momentum is conserved. The intrinsic
coordinate system Is located with respect to the reference coordinate
system by a control volume analysis utilizing the conservation of
momentum principle for thls constant pressure mixing region. The
relationship between the coordinate systems normal to the Jet Is:
with
X-Momentum Equation
The steady flow
Ym(x) = y-Y
Ym(O) - O.
x-momentum equation for the Jet flow shown In
figure 12, written for the reference coordinate system and expressed in
the previously defined dlmenslonless variables Is:
R R
S pu' I - $ pu' dY I (25)0 X'O -o X'X
For the momentum equation, the lower
Thls equation contains no surface
labyrinth seal if location R is far from of the stator wall.
equation (25) for the intrinsic coordinate system:
control surface is located at --.
forces. This is realistic for a
Rewriting
22
23
y6pu,dy I" #p"'dy I . )'gl),dy I0 x=O 6 x-O 0 x-x
(26)
Introducing the previously defined dimensionless variables, equations
(13) and (22), equation (26) becomes:
1.0 • nR_nm
np S (p/P_)¢o dE + nR - np- $ (P/P=)¢= dnO -R
(27)
Distance R is chosen such that=
I - ¢(nR) <<< i.O
Equation (27) becomes=
1.0 = nRnp $ (P/Pz)¢e dE + nR - np = $ (p/pz)¢ = dn + qm
0 --I=(28)
Applying the condition of no Inltlal boundary condition
equation (28) is:
nRnm- nR- I (p/p,)¢z dn
(np --> 0),
(29)
Continuity Equation
The steady flow
coordinate system, Is:
continuity equation, written for
R R
y pu dY I -.rpu dY l0 XmO YJ-Ym X-X
the reference
(30)
For the continuity equation, the lower control surface is coincident
with the Jet dividing streamline. Rewriting equation (30) for the
intrinsic coordinate system:
Introducing the
6 Ry pu dy I " Y pudy l " "fR;_mdy I
0 x-O 6 x=O yj x=x
prevlously defined dimensionless
(31)
coordinates and
muitJplying equation (31) by np/6:
24 I
Substltutlng
1.0 nR
np / (P/Pl)_o d_. + nR - np - $ (p/p_)_ dq ._ rim
0 nj
(32)
the results of the momentum equation, equation (29), into
equation (32) yields:
nR 1.0 nR
$ (P/Pi)_ dn - $ (p/pl)(1-_)o d_ + $ (p/p_)_= dn
nj --
Making the assumption
(33) becomes :
0
of no initial boundary
(33)
layer (np-->O), equation
nR nR$ (p/p=)$ dn- ./ (p/p=)_= dn (311)
nj -=
The density ratio, (p/p=), for lsoenergettc flow (constant temperature)
is given as:
P/Pi = (1-Ca=)/(1-Ca=¢ =) (35)
The ftna/ form of the continuity equation becomes:
nR nR$ (#/[1-Ca=_=J) dn = J (_:/[1-Ca=#=]) dn (36)
nj -=
Where Ca is the Crocco number. The Crocco number is defined as:
Ca = - (T-1)M=/(2+(_-I)H =)
The Crocco number
(37)
Is a dimensionless velocity similar to the Mach
number. The Crocco number uses the maximum Isentroptc speed of a gas
while the Math number uses the local speed of sound. The Math number
varles between 0 and -whlle the Crocco number has a range of 0 to 1.
The solution to equatlon (36),the locatlon of the dlvldlng
streamline, can be obtained by the following steps:
O) Calculate the Math number using the zeroth-order leakage value.
The zeroth-order leakage Is discussed in the next section.
1) Calculate the Crocco number using equation (37).
sis. Since equation
obtained. However,
2) Substitute equation (24) into equation (36) and integrate the
error function. The value of the error funetion at the llmlts R and --
Is 1.0, leaving an equation in nj only. This is solved for nj, whlch
is the dimensionless location of the dividing streamline.
3) Use the straight llne approximation, equation (22), to flnd y
as a function of x.
4) Insert y(x) from step 3 into equation (5) and calculate the
areas of the control volumes.
The above procedure yields the zeroth-order (centered) value for
the areas. The problem is to find the values for a perturbation analy-
(24) Is an error function, an explicit equation
_'ea for a pert_-batlon ,,,'-el_a,_u,ce cannot be
the above procedure could be carried out for a
range of clearances in the neighborhood of the nominal clearance and an
approximation for the change In area and a flnal result could be
obtained.
As noted at the beginning of this discussion, the advantages of
the dlvldlng streamline approaeh are that the free shear stress and
reeireulatlng velocity equations may be derlved along the dlvldlng
streamline, and the governing equations are simplified by the condition
of no mass flow across a streamline. The above solution procedure
yields only an approximation for the location of the dividing
streamline for a simplified (two-dimensional) flow while increasing the
difficulty in obtaining a solution, Therefore, the advantages of the
dividlng streamline approach are outweighed by the difficulty In
obtaining a solution. The geometric boundary approach and a
complete solution will now be presented.
25
• 26
GEOMEIHICBOUNDARYAPPROACH
Procedure
The analysls presented here Is developed for the teeth-on-rotor
"see-through" labyrlnth seal shown In figure 5. The equlvalent
equations for the teeth-on-starer labyrinth meal are given in Appendix
A. The continuity and elreumferential momentum equations will be
derived for the two-control-volume model shown in figures 13,14,15, and
16. A leakage model will be employed to account for the axlal flow.
The governing equations are linearlzed using perturbatlon analysis for
small motion about a centered position. The zeroth-order eontlnulty and
momentum equatlons w111 be solved to determine the steady state
pressure, axlal and eireumferentlal velocity for each cavity. The
flrst-order eontlnuity and momentum equations will be reduced to
llnearly independent, algebraic equations by assuming an elliptlea1
orbit for the shaft and a eorrespondlng harmonle response for the
pressure and veloelty perturbations. The force eoefflelents for the
seal are found by Integratlon of the first-order pressure perturbation
along and around the shaft.
Continuity Equatlons
The control volumes of figures 13 and 14 have a unity
circumferential width. Their continuity equations are:
I:
apA,
@t
II:
@pW,A, . . .
+ mi+1 _ mi + mr - O (38)
Rs,BB
@pA= @pW2A = •
"---- + mr - O (39)
at Rs,_)B
For the teeth-on-rotor ease, A_-LCr, A2-LB, Rs,=Rs, and Rsl-Rs÷B.
• 27
S TA70R
,e,,,,l""-'-p w U,, '_>"-1"11 ,, "'h I
J__J__ __ _ _ ,=_1
__-- ......... -jr--i,---7_ -
Flg. 13 Control Volumes with
geometric boundary.
Flg. 14 Isometric vlew of Control Volumes
wlth geometric boundary.
28
Momentum Equations
The following momentum equations for' control volumes I and II are
derived using figures 15 and 16 whieh show the pressure forces and
shear stresses acting on the control volumes.
apWIA, 2pWIAI aWl PWI aAi WIA l ap
I : . _--___ l . * ÷ mrWoi (40)at Rs= ae Rs= ae Rs= ae
• Az aPl
ml+iWll - miWii- I -Rs, aB
+ xJlLl- xslasiL1
II:apWzA = 2pW=A= aW= pW=
at Rs= a6 Rs=
aA= WzA• ap
ae Rs= ae
- mrWo! m
(41)
A= aPl
Rs• aezJlLl ÷ xrlarlLl
where ar
stresses act and are defined for the teeth-on-rotor labyrinth by
asl = I arl - (2BI ÷ Li) /hl •
Wo Is the clrcumferentlal velocity between the control volumes.
and as are the dimensionless length upon which the shear
(42)
Various models for the stator wall shear stress were evaluated by
comparison to CFD results of Rhode [23]. For a teeth-on-rotor labyrinth
meal, the optimum model for the stator shear stress (rotor shear stress
for a teeth-on-stator meal) was obtained by uslng the equation of
Olauert [30] for wall shear stress of a plane Jet issuing fore a slot.
However, this relation requires knowledge of the maximum axial velocity
and Its displacement from the wall• This information is not available
In a control volume analysis. The next best model, by comparison to
[23], Is Colebrook's formula [31], but thls equation Is not explicit In
the friction factor and cannot be perturbed. Experience [32] has shown
that the perturbation of the friction factor Is important in stiffness
29
' Ts as L, Rs, de_i Li Rs,dO_ 's°° L
W
Flg. 15 Forces on control volumes
with geometric boundary.
8PP,+ _ de
l_+___p _I. _, - _- r
.'--_ ; -| _,"
• "IIIIfll I
RsdE) L.
Flg. 16 Pressure forces on control volume I
of geometric boundary model.
30 •
calculations. The next best shear stress model is based on the assump-
tion that the shear stresses (for rotor and stator surfaces) are
similar to those found in the pipe analysis of Blaslus [33]. Blaslus
determined that the shear stresses for turbulent flow in a smooth pipe
could be written as
= mO
2
where Um is the mean flow velocity relative to the surface upon which
the shear stress is acting. The constants mo and no can be empirically
determined for a given surface from pressure flow experiments.
However, for smooth surfaces the coefficients given by Yamada [34] for
turbulent flow between annular surfaces are:
mo - -0.25 no - 0.079
Applying Blasius' equation to the labyrinth rotor surfaces yields the
following definitions for the rotor shear stress in the circumferential
direction. Note that the recirculatlon velocity, U=, is included in
the definition of the total velocity acting on the rotor.
I _(Rs=w-W,)Z+U=Z Dhl)mrIr - _ p/(Rs,w-W,)'+O," (Rs,w-W=) n . ' (43)V
where Dh21 is the hydraulic diameter of control volume II, defined by
Dhzi - 2BL/(B+L) (44)
Similarly, the stator shear stress in the circumferential direction is:
1 (vqdIz+O lI Dh_ msXs = _ p/WI'+U ," W l ns • (45)V
where Dh_i is the hydraulic diameter of control volume I, defined by
Dhll = 2CrL/(Cr+L)
and the axial velocity U_ Is
U i = _/pCr
(46)
(47)
31
Figure 17 shows a comparison of the predictions from equation (_5)
and CFD results for stator wall shear stress for seal A of table I.
The recirculatlon velocity, U2, is undefined at this point. It wlll be
discussed In the following section. Table I shows the seal geometries
calculated by Rhode [23]. The figure shows that the comparison Is very
good. Similar results are obtained for the other seals of table 1.
Figure 18 shows a comparison of rotor wall-shear-stress predictions
from equation (q3), CFD, and averaged CFD results for rotor wall shear
stress for seal A of table I. The averaged CFD result is used here for
comparison since the bulk flow model yields a single averaged result
for cavity shear stress and is not capable of modelling the complex '
flowfleld. The figure shows that the prediction of equation (43) is
close to the CFD results. The dlps in the CFD results are the lower
corners of the cavity. Similar results are obtained for the other
seals of table I.
The flow across a labyrinth tooth is very similar to the flow of a
turbulent jet issuing from a slot. The problem wlth using Jet-flow
results for labyrinth seals is that current Jet-flow theory only
considers the flow of a Jet with a coflowlng stream or a crossflowing
stream, not both. In the following derivation, the relations given by
Abramovich [35] for the velocity profile of a seml-contalned, one-
dimensional, turbulent Jet wlth a coflowing stream are assumed to apply
for the two-dlmenslonal labyrinth seal flow. According to Abramovlch
[35], the velocity profile for such a flow can be shown to flt the
following function when compared to experimental results:
1.5 2
32
6O
"_ 54n
_ 48
42
_) 36
3o
24J 18
12
8
0
SEAL A
--C]_) Resu]ts
---- --Theory
0c I Q e t e : -- o I
.2 .4 .8 .8 !.1 .3 .5 .7 -.9
XA.
Fig. 17 A comparison of Theoretical and
CFD results for stator wall shear
stress. See table 1 for seal
geometry,
SEAL A200
TCFD Avg.
o 180 !_--CFD16o -----
Results
140
I00
8O
40
zo
0 .2 .4 .O .8 l
• ! .$ .5 .? .eX/(L-28)
Fig. 18 A comparison of Theoretlcal and
CFD results for rotor wall shear
stress. See table 1 for seal
geometry.
33
/
V
////L.--.J
Fig. 19 Model of semi-contained turbulent Jet.
_8
B
L
Tp
Cr
Table 1. Seal i;eometrle8 calculated by Rhode.
Sealm
A B C D
72.0.5113mm 72.05113mm 72.05 JI3mm 1ti .780mm
3.175mm 3.175mm 3.175mm 0.889mm
3.175mm 3.175m" 3.969mm 0.8585mm
0.35ram O. 35ram 0.35ram 0.15ram
O.II06_mm 0.5OBtain 0.508,,,, 0.2159mm
°
34
where the coordinate y, the mlxtng thickness b, and the boundary layer
thickness Y= are defined In figure 19. The relationship between the
boundary layer thickness and the mixing thickness was found [35] by
comparison to experiment to be:
y=Ib - 0.584 - 0.134(v=/v !) (_9)
Once the velocity ratlo across the dlvldlng streamline, v=/v:, Is
found, equation (49) reduces to a constant. The total free shear stress
Is found using Prandtl's mixing length hypothesis [36]:
_Jt " pL" l_vlgy{(_vl_y) (50)
where the mixing length, _, for a labyrinth
from the calculations of Rhode [23] to be:
L = O.275b
seal, has been determined
(51)
Table I shows the seal geometries calculated by Rhode [22]. The mixing
length, E, given In equation (51) Is the most sensitive factor in this
solution. The large magnitude of the mixing length shows the high
turbulence level of the labyrinth flow as compared to similar flows.
The typical values given for the mixing lengths of rectangular and
round Jet flows, in one dimension, are In the range of 0.07 to 0.09.
Without the CFD results, one of these values would have to be used and
the results of using L in the range [0.07,0.09] would have been disap-
pointing.
