nonaxisymmetric incompressible hydrostatic pressure

N A S A TECHNICAL NOTE

NONAXISYMMETRIC INCOMPRESSIBLE HYDROSTATIC PRESSURE EFFECTS I N RADIAL FACE SEALS

Lewis Resemcb Center Cleuehnd, Ohio 44135

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , 0. C. NOVEMBER 1976

TECH LIBRARY KAFB, NM

19. Security Classif. (of this report)

Unclassified

1 . Report No, 2. Government Accession No.

NASA TN D-8343 4. Title and Subtitle

NONAXISYMMETRIC INCOMPRESSIBLE HYDROSTATIC PRESSURE EFFECTS IN RADIAL FACE SEALS

20. Security Classif. (of this page)

Unclassified

7. Author(s)

Izhak Etsion ~ ~~

9. Performing Organization Name and Address

Lewis Research Center National Aeronautics and Space Administration Cleveland, Ohio 44135

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D. C. 20546

0334074

5. Report Date November 1976

6. Performing Organization Code

8. Performing Organization Report No.

~ _ _ E-8817 10. Work Unit No.

. .-

505-04 11. Contract or Grant No.

13. Type of Report and Period Covered

-___ Tec'hnical Note

14. Sponsoring Agency Code

5. Supplementary Notes

~ ~~

6. Abstract

A flat s ea l having an angular misalinement is analyzed, taking into account the radial variations in sea l c learance. An analytical solution for axial force, tilting moment, and leakage is presented that covers the whole range f rom zero to full angular misalinement (surfaces in contact). Nonaxisymmetric hydrostatic p re s su res due to the radial variations in the f i lm thickness have a considerable effect on sea l stability. When the high p r e s s u r e is on the outer per iphery of the seal, both the axial force and the tilting moment a r e nonrestoring. seals where cavitation is eliminated is discussed, and the possibility of dynamic instability is pointed out.

The case of high-pressure

7. Key Words (Suggested by Authorls))

Seals Dynamic seals Radial face seals

~~ ~

18. Distribution Statement

Unclassified - unlimited STAR Category 37

NONAXISYMMETRIC INCOMPRESSIBLE HYDROSTATIC

PRESSURE EFFECTS IN RADIAL FACE SEALS

by lzhak Etsion*

Lewis Research Center

SUMMARY

A flat seal having an angular misalinement is analyzed, taking into account the radial variations in seal clearance. ment, and leakage is presented that covers the whole range from zero to full angular misalinement (surfaces in contact). pressures due to the radial variations in the film thickness have a considerable effect on seal stability. the axial force and the tilting moment are nonrestoring. to wear at the outer diameter. when angular misalinement is combined with radial distortions and the high pressure is on the inner periphery.

The case of high-pressure seals where cavitation is eliminated is discussed, and the possibility of dynamic instability is pointed out.

An analytical solution for axial force, tilting mo-

It is shown that nonaxisymmetric hydrostatic

When the high pressure is on the outer periphery of the seal, both This causes the seal surfaces

Instability and wear at the inner diameter can occur

INTRODUCTION

Axial forces in excess of those theoretically predicted for alined flat seal faces a re now well established by many experiments (e. g. , refs. 1 to 3). These axial forces are crucial in providing the mechanism that separates the seal faces and controls the gap between them.

A considerable amount of fundamental research has been done in the last two decades in an attempt to understand the origin of these axial forces and how they relate

National Research Council - National Aeronautics and Space Administration * Research Associate.

to the mechanism of seal operation. no generally accepted mechanism, theories a re listed. It seems that most of them relate the separating axial force to hydrodynamic effects.

