Squeeze effects in misaligned radial face seals with coning

Wear, 85 (1983)

143

143 - 149

SQUEEZE EFFECTS IN MISALIGNED RADIAL FACE SEALS WITH CONING PRAWAL SINHA and T. S. NAILWAL Department (India)

of Mathematics,

(Received September 4,1981;

Indian Institute

of Technology,

Kanpur,

Kanpur

208016

in revised form June 28,1982)

Summary Squeeze effects in misaligned radial face seals with coning significantly increase separating forces and tilting moments. The variation in the seal characteristics with coning parameter, tilt parameter and radius ratio is presented. Coning reduces the values of the separating forces and tilting moments but squeezing increases them significantly.

1. ~trodu~tion Considerable effort has been expended to predict and overcome seal problems, particularly premature failure and leakage. There is considerable scatter in the life of radial face seals. In an attempt to visualize the crux of the problem, Sharoni and Etsion [l] considered the effect of coning on seal characteristics. Etsion [2] analysed radial face seals by treating the film thickness as time dependent. The analysis of ref. 1 has indicated that high pressure seal stability may be improved by coning. In the present paper the squeeze effect in a misaligned high pressure radial face seal with coning is analysed. The film thickness was regarded as time dependent and expressions for forces and moments have been obtained. Etsion’s analysis [2] may be obtained as a special case of the present analysis. 2. Analysis

The Reynolds equation for misaligned radial face seals using an incompressible fluid is

where the film thickness h is (Fig. 1) h=c+yrcos@

+P(r--rl)

0043-1648/83/0000-OOOOt$O3.00

(2) 0 Eisevier Sequoia/Printed in The Netherlands

9=

Fig. 1. Misaligned

face seals.

where c is the seal clearance along the central line, y is the angle of tilt and (r, 0) is the coordinate of the point through which the thickness h is measured. Equation (1) may be solved separately for pure hydrodynamic, pure hydrostatic or pure squeeze effects but only the latter are considered here. With a narrow-seal approximation [ 31, eqn. (1) reduces to (3) where ah/at is obtained ah _= at

-u

+ +r,

cos

from eqn. (2):

19

where u is the velocity at the centre, i.e. u = &z/at, and f = &y/at. The symbol r, denotes the midradius, i.e. r, = (r, + r2)/2, where rl and r2 are the inner and outer radii respectively of the seal. By combining eqn. (3) and eqn. (4) we obtain

a

= 12~(-u

ar To integrate

p = 0 at r=rl aplar=

+ fr,

cos 0)

eqn. (5), the pressure boundary

conditions

are

and r=r2

0 at some

r=r’

where r’ is the local radius corresponding to the pressure extremum. Integration of eqn. (5) and use of the boundary conditions (6) gives ap = 12p(--u ar

+ $r,

cos 0) $

(6)

145 Integration

of eqn. (7) gives

p(r) = 12/4--u

+ fr,

cos O)J

r r-r’ h3

dr + c1

(8)

rl Equations

(8) and (6) together

give cl = 0 and

(9) By solving eqn. (9), substituting manipulations we obtain

in eqn. (8) and carrying

out some algebraic

6~(u - yr,,, cos O)(r - rl)(rz -r) p={

c + Y r cos 8 + fl(r - rl)}*{c

+ yr,

The axial force and tilting moment analytical integration of eqn. (10): F=

2r,

cos 8 + &r,

(10)

-r,)}

due to the squeeze effect

are obtained

n Tz p dr d6’ J.l0 r1

by

(11)

and 97

M = -2r,*

rz

ss 0 J.1

p

cos 8

de dr

(12)

Only the moment about the x axis is considered metry about the z axis.

