Centrifugal inertia effects in misaligned radial face seals

Wear, 129 (1989)

319

- 332

319

CENTRIFUGAL INERTIA EFFECTS IN MISALIGNED RADIAL FACE SEALS R. S. GUPTA

Department (Received

and L. G. SHARMA*

of Mathematics, Manipur University, Imphal 795003 (India)

July

18,1988;

accepted

October

6,1988)

Summary This investigation presents the analysis of customarily neglected centrifugal inertia effects in fluid-film-lubricated misaligned radial face seals. The forces and moments that act upon the primary seal ring due to axial translation and angular rotation are obtained for the complete range of angular misalignment. Lubricant inertia is found to be helpful in attaining the dynamic stability in the performance of misaligned face seals. It is also helpful in minimizing the wobbling of the primary seal ring and the leakage across the seal.

1. Introduction Mechanical face seals are employed to control fluid leakage in a wide range of devices: pumps, gearboxes, bearings, turbines, etc. These consist of a rotating surface and a stationary surface operating at relatively close clearance. The clearance for such seals is generally kept as small as possible to minimize the leakage rate and thereby to maximize the efficiency of the components. The earlier works have explained the mechanism of generation of pressure and load in mechanical face seals. Denny [ 11, Batch and Iny [ 21 and Nau [3, 41 have shown that hydrodynamic pressure, larger than hydrostatic, is generated in the fluid film because of the effects of waviness, misalignment and vibration of the faces. Following Denny’s findings, many investigators have tackled this problem and there have been several hypotheses put forth to explain the mechanisms responsible for the development of the lubricating film pressure which acts to separate the primary seal faces. So far, not much attention has been paid to the study of seal dynamics, which is thought to be of major importance. In some work [ 5,6] the dynamic response of an angular misaligned seal is considered to be due to a restoring moment that coincides with the angular

India, India.

*On study leave from Department under the Faculty Improvement

0043-1648/89/$3.50

of Mathematics, D.M. College of Science, Imphal, Programme of the University Grants Commission,

@ Elsevier

Sequoia/Printed

in The Netherlands

320

misalignment vector. A transverse moment, that leads the angular misalignment vector by 90” and is generated by hydrodyn~ic effects, has also been pointed out [ ‘71. This transverse moment may be the origin of dynamic instability and has to be considered in any dynamic analysis of a realistic seal model. Forces and moments acting on a flexibly mounted seal ring were solved analytically by Etsion and coworkers [8 - lo] for hydrodynamic, coning and squeeze effects in the sealing gap of a radial face seal. In 1981, Etsion and Dan [ll] used the results obtained from their earlier studies [8 - 101 to derive the equations of motion of the seal ring in three degrees of freedom namely, one axial and two rotational. The equations of motion include fluid film as well as flexible support forces and moments. Etsion [12] extended his previous work [ll] to the more general case where the rotating seal has a certain amount of axial runout and investigated the dynamic response of the flexibly mounted ring under such conditions. Later he has also described a new concept of zero leakage in a non-contacting mechanical face seal by optimization of the seal ring geometry [ 131. Recently, Lebeck [14] developed a two-Dimensions theory of hydrodynamic lubrication in contacting face seals by considering circumferential waviness. However, the problems of hydrodynamic and hydrostatic lubrication of seals, where seal ring inertia and other restraints are accounted, have hardly been touched analytically till now. In the present paper the centrifugal inertia effects in fluid-film-lubricated misaligned radial face seals are analysed. The forces and moments that act upon the primary seal ring because of its axial translation and angular rotation are obtained for the complete range of angular misali~ment. The non-linear partial differential equations arising in the analysis are solved with the help of the “energy integral approach” [ 15 - 171 (an approximation technique). 2. Analysis The momentum equations for laminar flow in the narrow sealing gap of a radial face seal (Fig. 1) are -_

P2 r

=-_

3P

a3.4

ar+‘12

a2

(1)

(3) where the variation in pressure in the z direction is negligible and the equation of continuity is 1 a(ur) -~+-_+_= r ar

i

a~

r&3

aw a2

0

(4)

321

iz

Fig. 1. Face seal with angular misalignment.