To generate hydrodynamic pressures with relative circumferential motion, the geometry of the seal faces must deviate from being absolutely alined or perfectly flat. This deviation can be in the form of angular misalinement (ref. 5), microasperities (ref. 6), and various forms of waviness (refs. 2, 7, and 8). It was postulated (ref. 9) that cavitation must be associated with the production of net forces greater than the hydrostatic forces. In the absence of cavitation, a hydrodynamic pressure increase in the circumferentially converging portions of the seal clearance is counterbalanced by an equal pressure decrease in the diverging portions, with the result of zero net hydrodynamic force. Although it explains the existence of hydrodynamic generated forces, cavitation is not likely to occur in high-pressure seals. In some experiments (refs, 10 and 11) where regions void of fluid were observed, the applied pressure was usually less than 3 atmospheres. In reference 11, where temperatures in the seal clearance were measured, it was found that boiling rather than cavitation was the reason for the appearance of gas in the seal interface.

Considering all these facts, it is not likely that hydrodynamic effects are the only mechanism that is responsible for maintaining a lubricated gap in face seals. Some investigators (refs. 1, 12, and 13) studied the effects on seal behavior of hydrostatic pressures caused by radial variations in seal clearance. Such variations, which can be caused by thermal and mechanical distortions, generate a net force that is different from the hydrostatic one that is related to flat, alined seal faces.

When the clearance decreases in the direction of flow, the pressures a re greater than those for parallel surfaces. Also positive stiffness and hence a stable fluid film are related to such configurations. flow the result is a decrease in pressure, negative stiffness, and a statically unstable fluid film.

In all the literature dealing with hydrostatic effects, with the exception of reference 14, the deviation from the flat, alined geometry is always axisymmetric. The inevitable result is axisymmetric pressure distribution in the seal clearance. Some experimenters report surface rubbing in a form that is most unlikely to occur when pressure symmetry prevails. In reference 1 it is shown that the seal surfaces rub on either the outer or the inner boundaries, depending on the direction of flow. The wear is always such that film convergence in the direction of flow is produced. such wear could be a result of an angular misalinement combined with nonrestoring moments .

Many theories have been suggested, but there is In a recent report (ref. 4), the various models and

When the clearance increases in the direction of

Obviously,

2

In reference 14 a misalined hydrostatic seal with an orifice bypass is analyzed. For the case of zero orifice flow, which is no more than a conventional face seal, it is found that the seal always produces restoring tilt moments. This result contradicts the experimental evidence of surface wear (ref. l), which suggests nonrestoring tilt moments.

ance converges in the radial direction over one-half of the circumference, over the other half it diverges. Hence, when the flow is outward, the hydrostatic pressure distribution around the seal tends to reduce the tilt and restore alinement. On the other hand, when the flow is inward, the result is a nonrestoring moment, which tends to in- c Tease mis alinement .

moment, but also the separating force and the leakage across the seal. Such effects have been overlooked in other investigations. The objective of this work is to study these effects, aiming at a better understanding of radial face seal operation.

Considering the misalined face seal of figure 1, it is clear that, while the clear-

Nonaxisymmetric hydrostatic pressure distribution may affect not only the tilting

SYMBOLS

C

cl’ c2 F

Fd

FS

FS

-

H

h

I

M

M -

P

Q

seal clearance along centerline

constants of integration

axial force

hydrodynamic force

axial force contributed by nonaxisymmetric hydrostatic pressure

2 nondimensional force, F/n(pi - po) ro

nondimensional film thickness, h/C

film thiclmess

given by eq. (15)

tilting moment 3 nondimensional tilting moment, M/n(pi - po)ro

pres sure

leakage

3

- Q

R

r

P

Y E

E -

e P

w

nondimensional leakage, Q kl - e 1 :)r(ro + ri)

nondimensional radius, r/ro

radial coordinate

angle of distortion

angle of tilt

tilt parameter, y ro/c

angular coordinate

viscosity

rotational angular velocity

Subscripts:

i at inner radius

m at midradius

o at outer radius

ANALYSIS

Figure 1 describes the geometry of the misalined seal. radius ratio ri/ro is very close to unity. Hence, the narrow-seal approximation results in the one-dimensional incompressible Reynolds equation

