A theory of sealing with radial face seals

Wear -

Elsevier Seqmia

A THEORY

S.A., Lausmne - Printed in &heNetherlands

51

OF SEALING WITH RADIAL FACE SEALS

SUMMARY

It is now well established that, for reliable operation over a long period, a radial-face mechanical seal must operate with its faces separated by a film of fluid. A comprehensive theory, based on the principles of hydrodynamic lubrication, of sealing under these conditions is presented. A number of observations pertinent to the design of mechanical seals arising from the theory are given, and it is concluded that the hydrodynamic theory can be applied to the desi&.nof mechanical seals for considerably more severe duties than those for which they are used at present. 1. INTRODUCTION

Batch and InyrV2 have shown that radial-face seals are lubricated by a hydrodynamic pressure generated at the interface by surface waviness, and a general theory based on classical hydrodynamic principles was developed to explain the behaviour of such seaW. Xnthe case of the typical carbon-face mechanical seal it was shown that the surface wave must comprise two equally spaced humps of nearly equal amplitude which are always in phase with the relative face misalignment. This paper extends the hydrodynamic analysis to a specific treatment in depth of the two-humped seal, the model configuration for the commercial seal> and gives guidelines for the design and operation of seals with predictable working characteristics. 2. ANALYSIS

2.1. The fluctuation in film thickness An essential feature of successful operation of mechanical seals is the development of a significant two-humped profile> which is always in phase with the relative face misafignment, on one of the co-acting seal faces. Thus if the co-ordinate system is fixed relative to the wavy face and the other face is assumed plane the mechanics of film thickness fluctuation may be represented schematically as in Fig. 1. If the relative tilt is described by a function of sin # and the face profile by a function of sin2 4, then the film thickness fluctuation beneath the tilted wavy face may be expressed as :

h= ho-l-h, sin$+h,

sin2(4+$r)

(1)

where h, is the amplitude of the relative face run-out (i.e, the difference between the run-out of the rigidly mounted face and the induced oscillation of the flexible eounterWW, 18 (1971) 51-69

52

Ii. II.

Fig. 1. The mechanics

of film thickness

IN1

fluctuation

face as it tries to follow the movement) and hz is the amplitude of the surface wave; the latter is described by a function of sin’& rather than a function of sin 24 for ease of application to seal design (section 3.1). 2.2. The interfacial film pressure Batch and Iny’ derived a general expression for the hydrodynamic sure generated at the seal interface: P=

1

(Yp, + $.$

[r2 ln(R,/R,)-Ri

ln(r/R,)-Rf

film pres-

ln(R,/‘)lj

ln(RJRJ where y = In (R2/r) for an inside seal, y = In (r/RI) for an outside seal and showed that it gives a close approximation to the measured film pressure if all predicted sub-ambient pressures are ignored. This is because the interfacial fluid was seen to cavitate in the diverging regions of the annulus thus curtailing the negative pressures at just below the ambient without materially affecting the positive pressure distribution (cf: the half-Sommerfeld boundary condition in journal bearings). Substituting for dh/dr$ from equation (1) gives : 1

’ = ln(R,/R,)

yP -I- v ’

[R: In(r/R,)fRf

ln(R,/r)-r2

ln(R,/R,)]}

0 (4

where f(4)=

-(1+

cos 4(e, - 2c2 sin 4) .a1 sin #+.s2 cos2 4)”

Inspection of eqn. (2) shows that the radial pressure distribution Wear, 18 (1971) 51-69

comprises

THEORY OF SEALIXG WITH RAESAL FACE SEALS

53

two components, an almost linear hydrostatic component due to the steady decrease of the supply pressure across the face and a hydrodynamically generated somewhat asymmetric, parabohc component which provides the film stifness; the latter varies cyclicly round the seal face under the inthren~ of the te~~~~~ being positive when f(rfr) is negative and conversely. 23. Applied load The force that must be applied to the seal faces to form an effective sea1 is equal to the opening force due to the total of the interfacial film pressures, or x