Jenny et al. [21] used a 2-D CFD code to obtain a correlation for
L/b as a function of clearance and tooth geometry.
shown below for the teeth-on-rotor case:
Lib = O.055(1+1.03CrlL+O.Oevr_sTE)
However, their shear stress relation neglected
velocity component, U 2. Upon comparison
Their relatlon Is
(52)
the reeirculatlng
with the data of Rhode [23],
35
the mixing length ratio, £/b, was found to be relatively constant when
the shear stress Is calculated using all velocity components.J
Substituting the differentiated version of equation (48) and equa-
tion (51) into equation (50) yields an expression for the total free
shear stress. At the Interface of the two control volumes (y-O), the
total free shear stress Is:
'Jt - 0.68 p Jv,-v, J(vt-v,)[1-(y,/bl'5]2(y,/b) (53)
The clreumferentlal component of the free shear stress ls:
• J - 0.68 p4(Wz_Hl)z+(U2-Ul) z (N2-Wl)[1-(ya/b_'5]2(yl/b) (54)
The elrcumferentlal component of the veloclty at the interface, Nol ' is
obtained from equation (48).
No! " Wz + (N2-N_)[1-(y2/b!'5]2(Ya !b) (55)
Equations (53,54,55) are all valid along the dlvldlng streamline.
SinGe the control volumes are defined geometrically and not by the
dividing streamline, the shear stress calculated using the above
equations is assumed to be close to that existing along the geometric
boundary line. This Is a good assumption considering that the angle of
the dlvldlng streamline from the horizontal has been found
experimentally to be on the order of 6 degrees by several Investigators
[37,38].
The analysls to this polnt ls Incomplete In that the reclrculatlon
veloclty, U,, and the relatlonshlp between the mixing thickness and the
boundary layer thickness, y21b, are undefined. In order to determine
the reclrculatlon veloclty, Uz, and subsequently y,lb, the analysls
presented In the prevlous section deallng wlth the DIVIDING STREAMLINE
APPROACH Is used. Again, this analysls Is valid along the dividing
streamllne, but Is considered close enough to the values along the
36 |
geometric boundary llne. The final form of the contlnuity equation,
equation (36), is rewritten here:
nR nR$ (t/[1-Ca=$2]) dn - I (¢=/[I-Ca=¢=]) dn (56)
nj -m
where Ca Is the Crocco number, n is the dimensionless coordinate, and ¢
is the velocity ratio U/U_. The solution to this equation Is obtained
by substituting equation (24) Into equation (56) and solving for the
dividing strea_tllne coordinate, nj, for a given Crocco number. This is
then inserted back into equation (24) and a value of CJ is obtained.
The results of this solution procedure are tabulated in table 2, for
alr. For air (T=I._) flowing In a labyrinth seal, the maximum possible
Math number Is 1.0. Therefore, the maximum possible Crocco number Is
0.408 or Ca==O.167. The range of solutions is:
0.61632 < Cj < 0.6263
Using an average solutlon of Sj = 0.62 gives a maximum error of less
than ± I%. The reelrculatlon veloclty at the interface Is:
U=j . 0.62UI (57)
The only remaining problem is the numerlcal definition of y=/b.
Looking back, equations (48) and (24) both describe the axlal velocity
proflle In the Jet flowfleld. If the following observation Is made
Va/V I _ CJ
then equation (57) can be substituted back into equation (49) yielding
the following numerical definition for ya/b:
yffi/b - 0.584-0.134¢j . 0.50
It Is interesting to note that Jenny etal. [21] assumed that yffi/b-0.5.
Figure 20 shows a plot of the dimensionless axial velocity profile
in the reclrculatlon region for seal A of table 1 as calculated by
37
I
.9
.8
.7
.6
.5
.4
.3
.2
p,-n_ 0
SEAL A
J_/ / " " " // I
] '*- ! | I I .* a • [. |
0 • 14 . 28 . 42 . 58 . 7• 07 . 21 • 35 . 4g
U/UI.63
Flg. 20 CFD calculation of dlmenelor_eee reclrculatlonvelocity.
!
38 Q
Ca 2
0.00000
0.05000
0.]0000
0.15000
0.20000
0.24000
0.28000
0.32000
0.36000
0.40000
0.44000
0.48000
0.52000
0.56000
0.60000
0.64000
Table 2. Tabulated solution to
0.61632
0.61915
0.62211
0.62523
0.62848
0.63129
0.63405
0.63725
0.64047
0.64387
0.64748
0.65132
0.65543
0.65979
0.66462
0.66982
equation
Ca _
0.68000
0.72000
0.76000
0.80000
0.84000
0.86490
0.88360
0.90250
0.92160
0.94090
0.96040
0.98010
0.992016
0.998001
1.000000
(56).
0.67553
0.68188
0.68903
0.69724
0.70689
0.713944
0.719944
0.726834
0.734949
0.744883
0.757869
0.777432
0.798766
0.823427
1.000000
39
clty as
dividing streamline makes an angle of 6°
agreement Is excellent• Equation (57) Is
interface of the two control volumes. The
Rhode [23]. This profile is for the center of the recirculation region
to the top or the labyrinth tooth. The intersection of the two dashed
lines is the location and value of the theoretical recirculation velo-
calculated using equation (57) and the assumption that the
with the horizontal. The
actually the velocity at the
velocity components used in
the shear stress equations are all average velocity components• To be
consistentt the average reclrculation velocity must be used. The CFD
results show that the
Integrating this yields:
Reduced Equations
The solution of
velocity distribution Is parabolic in nature.
" 0 "n:" t=_)U2 i *_VUU | %jr
the governing equations can be simplified by
continuity equation for control volume I becomes:
_PAI _pW*AI • _pA, _pWaA ,-- + -- + mi+1 - ml • -- + - 0
If equation (59) times the circumferential veloolty,
subtracted from equation (_0), the following reduced
momentum equation for control volume I is obtained:
pA, --- • --- • • (WoI_WI i))t Rs, )e L )t Rs,36J
• An _Pi• ml(wil-W,l_l) - .
Rs _ 3e+ xJlLl - xsiaslLi
Similarly, if equation (39) times the circumferential velocity, W,, Is
(59)
W,, IS now
form of the
(60)
t
by using equation (39) to eliminate mr from the other equations. The
reducing the number of equations by one. This reduction Is accomplished
40 •
subtracted from equation (41), the reduced momentum
control vol_ane II is obtained.
pA 2- • -- -- + + (WII-Wol)
Bt Rs= @@ L Rs, ej
+ XJILi - xrlarILi
The number of variables is
eliminate the density terms.
This concludes the development
analysis presented in this report.
the analysls of Jenny et al. [21].
The theory of Jenny etal. [21]
The theory of Jenny et
agreement with measured test
equation for
(61)
reduced by using the Ideal gas law to
Pi = pIRT (62)
of the governing equations for the new
The following is a discussion of
al. [21] has shown consistently good
results [39] in predictions of cross-
coupled stiffness and direct damping. The author had hoped to program
their solution and make direct comparison to the present theory;
however, ae outlined below, unresolvable difficulties arose in deriving
the published equations of [21].
The theory of Jenny et al. [21] was derived for the Wbox-ln-a-box"
control volume configuration illustrated in figure 5, Thus, a direct
comparison of their equations with those presented in this report is
not feaslble. However, a review of the development of their governing
equations is of Interest.
The following convention will be used for the control volumes in
figure 5: the large control volume is control volume I and the small
41
0
Q
e
control volume is control volume If. The continuity equations for the
control volumes shown in figure 5 are:
Contlnult_ I
_pW:AI _}pWiAl _p(AI+A2) .+ "--'-'--+ + ml+1 - mi - 0
Rs_e Rs_O 3t(J1)
Contlnult_ II
_PWzAz aPAz
+ mrl - 0Rs_e _t
(J2)
The following assumptions are used by Jenny et al. [21] ¢o slmpllfy
equations (Jr) and (J2):
a) the flow Is Incompressible (p = constant),
b) _i+1 - ml, and
-) the area of _h. control vol_ge _T" is constant
The first assumption seems questionable, since thls is a compressible
flow solutlon, and qulte often the flow in a labyrInth seal achieves
Hach I at the exit. Assumption (b) Is a valid assumption for the
zeroth-order, steady flow 8olutlon, but It Is questionable for the
first-order, unsteady flow solution for an orbiting rotor. Uslng the
chaln rule for the expansion of partial derivatives and the above
assumptions, equations (J1) and (J2) become:
Contlnult_ I@W• _W i _Cr
Az _ + A: _'- + WIL _ + RS_e ae @e
- 0 (J3)
Continuity II
pA_--'- - RSmrl . O_e
(J4)
The equations given by Jenny et al. [21] are:
Continuity I)Wi @W i @Cr @(A ,+A= )
A= "-- + A, -- - W,L RS ' - 0 (J5)
42 •
Contlnult_ II@W:
PA: m - mrl = 0 (J6)_e
The difference between equations (J3) and (JS) Is In the sign of the
third and fourth terms. The second and third terms in equations (J3)
and (JS) originate from the same partial derivative, but have opposite
signs. This author could not arrive at the same conclusion using the
chain rule. The difference between equations (J4) and (J6) is the
radius, Re, In the aecond term. This may or may not be a problem since
the radial mass flow term, mrl, Is not defined by Jenny etal. [21].
The author agreed with the derivation of the momentum equations
for the control volumes shown In figure 5 except for the aforementioned
assumptions and the following discrepancies:
(a) the axial velocity component Is incorporated Into the defini-
tion of the etator wall shear stress, but neglected In the definition
of the Reynoldts number which is used to calculate the friction factor
term In the shear stress relation.
(b) the perturbation of the friction factor is ignored. This term
has been shown [32_ to be important in the solution for rotordynamic
coefficients.
(c) the leakage equation is a global leakage equation. This means
that local perturbations for a cavlty can not be found from thls equa-
tion. Jenny etal. [21] perturb thls global equation for clearance.
(d) the carryover eoefflclent deflnltlon used in the leakage
equation Is a global equatlon and cannot be perturbed.
(e) the flow coefficient used In the leakage equation was obtained
from a plot of empirical data. No explanation was given for the method
43
used to obtain the dervatlves of the flow coefficient used in the per-
turbation equations.
The aforementioned problems prevented the author from obtaining a
solution based on the theory of Jenny et al. [21]. Regrettably, no
direct comparison between It and the theory presented in this paper was
possible. This completes the discussion of the theory of Jenny et al.
[21]. The following Is a discussion of the solution procedure for the
new analysis presented in thls report.
Leakage Equation
To account for the leakage mass flow rate in the continuity and
momentum equations, the following model was chosen.
Pi-I - PI
mi " P,i P= Hi RT (63)
where the kinetic energy carryover coefficient p= is defined by
Vermes [40] for straight through seals as=
I12_= = I/[I-a3 (64)
where
a = 8.521((Li-TPi)ICr÷7.23)
definition, for the first tooth of any seal
interlocking and
and is unity, by and all
the teeth in combination groove seals. This
definition of the carryover coefficient is a local eoefflclent which
can be perturbed in the clearance. The previous analyses by Childs and
Scharrer [6] and Jenny et aL [21] used a global definition which could
not be perturbed.
The flow coefficient is defined by Chaplygln [_1] as:_-I
Ull - = where, st - -I (65)w+2-5Sl+2s i \ Pi /
44 •
Thls flow coefficient yields a different value for each tooth along the
seal as has been shown to be the case by Egli [42]. For choked flow,
Fllegner's formula [43] will be used for the last seal strip. It is of
the form:
• 0.510p2
mNC - PNC HNC (66)J_T
PERTURBATION ANALYSIS
For cavity i, the continuity equation (59), momentum equations
(60,61) and leakage equation (63) are the governing equations for the
variables N_i , N2i , Pi, mi" A perturbation analysis of these equations
Is to be developed wlth the eccentricity ratio, ¢ - eolCr , selected to
be the perturbation parameter. The governing equations are expanded in
the perturbation variables
PI " Poi ÷ e Pzi Hi = Crl + ¢ H_
Nil - Wlol ÷ c Will AI " AO ÷ c LH i
W21 - W20i + c W21i
where ¢ = eolCr is the eccentricity ratio. The zeroth-order equations
define the leakage mass flow rate and the circumferential velocity
distribution for a centered position. The first-order equations deflne
the perturbations in pressure and circumferential velocity due to a
radial position perturbation
of a first order analysis are
centered position.
Zeroth-Order Solution
and
of the rotor.
only valid
Strictly speaking, results
for small motion about a
The zeroth-order leakage equation Is
@i+I " mi = mo (67)
is used to determine both the leakage-rate mo and pressure
45
dlstributlon for a centered position. The leakage-rate and cavity
pressures are determlned Iteratively, In the following manner. First,
determlne whether the flow Is choked or not by assuming that the Mach
number at the last tooth Is one. Then, knowing the pressure ratio for
flow at sonic conditions, the pressure in the last cavity is found.
The mass flow can be calculated using equation (66). Working
backwards towards the first tooth, the rest of the pressures can be
found using equation (63). The final pressure calculation will result
in the reservoir pressure necessary to produce the sonic condition at
the last tooth. If the actual reservoir pressure Is less than this
value, then the flow is unchoked. Otherwise, it is choked. If the
flow is choked, a similar procedure is followed, but now the pressure
in the last cavity is guessed and a mass flow rate calculated using
equation (66). The remaining pressures are calculated using equation
(63). This is repeated untll the calculated reservoir pressure equals
the actual reservoir pressure.
In the first cavity is guessed and a mass
equation (63). The remaining pressures are calculated with the
equation. This procedure Is repeated until the calculated
pressure equals the actual sump pressure.
The zeroth-order circumferentlal-momentum equations are
I
mo(Wioi-Wioi-1) = (zJlo-xslo asi)Li
From calculated
If the flow Is unchoked, the pressure
flow rate calculated using
same
sump
XJoiLi = xrol arlLi
pressures, the densities
(68)
(69)
can be calculated at each
cavity from equation (62), and the only unknowns remaining in equations
(68) and (69) are the circumferential velocities W_o I and Waol. Given
an inlet tangential velocity, a Newton-root-flnding approach can be
Q
46 •
used to solve equations (68) and (69) for the i-th velocities, one
cavity at a time; starting at the first cavity and working downstream.