In most applications the

The film thiclmess distribution for the misalined seal is given by

where y is the angular tilt and 8 is measured from the point of maximum clearance. The boundary conditions for equation (1) a re

4

at r = r . P = Pi 1

P = Po at r = ro

Because of its linear nature, equation (1) can be solved separately for the hydrostatic and hydrodynamic pressure components. Hence, the hydrostatic component is obtained from the solution of

A common assumption for narrow seals is that curvature effects may be neglected. When applied to equation (3), this must be carefully examined. In some works this assumption results in a modified expression for the film thickness having the form

h = C +yr,cos 8

This is, of course, an unrealistic, warped surface that if substituted into equation (3) will result in hydrostatic pressure distribution identical to that in the case of parallel seal surfaces.

Hence, when neglecting curvature effects, equation (3) becomes

but the film thickness h remains a function of both 8 and r, as given by equation (2). Integrating twice with respect to r gives for the pressure

For the boundary conditions, the constants of integration become

and

c1 c2 = Po - - h:

5

Hence, the hydrostatic pressure in the seal clearance is

ho 2 2 - hi

Using the film thickness expression given by equation (2) yields

h: - h2 ro - r ho + h - -

Substituting this into equation (5) yields

P = (Pi - Po) ro - ri ho +hi

The first term in the braces of equation (6) represents the well-known alrisymmetric solution for parallel surfaces. The second term is the contribution of misalinement. This contribution is not axisymmetric and depends on the gap.

The axial force, which tends to

F = 2

The nondimensional film thickness

Axial Force

separate the seal surfaces, is obtained from

f ro l ' p r dr de 'i

h C

H = - = 1 + E R C O S e

is introduced where

(7)

6

and the tilt parameter E is given by

Equation (5) can be written in the form

P = (Pi - Po) 2 2 H; ($- I) +PO Ho - Hi

(9)

A direct integration of equation (7) is very difficult. A perturbation technique (ref. 5) could be used where the pressure is expanded in powers of E. However, such a solution is restricted to very small values of the tilt parameter E and is inadequate when nonrestoring moments exist since this situation may lead to surface contact. In that case the solution has to cover the whole range of misalinement through E = 1.

With the assumption of negligible curvature effect, that is, r 3 rm, equation (7) becomes

1 F = 2rmr0 l'f p dR de

Ri

Integrating by parts we have

JR. 1

From equations (9) and (8) we have

aR

Substituting equation (12) in equation P

2 2 2HiHo E cos

HE- H: H3 - Po)

(11) and using the integral

1 dR= E COS e

we have

7

where

1 I =

Since

m H, + Hi = 2H

and

1 + R i = 2 R m

the expression for I becomes (by using eq. (8))

R~ + E ~ i COS e 1 +:cos e

I =

where

- E = €Rm

The axial force thus becomes

From the journal bearing integrals (ref. 15),

K

- 7r - de 1 +: COS e

and

8

.. . . . .

T

cas 6 de - T

1 +E cos e E: - - - -

Hence, from equations (15) to (18)

r - p.R. + <pi - PJ

1 1

F=2"ror0

.[ ( - 2 y -j 1- E

This can be further developed into

+%I] Rm

Noting that

and that

-%=l [(TT-R]= 1 + Ri (1 - Ri)' Rm

Rm Rm 4Rm

we finally have for the axial force

The first term on the right side of equation (21) is the axial force for the case of parallel flat surfaces. The second term presents the contribution of the nonaxisymmetric hydrostatic pressure distribution due to seal misalinement. Denoting that con-

9

tribution by Fs, we have

- 2 Fs = 7rr 0 (p. l -Po)Fs

- where Fs is a nondimensional force given by

From equation (22) it is clear that, for any angular misalinement, Fs is positive. Hence, the deviational force Fs depends on pi - po. For pi > p Fs is positive and thus yields axial forces in excess of the hydrostatic force for alined flat surfaces. Since E is given by yro/C, it can also be seen that a decrease in C increases E

and hence increases Fs. clearance decreases, producing positive stiffness and a statically stable film. On the other hand, when pi e po, the axial force Fs and the axial stiffness become negative and thus result in unstable operation.