W=

Prd$dr li within the appropriate circumferential and radial limits. As shown by Batch and InyZ the sealing load may be computed by integrating the pressure wave over its positive portions only: for all practical purposes the condition P 30 may be replaced byf@) XI (see Appendix) making the load requirement of the seal

where fi (R) = R: -R: - 2R: In (R,/R J fi (R) = Ri -R$ + 2R: In (R,/R,)

for an inside seal for an outside seal

and fi(R)=(R;-R~)ln(R,/R,)-(R$-R~)2 are bin-djmensionali~ed

and plotted against aspect ratio in Fig 2 Solving the above

Fig. 2. Variation of geometry parameters with aspect ratio. Wear, 18 (1971) 51-69

equation with the appropriate yields : w=

values of h andf‘(4) from eqns. (1) and (2) respectively

TR%f,(R) +

3p+f2 2h2

(R)

(3)

where the dynamic load coefficient

___2&l x = (1 --Ey

for

c1 >2c,.

Note that when s1 = 1.0 the mechanism of hydrodynamic pressure generation breaks down, and the load is carried by surface asperities in solid contact. Also when c1 > 2~~ there is little contribution to the load-carrying capacity from the profile.

E,=06

02

04

O-6

O-8

DIMENSIONLESS

I.0

I.4

I -2

AMPLITUDE,

E:!

Fig. 3. Effect of gap fluctuation on dimensionless load.

Hence the profile induced stable working range, as defined by Batch and Iny2, must satisfy the conditions .sl ~0 and s1 ~2s~. The variation of X with e1 and s2 over this range is shown in Fig. 3. 2.4. Radial flow Radial flow through the annulus is given by2: +;R(R)j:$d$j

fbP)ao where B(R)= R:-R:-2R,R, Weur, 18 (1971) 51-69

ln(R,/R,)

THEORY OF SEALING WITH RADIAL FACE SEALS

55

and I( = + 1 for an inside seal, = - 1 for an outside seal. The sign convention is such that positive flow is radially outward. Solving the above equation for the flow across the seal face gives : 7EP,k3W Q = 192 p ln(R,/R,)

(KZ

- WA)

(41

where G =&z (16 + 24~: + 6~: .s2+ 24~ + 18~2,+ 5$) is a non-dimensional flow coeffrcient showing the effect of gap ~uctuation on the flow induced by applied pressure, &:+4&: M=--.---2% El6 2&Z is a non-dimensional flow coefficient characterizing pressure fluctuations, r = [(RJR,)‘-

radial flow induced by dynamic

1 -2R2/R,~ln(R,/R,)]

is a non-dimensional positive sealing term depending on the geometry of the annulus, and A = pNR:/P,ht is a duty parameter akin to the bearing number in lubrication terminology. The variations of I’ with aspect ratio and of (G/~) with dimensionless amplitudes s1 and e2 are shown in Figs. 2 and 4 respectively.

201 0

o-2

04 0.6 0.8 I-O DIMENStONLESS AMPLITUDE,

I.2 f: 2

1.4

Fig. 4. Effect of gap fluctuation on sealing parameter. Wear, 18 (1971) 51-69

56

I:. I-1. IN\

Equation (4) shows that the radial flow through the annulus is made up of two components, one due to the applied pressure gradient and the other due to the hydrodynamic action of the seal and the particular geometry of the annulus. In an outside seal the two components are additive, giving rise to an ever present leakage under hydrodynamic lubricating conditions. In ‘in inside seal, however, the two components are in opposition, and absolute sealing occurs when they are exactly balanced, i.e. when

G/M = 0.8 r/i .

(51

When G/M > 0.8 rn the flow will be positive constituting a leakage ; when G/M < 0.8 rn there will be a net inflow at the inner boundary, and the seal will act as a pump transferring fluid from the outer boundary to the inner reservoir. A sealing condition is achieved by applying a sufficiently high closing force to induce a hydrodynamic pressure of magnitude such that the inward flow induced by this pressure just balances the outward flow due to the supply pressure. This unique relationship between applied load and sealed pressure is obtained, in terms of the dimensionless design parameters, G, M and X, by eliminating h, between the sealing equation (5) and the load equation (3) and rearranging, thus 16TR: ln(R,/R,)

GX M

-=

I

L(R)

where GXfM is a function of sI and s2 only as shown in Fig. 5. Hence for a given size

01 0

0.2

04

06

DLMENSIONLESS

08

I.0 AMPLiTUDE,

I.2

I .4 E2

Fig. 5. Effect of gap fluctuation on sealing coefficient.