The film thickness h for a misaligned seal (Fig. 1) is given by h = C + yr cos 8

(5)

where C and y are the seal clearance along the centre line and the angle of tilt respectively. In contrast to the film thickness, its circumferential gradient is almost unaffected by the radius and hence may be approximated by [8] ah = -yr,

ae

sin 8

where r’, is the mean radius. The radia1 fluid velocity distribution without centrifugal effects satisfying eqn. (4) under the following boundary conditions: u=0 is

at

given by

z =0

and

h

where r’ is a constant of integration corresponding to the radius where the pressure has an extremum. The tangential fluid velocity u can be found by inte~ating eqn. (2) twice, subject to the boundary conditions that u = 0 at z = h and u = wr at z = 0, we obtain

where w is the rotational angular velocity of the seal seat. Multiplying eqn. (1) by u and integrating the resulting expression from z==Otoz=h,wehave

h a%

hudz=p

aP

s

;lr*.I-

us

quv2d.z

dz+

(8)

I”0

0

From eqns. (6) - (8), we get

aP

z

= (3/10)po2r -

i

Gpwyr, sin % h3 (r-r’)

(9)

Integration of eqn. (9) subject to the boundary conditions P=Pi

at

r=ri

and

p=po

at

r=r,

(10)

yields (11)

P=Pi+Pz where PI =Po +

oli

-PO)

hohilho + h) ho + hi

I

and ‘(ho2 - h2)(ro2 - ri2) p2 = -(3/20)pw2

I

(ro2 - r2) -

ho2 - hi2

IIere, p1 presents the cont~butions of hydrostatic and hydrodyn~i~ pressure (7,8] and p2 is the contribution due to the centrifugal inertia of the lubricant.

323

From eqn. (5) the dimensionless film thickness is given by h - =l+eRcost3 c

H=

(12)

where R=

i ro

and the tilt parameter,

Substituting eqn. (12) in eqn. (11) we get (H,,*/H* - 1) H 2 _H.2 1 0

P =Po + (Pi -Po)Hi* -

HoHi(l/H*

-

l/H,*)

Ho +Hi

-

-

(3/20)pw*r,*

(1 -R*)

(1 - Ri*)Hi*(Ho*/H - 1)

(13)

H,* -Hi*

3. Axial force For high pressure misaligned seals, the axial force which tends to separate the seal surfaces is obtained from ro F=

$.I-

‘i

277

Pr dr d6’

(14)

0

Neglecting the curvature effects, eqn. (14) reduces to 2n

F = r,,,r,,

1

p dR d&’

ss 0

(15)

Ri

Substituting the value of p from eqn. (13) in eqn. (15) and simplifying, the axial force may be reduced in the following form: n F = 2r,,.,r,

$ {(Pi-Po)l+P,-PpiRi}de-(3/10)po2r,r,3 0

(1

-Ri)

-

(’ iRi3)~(1

-Ri*)(I-RR,)

(16)

324

where

I= [{woHi/(Ho +Hi)I-ll E cos 8 Taking H,+Hi=Wm and

+ Ri

1

= 2R,

we have R,

I=

+ ERi

COS 8

1 + ER, cos 8 R,

=

’ ERi

COS 8

(17)

1 + E cos 8 where

Utilizing the results [ 181 ?T

1

J o 1 +E case

d8=

(1 -

=

31/2

and 7T

s

o

1

cos 8 (1 _

1+ c case

F2)1/2

(18)

-l

we get n

Rm- (RilRm)

s

Ide=n

(1 - E*R,~)~‘~

0

From

Ri

(19)

+ R, I

eqns. (16) - (19), we get

F = B?rr,r,

p. -PiRi [

-(3/10)~p~*r,ro3

- (’ - Ri)

Rrn - (RilRrn) (1 _ e2R,2,1,2

+ @i -PO)

Ri + R,

i

[

(1 -Ri)

-

l--i3 3

11

+ (1 -Ri*)Ri

(20)