0’

Thus, when pi > po, the axial force Fs increases when the

Tilting Moment

The tilting moment due to the nonaxisymmetric component of the hydrostatic pressure is

The (-) sign is used in order to obtain restoring moments as positive values. Again, when the curvature effect is neglected, the moment becomes

The same procedure as for the axial force is repeated here, giving the tilting moment in a form similar to equation (17):

M = -2r 2 lllo r (p. - p ) l l i I c o s 8 d 8

10

From reference 15 we have

Substituting equation (15) into equation (24) and using equations (16), (18b), and (25) give the tilting moment as

where the nondimensional tilting moment M is given by

As in the case of the axial force, the nondimensional moment %i is always positive and vanishes only for E = 0, as expected. For pi > po the tilting moment as given by equation (26) is positive and hence is a restoring moment. For pi < po the tilting moment is a nonrestoring one, the angular stiffness is negative, and the seal is unstable.

Leakage

The leakage across the seal is obtained from

If we use the nondimensional terms H and R, substitute equation (12) into equation (28), noting that H: - Hi = (Ho + Hi)(l - Ri)e cos 8, and neglect the curvature effect, equation (28) becomes

2

11

+ z ~ e + i + R ~ ) E 9 3 COS e + R ~ ~ 2 4 COS 4 e (30)

Again, from reference 15 we have

and

(3 la)

Using equations (18), (25), (311, and (30), we have from equation (29)

After rearranging this becomes

where the nondimensional leakage is given by

12

From equation (20b). can be written in the form

From equation (33) it is clear that for the alined seal, where E . 0, a will equal 1, as expected. As the misalinement increases, the leakage increases too. For very narrow seals, where 1 - Ri < 0. 1, it is seen from equation (33) that the increase is most affected by the term (3/2,t2Ri. The nondimensional leakage, given in the form

is obtained when the radial variations in the film thickness h (eq. (2)) are neglected. Hence, it can be seen that, with regard to leakage, the simpler expression h = C i - y r , cos 8 can be used in the analysis.

RESULTS AND DISCUSSION

Values of the nondimensional parameters Fs, E, and a are presented in table I and figures 2 to 4 to cover the whole range of tilt parameters from E = 0 to E T. 1. An interesting result is the axial force at E = 1. this force adds to any unbalanced closing force and increases substantially the friction between the rubbing surfaces. A seal of 10-centimeter outer diametcr, 0.9 radius

force of 86 newtons and a tilting moment of 430 newtons per centimeter.

this ratio decreases, the axial force and tilting moment increase quitc rapidly.

equation (33) and those calculated by equation (34) are ncgligiblc. expression (eq. (34)) can be used for any practical radius ratio.

In the case of radially inward flow,

* ratio, and 10-atmosphere seal pressure, will experience at E ~7 1 an unbalanced axial

The radius ratio ri/ro affects very much the a.xial force and tilting moment. As

As can be seen from table I the differenaes between thc values of a calculated by

*

Hencc. the simpler

13

The analysis presented in the previous section has indicated unstable seal opera- Such i~ is tab i I j t \~ is confirmed by the ex-

The nonrestoring moiiient i n t h c case of radially inward How-

tion under radially inward flow when pi < pO. periments of Denny (ref. 1). flow causes the surfaces to contact and wcar at the outer diamctcr of the seal. ever, Denny also found that with the flow radially outward the wear was greatest at the inner diameter. be a result of combined mislinciiient and coning due to thermal and mechanical distortions.