3. Nondimensional

substitutions

are used:

= rm/r2 = (1 + R ,)/2

R = r/r,

R,

F+ = FL,=,

M, =Mqco (r2/c)3

By using this scheme,

E = yr,/c

R 1 = rl/r2

M+ =M,zo

fi+ = M+ /24pfr23(r2/c)3

F,j =F+co

E, = Fu/24/..wr2(r2/c)3 (13)

&, = M, /24pfr2* (r2/c)3 6 = pr,/c

the dimensionless

6p(r2*/c3)(u - fr, ‘=

of the pressure sym-

scheme

The following

?? = Fi /24p+r,*

because

pressure is obtained

cos @(R

{1+~Rcos~+6(R-R,)}*{1+~R,cos8+6(R,-R,)}

-R

i)(l

-R)

:

(14)

146

Equation I=

with respect to r and then factorized:

(14) is first integrated l+ecosO

1 (E cos 8)3

+6(1-R,)

(1 -R,)(E

-

case + 6)

l+eR,cos8+6(R,-RR,)

1+eR,cos8

I (15)

7r

F,,=R,

s

Id8

(16)

0

& = ii& = -&,2

fI COS8d8

(17)

0

Mq

=

R,3j?

de

cOs*e

(18)

0

Equations (16) - (18) are integrated numerically are presented in the form of graphs.

‘O-1.00

1

02

1

I

0.L Ed

06

I

08

‘O-61 00

Fig. 2. The translational damping coefficient various values of the coning parameter and rl/r2

and the values thus obtained

02I

as a function = 0.80.

Fig. 3. The translational damping coefficient as a function various values of the coning parameter and rI/r2 = 0.98.

OL I

061

0.6 I

0: the

tilt

parameter

for

of

tilt

parameter

for

the

147

4. Results and discussion The dimensionless damping coefficient F, is shown in Figs. 2 and 3. The cross-coupled damping coefficient M, is shown in Figs. 4 and 5 and the rotational damping coefficient I@.+in Figs. 6 and 7. Each of the characteristics is plotted against the tilt parameter e. 6 = 0 corresponds to the case when no coning is considered [ 21. All the graphs indicate that as 6 increases the values of the forces and moments decrease whereas they increase with the tilt parameter E. However, the seal characteristics decrease with increase in the value of the radius ratio. Each curve for 6 > 0 lies below the curve which corresponds to 6 = 0, indicating that seal coning results in forces and moments lower than those resulting with no coning and confirms earlier findings [ 11. The normalizing factors by which forces and moments are obtained from their corresponding non-dimensional forms in ref. 1 are similar to those

I ~0“0.0

I

0.2

I

I

0.L

0.6

1

06

‘0 -7

00

I 02

E_

I 04

I 06

, 06

f--,

Fig. 4. The cross-coupled damping coefficient as a function various values of the coning parameter and rl/rz = 0.80.

of the

tilt parameter

for

Fig. 5. The cross-coupled damping coefficient as a function various values of the coning parameter and t-,/r2= 0.98.

of the

tilt parameter

for

‘O-5

1

00

02

L

OL E-

I

0.6

I

08

,.,i,

02

0.4

06

08

E---

Fig. 6. The rotational damping coefficient as a function values of the coning parameter and rr/rz = 0.80.

of the tilt parameter

for various

Fig. 7. The rotational damping coefficient as a function values of the coning parameter and rl/rz = 0.98.

of the tilt parameter

for various

used in the present paper. These factors are 2_4pur,(r,/c)3 and for the axial force E, and the tilting moment M+ respectively. in a way similar to that adopted by Etsion [2] , squeeze shown to have immense significance because they are usually hydrodynamic effects.

24pujr,3(rz/c)3

By proceeding effects can be larger than the

References A. Sharoni and I. Etsion, Performance of end-face seals with diametral tilt and coning, ASLE Trans., 24 (1) (1979) 61 - 70. I. Etsion, Squeeze effects in radial face seals, J. Lubr. Technol., 102 (1980) 139 - 144. I. Etsion, The accuracy of narrow seal approximation in analysing radial face seals, ASLE Trans., 22 (2) (1978) 208 - 216.

149

Appendix A: nomenclature seal clearance along the centre-line axial force dimensionless translational damping coefficient film thickness tilting moment dimensionless cross-coupled damping coefficient dimensionless rotational damping coefficient pressure radial coordinate of extremum pressure inner radius of the seal outer radius of the seal angle of coning angle of tilt flr2/c, coning parameter -yr2/c, tilt parameter viscosity