Flrst-Order Solution
The governing flrst-order equations (70,71,72), define the
pressure and velocity fluctuations resultlng from the seal clearance
function. The continuity and momentum equations follow In order:
BPtl BP:I BWt II BWa tl
O:l --- + Oal ---- ÷ Gel -- + G_I -- + Gel PtlBt Be Be Be
BH, I
+ G,I P,I-] ÷ GTI P,I+I - - Gol Hll - GolBt
BW,,I X,lW,ol BW,,I _ X,lW,ol__ BP,I-----+ + ,I+R_ ]X,l Bt Rsl Be Be
X ,IPol BW2,1
BH,1
G:°l B'-_-
BPII+ X,I --
Bt
+ + X_l P:I + Xsl P,I-1 ÷ X,I Will + XTl W2,1Rs 2 Be
"SWill-1 = X,l Hit
T,I Bt Rs2 Rs2 j Be Be
BPI1
-- + ¥,,1 Pll + ¥sl W_,l + 16,I PII-I + ¥71 W,,l = ¥ml H:I*Y21 @t
(70)
(71)
(72)
where the Xl,s, ¥1's, and Gl's are defined In Appendix B. These
perturbation equations are very different from those of Jenny et aL
[21], because their analysis neglects pressure perturbations In the
leakage and shear stress equations and assumes that the density Is
constant.
If the shaft center moves in an elliptical orblt, then the seal
clearance function can be defined as:
cH, - -a coswt oose -b slnwt slne
= -a [cos (e-wt) + cos (e+wt)] - b [cos (e-wt) - cos(e+wt)]
(73)
The pressure and velocity fluctuations can now be stated In the
associated solution format:
47
Psl" PclCOS(8+wt) + Pslsin(8_wt) + PclCOS(9-wt) • Psisin(6-wt) (74)
W_I i = WiciCOS(8+_t)*W_sisin(8*_t)+W_cicos(e-_t)+W_sisin(8-_t) (75)
W2, I = W_clCOS(8+wt)+W_slsln(8+wt)+W2clCOS(8-wt)+W2slSln(8-wt) (76)
Substituting equations (73), (7_), (75) and (76) Into equations (70),
(71) and (72) and grouping llke terms of sines and cosines (as shown In
Appendix C) eliminates the time and theta dependency and yields twelve
linear algebraic equations per cavity. The resulting system of
equations for the l-th cavity can be stated:
[Ai-1] (XI-1) + [ALl (Xl) ÷ [AI+I] (Xl+l) - a (BI) + b (Cl) (77)E C
where
4" dk -- -- 4_ + --
(Xi-1) = (Psi-l, Pci-1, Psi-l, Pci-1, W_SI-1, W_cl-1, W_sl-1,
- + + - _ )TWIcl-1, W_SI-1, W2st-1, W2sl-1, W c1-1
(XI) = (Psi, Pcl, Psl, PCl, W_sl, W,cl, W,sl, W_cl, W2sl, W2cl,
w si,w oi)z_ -- -- ÷ • --
(Xl+l) " (Psi+l, PcI+I, Psi+l, Pcl+I, Wlst+l, Wicl+l, WIsl+l,-- ÷ ÷ r- '- T
Wlcl+l, W=SI+I, N=ci+l, W=sI+I, W2ci*l)
The A matrices and column vectors B and C are given in Appendix C. To
use equation (77) for the entire seal solution, a system matrix can be
formed which Is block trldlagonal In the A matrices. The size of this
resultant matrix is (12NC X 12NC) since pressure and veloclty
perturbations at the inlet and the exit are assumed to be zero. This
system is easily solved by various linear equation algorithms, and
yields a solution of the form:
48 •
+ a + b +Psi = - Fasi + - Fbsl
E £
- a - b -Psi = - Fasi + - Fbsi
E C
a + b +Pcl = - Faci ÷ - Fbcl
C C
- a - b -Pcl = - Facl + - Fbcl
C C
(78)
Determination of D_namic Coefficient
The force-motion equations for a labyrinth seal are assumed to be
of the form:
(79)
The solution of equation (79) for the stiffness and damping
coefficients is the objective of the current analysis. For the assumed
elliptical orbit of equation (73), the X and Y components of
displacement and velocity are defined as:
X = a coswt X m -am sinwt
Y = b sinwt Y = bw coswt
Substituting these relations into equation (79) yields:
F x = -Ka coswt - kb sinwt + Caw sinwt - cbm coswt
Fy = -ka coswt - Kb sinwt - caw sln_t - Cbm cos_t
(80)
Redefining the forces, Fx and Fy, as:
Fx " Fxc cos_t + Fxs sinwt (81)
Fy = Fy c cosmt + Fys sln_t
and substituting back into equation (78) yields the following
relations:
-Fxc = Ka + cbw
-Fyc - ka + Cb_
The X and ¥ components
-Fxs = -Caw + kb
-Fys - Kb + caw
(82)
of force can be found by integrating the
pressure around the seal as follows:
49
NC 2_
Fx - -Rsc _ I PII Li cosO dO (83)1-1 0
NC 2w
Fy- Rsc __ I PII LI sin6 d6 (84)1-1 0
Only one of these components needs to be expanded in order to determine
the dynamic coefficients. For thls analysis, the X component was
chosen. Substituting equation (7_) into (83) and integrating ylelds:
NC @ _ @ _
Fx- -cwRs _ Li [(Psi - Psi) slnwt + (Pcl + Pcl) coswt] (85)I-I
substituting from equations (78) and (80) into equation (85) and
equating coefficients of sinwt and cos_t yields:
NC
FXS - -wRS
I=I
Fxc - -_Rs N_-i=I
÷ -- ÷
LI [a(Fasl -Fast) + b (rbsi -Fbsl)]
F ÷Li [a(F_ci + Facl) + b ( bcl + Fbci)]
(86)
Equating the alternatlve definitions for Fxs and Fxc provided by
equations (82) and (86) and grouping like terms of the linearly
independent coefficients a and b ylelds the final solutions to the
stiffness and damping coefficients:
NC + _
K - wR __ (Fact ÷ Fact ) LiI=I
NC + -k - wR _. (Fbsl - Fbsl ) LI
I-I
-wRs NC + -
C = _ __ (Fast - Fast ) LII-I
wRs NC ÷ _
--Ic = _ i- (Fbcl + Fbel) LI
(87)
50
Data Requirements and Solution Procedure Summary
The required input for the analysts presented is as follows:
a) Reservoir pressure, temperature, and kinematic viscosity.
b) Sump pressure.
c) Gas constant and ratio of speclflc heats.
d) Inlet circumferential veloclty and rotor speed.
e) S_al radius, radial clearance, tooth pitch, height and tip
width.
f) Rotor and stator friction coefficients (mr,nr,ms,ns).
g) Number of teeth.
In review, the solution procedure uses the following sequentlal steps:
a) Determination of whether flow is choked or not using equations
(63) and (66).
b) The steady-state pressure distribution and leakage are found
using equation (63)and/or (66).
c) The steady-state clrcumferentlal velocity distribution is
determined using equation (68).
d) equation is formed for the flrst-order
variables and solved using the cavity equation
e)
A system
perturbat Ion
(77).
Results of thls first-order perturbation solution, as defined
In equations (78), are inserted Into equation (87) to
define the rotordynamic coefficients.
51
CHAPTER III
TEST APPARATUS AND FACILITY
TESTING APPROACH
The testing method employed at the TAHU facility ls the same as
that used by Ilno and Kaneko [4_]. An external hydraullc shaker Is
used to impart translatory motion to the rotating seal, while rotor
motion relative to the stator and the reaction force components acting
on the stator are measured.
Figure 21 shows the manner in which the rotor could be positioned
and osclllated in order to identify the dynamic coefficients of the
seal for small motion about an eccentric posltion, eo. Equation (I) is
rewritten here
{"'<i {;ircxx<<°>Jill "il"
F i L_X(¢ o) K_ii(¢o)J LCyx(co) Cyy(co)J
First, harmonic horizontal motion of the rotor is assumed, where
X - •o + A sin(Rt) + B cos(Rt)
- A_ cos(_t) - BO sin(nt)
¥ = ! - 0
This ylelds small motion parallel to
where Q is the shaking frequency. In
direction force components can be expressed
FX - FXS sin(_t) + FXC cos(_t)
Fy . FYS sln(_t) + F¥C cos(_t)
Substituting these expressions into equation
coefficients of constant, sine, and cosine terms
four equations for the dyn_mlc coefl'Iclents
(88)
the static eccentricity vector,
a similar fashion, the X and ¥-
(89)
(88) and equating
yields the following
52 •
Y
X
W18. 21 External shaker method used for ooefflolentldentlfloatJon. ;
53
Solvlng this system of
dynamic coefficients as
FXS= KXX A - CXX B
FXC - KXX B * CXX A
F¥S = KyX A - Cyx B
F¥C - Ky]( B * Cfx A.
four equations in four
(9O)
unknowns defines the
KXX(C o) - (Fxc B • FXS A) / (A 2 • B 2)
KrX(¢ o) - (Fxs A • FXC B) / (A 2 • B')(91)
CXX(¢ O) ,,,(Fxc A - FX.S B) / Q(A 2 + B2)
Cyx(¢ o) - (F]( c A - FyS B) / Q(A 2 + B2)
Therefore, by measuring the reaction forces due to known rotor
motion, determining the Fourier coefficients (A,B,Fxs,Fxc,Fys,Fyc), and
the above definitions, the indicated dynamicsubstituting into
coefficients can
centered position
be identified. If the rotor is shaken about a
(eo=O), the process is complete. Since the
llnearlzed model has skew-symmetrlc stiffness and damping matrices, all
of the coefficients are identified. If, however, the rotor is shaken
about an eccentric position as Inltlally postulated, then it must be
shaken vertlcally about that same point in order to complete the
identification process.
Assuming harmonic vertical motion of the rotor, as defined by
X _ eo, X - O,
Y - A sin(gt) + B cos(Qt), and
a
¥ = A_ cos(Qt) - Bg sin(gt),
yields oscillatory motion that is perpendicular to the assumed static
eccentricity vector. A similar process as before results in the
54 •
coefficient definitions
KYy(¢o) - (Fxs A 4 FXC B) / (A 2 + B 2)
KXy(¢o) - -(FYC B * FYS A) / (A 2 + B 2)(92)
Cyy(Eo) - (Fxc A - FXS B) / Q(A 2 + B 2)
CXy(¢o) = (FYs B - FYC A) / O(A 2 + B=).
All eight dynamic coefficients are thus determined by alternately
shaking the rotor at one frequency g in directions which are parallel
and perpendicular to the static eccentricity vector.
APPARATUS OVERVIEW
Detailed design of the TAHU gas seal apparatus was carried out by
J.B. Dressman of the University of Louisville. It Is of the external
shaker configuration, with the dynamic-coefflcient-identiflcation
process described in the preceding section.
Considerlng both the coefficient identification process and the
analysis,
apparent.
must provide
(b) measurement of
motion. Secondly,
comparison) tf the
some objectives for the design of the test apparatus are
First, to determine the dynamic coefficients, the apparatus
for (a) the necessary rotor motion within the seal, and
the reaction-force components due to this
parameters
It would be
apparatus could
afforded by the analysis
advantageous (for purposes of
provide the same variable seal
(i.e., pressures, seal geometry,
rotor rotational speed, fluid prerotatlon, and rotor/stator surface
roughness). Wlth this capability, the Influence of each independent
parameter could be examined and compared for correlation between
theoretical predictions and experimental results.
With these design objectives in mind, the discussion of the test
@
@
• 55
apparatus Is presented In three sections. The first section, Test
Hardware, describes how the various seal parameters are physically
executed and controlled. For example, the manner In which the dynamic
"shaklng" motion of the seal rotor Is achieved and controlled is
described In thls section. The second section, Instrumentation,
describes how these controlled parameters, such as rotor motion, are
measured. Finally, the Data Acquisition and Reduction section explains
how these measurements are used to provide the desired information.
TEST HARDWARE
This section deals only with the mechanical components and
operation of the test apparatus. It provides answers for the following
questions:
I) How is the static position of the seal rotor controlled?
2) How Is the dynamic motion of the rotor executed and
controlled?
3) How is compressed air obtained and supplied to the apparatus,
and how is the pressure ratio across the seal controlled?
4) How Is the incoming alr prerotated before it enters the seal?
5) How are the seal rotor and stator mounted and replaced?
6) How is the seal rotor driven (rotated)?
Recalling the rotordynamlc-coefflclent-ldentlficatlon process
described earlier, the external shaker method requires that the seal
rotor be set In some static position and then oscillated about that
point. The test apparatus meets those requirements by providing
independent static and dynamic displacement control, which are
described below.
56 •
Static Dlsplacement Control.
The test apparatus Is deslgned to provlde control over the static
eccentricity position both horizontally and vertically wlthln the seal.
The rotor shaft 18 suspended pendulum-fashlon from an upper, rigidly
mounted pivot shaft, as shown in figures 22 and 23. This
arrangement allows a side-to-side (horizontal) motion of the rotor, and
a cam within the pivot shaft allows vertical positioning of the rotor.
The cam which controls the vertical position of the rotor is
driven by a remotely-operated DC gearhead motor, allowing accurate
positioning of the rotor during testing. Horizontal positioning of the
rotor is accomplished by a Zonlc hydraulic shaker head and master
controller, which provide independent static and dynamic displacement
or force control. The shaker head is mounted on an I-beam support
structure, and can supply up to 4450 N (1000 Ibf) static and 4450 N
dynamic force at low frequencies. The dynamic force decreases as
frequency Is increased. As illustrated in figure 22, the shaker head
output shaft acts on the rotor shaft bearing housing, and works against
a return spring mounted on the opposite side of the bearing housing.
The return spring maintains contact between the shaker head shaft and
the bearing housing, thereby preventing hammering of the shaker shaft
and the resulting loss of control over the horizontal motion of the
rotor.
57
58I
Zmm
o
\
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D.
I06JGJ
t-4
CO
59
D_namle Displacement Control.
The dynamic motlon of the seal rotor within the stator Is
horlzontal. In addition to contro111ng the static horizontal posltton
of the rotor, the ZonJc shaker head moves the rotor through horlzontal
harmonic osclllatlons as the test Is run. A Wavetek funetlon generator
provldes the slnusoldal input signal to the Zonlc controller, and both
the amplitude and frequency of the rotor oscillations are controlled.