325

Equation

(20) can be simplified

F = m,(r,

-ri)(pi

-p,)

further

+

+ (3/20)7rpo2r,ro3(1

ro2@i

as follows: -p,)(l

1

--Ri)2

(1 -

E2R,2)l/2

1

--Ri)3

(1

_e2j3,2)1/2

-

l

i

u3

1

-

l

t

(21)

In eqn. (21), the first term, on the right-hand side, is the axial force for the case of parallel flat surfaces, while the second and the third terms present the contributions of the non-axisymmetric hydrostatic pressure and the centrifugal inertia effects respectively. Denoting the last two terms contribution by Fsi, the dimensionless force F,i is given by

+ MR,(

1

I

.Fsi= (1 -Ri)?

(1

-

e2J3,2)1/2

-

l

i

1

1 - Ri)3 [I

(I_

E2R,2)1/2

-

l

t

1/3

1

(22)

where

Fsi

F,i =

and

M=

( 3/20)pw2r02 Pi -PO

kr02(Pi -PcJ) 4. Restoring

moment

The restoring 277

M,=-

Neglecting

cos e dr de

pr2 ri

the curvature 2s

M, = -rorm2

about the x axis (Fig. 1) is given by

ro

ss

0

moment

the moment

becomes

1

p cos e m

ss

0

effect,

(23)

de

4

The sign convention in eqn. (23) ensures that a restoring moment will have a positive value. Integrating eqn. (23) with the same procedure as in the case of axial force we get M, = -2r02rm2(pi

-p,)

cos 8 de

11

0 - (3/10)pu2r03rm2(1

-

Ri2 ,j, 0

~0~8 de

(24)

326

Substituting Cl31

the value of I from eqn. (17) into eqn. (24) and using the result

cos*e

77

s

1-ecoSe

o

?I

d8 = 2

1

(1 -

we have the dimensionless = (1 -_Ri)2{l/(l

~

1 E2)1/2

restoring

-e2R,2)1’2-

x

- 1

(25)

t moment

as follows:

l}

2E +

MR,(l

-Ri)3{lj(l

- E2R*2)“2 - ‘1

(26)

E where fi, =

Mx

-PO)

71r03fPi

moment

5. Transverse

The transverse

moment

about the y axis (Fig. 1) is given by

277 ro pr*

w=JJ 0

drdtl

sin0

‘i

which after neglecting 2n

M, = r,r,,.,2

X

effect becomes

1

p sin 8 dR d8

fJ

0

Substituting

the curvature

(27)

Ri

the value of

p

from eqn. (13), we get

e sin*8

(1 + ER, cos r?)(l + ER cos8)2

dB do

(28)

which is the same as given by Etsion [8]. From eqn. (28) the non-dimensional transverse moment can be written as

w, =

Sin-‘( ERi) - sin-‘(e) E2 +

Ri(f

-

(1 -Ri)(l

E~R$)“~ - (1 - E*)~‘~ E

J&II3

-E*R,*)"~

(29)

321

where

MY

MY = 67rpo

0

2

s

ro3

6. Leakage The leakage across the seal is given by (30) Neglecting the curvature effect and R, eqn. (30) becomes

and using the non-dimensional

parameters H

(31) Utilizing the results [ 181 n J o

c0s2e 1 +ECOS~

and 77

c0s3e

JI

(32)

+ 2 ~0~8

o

t

in eqn. (31), the non-dimensional B’Qo+M[2R,{Q.-

(I+

leakage Q is given by

%)I]

where 12pr,

a= Nr, +

ri)C3(Pi

Q -PO)

and 2

Q, = 1 + (3/2)E2Ri + f

Ri

+ ((1 -

E~R,~)~‘~ -

1)

‘7. Results and conclusions This investigation presents the analysis of customarily neglected centrifugal inertia effects on the performance of misaligned radial face seals. The