Such a wear appearance is unlikely for a flat misalined seal but can

In figure 5, a misalined distorted seal is shown. It is clear from the figure that, whenever the distortion angle p is larger than the tilt angle y, the seal surfaces can come into contact at the inner diameter. Morem-cr, since an unstable film is related to divcrging clcm-ances in the direction of flow, it can be seen that, when p > y, an instability will rcsult f rom a radially outward flow.

generated due to angular misalinement. force Fd for a misalined seal. metric about the line BB and thercfore produces moincwts only about the line A4, this is not the case with the hydrodynamic pressure. I.f the scaled pressure is high enough to eliminate any cavitation, the hydrodynamic press~iiw distribution becomes antisym- metric about the line BB. duces a couple that lags the hydrostatic tilt moment vector by 90°. prone to dynamic instability that can damage the seal.

A word should bc added with reg8rd to the effect of the hydrodynamic pressure Figure 6 shows the resultant hydrodynamic

While the hydrostatic pressure distribution is sym-

This results in a net zero hydrodynamic axial force but pro- Such a system is

CONCLUDING REMARKS

Contrary to the common belief that hydrodynamic effects arc mainly responsible for proper seal operation, i t is shown that hydrostatic effects play an important role in seal performance. variations in film thickness. Hence, these variations cannot be neglected even when dealing with radius ratios close to unity.

shows that hydrostatic a,xial forces and tilting moments can be generated because of the misalinement. These forces and moments can be either restoring o r nonrestoring, depending on the location of the high pressure with respect to the seal boundaries. When the high pressure is on the inside, the seal is stable. When the high pressure is on the outside, static instability occurs.

The hydrostatic pressure distribution is very sensitive to radial

The analytical solution for a flat radial face seal having an angular misalinement

14

Thc stability condition can be revcrsed whcn coning, due to thermal and nicchani:. cal distortions, is combined with angular misalinement. Finally, hydrodynamic pressures due to seal misalinement can well be the origin of dynamic instability.

Lewis Research Center, National Aeronautics and Space Administration,

Cleveland, Ohio, August 30, 1976, 50 5- 04.

REFERENCES

1. Denny, D. F. : Some Measurements of Fluid Pressures, between Plane Parallel Thrust Surfaces with Special Reference to Radial- Face Seals. Wear, vol. 4,

1961, pp. 64-83.

2. Pape, J. G. : Fundamental Research on Radial Face Seals. ASLE Trans., vol. 11, no. 4, Oct. 1968, pp. 302-309.

3. Orcutt, F. K. : An Investigation of the Operation and Failure of Mechanical Face Seals. J. Lub. Tech., Trans. ASME, Ser. F, vol. 91, no. 4, Oct. 1969, pp. 713-725.

4. Ludwig, Laurence P. : Face-Seal Lubrication. I - Proposed and Published Models. NASA TN D-8101, 1976.

5. Sneck, H. J. : The Effect of Geometry and Inertia on Face Seal Performance - Laminar Flow. J. Lub. Tech., Trans. ASME, Ser. F, vol. 90, no. 2, Apr. 1968, pp. 333-341.

6. Anno, J. N . ; Walowit, J. A.; and Allen, C. M.: Microasperity Lubrication. J. Lub. Tech., Trans. ASME, Ser. F, vol. 90, no. 2, Apr. 1968, pp. 351-355.

7. Kuzma, Dennis C. : Theory of the Mechanism of Sealing with Application to Face Seals. J. Lub. Tech. , Trans. ASME, Ser. F, vol. 91, no. 4, OCt. 1969, pp.

4 704-7 12.

8. Sneck, H. J. : The Eccentric Face Seal with a Tangentially Varying Film Thick- I ness. J. Lub. Tech., Trans. ASME, Ser. F, vol. 91, no. 4, Oct. 1969, pp.

74 8- 7 55.

9. Findlay, J. A. : Cavitation in Mechanical Face Seals. J. Lub. Tech., Trans. ASME, Ser. F, vol. 90, no. 2, Apr. 1968, pp. 356-364.

15

10. Findlay, J. A. : Measurements of Leakage in Mechanical Face Seals. J, Lub. Tech,, Trans. ASME, Ser. F, vol. 91, no. 4, Oct. 1969, pp. 687-694.