Although the test-rlg des18n provldes for dynamic motlon of the
rotor only In the horlzontal X-directlon, all of the coefficients for
elther seal model (equation (I) or (2)) can st111 be determined. As
flgure 24 shows, the requlred rotor motion perpendleular to the static
eecentrlclty vector can be accomplished In an equlvalent manner by
statlcally dlsplaclng It the same amount (eo) In the vertlcal directlon
and oontlnulng to shake horlzontally.
In addltlon to providing control over the rotor's statlc posltlon
and dynamlo motlon, the test apparatus allows other seal parameters to
be controlled Independently, provldlng insight lnto the Influence these
parameters have on seal behavior. These parameters eolnclde wlth the
varlable input parameters for the analysls, and they Include:
I) pressure ratio across the seal,
2) prerotatlon of the incoming fluid,
3) seal configuration, and
t) rotor rotational speed,
60
•
••
i
Pressure Ratio.
The inlet air pressure and attendant mass flow rate through the
seal are controlled by an eleetrle-over-pneumatlcally actuated
Masoneilan Camflex II flow control valve located upstream of the test
section. An Ingersoll-Rand SSR-2000 slngle stage screw compressor
rated at 34 m'Imin @ 929 kPa (1200 scfm @ 120 psig) provides compressed
air, which is then filtered and dried before entering a surge tank.
Losses through the dryers, filters, and piping result in an actual
maximum inlet pressure to the test section of approximately 825 kPa
(105 psig) and a maximum flow rate of 10 m'Imin (350 scfm). A
four-lnch inlet pipe from the surge tank supplies the test rig, and
after passing through the seal, the air exhausts t_ atmosphere through
a manifold wlth muffler.
Inlet Circumferential Velocit_ Control.
In order to determine the effects of fluid rotation on the
rotordynamlc coefficients, the test rig design also allows for
prerotatlon of the incoming air as it enters the seal. This
prerotation introduces a circumferential component to the air flow
direction, and is accomplished by guide vanes which direct and
accelerate the flow towards the annulus of the seal. Figure
25 illustrates the vane configuration. Five sets of guide vanes are
available; two rotate the flow in the direction of rotor rotation at
different speeds , another introduces no fluid rotation, and two rotate
the flow opposite the direction of rotor rotation at different speeds.
The important difference between the vanes is the gap height, A. The
vanes with a small gap height produce the highest inlet tangentlal
velocity.
61
62 II
(1")
_1011 VITII ItOTATIOll
II
J/J
¢
PIU['--_'_XOH AGAZHST ROTATION
Pi8._ Inlet juide vnne detni2.
I
Q
Q
4
r
63
Seal Conflsuratlon.
The design of the test rlg, flgure 26, permits the installation of
various rotor/stator combinations. The stator is supported in the test
section housing by three Kistler quartz load cells in a trlhedral
configuration, as shown In figure 27. Different seal stator designs are
obtained by the use of Inserts. The smooth and labyrinth Inserts used
for the .4mm (.016_In.) radlal clearance seal tests are shown In figure
2B. The labyrinth rotor and the tooth detall are shown In figures 29
and 30. Seals wlth different geometries (i.e., clearances, tapers,
lengths) can be tested, as well as seals wlth different surface
roughnesses.
Rotational Speed.
A Westinghouse 50-hp varlable-speed electric motor drives the
rotor shaft through a belt-drlven jackshaft arrangement. Thls shaft is
supported by two sets of Torrington hollow-roller bearings [45]. These
bearings are extremely precise, radially preloaded, and have a
predictable and repeatable radial stiffness. The shaft bearings are
lubricated by a posltlve-dlsplacement gear-type oll pump.
Different Jackshaft drlve-pulleys can be fitted to provide up to a
4:1 speed increase from motor to rotor shaft, which would result In a
rotor shaft speed range of 0-21,200 cpm. Previously, the maximum
posslble test speed was 8500 cpm. Hlgh bearing temperatures and the
reduction of interference in the rotor-shaft fltment wlth increasing
speed had served to limit shaft speed. These problems have been
addressed by some specific design modifications which are discussed
below.
64•
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69
In the past, the seal rotor was press-fitted and secured axially
by a bolt circle to the rotor shaft. As the running speed is
increased, however, the Inertla-induced dlametral growth of the rotor
exceeds the growth of the shaft. By increasing the interference In
stationary rotor-shaft flt, a greater allowance for thls growth
difference has been provlded. Figure 31 shows the present rotor-shaft
design, a tapered rotor which Is hydraulically expanded during
Installation. The rotor Is Inserted over the end of the tapered shaft
and a large nut Is used to pull the rotor onto the shaft. Fluid Is
pumped between the shaft and rotor, causing the rotor to expand. Thls
separating force allows the rotor to be pulled onto the shaft until the
desired Interference flt Is achieved.
The problem of high bearing temperatures has been eliminated by
replacing a roller-type thrust bearing and modifying the lubricant
flow. A Torrlngton Hydraflex thrust bearing, consisting of eight one-
Inch rubber-faced pads which are water lubricated, Is now In place at
the rear of the rotor. In addition, the lubricant for the Torrlngton
hollow-roller bearings which support the shaft has been changed to
light turblne oll with a maximum temperature of 270eF. The hollow-
roller-bearing caps have been modlfled to direct the oll flow to the
regions of heat buildup. These modifications are shown in figure 31.
The flnal modification to allow operation of the TAMU gas seal
test apparatus at hlgh speeds was the installation of Koppers
circumferential seals for the hollow-roller and thrust bearing
lubrication systems. At 16,O00 rpm, the surface speeds of the shaft
and rotor (170 and 350 ft/sec, respectively) exceed the limits of llp
7OII
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1-1
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,-.IEl_l
1,1-1
Ioo
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71
seals, whlch had been used on the TAHU apparatus. The Koppers seals In
flgure 31 were deslgned [or gas applications. The seallng mechanism Is
a segmented carbon seal rlng.
To conclude thls dlscusslon ot the test hardware, two vlews or the
oomplete test apparatus are Included. Figure 32 shows the assembled
rlE. while an exploded vlew Is provided In £1guure 33.
72•
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73
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INSTRUMENTATION
Having discussed the seal parameters that can be varied, and how
the variations are implemented, their measurement will now be
described. The types of measurements which are made can be grouped Into
the following three categories:
I) rotor motion,
2) reactlon-force measurements, and
3) fluid flow measurements.
These categories are descrlbed Indivldually in the sections that
follow.
Rotor Motion Measurements.
The position of the seal rotor within the stator is monitored by
four Bently-Nevada eddy-current proximity probes, mounted in the test
section housing. These probes are located 90 degrees apart, and
correspond to the X and Y- directions. The proximity probes are used
to determine the static position and dynamic motion of the rotor, and
their resolution Is 0.0025 mm (0.1 mll).
Reactlon-Force Measurements.
Reaction forces arise due to the motion of the seal rotor
within the stator. The reaction forces (Fx, FX) exerted on the stator
are measured by the three Kistler quartz load cells which support the
stator In the test 8ectlon housing. When the rotor is shaken, vlbratlon
is transmitted to the test 3ectlon housing, both through the thrust
bearing and through the housing mounts. The acceleration of the
housing and stator generates unwanted inertial "ma" forces which are
sensed by the load cells, in addition to those pressure forces
74
75
0
t
g
developed by the relative motion of the seal rotor and stator. For
this reason, PCB piezoelectric accelerometers with integral amplifiers
are mounted in the X and Y-directions on the stator, as shown in
figure 27. These 4ccels allow a (stator mass) x (stator acceleration)
subtraction to the forces (Fx, Fy) indicated by the load cells. With
this correction, which Is described more fully in the next section,
only the pressure forces due to relative seal motion are measured.
Force measurement resolution is a function of the stator mass and
the resolution of the load cells and accelerometers. Accelerometer
resolution is 0.005 g, which must be multiplied by the stator mass in
order to obtain an equivalent force resolution. The masses of the
stators used in the test program reported here are 11.5 kg(25.3 Ib) and
11.0 kg(24.2 Ib), corresponding to the smooth and labyrinth stators,
respectively. Hence, force resolution for the accelerometers is 0.560
N (0.126 Ib) and 0.538 N (0.121 ib), for each stator, respectively.
Resolution of the load cells is 0.089 N(O.02 Ib). Therefore, the
resolution of the force measurement is llmlted by the accelerometers.
With a stator with less mass, and/or accelerometers with greater
sensitivity, force resolution could be improved.
Fluid Flow Measurements.
Fluid flow measurements Include the leakage (_ss flow rate) of
air through the meal, the pressure gradient along the meal axis, and
the inlet fluid clrcumferential velocity.
Leakage Js measured with a Flow Measurement Systems Inc. turbine
flowmeter located in the piping upstream of the test section.
Resolution of the flowmeter is 0.0005 act, and pressures and
76 •
measured for mass flowtemperatures up and downstream of the meter are
rate determination.
For measurement of the axial pressure gradient, the stator has
pressure taps drllled along the length of the seal In the axlal
dlreetlon. These pressures, as well as all others, are measured with a
0-1.034 MPa (0-150 pslg) Scanlvalve dlfferentlal-type pressure
transducer through a t8 port, remotely-controlled -Scanivalve model J
scanner. Transducer resolution is 0.552 kPa (0.08 psi). Overall
accuracy of the pressure measurements Is limited by the resolution of
the 12 bit A/D converter which can only resolve the pressure slgnal to
+ 0.62 kPa (0.09 psi). Combined linearity and hysteresis error for the
pressure transducer is 0.06%.
In order to determine the clrcumferentlal veloclty of the air as
it enters the seal, the static pressure at the guide vane exit is
measured. This pressure, in conjunction wlth the measured flowrate and
inlet air temperature, is used to calculate a guide vane exit Mach
number. A compressible flow continuity equation
e
m = Pex Aex Mex [(X/RTt) (I + (X-1)Mex 2 I 2)] I/2 (93)
Is rearranged to provide a quadratlc equation for Mex
Mex 2 = {-I + I + 4((_-I)/2Y) (_ RTt /Pex Aex )2} / (_-I) (94)
where X Is the ratio of specific heats and R is the gas constant for
alr, Tt Is the stagnation temperature of the air, Pex is the static
pressure at the vane exit, and Aex is the total exit area of the guide
vanes. Since all of the varlables in the equation are either known or
measured, the vane exit Mach number, and therefore the velocity, can be
found.
77
In order to determine the circumferential component of this inlet
velocity, a flow turning angle correction, in accordanee with
Cohen [46], Is employed. The correction has been developed from guide
vane cascade tests, and accounts for the fact that the fluid generally
Is not turned through the full angle provided by the shape of the guide
vanes. Wlth this flow deviation angle calculation, the actual flow
direction of the air leaving the vanes (and entering the seal) can be
determined. Hence, the magnitude and direction of the Inlet velocity
Is known, and the appropriate component Is the measured Inlet
circumferential velocity.
DATA ACQUISITION AND REDUCTION
With the preceding explanations of how the seal parameters are
varied, and how these parameters are measured, the discussion of how
the raw data is processed and implemented can begin. Data acquisition
Is directed from a Hewlett-Packard 9816 (16-blt) computer wlth disk
drive and 9.8 megabyte hard disk. The computer controls an H-P 69_0B
multiprogrammer which has 12-bit AID and D/A converter boards and
transfers control commands to and test data from the instrumentation.
As was previously stated, the major data groups are seal
motlon/reactlon force data and fluid flow data. The motlon/reaction
force data are used for dynamic coefficient Identification. The
hardware involved Includes the load cells, accelerometers, X-direction
motion probe, a Sensotec analog fllter unit, a tuneable bandpass
filter, and the A/D converter. The operation of these components Is
illustrated In flgure 34, and their outputs are used in a serial
sampling scheme which provides the computer with the desired data for
78•
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79
reduction. Recalling the discussion of the reaction force measurements
In the preceding section, a (stator mass) x (stator acceleration)
subtraction from the Indicated load cell forces is necessitated due to
vibration of the stator and test section housing. This subtraction Is
performed with an analog circuit, and results in corrected Fx and Fy
force components due to relative seal motion. The forced oscillatory
shaking motion of the seal rotor is the key to the operation of the
serial synchronous sampling (SSS) routine which is employed. The
frequency of the rotor oscillation is set by a function generator, and
rotor motion Is sensed by the X-directlon motion probe. The motion
signal is filtered by the narrow bandpass filter, and is used as a
trigger signal for the SSS routine. Upon the operator's command, the
SSS routine is enabled, and the next posltlve-to- negative crossing of
the filtered motion signal triggers a quartz crystal clock/tlmer. Ten
cycles of the corrected Fx(t) signal are sampled, at a rate of 100
samples/cycle. The second positive-to-negative crossing of the
filtered motion signal triggers the timer and initiates the sampling of
ten cycles of the Fy(t) signal. Finally, the third posltive-to-negative
crossing triggers the timer again, and ten cycles of the corrected X(t)
signal are sampled. Thus, at every test condition, 1000 data points
are obtalned for Fx(tl),Fy(tl), and X(tl), and the data arrays are
stored in computer memory.
Some Important points need to be stressed concerning this
force/motion data acquisition. First, the bandpass filter is used only
to provide a steady signal to trigger the tlmer/clock. Any modulation
of the motion signal due to rotor runout is eliminated by this filter,
80 •
as long as the rotational frequency and shaking frequency are
adequately separated, and the shaking frequencies are selected to
provide adequate separation with running speeds. However, the rotor
motion and corrected force slgnals which are sampled and captured for
coefficient Identification are filtered only by a low-pass filter (500
Hz cutoff), and the effects of runout as well as shaking motion are
present in the recorded data. A second point worth noting is that the
sample rate is directly dependent on the shaking frequency. As the
shaking frequency is increased, the sample rate (samples/second) also
increases. In order to get the desired 100 samples/cycle, shaking
frequencies must be chosen to correspond to discrete sample rates which
are available. Hence, the frequency at which the rotor is shaken is
carefully chosen to provide the desired sampling rate and a steady
trigger signal. The uncertainty In the shaking frequency is 0.13 Hz
for the 74.6 Hz case.
Most of the fluid flow data are used
required by the analysis. The upstream
for the input parameters
(reservoir) pressure and
temperature, downstream (sump) pressure, and the inlet circumferential
velocity (determined as outlined earlier) are provided directly. The
frlctlon-faetor values of the rotor and stator are supplied in the form
of coefficients, which are obtained from the pressure distribution data
for the smooth annular seals, see Nicks [473 and Nelson et al. [48],
and are assumed to be the same for the labyrinth surfaces.