328

non-dimensional forms of the axial force Fsi, the restoring moment M,, the transverse moment M, and the fluid leakage & for high pressure seals are calculated from eqn. (22), eqn. (26), eqn. (29) and eqn. (33) respectively. The analytical expressions of these characteristics contain two components; the first group of terms in each expression gives the value when the effect of lubricant inertia is not considered, while the second group represents the modification caused by rotational inertia. However, the transverse moment M,, is found to be independent of rotational inertia. The numerical results for the classical case [ 7, 81 can be obtained from the expressions as M + 0, i.e. when the lubricant inertia is neglected. Figures 2 - 4 are plotted to show the effect of the speed and tilt parameters on the non-dimensional axial force FSi, the restoring and the transverse moments of the misaligned radial face seals respectively. The radius ratio has a significant influence on these characteristics. As the radius ratio decreases, the force and the moments increase rapidly. For a particular radius ratio, the

-2 IO

I

/

‘*

,

R,

-

=0

90

= 0

96

/ ‘/ :

0

I.

04

02 Tdt

Fig. 2. Non-dimensional and rotation parameters.

---

f/

06 parometar,

force

08

IO

E F,i

as a function

of tilt parameter

for various

radius ratios

I

I

E

0.6

IL,,

transverse

parameter,

I 0.4

___

---Ri

Fig. 4. Non-dimensional

Tilt

I

I

restoring

0.2

I

‘5

’

/, j,-

Fig. 3. Non-dimensional

///

,%

//,

= 0.90

moment

moment

0.8

= 0.99

= 0.96

as a function

as a function

1.0

of tilt parameter

of tilt parameter

0

8

I

I

1

for various

I

1

I IO

and rotation

I 0.6

ratios.

ratios

,6

I 06

radius

radius

parameter

0.4

for various

Tilt

0.2

parameters.

330

axial force decreases with the increasing inertia parameter, while the restoring moment increases with the increasing inertia parameter. These characteristics are also increasing when the misalignment is greater. Figure 5 shows the variation in the ratio of the transverse to the restoring moment My/M, due to the tilt parameter for different values of the inertia parameter. It is observed that My/M, varies greatly with the tilt parameter E. At E = 0.1, MY is 2% - 3% of iM, whereas when E = 1.0, MY becomes approximately 30% of aX. This indicates a strong coupling between the transverse moment and the angular misalignment for higher misalignment. Such coupling is a possible source of dynamic instability and can result in a wobbling of the primary seal ring. These results are in good agreement with the findings of Etsion [ 7, 81. By considering the lubricant inertia, the ratio My/Mx tends to a constant value, Fig. 5, which shows a reduction in the coupling between the transverse moment and the angular misalignment. It can be seen from Table 1 that the fluid inertia also helps the reduction in leakage across the seal.

0

04

0.2 Tilt

06

0.8

IO

porometer, E

Fig. 5. Ratio of transverse to restoring radius ratios and rotation parameters.

moment

as a function

of tilt parameter

for various

Hence it can be concluded that the lubricant inertia is very helpful in attaining dynamic stability in the performance of misaligned radial face seals. It also helps in minimizing wobbling of the primary seal ring and leakage across the seal.