11, O~cutt , F. K. : An Investigation of the Operation and Failure of Mechanical Face Seals. J, Lub. Tech., Trans. ASME, Ser. F, vol. 91, no. 4, Oct. 1969, pp. 7 13-725,

12. Cheng, H. S.; Chow, C, Y.; and Wilcock, D. I?. : Behavior of Hydrostatic and Hydrodynamic Noncontacting Face Seals. J. Lub. Tech, , Trans. ASME, Ser. F, vol. 90, no. 2, Apr. 1968, pp. 510-519.

13. Zuk, John: Analytical Study of Pressure Balancing in Gas Film Seals. Trans. ASLE, vol. 17, no. 2, 1974, pp. 97-104.

14. Adams, M. L, ; and Colsher, R. J. : An Analysis of Self-Energized Hydrostatic Shaft Seals. J. Lub. Tech., Trans. ASME, Ser. F, vol. 91, no. 4, Oct. 1969, pp, 658-667.

15. Booker, J. F. : A Table of the Journal-Bearing Integral. J. Basic Eng. , Trans. ASME, Ser. D, vol. 87, June 1965, pp. 533-535.

16

TABLE I. - MISALJNED-SEAL PERFORMANCE PARAMETERS

c

~

Ratio of inner to

outer rzdi\ie,

ri'ro

0.X0

-

0.85

0.90

__ 0.92

~~~ ~

Tilt parameter,

€

0.1000E 00 0 .2000 0.3000 0.4n00 0, snan a.6not-i

r) . ioon i.onsn

0.7000 0.X000

o . i o o n ~ no

0,4000

0.7n0n

0.9000 1, aeon

0.?000 0.3000

0.5000 0,5000

0 .8000

o . i n o n ~ no 0.2000

0.400n

o.fioon

0. soan o.9nm

0. 3000

0 .5000

0 .7000

1 .0090

~~ ~~

Nondimensional force,

r). 41143 ~ - n 4 0.197RE-03 0.4.599c-03 o . n 5 9 4 ~ - n 3 0,?.439E-0? 0.2274E-09 0.3.513E-02 0.5476E-02 0.10SfiE-011 0.ln36e-01.

1 .

Nondimensional tilting moment,

M -

_ " . . .

Non- dimensioaal

leaKage, - Q

i . n i z l.r)4R 1.108 1,lQt 1.301 1.434 1 . 5 9 1 1 . 7 7 1 1 ; 9 7 6 2 , 2 0 5

1,014 1 . 0 5 4 1 .122 1,216 1.3311 1,4116 1.662 1.1165 2 .095 2 . 3 5 1

1.1114 1.055 1 , 1 2 4 1.112'. 1 . 3 4 5 1 . 4 9 7 1 , 6 7 7 1,5184 2 , 1 1 1 2.3R1

Simplified nondimensional

leakage,

1 + 2 c2Ri 2

1.n12 1.043 1 . 1 0 3 1 , 1 9 2 I.ZOO 1 , 4 3 2 1 .503 1 , 7 6 3 1 , 1 7 2 2,200

1 , 0 1 ? 1 ,ns i . 1,?.25 1 , ? 0 4 1 . 3 1 9 1 , 4 5 9 1 . 6 2 5 i . n i t 2.1135 2 , 2 7 5

1 . 0 1 3 1 . n ~ : 1 .12 , 1 , 2 1 G 1 .737 1 . 4 8 6 1.661 1 . R t 4 2,1193 2.750

i , n i s 1,055 1,124 I '2, 1:145 1 , 4 9 7 1 , 6 7 6 1 , 8 8 3 2.1.13 2 , 7 8 0

17

TABLE I. - Concluded.

Ratio of inner to

outer radius, ri/ro

0.94

0 . 9 6

0.98

0.99

18

Tilt parameter,

E

0 . i n o o E on o.2non 0 . 3 0 0 0 0 .4000 0. so00 o.finon 0 . 7 0 0 0 n.xnoo n.1noo 1.0000

0.1000E 00 n .2000