PROCEDURE
At the start of each day's testing, the force, pressure, and
flowmeter systems are calibrated. The total system, from transducer to
81
computer, is calibrated for each or these variables. The force system
calibration utilizes a system of pulleys and known weights applied in
the X and Y-directlons. An alr-operated dead-weight pressure tester is
used for pressure system calibration, and flowmeter system calibration
is achieved with an internal precision clock which almulates a known
flowrate.
All of the tests performed to date have
executing small motion about a centered
begins by centering the seal
capability of the Zonlc hydraulic
been made with the rotor
position. A typical test
rotor in the stator with the static
shaker, starting the airflow through
the seal, setting the rotational speed of the rotor, and then beginning
the shaking motion of the rotor. Data points are taken at rotational
speeds of 3000, 6000, 9500, 13000 and 16000 cpm with a tolerance of +u
10 cpm. At each rotational speed, data points are taken at pressures
of 3.0B bar (30 palg), 4.46 bar (50 pslg), 5.B_ bar (70 pslg), 7.22 bar
(90 psig), and B.25 bar (105 psig), as measured upstream of the
flowmeter with a tolerance of + O.069 bar (I.0 palg). For each test
case (1.e., one particular running speed, shaklng frequency, inlet
pressure, and prerotatlon condition), the measured leakage,
rotordynamlc ooefflclents, and axial pressure distribution are
determined and recorded.
This test sequence Is followed for each of two different shaking
frequencies, and for five inlet swirl directions. Therefore, twenty-
five data points are taken per test with a total of ten tests per seal
for shaking about the centered position. Shaking a seal about an
eccentric position would require more tests.
B2
TEST RESULTS: INTRODUCTION
The results reported here are from tests of six "see-through"
labyrinth seals, three with teeth on the rotor and three with teeth on
the stator, each with different radial clearances. Tables 3, _ and 5
show the pertlnant data for each seal configuration. For the remainder
of this report, the seals will be referred to as seal 1, seal 2, and
seal 3, as given In table 5, in addition to thelr respective
configuration.
The test program had the following objectives:
I) Acquire leakage, stiffness, and damping coefficients as a
function of rotor speed, pressure drop, and lnlet elrcumferentlal
velocity for three teeth-on-rotor and three teeth-on-stator
labyrinth seals wlth different radial clearances.
2) Compare the effect of varying the radial seal clearance
on the experlmentally determined rotordynamic coefflclents.
3) Compare test results to the predictions of the new analysis
presented In thls report.
When shaklng about the eentered position, the test apparatus can
be used to control the rotor speed, reservoir pressure (i.e. supply
pressure), circumferential velocity of the lnlet air, and the
frequency and amplitude of translatory rotor motion. Two shake
frequencies, 56.8 and 74.6 Hz, were used during testlng wlth
essentially the same results. The results plotted here were obtained
83
Seal 1
Diameter:
upstream
downstream
Haterlal:
Table 3. Test stator specifications.
Smooth Stator Labyrinth Stator
15.197 cm (5.983 in)15.197 om (5.983 In)
15.202 om (5.985 In)15.202 cm (5o985 in)
aluminum brass
Seal 2
Diameter :
upstreamdownstream
Material :
15.217 cm (5.991 In)15.217 em (5.991 In)
15.217 em (5.991 In)15.217 om (5.991 In)
aluminum brass
Seal 3
Diameter:
upstream
downstream
Haterlal:
15.245 cm (6.002 In)
15.247 cm (6.003 in)
15.237 cm (5.999 In)15.237 cm (5.999 in)
brass brass
Seal
Diameter:
upstreamdownstream
Material:
Table 4. Test rotor speolfleatlons.
Labyrinth Rotor Smooth Rotor
1,2,3 1,2,3
15.136 cm (5.959 ln)15.136 em (5.959 In)
304 stainless steel
15.136 om (5.959 In)15.136 c_a (5.959 In)
304 stainless steel
84 •
Seal 1
Radial Clearance :
upstreamdownstream
Seal Length:
Number of teeth:
Seal 2
Rad la 1 Clearance :
upstreamdownstream
Seal Length:
Number of teeth:
Seal 3
Radial Clearance :
upstreamdownstream
Seal Length:
Number of teeth:
Table 5. Test seal specifications.
Teeth-On-Rotor Teeth-On-Stator
0.3048 cm (0.012 In)0;3048 cm (0.012 in)
5.080 cm (2.000 in)
16
0.3302 om (0.013 in)0.3302 cm (0.013 in)
5.080 cm (2.000 in)
16
0.4064 cm (0.016 in)
0.4064 cm (O.O16 in)
5.080 cm (2.000 in)
16
0.t064 cm (0.016 in)
0.4064 cm (0.016 in)
5.080 cm (2.000 in)
16
0.5461 cm (0.0215 in)0.5588 cm (0.022 in)
5.080 cm (2.000 in)
16
0.5080 cm (0.020 in)0.5080 cm (0.020 in)
5.080 cm (2.000 in)
16
Pressures
1 p 3.08 bar
2 - 4.46 bar
3- 5.84 bar
4 - 7.22 bar
5 - 8.25 bar
Table 6.
Rotor speeds
I D
2-
3-
Definition of symbols used in flgures.
Inlet circumferential velocltles
3000 cpm 1 -Hlgh velocity against rotation
6000 cpm 2 - Low velocity agalnst rotation
9500 cpm 3 - Zero circumferential velocity
4 - Low velocity with rotation
5 - High velocity with rotation
4 - 13000 cpm
5 - 16000 cpm
The pressure for each test is set at the flowmeter of Figure 12.
85
g
@
O
by shaking at 7_.6 Hz ut an amplitude between 3 and _ mils. The actual
test points for each of the other three independent varlables are shown
in Table 6.
Figures 35-37 show the inlet circumferential velocity of the air
(Us) for the configurations described In table 6 for the seals reported
on here. The equation for U e Is
U e - m sin ^ / p Av
where m ls the fluid mass flow rate, p ls the fluid density, A v is the
exit area of the fluid turning vanes, and A Is the fluid swirl angle at
the turning vanes exit as measured from the axial direction. The
method used to determine A is described In the TEST APPARATUS AND
FACILITY chapter of_ this report.
represent velocities opposed to
l
Negative circumferential velocltles
the direction of rotor rotation.
Positive velocities are In the direction of rotor rotation. Note that
curve 3 (representing zero inlet circumferential velocity) lles on the
horizontal axis in each figure. The Inlet circumferential velocity
ratio, the ratio of inlet circumferential velocity to rotor surface
velocity, ranged from about -6 to about 6. When reviewing the
following figures, table 6 and figures 35-37 should be consulted for
the definitions of symbols used•
NORMALIZED PARAMETERS
Before the tests descrlbed hereln were performed, the TAMU gas
seal test apparatus was modified as described in the TEST APPARATUS AND
FACILITY chapter to allow operation at running speeds up to 16,O00 cpm.
As expected, subsequent tests revealed a dependence of the rotor
diameter on running speed due to inertia and thermal effects. The
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89
Table 7. Growth of rotor with rotational speed.
Rotor speed
(rpm)
Diametrical growth
(ms) (inches x 1000)
3o0o 0.01 0.46000 0.02 o.79500 0.03 1.2
13000 0.05 1.916000 0.11 #._
Table 8. Normalized coefficients.
Ho
DL(AP )
Ho_-C
DL (AP)
K = stiffness (N/ms)
Ho = seal exit clearance (ms)
L = seal length (m)
(nondim)
(see)
C - damping (N sec/mm)
D - seal diameter (m)
AP - pressure drop across
seal (N/m t )
@
@
9O (
rotor growth data, shown in table 7, were obtained from eddy
current motion probes positioned at the mldspan of the seal. Thus, as
the rotor turns faster, the forces in the seal are affected not only by
the Increased surface speed of the rotor (drag) but also by a change in
clearance (friction factor). See table B for the definitions of the
normallzed parameters. Theoretlcally, normalization would collapse the
data and make the presentation simpler and more straight forward.
However, this is not the case with the labyrinth seals tested in this
study. Figure 38 shows a comparison of dimensional and nondimenslonal
direct stiffness versus rotor speed for the inlet pressure set of tablet
6. The data did not collapse to a single curve and shows increased
irregularity.
Figure 39 shows a comparison of normalized and dimensional direct
damping for a teeth-on-stator labyrinth seal versus clearance for the
inlet clrcum£erentlal velocity set of table 6. The normalized results
lead one to believe that the direct damping coefficient increases as
clearance increases. However, the dimensional results show that the
direct damping coefficient decreases as clearances increases. To avoid
this type of confusion, the rotordynamic coefficients presented in this
study have not been normalized.
RELATIVE UNCERTAINTY
Before the test results are given, a statement about the
experimental uncertainty is needed. The method used is that described
by Holman [49] for estimating the uncertainty in a calculated result
based on the uncertainties in primary measurements. The uncertainty wR
in a result R which is a function o£ n primary measurements
91
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• 93
@
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0
xl,x2,x3,...,x n with uncertainties Wl,W2,W3,...,Wn is
-,-In this case,
equation (91).
2 2 2_ 1/2
w,) + (_)R w,) + ... • (_R wn) J (95)
the rotordynamlc coefficients are calculated using
The primary measurements are forces, displacements, and
frequency. The
apparatus are 0.89
respectively. For
the stiffness and
0.0875 N-s/mm (0.5
uncertainty in these measurements on the TANU test
N (0.2 lb), 0.0013 mm (0.05 mils), and 0.13 Hz,
the six seals tested, the estimated uncertainty in
damping coefficients were 7 N/mm (40 Ib/in) and
Ib-s/In), respectively. The uncertainty in the
cross-coupled damping coefficients were of the same order of magnitude
as the coefficients themselves. Since the uncertainties in the cross-
coupled-damping values were so high, and since the cross-coupled-
damping forces are of minor significance compared to the other damping
and stiffness forces, comparisons of the cross-coupled-damplng
coefficients have been omitted from this report.
SELECTION OF REPORT DATA
For each of the six seals tested, there were 125 test points for
leakage, direct and cross-coupled stiffness, and direct damping at the
74.6 Hz shake frequency. Generally, a ranking of the three independent
variables of the test apparatus in order of the relative effect on the
rotordynamlc coefficients of a seal is: inlet circumferential velocity,
pressure ratio, running speed. The previous report of Scharrer [22]
thoroughly catalogued the results for the effects of pressure ratio,
rotor speed up to 8000 cpm and inlet circumferential velocity on the
rotordynamic coefficients. Since the rotor speed capability of the
Q
94 •
test apparatus has
concerning rotor
chapters show the
been changed,
speed will be reviewed.
dependence of leakage and
results which show new information
Figures in the next two
rotordynamlc coefficients
on radial
Figures in
gradients
seal clearance for inlet swirl conditions of table 6.
Appendix D show additional information on leakage, pressure
and rotordynamlc eoefflclents. Generally, solid lines in a
figure represent experimental results, and broken lines represent the
predlotlons of the new analysis presented In thls report. These figures
will be used to compare the effect of radial seal clearance on seal
performance, and to evaluate the new analysis presented in this report.
95
CHAPTER V
TEST RESULTS: RELATIVE PERFORMANCE OF SEALS
This evaluation of the effect on seal performance of varying the
radial seal clearance requlres frequent use of the Information In
table 6 and figures 35-37. It might seem obvlous, since this report
evaluates the effect of radlal seal clearance, that the data should
be presented as a function of clearance(clearance being on the x-
axis). However, since the lnlet circumferential velocity Is dlreetly
dependent on the seal leakage and the seals leak at different rates
due to differing cross-sectional areas, inlet circumferential
veloclty 5 (swirl 5) for seal I Is less than those for seals 2 and 3.
This Is a problem because the rotordynamic coefficients are very
sensitive to the Inlet clrcumferentlal veloclty. Therefore, the
dynamic data w111 be presented as a function of Inlet circumferential
veloclty at one inlet pressure and one rotor speed. Comparisons of
the leakage, direct stiffness, cross-coupled stiffness, dlrect
damplng, and stability of the six seals follow.
LEAKAGE
The flow rate of alr through each seal was measured wlth a
turbine flowmeter located In the piping upstream of the test sectionQ
(see figure 32). Figures 40-_3 show seal leakage as a Function of
radlal seal clearance For the inlet circumferential velocity set of
table 6 For teeth-on-rotor and teeth-on-stator labyrinth seals,
respectively. The plot on the left slde of the page Is for the
teeth-on-rotor seal and the one for the teeth-on-stator seal is on
96I
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I00 •
the right. This convention will be followed for the remainder of the
presentation. A comparison of the leakage of the six seals reveals, as
expected, greater leakage occurs for greater radial clearances. The
difference in the shapes of the curves, between the two seal types, can
be attributed to the difference in the performance characteristics of
the two seals.
DIRECT STIFFNESS
Figures _q-_7 show the dimensional direct stiffness versus inlet
circumferential velocity ratio for the clearances of table 5. These
plots show that the direct stiffness decreases in magnitude as
clearance increases. One would expect zero direct stiffness values at
sufficiently large clearances. Figures _8-50 show the dimenslonal
direct stiffness versus rotor speed for the pressure ratios of table 6
and Inlet circumferentlal velocity 3. The figures show that direct
stiffness becomes increasingly negative as rotor speed increases, for
the teeth-on-rotor seal, and is unchanged for the teeth-on-stator seal.
This effect was not noticeable in the results from the low speed test#
rig and could be a result of clearance change due to rotor growth. The
dimensionless direct stiffness coefficient, defined in table 8, removes
the effect of clearance change due to rotor growth from the plot.
Figure 51 shows the dimensionless direct stiffness versus rotor speed
for seal I (minimum clearance seal) at the pressure ratios of table 6
and Inlet circumferential velocity 3. The figure shows that the dimen-
sionless direct stiffness increases in magnitude as rotor speed
increases, for the teeth-on-rotor seal, and Is inconclusive for the
teeth-on-stator seal. Associated direct stiffness plots can be found in
Appendix D.