331 TABLE

1

Variation of dimensionless parameters M

E

leakage

with radius ratio for various values of tilt and rotation

Ri = pi/r0 0.90

0.93

0.96

0.99

0.5 1.0 1.5

1.013512 1.013489 1.013465 1.013441

1.013956 1.013944 1.013932 1.013920

1.014402 1.014398 1.014394 1.014390

1.014850 1.014850 1.014850 1.014845

0.4

0.0 0.5 1.0 1.5

1.216200 1.215820 1.215440 1.215060

1.223298 1.223109 1.222920 1.222273

1.230432 1.230369 1.230306 1.230244

1.237602 1.237598 1.237594 1.237590

0.7

0.0 0.5 1.0 1.5

1.662113 1.660950 1.659788 1.658625

1.683850 1.683271 1.682692 1.682113

1.705698 1.705506 1.705314 1.705122

1.727656 1.727644 1.727631 1.727619

0.0 0.5 1.0 1.5

2.351263 2.348901 2.346538 2.344176

2.395616 2.394438 2.393260 2.392081

2.440200 2.489809 2.439417 2.439026

2.485012 2.484987 2.484962 2.484938

0.1

0.0

.O

References 1 D. F. Denny, Some measurement of fluid pressures between plane parallel thrust surfaces with special reference to the balancing of radial-face seals, Wear, 4 (1961) 64 - 83. 2 B. A. Batch and E. H. Iny, Pressure generation in radial face seals, 2nd Int. Conf. on Fluid Sealing, BHRA, 1964, Paper F4. 3 B. S. Nau, An investigation into the nature of the interfacial film, the pressure generation mechanism and centripetal pumping in mechanical seals, BHRA Rep. RR 754, 1963. 4 B. S. Nau, Film cavitation observations in face-seals, 4th Conf. on Fluid Sealing, BHRA, 1969, Paper 20. 5 D. S. Kupperman, Dynamic tracking on non-contacting face seals, ASLE Trans., 18

(4) (1975)

306 - 311.

6 L. P. Ludwing and G. P. Allen, Face seal lubrication II, Theory of response to angular misalignment, NASA TND 8102, 1976. 7 I. Etsion, Non-axisymmetric incompressible hydrostatic pressure effects in radial face seals, J. Lubr. Technol., 100 (3) (1978) 379 - 385. 8 I. Etsion, Hydrodynamic effects in a misaligned radial face seal, J. Lubr. Technol., 101 (3) (1979) 283 - 292. 9 I. Etsion and A. Sharoni, The effect of coning on radial forces in misaligned radial face seals, J. Lubr. Technol., 102 (2) (1980) 139 - 144. 10 I. Etsion, Squeeze effects in radial face seals, J. Lubr. Technol., 102 (2) (1980) 145 152.

332 11

I. Etsion

and Y. Dan, An analysis of mechanical face seal vibrations, J. Lubr. Tech428 - 435. 12 I. Etsion, Dynamic response to rotating-seat runout in non-contacting face seals, J. Lubr. Technol., 103 (4) (1981) 587 - 592. 13 I. Etsion, A new concept of zero-leakage nonconducting mechanical face seal, J. Tribal., 106 (3) (1984) 338 343. 14 A. 0. Lebeck, Hydrodynamic lubrication in wavy contacting face seals: a two dimensional model, J. Lubr. Technol., IO3 (4) (1981) 578 586. 15 A. F. Elkouh, Inertia effects in laminar radial flow between parallel plates, Int. J. Mech. Sci., 9 (1967) 253. 16 V. K. Kapur and K. Verma, Energy integral approach for hydrostatic thrust bearing, Jpn. J. Appl. Phys., 112 (7) (1973) 1070. 17 V. K. Kapur and K. Verma, Energy integral approach for MHD hydrostatic thrust bearing, J. Lubr. Technol., 97 (4) (1975) 647. 18 J. F. Booker, A table of the journal bearing integral, J. Basic Eng., 87 (1965) 533 535.

nol., 103 (3) (1981)

Appendix

c F Fsi Fsi

h H M MX Kc M _y M, P

Q a

R r

r’ 4 u, w x, Y, 2

A: Nomenclature seal clearance along centre line axial force axial force contributed to by non-axisymmetric hydrostatic sure and centrifugal inertia non-dimensional force, F,i/{Tr,*@i -p,)} film thickness non-dimensional film thickness, h/C rotation parameter, (3/20)pU2r0*/@i -p,) restoring moment non-dimensional restoring moment, M,/{ Tro3@i -p,)} transverse moment non-dimensional transverse moment, M, = MY/{6~~~(ro/C)*r,*} pressure leakage nondimensional leakage, 12pr,Q/{n(r, + ri)C3@i -p,)} nondimensional radius, r/r, radial coordinate radial coordinate of extreme pressure radial, tangential and axial fluid velocities orthogonal axes, see Fig. 1 angle of tilt tilt parameter, r-,/C angular coordinate viscosity density of the lubricant rotational angular velocity

Subscripts i at inner radius at mid-radius m 0 at outer radius

pres-