A Comparison Of Experimental And Theoretical Results For ...€¦ · A comparison of experimental...

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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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 ..............................

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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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27

27

29

29

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37

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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. . .


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.


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 ................



rotor speed of 16000 cpm ...............



rotor speed of 3000 epm ...............




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Fig. 56 Cross-coupled stiffness versus rotor speed for seal 1

and inlet circumferential velocity 3. • ......


and inlet clrcumferential velocity 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


of 16000 cpm ....................

Fi g. 62 Direct damping versus inlet circumferential velocity


of 3000 cpm ......................

Fig. 63 Direct damping versus inlet circumferential velocity


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


Fig. 70 Whirl frequency ratio versus inlet circumferential



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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...........


gradients of seal 2 for a rotor speed of 3000 cpm andInlet circumferential velocity 3 ..........


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


A comparison of experimental and theoretical leakage

versus inlet circumferential velocity ratio for seal 3


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.......................



at an inlet pressure of 3.08 bar and rotor speed of16000 cpm .......................




3000 cpm .......................



at an Inlet pressure of 8.25 bar and rotor speed of

16000 cpm .......................

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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


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


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


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


3000 cpm .......................

Page

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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


3000 epm .......................


damping versus inlet circumferential velocity ratio


I6000 cpm .......................


damping versus rotor speed for seal I and Inlet


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 .............


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


Fig. D2 Direct stiffness versus rotor speed for seal 2 and


Fig. D3 Direct stiffness versus rotor speed for seal 3 and


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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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


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..........


gradients of seal I for a rotor speed of 3000 cpm andinlet circumferential velocity 2..........


gradients of seal I for a rotor speed of 3000 cpm andinlet circumferential velocity t ...........


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 ...........




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203

204

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206

207

@ xvll


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


• inlet circumferential velocity 1 ........... 209




Fig. D23 A comparison of experimental and theoretlcal pressure



;)10

211



• 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;


as Dimensionless length upon which shear stress acts on stator;


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

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Zmm

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D.

D.

I06JGJ

t-4

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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.

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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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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

86g

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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

nt0F

-0IZ

-J

IW

,.,/W

I'

II

II

II

I_-gO

03I_

=-'O

JP

')(_

(_(_

---,...,

...,...,

.-,iS

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QE

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QQ

••

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e•

••

II

II

II

II

(=!puou)

xx_

IS)

0

Q_

_•

ellip

c¢,'_om

.8)

•_tp

I..-.-4

tD

IS)

IS)

IS)

(Sl

I_IS

)IS

)'_"

CO

¢Z)

(_I'U

_"tO

II

I--.

---,--.

--.I

II

I

(_/N)

xx_

92•

tr_t_

It)C

Dtn

_In

_w

-_

tL_

trot

Od

OJ

e-t

o

.(oes)

000

l,xx__3

,S_'

ug

_

(w/o

oS

-N,)

xxs)

• 93

@

@

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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118 Q

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.

11g

133Q

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model of

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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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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

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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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_ 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=


AI+I MATRIX

A1,2 = A2,1 = A3,4 " A4,3 = G,


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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