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Figure 5g shows
speed for seal
CROSS-COUPLED STIFFNESS
Figures 52-55 show cross-coupled stiffness versus inlet
circumferential velocity ratio for the radial clearances of table 5.
The plots show, in all cases, that the cross-coupled stiffness Is a
somewhat linear function of inlet circumferential velocity ratio. The
figures do not show any consistent trend with respect to radlal seal
clearance, for either teeth-on-rotor or teeth-on-stator seals. Figures
56-58 show cross-coupled stiffness versus rotor speed for the pressure
ratios of table 6. The figures show that cross, coupled .stiffness
increases wlth increasing rotor speed, for the teeth-on-rotor seal, and
decreases with increasing rotor speea for the teeth-on-stator seal.
This effect was not evident in the results from the low speed test rig.
The dlmenslonless cross-coupled stiffness coefficient, defined in table
effect of change of clearance due to rotor growth.
dimensionless cross-coupled stiffness versus rotor
circumferential
I for the pressure ratios of table 6 and Inlet
velocity 5. The figure shows that cross-coupled
stiffness Increases for Increasing rotor epeed, for the teeth-on-rotor
seal, and decreases with increasing rotor speed for the teeth-on-stator
seal. The decrease in cross-c0upled stiffness with rotor speed, for a
teeth-on-stator seal, was also evident in tests of an 11 cavity seal
for Sulzer [50] and in the steam tests of I-3 cavity seals by Hisaet
al. [8]. Associated cross-coupled stiffness plots can be found in
Appendix D.
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DIRECT DAMPING
Figures 60-63 show the direct damping versus inlet circumferential
velocity ratio for the radial seal clearances of table 4. The data
show that the direct damping for a teeth-on-rotor seal increases as
clearance Is increased while the direct damping for a teeth-on-etator
seal decreases as clearance Is increased. Figures 64-66 show direct
damping versus rotor speed for the pressure ratios of table 6 and Inlet
circumferential velocity 3. These figures show that the direct damping
increases slightly as rotor speed increases, for teeth-on-rotor seals,
and decreases slightly as rotor speed increases for teeth-on-stator
seals. These results are deceiving, since direct damping is very sen-
sitive to clearance change. The normalized direct damping coefficient,
defined In table 8, removes the effects of clearance change due to
rotor growth. Figure 67 shows normalized direct damping versus rotor
speed for seal I for the pressure ratios of table 6 and inlet
circumferential velocity 3. This figure shows that direct damping
decreases as rotor speed Increases for both teeth-on-rotor and teeth-
on-stator seals. The result for normalized direct damping versus rotor
speed for the teeth-on-rotor seal is inconsistent wlth the dimensional
data. If dimensional direct damping, for a teeth-on-rotor meal,
decreases as clearance decreases and increases as rotor speed increases
then the normalized value should increase as rotor speed increases
because the seal clearance decreases as rotor speed increases. Thls
inconsistency is not readily explalnable. Associated direct damping
plots can be found in Appendix D.
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parameter,
frequency
model of
forces
STABILITY ANALYSIS
One further parameter of comparison among the test seals is the
dimensionless whirl frequency ratio. To understand the value of this
consider a rotor In a circular orbit of amplitude A and
w (Fig. 68). The X and ¥ components of force In the seal
equation (79) may be resolved Into radial and tangential
Fr = Fx cos _t + Fy sin mt
Ft = -Fx sin _t + Fy cos wt
Expressing the rotor motion as
X - A cos _t X - -A_ sin wt
Y - A sin _t Y = A_ cos _t
and using equation (79), the resultant radlal and tangentlal forces are
illustrated in the figure and are defined by
-Fr/A = K + c_
FtlA l k -- CW
If Ftl A is a positive quantity, the tangential force is destablllzing
since it supports the whirling motion of a forward whirling rotor.
Conversely, of Ft/A is negative, it opposes the whirling motion of a
forward whirling rotor, and is therefore stabilizing. The whirl
frequency ratio is defined by
Whirl frequency ratio - klCw .
From the above discussion, If the whirl ratio is less than one, the
tangential force on the rotor is stablllzlng. A minimum value of the
whirl frequency ratio is optimum for stabillty.
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128 •
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Fig. 68 Forces on a synchronously preceeeing seal.
129
Figures 69 and 70 show the whirl frequency rBtios at a running
speed of 16000 cpm and a shake frequency of 74.6 Hz. For teeth-on-rotor
seals, the figures show that as clearance increases the meal becomes
more stable. For teeth-on-stator seals the opposite is true; as
clearance increases the seal becomes less stable, for the positive
inlet circumferential velocity case. The figures also show that the
teetb-on-stator seals are more stable than the teeth-on-rotor seals for
positive inlet circumferential velocity ratio, as was found previously
[6]. Figure 71 shows the whirl frequency ratio versus rotor speed for
the seals of table 5. The figure shows that as speed Increases, both
the teeth-on-rotor and teeth-on-stator seals become more stabie.
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CHAPTER ¥I
TEST RESULTS: COMPARISON TO THEORETICAL PREDICTIONS
In this chapter, the experimental results from the tests of three
teeth-on-rotor and three teeth-on-stator labyrlnth seals are compared
to the new analysis presented in the Geometric Boundary Approach
section in the THEORETICAL DEVELOPMENT chapter of this report. The
seals tested are described in tables 3-5. Tables 5 and 6 and figures
35-37 define the symbols used in the figures, Generally, the solid
lines are the experimental points and the broken lines are the
predictions.
STATIC RESULTS
Before proceeding with the comparison to the theory, some
necessary input parameters to the model must be given. Table 9 shows
the varlables used as input to the program for the comparisons shown
here. The temperature given was fairly constant for all of the tests.
The viscosity was calculated for each case using Sutherland's formula
[51]. The pressure gradients for the five rotor speeds of table 6 are
shown in figure 72 for a single inlet pressure and Inlet
circumferential veloclty. The curves show that the pressure gradient
has little or no sensitivity to rotor speed. Any slight differences In
the curves are due to variations in the actual points taken.
Therefore, only one rotor speed will be used for comparison of the
pressure gradients.
Figures 73-75 show a comparison of the experimental and
134 I
Table 9. Input parameters for seal program.
Inlet temperature
Ratio of sp. heatsGas constant (air)
Compressibility factor
Rotor friction exp.(mr)Rotor friction const.(nr)
Stator friction exp.(ms)
300K
287°06 J/(kgK)1.0
-0.250.079
-0.25
Stator friction const.(ns) 0.079
Number of teeth 16
135
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theoretical pressure gradients for the no-inlet-clrcumferentlal-
velocity case. The Figures show that the theory underpredlcts the
cavity pressures for the teeth-on-rotor case and overpredlcts the
cavity pressures for the teeth-on-stator case. This difference is due
to the difference in the lnlet losses for the two seal types. The
theory accurately predicts the inlet loss for the teeth-on-rotor case.
However, the teeth-on-stator seal has a much larger inlet loss followed
by a pressure recovery. Thls positive pressure recovery cannot be
modelled by a simple leakage equation. The remainder of the pressure
gradient comparison plots can be found In Appendix D.
Figures 76-78 show a comparison of experimental and theoretical
leakage versus inlet circumferential velocity ratio for the inlet
pressure set of table 6. The plots show that the theory underpredlcts
the leakage for both seal types by about 255. This is much worse than
the 55 error for ,the theory of Childs and Scharrer [18]. This
difference Is due to the change in the equation for the kinetic energy
carryover coefficient, uz- The change In the coefficient was made in
order to obtain a local equation which would yield a clearance
perturbation. The former coefficient was a global equation and could
not be perturbed. The contribution of thls coefficient to the first-
order equations and the subsequent solution Is very substantial. In
effect, the leakage calculation was sacrificed in order to improve the
calculation of the dynamic coefficients.
DYNAMIC RESULTS
The experimental and theoretical results to be compared include
the direct and cross-coupled stiffness and direct damping coefficients.
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143
A cross-coupled damping
uncertainty present in
Uncertainty section). Of
comparison has been omltted because or the
the experimental values (see the Relative
the remaJnlng three coefflclents, the direct
stiffness comparison will be
comparison wlll be last.
Direct stiffness
Figures 79_2 show a
direct stiffness versus
seals defined In table 5.
presented first, and the direct damping
comparison of experlmental and theoretlcal
inlet circumferentlal velocity ratio for the
The figures show that the theory correctly
predicts a decrease In direct stiffness as clearance increases, for
both seal types. The figures also show that for both inlet pressures,
the theory overpredlcts the direct stiffness at low rotor speeds and
underpredlcts the direct stiffness at high rotor speeds. This trend is
made clearer by figures 83-85. Figures 83-85 show a comparlson of
experlmental and theoretical direct stiffness versus rotor speed for
the inlet pressure set of table 6 and inlet clrcum/erential velocity 5.
The figures show that the theory is oversensitive to rotor speed. Some
of the test data dld show an decrease in direct stiffness with rotor
speed, but not wlth the sensitivity predicted by the theory. Thls Is
an improvement over the previous theory of Childs and Scharrer [18]
which consistently underpredlcts direct stiffness.
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151
Cross-coupled stiffness
Figures 86-89 show a comparison or experimental and theoretical
cross-coupled stiffness versus inlet circumferential velocity ratio for
the seals defined in table 5. The figures show that, llke the test
results, there is no consistent trend wlth clearance. The figures also
show that the theory does an excellent Job or predicting the cross-
coupled stiffness for the low rotor speed results and a reasonable Job
for the high rotor speed results. Rotor speed effects are much clearer
in figures 90-92. Figures 90-92 show a comparison of experimental and
theoretical cross-coupled stiffness versus rotor speed for the inlet
pressure set of table 6 and inlet circumferential velocity 5. These
figures show that
speeds are reached.
the theory predicts reasonably well until higher
The theory then predicts a sharp upswing in the
cross-coupled stiffness at high speeds. This effect is shown by the
experimental data in figure 90. The larger clearance seals do not show
this effect at the speeds tested. Perhaps at higher speeds, larger
clearance seals will show this effect.
152
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Direct damping
Figures 93-96 show a comparison of experimental and theoretical
direct damping versus inlet circumferential velocity ratio for the
seals defined in table 5. The figures show that the theory correctly
predicts an increase In direct damping for an Increase in clearance for
the teeth-on-rotor seal. However, the theory incorrectly predicts the
same trend for teeth-on-stator seals. This error renders the theory
suspect when used for teeth-on-stator seals whose geometry differs
significantly from those tested In thle study. However, the theory
averages an error of _05 for the teeth-on-rotor seals, which is a great
improvement over the 755 error of the previous theory of Chllds and
Scharrer [18]. Figures 97-99 show a comparison of experimental and
theoretical direct damping versus rotor speed for the inlet pressure
set of table 6 and inlet circumferential velocity 5. These figures
show that the theory predicts more speed sensitivity than Is shown by
the experimental data. Perhaps at higher speeds, the test data will
show the same trends.
.Comparison to theor_ of [18]
Figures 100 and 101 provide a brief comparison of the present
theory to the theory of Chllds and Scharrer [18]. Figure 100 shows
cross-coupled stiffness versus pressure ratio for a teeth-on-rotor
labyrinth seal at 16000 cpm. The figure shows that the present theory
follows the experimental data closely while the former theory deviates
as pressure ratio Is increased. Figure 101 shows direct damping versus
pressure ratio for a teeth-on-rotor seal at 16000 epm. The figure
shows that the present theory follows the experimental data more
closely than the former theory.
160
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Q
O
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400
_. _o
..=,
u'l
_ 0
TEETH-ON-ROTOR LABYRINTH SEAL
SPEED = I6000 CPM
O EXPERI MENTAL DATA0 NE_ PROGRAM RESULTS& IWATSUBO PROGRAM RESULTS
C $ 8 $ $ l o I ! , j
0 1.6 3.2 4. B E. 4 O.B 2.4 4 5.6 7.2
PRESSURE RATIO
Fig. 100 A comparison of experimental and theoretical
results of this report with those of [18]
for cross-coupled stiffness.
C3
F-uWG_
C3
500
450
400
350
3OO
250
2OO
150
IOO
5O
0
TEETH-ON-ROTOR LABYRINTH SEAL
SPEED- 16000 CPM
0 1.6 3.2 4.8 6.4• 8 2.4 4 5.6 7.2
PRESSURE RAT TO
Fig. 101 A comparison of
results of this report
for direct damping.
I
0
experimental and theoretical
with those of [18]
168 •
CHAPTER VII
CONCLUSIONS
A new analysis utilizing a two control volume
incorporates a solution for the reclrculatlng velocity In the
and information from a 2-D CFD calculation, has been presented
problem of calculating rotordynamic coefficients for labyrinth seals.
This analysis was developed to provide both an improved prediction for
the rotordynamlc coefficients and a more detailed model for the flow in
a labyrinth seal. A seal-test facility has been developed and modified
for high speed testing for the study of various types of gas seals. A
method for determining rotordynamic coefficients from experimental data
has been established, and consistent, repeatable results have been
obtained.
A comparison between the CFD results of Rhode and the results of
new analysis presented in this report support the following
conclusions:
(1) The new two-control_volume model accurately predicts the
stator wall shear stress for a teeth-on-rotor labyrinth seal cavity.
(2) The analysis predicts the cavity wall shear stress of a
teeth-on-rotor seal within 25_ of the average of the CFD result.
(3) The 2-D Jet flow theory used in this analysis accurately
predicts magnitude of the reclrculatlon velocity along the dividing
streamline.
(4) The CFD results show that the mixing length parameter, E,
used in the equation for the free shear stress is relatively constant,
model which
cavity
for the
169
for teeth-on-rotor seals, and need not be considered a function of seal
geometry, as was assumed by Jenny et al. [21].
The experimental results of the previous section support the
following conclusions:
(1) For teeth-on-rotor seals the direct damping Increases as
clearance Increases; for teeth-on-stator seals, the direct damping
decreases as clearance Increases.
(2) Direct stiffness and direct damping show little or no
sensitivity to rotor speed up to {6000 cpm. Cross-coupled stiffness
shows a sharp upswing at higher rotor speeds, for a teeth-on-rotor
seal. Cross-coupled stiffness decreases as rotor speed increases, for
a teeth-on-stator seal.