Wear, 18 (1971) 51-69

57

THEORY OF SEALING WITH RADIAL FACE SEALS

of seal the load required to achieve total sealing is directly proportional pressure.

to the supply

2.5. Friction The fundamental analysis of Batch and Iny on which the present treatment is based assumed the circumferential flow induced by the film pressure to be negligible compared to that due to the motion of the surface (c$ the narrow bearing approximation of Dubois and Ocvirkj) thus allowing the assumption of a linear velocity gradient across the fluid film. The friction force is then given by:

This expression may be readily evaluated over the positive pressure regions, but the mixture of liquid and gas in the cavitated regions renders the effective vicosity of the fluid indeterminable. It was found experimentally, however, that calculations based on a homogeneous fluid having the single phase (liquid) viscosity overestimate frictional drag by no more than 15 ‘A,which is sufficiently accurate for design purposes. Indeed it is sensible to afford the designer a small margin of safety, because overheating can cause serious damage to the seal faces and the usual formulae used for heat transfer calculations are only accurate to about ,20x (ref. 4). Thus, assuming constant viscosity : F =

F(j+R;)

(7)

0

where

working range (i.e. .sl < 1.0 and c1 < 2~~)

Within the stable hydrodynamic

where

B=i-J

Ef+4E&+EJ 2E

.

2

Y can be regarded as a friction coefficient which modifies the Petroff value of friction between plane parallel faces to allow for the effects of the film thickness fluctuations. For a practical seal Y rarely exceeds unity (Fig. 6). The heat generated in the fluid film is then given by : H=

~(R,+R,)(R:-R:) 0

(8)

2.6. Fluid Film Sti@kss

For stable operation

the fluid film should possess adequate stiffness, i.e. a Wear, 18 (1971) 51-69

E,‘06 E,‘OS

E,=

0.4

Gzo.3 E’:“,.: E:.O

0

02

0.4

06

DIMENSWSNLESS

Fig. 6. Effect of gap fluctuation

0.8

I .o

l-2

AMPLITUDE,

14

C .J

on friction.

significant increase in load should be needed to cause a reduction in the mean film thickness. The curves of dimensionless load in Fig. 3 suggest that the film stiffness varies considerably. The nature of this variation cannot be deduced directly from the graph but may be obtained as follows: Fluid film stiffness = - g

= - g. m

2 0

-3p~[(Ri-R;)

In

ln(Ri-Rf)2]

d

g ln(R,IR,) Thus the variation of film stiffness with surface profile in the stable range (i.e. cl 628,) is given by:

~,LKOS(R’: - Rt) ln(R,/R,) 4h;

- (R: __. - Rf)2

ln(R21RI)

1

where

(9) whilst the variation with ho at constant surface profile is given by : (R~-Rf)ln(R2/R,)-(R~-Rf)2 ln(R,IRJ

l-

The variation of stiffness with gap fluctuation for the two cases is shown in Wear, 18 (1971) 51-69

THEORY OF SEALING WITH RADIAL

59

FACE SEALS

Figs. 7 and 8, respectively; the distinction between the two is impo~ant when considering the design of the seal. Whereas stiffness factor S, Fig. 7, describes the effect of changes in surface profile and flexibility of mounting on the film stiffness for a given minimmn face separation and is useful for optimizing the design of a single duty seal, stiffness factor E$, Fig. 8, outlines the variation in film stiffness of a given seal as the film thickness is varied, for example by changes in speed and pressure. 3.