o . i n o n E 00 0.2000 0 . 3 0 0 0 0 .4000 0.51300 o.soon 0 . 7 0 0 n o.nnon o . m o n i . o n o n

0.1000E 0 0

0 . 3 0 0 0

0 .5000

0.7n00

1.0000

o . ? n o n

o .soon

o.6oon

o . x m o.qnon

Nondimensional force, -

Fs

0 .3860E-0S

0 . 3 K 9 9 E-0 4 0.r ,960E-04 0 . 1 1 7 7 E - 0 3 0.18 9 0 f-0 3 0 .?91SE-03 0 . 4 3 8 7 6 - 0 3 O. f l97KE-03

0 . 1 5 8 2 ~ 4 4

0.3 2 2 n E-n 2

0 .9872E-0K 0.4 0 3 9 E-0s 0. ! l 4 5 1 E- 0 5 0.1.7 8 I E- 04 I I . ~ ~ . I X E - ~ ~ 0 ,4861E-04 0 . 7 7 4 2 ~ - n 4 0 . 1 2 7 6 5 - 0 3 0 . 2 4 0 5 E - 0 3 n . 1 2 i x E - n z

~ - -

Nondimensional tilting moment,

M -

- Non-

dimensional leakage,

Q -

1 . 0 1 4 1 . 0 5 6 1;127 1 . 2 2 6 1.353 1 . 5 0 8 1, K 1 1 1 .903 2 , 1 4 ? 2 .410

I, 0 1 4

1 . 7 3 0 i . 3 f i n

1 , 0 5 8 1 . 1 3 0

l is18 1,7116 1 .922 7 ,167 2,440

1 . 0 1 5 1.059 1 , 1 3 2 1,235 1 .36X 1 . 5 2 9

1,041. 2.191

1,720

2.4711

1 . 0 1 5 1 . 0 5 9 1 . 1 3 4 1 . 2 3 8 1 . 3 7 1 1 , 5 3 5 1 . 7 2 8 1 . 0 5 0 2 , zn- 2 . 4 8 5

Simplified nondi mensional

leakage,

1 + 3 c2R. 21

1,014 1 . 1 5 6 1;1.27 1 , 2 2 G 1 3 5 2 1:50R 1,691 1.'10? 2 , 1 4 2 2 .410

1 . 0 1 4 1 ,053 1,130 1 , 2 3 0 l.?GO 1:51n 1,70G 1 . 9 2 2 2, l G G 2 ,440

1.1115 1,059 1 , 1 3 2 1 , 7 3 5 1 . 3 6 7 1.527 1.720 1,141 2, 1 3 1 2 .470

1 ,115 1,059 1 . 1 3 4 1;?3R 1 . 1 7 1 1.535 1 .723 1 . 1 5 0 2 , 2 0 5 2 . 4 9 5

e -

f

s Figure 1. -Face seal with angular misalinement.

Ratio of inner to outer radius,

rilro

I 0.90

10-so ! Figure 2 - Nondimensional force as function of tilt param-

eter for various radius ratios.

19

Po 0

Ratio of i n n e r to outer radius,

ri/ro 0.90

l 0 - F L

10-3 - /

-

0 . 2 .4 .6 . 8 1.0 10-5

Ti l t parameter, E

- /

26

2.0

c 1.8

al- a- m Y m

m 0 v) c .-

.- V

1. 2 -

Ratio of i n n e r to outer radius,

rik0

0.99

Ti l t parameter, E

Figure 4. - Nondimensional leakage as func t i on of tilt parameter for radius ratios of 0.9 and 0.99.

Figure 3. - Nondimensional t i l t i ng moment as funct ion of tilt parameter for var ious radius ratios.

t

c

F igu re 5. - Face seal w i t h angular misal inement and coning.

Figure 6. - Hydrodynamic forces in face seal.

NASA-Langley, 1976 E- 8817 2 1

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nonaxisymmetric incompressible hydrostatic pressure

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