(3) Direct stiffness is negative and increases as clearance
increases, for both seal configurations. Cross-coupled stiffness
showed no consistent trend wlth respect to clearance changes.
(4) As clearance decreases, teeth-on-rotor seals become less
stable and teeth-on-stator seals become more stable.
The theoretical results of the previous section supportthe
following conclusions:
(I) Theoretical results for leakage underpredict the test
results presented in this report by about _S. Leakage increases as
clearance Increases for both seal types.
(2) Theoretical results for pressure gradient are underestimated
for teeth-on-rotor seals and overestimated for teeth-on-stator seals.
(3) The theory correctly predicts that direct stiffness is
negative and increases as clearance increases, for both seal
170 Q
configurations. The theory Incorrectly predicts an approximately
quadratic increase in the direct stiffness magnitude (becoming more
negative) as speed increases. Test results show scant sensitivity.
(_) The theory accurately predicts an increase in cross-coupled
stiffness at hlgh speeds, for a teeth-on-rotor seal.
(5) For teeth-on-rotor seals, the theory correctly predicts an
Increase in direct damping for an Increase in clearance. However, the
theory incorrectly predicts the same trend for a teeth-on-stator seal.
(6) The theory incorrectly predicts an approximately quadratic
increase in direct damping with running speed. Test results show no
systematic change In direct damping wlth running speed.
(7) A comparison wlth test results for a teeth-on-rotor seal
shows that the theory presented In this report does a better Job of
predicting direct damping and cross-coupled stiffness than does the
theory of Chllds and Scharrer [18]. A comparison with the theory of
Jenny et al. [21] was not possible, as discussed In the THEORETICAL
DEVELOPMENT chapter of this report.
In summary, the analysis presented here Is considered useful for
predicting cross-coupled stiffness
on-stator labyrinth seals directly. These
consistent for the various geometries
for both teeth-on-rotor and teeth-
results were reasonable and
and operating oondltlons
presented. However, the results for the direct damping coefficient for
teeth-on-stator labyrinth seals were not consistent wlth test results
for clearance change effects. Thls discrepancy renders the analysis
suspect for predlctlng these coefficients for teeth-on-stator seals
whose geometry differs significantly from the seals tested In this
171
study. Predictions for the remaining coefficients can be modified using
the appropriate correction factor from the comparison plots presented
in this study.
In the future, if any advances are to be made In the predlctlon of
rotordynamlc coefficients for labyrlnth gas seals, they wlll probably
Involve the perturbation of a flnlte difference aolutlon. The "bulk
flow" model presented in this report Is too crude to model the complex
flowfleld present in a labyrlnth cavlty,
I
$
I
@
172 ID
REFERENCES
I , Wachter, J., and Benokert, H., "Querkr'afte aus SpaltdJchtungen-Eine
mogl iche Ursache fur die baufunr uhe yon Tw'bomaschi nen ,"
Atomkernenergle Bd. 32, 1978, Lfg. 4, pp. 239-246.
. Wachter, J., and Benckert, H., "Flow Induced Spring Coefficients
of ILabyrinth Seals for Applications in RotordynamJc," NASA CP 2133
Proceedings of a workshop held at Texas A&M University 12-14 May
1980, Entitled Rotordynamic Instability Problems of High
Performance Turbomachlnery, pp. 189-212.
. Benckert, H. ,
Labyri nt hdi cht ungen ,"
Stuttgart, 1980.
"Stromungsbedi nte Feder kennwert e in
Doctoral dissertation at University of
Wright, D.V., "Labyrinth Seal Forces on a Whirling Rotor," Rotor
Dynamical Instability. Proceedings of the ASME Applied Mechanics,
Bioengineering, and Fluids Engineering Conference, June 20-22,
1983, Houston, Texas. pp. 19-3].
. Brown, R.D, and Leong, Y.M.M.S, " Experimental Investigation of
Lateral Forces Induced by Flow Through Model Labyrinth
Glands," NASA CP 2338, Rotordynamic Instability Problems in High
Performance Turbomachinery, proceedings of a workshop held at
Texas A&M University 28-30 May, 1984. pp. 187-210.
o Childs, D.W.
Coefficient
Labyrinth Gas
and Scharrer, J.K., "Experimental Rotordynamic
Results for Teeth-On-Rotor and Teeth-On-Stator
Seals," ASME Paper No. 86-GT-12.
, Kanemltsu, Y. and Ohsawa, M., "Experimental Study on Flow Induced
Force of Labyrinth Seal," Proceedings of the Post IFToMM
Conference on Flow Induced Force in Rotating Machinery, September
18-19, 1986, Kobe University, Kobe, Japan, pp. 106-112.
, Hisa, S., Sakakida, H., Asatu, S. and Sakamoto, T., "Steam Excited
Vibration in Rotor-Bearing System," Proceedings of the
International Conference on Rotordynamlcs, September 14-17, 1986,
Tokoyo, Japan, pp. 635-641.
. Alford, J. S., "Protecting Turbomachinery from Self-Excited Rotor
Whirl," Transactions ASME J. of Engineering for Power', October
1965, pp. 333-344.
10. Spark, J. H., and Keiper, R., "Selbsterregte Schwingungen bei
Turbomaschinen Infolge der Labyrinthstromung," lngerleur-Arehive
43_..__,1974, pp. 127-135.
11. Vance, J. M., and Murphy, B. T., "Labyrinth Seal Effects on Rotor
Whirl Stability," inst. of Mechanlcal Englneer, 1980, pp. 369-373.
173
Q
• "A Theoretical Analysls of the Aerodynamic Forces12. Kostyuk, A G.,
In the Labyrinth Glands of Turbomachlnes," Teploener_etlca, 19 .(11)O, 1972, pp. 39-44.
13. lwatsubo, T., "Evaluation of Instability Forces of Labyrinth Seals
in Turbines or Compressors," NASA CP 21 33 Proceedings or aworkshop at Texas A&M University 12-14 Hay 1980, EntitledRotordynamlc Instability Problems In High PerformanceTurbomachlnery, pp. 139-167.
14. Iwatsubo, T., Hatooka, N., and Kawal, R., "Flow Induced Force andFlow Pattern of Labyrinth Seal," NASA CP 2250 Proceedings of aworkshop at Texas A&H University 10-12 Hay 1982, EntitledRotordynamlc Instability Problems In High PerformanceTurbomachlnery, pp. 205-222.
15. Kurohashl, H., Inoue, ¥., Abe, T., and FuJikawa, T., "Spring andDamping Coefficients of the Labyrinth Seal," Paper No. C283/80delivered at the Second International Conference on Vibrations InRotating Machinery, The Inst. of Mech. Engineering.
16. Cans, B.E, "Prediction of the Aero-Elastlc Force in a Labyrinth
Type Seal and its Impact on Turbomachinery Stability," M.S. ThesisM.I .T. , 1983.
17. Martinez-Sanchez, M., Lee, O.W.K., CzaJkowskl, E., "The Prediction
of Force Coefficients for Labyrinth Seals," NASA CP 2338,
Rotordynamic Instability Problems In High Performance
Turbomachlnery, proceedings of a workshop held at Texas A&H
University 28-30 may, 1984. pp. 235-256.
18. Childs, D.M., and Scharrer, J.K., "An Iwatsubo Based Solution forLabyrinth Seals: A Comparison to Experimental Results," ASH___ETrans. Journal of En6ineerin _ for Gas Turbines and Power, April
1986, Vol. 108, pp. 325-331
19. Hauck, L., "Exciting Forces due to Swirl-Type Flow In Labyrinth
$eals," Proceedings IFTOHH Conference on Rotord_namlo Problems inPower Plants, 28 September-1 October 1981.
20. FuJikawa, T., Kameoka, T., Abe, T., "A Theoretical Approach toLabyrlnth Seal Forces," NASA CP 2338, Rotordynamlo Instabllity
Problems in High Performance Turbomachinery, proceedings of
workshop held at Texas A&H University 28-30 May, 1984. pp.173-186.
21. Jenny, R.J., Myssmann, H.P., Pham, T.C., "Prediction of Stiffness
and Damping Coefficients for Centrlfugal Compressor Labyrinth
Seals," ASHE 84-GT-86. Presented at the 29th Internatlonal GasTurbine Conference and Exhibit, Amsterdam, The "Netherlands, June
4-7, 1984.
174
22. Seharrer, J.K., "A Comparison of Experimental and Theoretical
Results for Rotordynamic Coefficients for Labyrinth Gas Seals,"
TRC Report SEAL-2-85, Texas A&M University, May 1985.
23. Rhode, D., Private correspondence, Texas A&M University, 1985.
24. Rhode, D., "Simulation of Subsonic Flow Through a Generic
Labyrinth Seal Cavity," ASME Paper No. 85-GT-76.
25. Stoff, H., "Incompressible Flow in a Labyrinth Seal," do_-nal o£
Fluid Meohanlos, Vol. 1OO, part 4, pp. 817_829, 1980.
26. Korst, H.H., Page, R.H., and Childs, M.E., Univ. of Illinois Eng.
Exp. Report ME TN 392-I, Urbana, Illinois, April 1954.
27. Liepman, H.W. and Laufer, J., NACA TN1257, 1947.
28. Korst, H.H. and Trlpp, W., "The Pressure on a Blunt Trailing Edge
Separating Two Supersonic Two-Dimensional Alrstreams of Different
Math N_iber and Stagnation Pressure But Identical Stagnation
Temperature," Proceedings of the 5th Midwestern Conference on
Fluid Mechanics, University of Michigan Press, Ann Arbor', Michigan
pp. 187-200, 1957.
29. Goertler, H., Z. Angew Math. Mech., 22, pp. 244-254, 1942.
30. Glauert, M.B., "The Wall Jet," Journal of Fluid Mechanics, I, pp.
625, 1956.
31. Schllchtlng, H., Boundary Layer Theory, McGraw-Hill, New York, NY,
pp.621, 1979.
32. Nelson, C.C. and Nguyen, D.T., "Comparison of llirs' Equation wlth
Moody's Equation for Determining Rotordynamic Coefficients of
Annular Pressure Seals," ASME Paper No. 86-TRIB-19, also accepted
for the ASME Journal of Tribology.
33. Blasius, H., "Forschungoarb", Ing.-Wes., No 131, 1913.
34. Yamada, Y., Trans. Japan Soc. Mechanical Engineers, Vol. 27, No.
180, 1961, pp. 1267.
35. Abramovich, G.N., The Theory of Turbulent Jets, MIT Press,
Cambridge, Massachusetts, 1963.
36. Schlichting, H., Boundary Layer' Theory, McGraw-Hill, New York, NY,
PP. 579, 1979.
37. Jerie, J., "Flow Through Stralght-Through Labyrinth Seals,"
Proceedings of the 7th International Congress on Applied
Mechanles, Vol.2, pp. 70-82, 1948.
175
38. Komotori, K. and Mori, H., "Leakage Characteristics or Labyrinth
Seals," Fifth Internatlonal Conference on Fluid Sealing, 197],
Paper E4, pp. 45-63.
39. Wyssmann, H.R., "Theory and Measurements of Labyrinth Seal
Coefficients for Rotor Stability of Turbocompressors," Proceedings
of a workshop at Texas A&M University 2-4 June 1986, Entitled
Rotordynami c Instability Problems in Nigh Performance
Turbomachlnery (in press).
40. Vermes, O., "A Fluld Mechanics Approach to the Labyrinth Seal
Leakage Problem," ASME Journal of Engineering for Power, Vol. 83,No. 2, Aprll 1961, pp. 161-169.
41. Gurevich, M.I., The Theory Of Jets In An Ideal Fluid, Pergamon
Press, London, .England, 1966, pp. 319-323.
42. Egli, A.: The Leakage of Steam Through Labyrinth Glands, Trans.
ASME, Vol. 57, 1935, pp. I15-122.
LJ_,I"1_. IJ I,J, ii, , Uib./_ l _._Q_ l_'a...... "S ,,j_ie, N _'_' v,-,,_t, _v 1o7(}
44. linG, T., and Eaneko, H., "Hydraulic Forces Caused by Annular
Pressure Seals in Centrifugal Pumps," NASA CP 2133, Rotordynamic
Instability Problems in High Perfor'manc@ Turbomaehi nery,
proceedings of a workshop held at Texas A&M Univ., 12-14 May 1980.
45. Bowen, W.L., and Bhateje, R., "The Hollow Roller Bearing," ASME
Paper No. 79-LUB-15, ASME-ASLE Lubrication Conference, Dayton,Ohio, 16-18 October 1979.
46. Cohen, H., Rogers, G.F.C., and Saravanamuttoo, H.I.H., Gas Turbine
Theory, Longman Group Limited, London, England, 1972.
47. Nicks, C.O.,"A Comparison of Experimental and Theoretical Results
for Leakage, Pressure Distribution, and Rotor'dynamic Coefficients
" M.S Thesis, Texas A&M University, 1984for Annular Gas Seals, .
48. Nelson, C.,Childs, D.W., Nicks, C.O., Elr'od, D.," Theory Versus
Experiment for the Rotordynamie Coefficients of Annular Gas Seals:
Part 2. Constant-Clearance and Convergent-Tapered Geometry," ASME
Journal of Tribology, Vol. 108, pp. 433-438, July 1986.
49. Holman, J.P., Experimental Methods for Engineers, McGraw-Hill,
New York, NY, 1978, pp. 45.
50. Chllds, D.W., Scharrer, J.K., and Hale, R.K., "Rotordynamic
Coefficients for Sulzer Teeth-On-Stator Labyrinth Gas Seal,"
Texas A&Id Univ. Turbomachinery Lab. Report TRC-SEAL-3-86, 1986.
51. Schllchting, H., Boundary Layer Theory, McGraw-Hill, New York, NY,
pp. 328, 1979.