DISCUSSION

The best seal for a given duty will be the one with the least leakage for a given film thickness or conversely the greatest film thickness for a given leakage. Equation (4) shows that for both inside and outside seal configurations this requires the sealing parameter G/M to be a minimum. Also in the interest of reliability G/M should not vary si~~~cantly over a range of film thicknesses or surface profIles. Figure 4 shows that this condition is best met when .eI is small, say less than 0.3, and gz between, say, 0.6 and 1.0. Thus one of the faces must be flexibly mounted and the surface profile should be between 600% and 100% of the minimum film thickness. This impligs imposing an appropriate profile during manufacture rather than allowing it to be produced by the lapping process or to develop by wear during the initial running-in period as is present practice. Note that when cl is very small the controlling design parameter 8x represents the ratio of surface profile height to minimum face separation, so that holding either

DiMENSIONLESS

AMPLITUDE.

f2

Fig. i’. Variationof ~~~od~na~ic filmstiffnesswithamplitudeof Facewavinessat constantgap.

Wear,18 (1971)51-69

60

Ii. H. 1x1

0

0

0.2

0.4

06

08

DIMENSIONLESS

Fig. 8. Variation

of hydrodynamic

IO AMPLITUDE.

.I 2

I-4 C2

film stiffness with dynamic

gap fluctuation

at constant

profile.

constant allows the designer to consider the effect of a change in the other on the operating characteristics of the seal. If a function of sin 24 had been used to describe the profile-induced film thickness fluctuation then dimensionless amplitude .s2 would have represented the ratio of the profile amplitude (i.e. half the height) to the mean face separation. Since the latter depends on both minimum face separation and amplitude of surface undulations, a consideration of the role of one in isolation of the other would have been rather involved. 3.2. Conditions,for sealing Because of the curvature of the annulus the hydrodynamic component of the film pressure becomes asymmetric in the radial direction thus causing more fluid to flow inwards than outwards (eqn. 2). At the same time the flow of fluid into the annulus over the cavitated region is less at the inner periphery than at the outer’. There is thus a net transfer of fluid from outside to inside (eqn. 4). This is the inward pumping effect observed by Denny’ and others and long regarded as an interesting phenomenon associated with radial-face seals; in truth it is an integral and very relevant part of seal behaviour, the very essence of the sealing mechanism. A mechanical seal having the sealed fluid on the inside can be made to seal perfectly whilst running with full hydrodynamic lubrication by utilizing the inward pumping effect to balance the sealed fluid pressure (eqn. (5)). However, since the factors influencing the behaviour of the seal cannot be closely controlled, this condiWeur,18 (1971)

5149

THEORY OF SEALING WITH RADIAL FACE SEALS

61

tion is impossible to achieve in practice; the seal will either leak slightly or pump the fluid from its faces. Since in all but the lightest duties the lack of adequate lubrication will result in excessive wear and seal failure, it is preferable in most cases to design the seal to leak slightly by selecting 0.8 FA -CG/h4 ; the amount of leakage can be calculated from eqn. (4). Although it does not seal perfectly this arrangement offers other considerable advantages: for a given leakage the seal operates at a larger clearance, generating less frictional heat and requiring less stringent limits on face ali~ment, flexibility and surface finish. In certain low-pressure, high-speed applications the inward pumping action might be too great to be overcome by the sealed fluid pressure, and a more satisfactory design would result from arranging the sealed fluid on the outside of the seal. Such a seal will always leak but will be reliable, and by selecting a suitable clearance the leakage can be kept small. Seals for hazardous or hot fluids are usually double-seal arrangements with a buffer ‘or cooling liquid between the product and atmosphere. By arranging the seal pair such that the intermediate fluid is on the outside of the inner seal, the inward pumping action may be harnessed to ensure that any flow is into the sealed fluid without the need for an intermediate pressure greater than that of the sealed fluid. In fact the buffer or coolant pressure can be quite low, easing the duty of the outer seal which will have its sealed fluid on the outside and will therefore leak slightly. Since this will be cool, innocuous fluid a slight leakage can be tolerated, 3.3. Seal face loading