176
knFENDIX A
GOVERNING EQUATIONS FOR TEETH-ON-STATOR SEAL
177
g
@
Q
Reduced Equations
The main dlfference between the teeth-on-stator equations and the
teeth-on-rotor equations occurs in the momentum equations. The shear
stresses acting on control volume I are now the rotor shear stress, ir,
and the free shear stress, xj. Similarly, the shear stresses acting on
control volume II are now the stator shear stress, 18, and the free
shear stress, xj. These difference are evident in the reduce form of
the continuity and momentum equations given below:
Contlnult_.l
_pAz
dr,,
_pW,A, _PA2 _pW2A2
_" + mi+ I - mI + _+ _- 0Rs,_6 at Rs=a6
(At)
Momentum I
awl pW,AI awl
pA, _ +
at Rss 3e
PA2 aWzAzP7+ -- + -- (Wol-Wil)
Lat Rs,ael (A2)
+ ml(W,l-W,l_ I) = + IjlL 1 + xrlariL i
Homentum II
aWz pWzA, aWz
pA:-- +
at Rs_ _e
+ _ apW'A"I+ l(W,i-Wol)L at Rs,aej
(X3)
As aPI
Rsz 36_jILI - xslaslL i
where as i and ar I are defined as
as i = (2B*L)/L ; ar I = 1.0
The rotor shear stress in the circumferential direction is now defined
using the smaller hydraulic diameter and the velocity components of
control volume I.
@
178 a
2
I (RS iw-W I )'+U,
I r = _ pJ(RS,w-W,IZ+U,Z (RS,w-W,) nV
Dh, l/mr (Aq)
where Dhli Is the hydraulic diameter of C.V. I, defined by
Dhll - 2CriL/(Cri+L) (A5)
Similarly, the stator shear stress In the circumferential direction is
now defined using the larger
components of control volume II.
I
"S " _ p,/W,'+U,' W2
hydraulic diameter and the velocity
V_!" +U," Dh,l) msns . (A6)V
where Dh21 Is the hydraulic diameter of C.V. II, defined by
Dh21 = 2BL/(B+L) (A7)
The definition of the free shear stress remains the same. However,
since CFD results were only available for the teeth-on-rotor
configuration, the sensitive mixing length ratio [/b may change for a
teeth-on-stator seal.
Zeroth-Order EQuations
Continuity: mol+ I = mol (AS)
Momentum I: mol(W_ol-W_ol-1) - (xjlo+mrloarl)Ll (A9)
Momentum II: mJloLl - -msloaslLl (A10)
Flrst-Order Equatl ons
The flrst-order equations remain exactly the same as before.
Since changes were made In the locations and definitions of the rotor
and stator shear stress terms, the followln8 changes In the
coefficients of the flrst-order equations are necessary:
- 'o(Wii-W,l-1)Poi 'o(W,I-W,I-I) -_Y -\ ,JILl
X_ - m 2 + P'l(4Sll-5)/_-)(S'l+l) - --Poi-1 - Pol wPoi Poi
179
X$
X 4
.Xri ( 1*mr )_ z1Ll arl2
L(Rs_-w,l)+ v,l
_Ji (UsI-U si)Li (¢-I)7+ 2 ;J(W21-WII)* (U=t_Uil
-1
mo(W,l-w,l-1)PoH =o(W,l-w,,-1) /x-lVeoH \
Po1-1 - Pol wPol _ "f
irl ( 1+mr)U zILlarl _Jl (Us t-U 11 )L1 ($-1
[°,,.o,-,X z 2 -
Pol-l-Pol :Pol \_-/\Pol / J
• *mr _ tn_-R si I'-la_l _jILI
m llO + 4 2 2 4
RSm-Nll (Rsm-N,I)+ Ull N=l-Nil
_Jl (W=l"W,l )LIi
2 2
(W=I-WII)+ (U=I-UII)
Xe" "m°(Wll-Wll-l')crl _ +
L Poi Pol-l-Poi
2 2
(Ll-_pi)(_2,-1/ ] ,rlarl'.l-rD,,l
Xjl (UzI-UII)LI (@-I) xsl (1+ms)@U:lasLi_x _ ., . . / ÷ _JlLI -_W:l-1411)+(Uel-Ult) W:I * u:l J Pol
xsl(1*ms)aslLl
Pol
xsLaslLl
U_I Pol-1
_o1-1-Pol
xsi (1+ms)NslaslLl xJlLl xJlLl (Wsl, N_l)4- 4.
Msi + UzL W:I-W_L (14sl-W_t)+ (UsI-U _1)-1
P, IU, 1 (_S, 1-5)t'-1_Po1-1/ _Pol \ X /\ Pol /
I_jI(U:I-:,I)LI(,-I): • ,sl(1"ms),U,lasLl],:L(w_i-W_l)+ (U=l-U _i) w_i * U:l
180 |
APPENDIX B
DEFINITION OF THE FIRST ORDER CONTINUITY
AND M(X_34TUM EQUATION COEFFICIENTS
181
G1 m
Ll(Cri_Bl)
RT; G= "
CriN_iLi BiLiNai PoiLiCri4 ; G i -
RTRs _ RTRs = RTRs
PoiBiLit G_ Im .
RTRs z
G s
mo Poi mo pli+14"
2 2
PoI-I -Pol s
-I
\_PoI+ I/ \Poi+ II
GI i=
mO P11 (Y-I_Po|-,_"Y"-
w \ Pol I
- mo Poi-1 mo Pli
2 I
Poi-1- Pol
- mo Poi+l mo pli+lG? m --2 2
Poi = PoI+I.
G I ==
mo(Cri-Cri*l)
Cricri+l
2 2
o(L,-,011z \-_ZL I +17.04Cri
mo Poi
B | -
Pol _ Poi+l-I
I (,o,-,?(5- _s11)_=_, I
\ Pol /\ ,rQxl_r-1
¢,-,(5- 4SIi+I) .P_I \Poi*!--!
2 2
mo(Li+ 1-Tpi+ 1 )_)17.Ol;Crl÷ 1 _ P21+l/
PolLi PoINziLi
Gj - ----- _ G1o = " ;RT RTRs=
l
t
PoiCriLi CriLiXz ," ; Xz'_ ;
RT Rs
BiLlX, -, -- (Woi-W_i)
RT
- dSo(Wil-Wzi-1 )Pol
PoI-1 - Pol
_o(W,i-w,l-1) /_-1\ _OiLt
_,i(4s,l-_)_ ,r }(s,i+,)• Poi Pot
Xsl ( 1+ms)U iiLlaSi _ji (U, t-U _t )LI (+-1)]
(w.i-w,i;+ (u,i-u,i;J
[- _-_XI) )I xsi (l+ms)Llx Uzi Ull Pol pzlU_I(IISz1-5) (Sll+l
-- • " I' + +
L Pol PoI-I"Pol 'RPoi Poi
Xl m
-1
'o(Wzl-Nzl-' )Pot-' mo(Nzi-Nzl-, ) Cf-'_O_i- I ) _"¢/X• . - ,.,,l(,s.i-s),--A-Po1-1 " Poi _Pol Poi
l
182
S1 :W! 1 + Uil
U_l Po1-11 II
Pol- 1-Pol
(l+ms)UllLlaSl_ lJl(U=l-Uil)Ll(¢-l___m_i.____..(W=l_Wjl)+(U=l_Uil
-1
p,lU,l(llStl-5) X-1 Pol-1 _-]
JwPol Y Pol
xslLlaSl zsl (1+ms)W_ILlasl xJILIXs m &O ÷ " 4 2 :e" • 4. , 4.
W,I W:l + Uil W:l-W,1
mj I (WtI-W, 1 )LI| 2
(Mtl'W,l)+ (Ut t'U ,1 )
-zJ ILl xJILi (W= I-W, I)X7 = a "2 ;
Wll-W,1 (Wel'Wll)+ (Uzt-Ull)
X. m°(W'l-W'l'l) [1 (LI-TpI) p_l-l']Or1 17.OtlCrl p21
PolLIBI BILI (W_l-Wol) BILlY, " ; Y= " ; Ym "
RT RT Rs 2
zslaSlLlmsDh,l2
2Crl
Y_
Y$
YI m
U_I U,1 Pol p_tU,t(qS,t-5) _r-1 )l+ : : -- (S,I+ 1
LPol Pol-1 -Pol wPol x
XI-_jI(V,l-U ,I)LI (¢-I)
L(w21-w,I)+ (u2t U,I)
tri (1+mr)¢U21arLiT+ _JILiI
(Rs2w-W,I) + U21 J Pol
•rlariLl + ,ri(1+mr)(Rs2_-Wai)arlLi + ,JILl_2 2
Rs=w-W21 (Rs=w-W21) + U=I W_l'W,l
mjILl (Wt l'-W_I)
2 2
(W, I-W il )+ (U 2I-U It )-1
U_I Pol-1 _ I_llUll(llSll'5) _-1 PoI___I _]t •
LPoI-1-Pol v Pol _' Pol
xJ 1 (u, l-U ,1 )LI (¢-1)
XL(w,I_W,I _+ (U,i_U, 1 _ " xrl(l+mr)¢U,larLl 1(Rs_w-W21) + Ull
trl(l+mr)arlLl
Pol
Y7
- XJILl
(WzI-W,I)
mJlLl (W:I_W,I)2 2
(W21-W,1)+(U21-U,1); ¥, - 0.0
183
APPENDIX C
SEPARATION OF THE COWFINUITY AND MOMENTUM EQUATIONS
AND DEFINITION OF THE STSTEM MATRIX ELEMENTS
184 •
CONTINUITY :
4. 4. 4. + +
cos(e+wt): (G_m * G,)Pai + G a Wls i + G, W,si * G= Pci 4 G, Pcl-1+
+ G7 PCI+I = Go (a-b)/2
4 "1" + @ +
sln(e+_t): -(Gi_ + G2)Pcl - G. Wtc I - G. W,cl + G, Psi + G, PSI-I
• .]+ G7 Psi+1 " + G l (b-a)
cos(e-wt): (-(]zw + G.)P;I ÷ G, Wls I + G_ W,sl + G i Poi + G, PcI-I
D
+ G7 PcI+I = G, (a+b)/2
sln(e-_t): (G,_ - Gz)Pcl - G, Wlc i - G_ W2c i + G s Psi + G, Psl-1
m
+ Gv PSI+I = - Gto (a+b)
MOMENTUM I :
+ X= ÷ X s Psl+ --COS(e+_t): X_ W,sl [Rsl + + Rs= Rs, W=sl
• I- + + + @
+ X_ Pcl + Xs PCI+I ÷ X, Wlc I +X 7 W=c I - moWlcl- I = Xo(b-a)/2
It1 + - + Xz + X s P I Rs, Wzcl
4. 4" ,I" ÷ 4-
+ X_ Psl + Xs PsI+I + X, Wls I + X T Was I - m o Wisl- I . 0
• _ .] o]- XaPol -
cos(e-.t,: X, W,s i +LR'F +x,-x, Ps, "..I
+ l.. Pcl + Xs PcI+I + X, Wlc I ÷X, W=c I - moW'cl-1 = -X,(a+b)/2
sln(e-wt): -X, W,cl - LRs_ + X2 - X, Pcl Rs= W=cl
* X_ Psl+ Xs PsI*I + X, Wis i * X 7 W,s I - m o W,si_ I = 0
185
HOHENTUM lI :
4. 4. 4" 4"
÷ ¥_ Pel ÷ ¥o Wzcl ÷ Yo PcI-1 ÷Y7 Wicl " ¥o(b-a)/2
• o]•_.(o.=_-Y,.w,o_-L_-."Y ""Y' P;_- rY'P°I Y,w,_l.+ Y_ Psi + Ys Wasl + Ys P81-1 + Y7 W:81 " 0
- [ -x'W'I w] - [Y'P°I ¥'W'll "cos(O-.t):-¥,u W.e i + L_ + Y.- X= PSI + L Rs= ÷ R"_"-_ W=sl
+ ¥_ PCl + ¥o W=cl + Yo Pc1-1 + ¥_ W_cl " -Xl(a+b)/2
o] r,.,o, ,o,781n(e-_t): Yn_ Wzcl- + Xs - X= Pcl - LRs= + Rs.J W=cl
• _, P;,.• Y,w;,: • _, P;:-: • Y,w_.,:- o
At-1 NATRIX
f
g
O
A1,2 " A2,1 " A3,4 " AZl,3 " G.
AS, 2 - A6,1 - A7,_ = A8, 3 = Xs
A5,6 " A6,5 = A7,8 " A8,7 = -_o
A9,2 " A10,1 " Al1,11 = A12,3 " YI
The remaining elements are zero.
AI MATRIX
A1,1 " -A2,2 " Gm co + G z
A3, 3 " -At,4 = Gj w + Gz
AI,2 " A2,1 = A3,4 = Aq,3 = G=
A5,2" A6,1 = A7,I_ " A8,3" Xw
A5,1 " -A6,2 " X. u + X= + X_W=I/RS =
AT, 3 - -As,t I --X= _ + X= + X=W=L/RS =
A9, 2 " A10.1 - Al1._ = A12.3 = Y..
@
186 9#
A9.1 " -A10.2 " Y= w "_ ¥1 + Y=Wzl/RSz
All.3 " -A12.tl " -Y= u + Ym * YzW=I/RS.
A1.5 " -A2.6 = A3.7 " -Atl.8 = G,
W"
A5.5 = -A6.6 = X= lu + RsW-_=tJ
A7,7 = -A8,8 = X,
A5,6 " A6,5 = A7,8 = A8,7 = Xo
A9,6 = A10,5 = All,8 ! A12,7 = Y7 .
A1,9 " A3,11 = -A2,10 " -A4,12 " G,,
A5, 9 = -A6,10 - A7,11 = -A8,12 = X s Poi/Rs,
A5,10 - A6, 9 " A7,12 = A8,11 = X7
A9,9 " -A10,10 = Yz co 4. Yz Wzl/Rsz * YzPoi/Rsz
Al1,11 = -A12,12 = -Yz (_ " Yz Wzi/Rs= 4- yzpot/Rs z
A9,10 - A10,9 = Al1,12 = A12,1 1 - Y=
The remaining elements are zero.
AI+I MATRIX
A1,2 = A2,1 = A3,4 " A4,3 = G,
The remaining elements are zero.
187
B S
G o
2
Gi
2
G 9 G| •
2 2
-Xo
2
0
B AND C COLUHN VECTORS
0
;G O
2
GI GIo
2 2
Go
grand
2
G9 GIO
2 2
X I
2
0
-X o
2
0
Yo
b
2
0
"YI
2
0
188 I
APPIDIDIX D
THEORY VS. EXPERIMENT
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