The “sealing” conditions discussed in the preceding section are realized by applying a sufftciently high closing force to the seal faces to cause the generation of a hydrodynamic pressure of a magnitude such that the centripetal flow induced by this pressure is in specific imbalance with the outward flow due to the sealed pressure gradient. Reference to eqn. (3) shows that part of the required face loading is proportional to the sealed fluid pressure and part is proportional to the speed of rotation. By suitable design of the seal components, the proportion of the load derived from the sealed fluid pressure can be varied and the seal characteristics can be tailored to suit the duty. For example, if the load derived from the sealed fluid pressure approaches the minimum required for the faces to remain closed at standstill, nefi (R)/2 In (RJR,), the seal will operate with a small variation in film thickness over a wide pressure range at the expense of a considerable increase in leakage with pressure. Conversely, as the proportion of the load derived from the sealed fluid pressure is increased, the rise in leakage with pressure can be reduced at the expense of a reduction in film thickness. Equation (3) also shows that if the applied closing force is constant the film thickness wili vary with the speed, and when starting and stopping the faces will rub. Because of the small amplitude of the face profile the allowable wear is very small, except perhaps for a carbon face which can wear constructively during running to repair any modi~cation to the profile by low-speed rubbing fcJ: Batch and Iny’). Careful selection of face materials is therefore needed for high duty seals or some means devised for relieving the face loading as the speed falls. 3.4. Function Viscous drag in the fluid between the faces generates heat which, because of the Wear, I8 (1971) 51-49

very small interchange of fluid between the clearance and the bulk of the sealed fluid. must be removed by conduction through the seal components. To minimize thermal distortion of the components, and to prevent vaporization of the fluid at the interface, the seal should be adequately cooled, and this would be facilitated if the frictional heat generated was as small as possible. Reference to Fig. 6 shows that for a given value of h,, representing a given duty, the least heat is generated by a seal witha very flexible face (ci very small) and a relatively large surface profile (Q maximum). It is fortunate that the optimum working range from considerations of sealing is also compatible with the lower values of friction. It should be noted that in deriving the friction equation the effects of cavitation in the fluid film were ignored; the actual friction will thus tend to be lower than estimated. 3.5. Size c~~seal It is common experience that scaling up a successful design of mechanical seal often results in an unsatisfactory seal. It would therefore be pertinent to look at the effect of seal geometry on the performance in the light of the present theory. Considering the condition of perfect flow balance with si =O, Ed= 0.8, G -= M

0.8 pNR: r

= constant

Ps%

and for a given seal duty the sealing gap is proportional to R, r”. The coefficient r is a function only of the seal face aspect ratio b/R I so that if all the seal face dimensions are increased in proportion, sealing will occur at a proportionately larger film thickness. Conversely, the sealing film thickness of a given seal can be increased by increasing the face aspect ratio (Fig. 9). The increase in film thickness with seal size moderates the considerable increase in heat generated due to the increased peripheral speed.

ASPECT

RATIO,

b/R,

Fig. 9. Effect of size on seal performance. Wuar. 18 (1971) 51-69

THEORY

OF SEALING

WITH

RADIAL

63

FACE SEALS

However, in most commercial seals the face aspect ratio tends to be reduced in the larger sizes, and the benefit of an increased Iihn thickness is thus partly lost. Indeed, if the seal size is increased with a constant face width the film thickness for sealing is actuahy reduced. Effective dissipation of frictional heat is essential for reliable operation, and it is therefore worth taking a closer look at the effect of seal size on heat generation. Substituting for h, from the sealing condition (eqn. (5)) in eqn. (7) the heat generated in the fluid film can be expressed in terms of the face aspect ratio as :

It is immediately apparent that for a given sealing duty the heat generated increases with the cube of the size for the same aspect ratio. The in~uence of aspect ratio is shown in Fig. 9 where the function f, (b/R,) is plotted. This exhibits a definite minimum at an aspect ratio of about 5, although the overall variation in heat generated is not large. A rather better idea of the importance of choosing the correct aspect ratio is obtained by considering the heat generated per unit area of the seal face, since this determines the temperature gradients in the seal components. Dividing the expression for heat generated by the face area gives:

As would be expected from the previous discussion the heat generated per unit area is directly proportional to the seal size if the aspect ratio is constant. Increasing the aspect ratio produces a considerable reduction in the heat flux at the lower values (Fig 9) but further increases produce diminishing returns and there would seem little point in proceeding beyond the point of minimum total heat generation. From heat dissipation considerations, therefore the optimum aspect ratio would appear to be 3. This represents a somewhat wider face than is generally found in mechanical seals, particularly in the larger sizes, which may have aspect ratios as low as $. It must be emphasized that these observations apply only to seals having a face profile within the optimum range of 60 to 100% of the minimum film thickness and operating fairly close to the absolute sealing condition. However, these conditions embrace the majority of mechanical seals likely to be designed using hydrodynamic lubrication principles. 3.6. Limitations of the theory The limitations imposed by the two principal simplifying assumptions made in developing the theory were examined and found to be of little practical consequence (see Appendix). The narrow bearing approximation gives an adequate estimate of the film pressures for most practical values of face width, and in extreme cases the computed load capacity could be corrected using the data provided. Neglecting the net positive film pressure over the “negative” half of the generated pressure wave also Wear,18(1971)

51-69

64

E. H.

IN’L

affects only certain extreme cases; in the majority of seal designs the load capacity can be computed with sufficient accuracy using the equations derived from the approximate integration by ensuring that the parameter ,4 is greater than 103. Extreme cases could again be corrected if necessary by reference to Fig. 10.

I.0

r

lo-’

IO

IO DUTY

IO‘

IO’

lo-

IO'

PARAtv4ETER.A

Fig. 10. Comparison between approximate analytical and numerical integration of film pressure 1 in. < 6 10 in.; 0.05 4 b/R, < 0.5; 0~ a < n. Band shows variation with dimensionless amplitudes over the range O< s1 < 0.5; 0.4
R,

In the majority of cases the lower limit on ,4 will not impose any practical limit on the design, these limits being imposed by other considerations, for example, the upper limit on film thickness is more likely to be imposed by the transition from laminar to turbulent flow than by the restriction on the value of LLThe film stiffness also reduces fairly rapidly as n is reduced much below 103. The present theory can thus be applied to the design of a mechanical seal for any practical duty, those duties outside the scope of the theory being also outside the capabilities of the seal itself. 3.6.1. Practical implications The fundamental principle behind mechanical seals with hydrodynamic lubrication is the application of sufficient closing force to the faces to generate film pressures which oppose the flow of fluid outwards across the face. The limit imposed by the method of integration can therefore be represented as a lower limit on the ratio of closing force per unit face area to sealed fluid pressure; the upper limit of this ratio has already been established as the point at which the inward pumping action begins to drain fluid from the clearance. These limits are shown in Fig. 11 plotted against aspect ratio, and it is clear that the narrower the face the greater is the range of face loading, and hence the tolerance of the seal to changes in its design parameters, and the smaller is the face loading at which the analytical solution becomes inaccurate. At the optimum face width from heat dissipation considerations there is a useful range of face loading of almost 3 : 1 but beyond this the margin decreases steadily until at an aspect ratio of just over 0.5 the seal pumps fluid from its faces at the minimum loading Such face widths are unlikely to be considered in practice, but the graph suggests that the designer should select as narrow a face as possible consistent with adequately dissipating the heat generated. Wear,18 (1971) 51-69

THEORY

OF SEALING

WITH

RADIAL

FLUID FROM THlS

ASPECT

Fig. 11. Limits

4.

RATIO.

of application

65

FACE SEALS

PUMPED FACES IN REGION

b/R,

of the theory

in terms of pressure

CONCLUSIONS

The theoretical background for the design and operation of wavy-faced mechanical seals is outlined; a two-humped surface wave is selected because it represents the great majority of mechanical seals in current use. It is shown that the seal can operate with its faces fully lubricated and with very little leakage. The separation of the faces is controlled by hydrodynamic fluid pressures induced in the lubricating film by the face waviness; the generated pressures combine with the curvature of the ammlus to induce a net inward flow of fluid across the face. If the sealed fluid is arranged on the inside of the annulus this inward pumping effect can be utilized to reduce leakage, although for reliable operation seals should always be designed to just leak. Optimum performance requires near parallelism of the sealing faces at all times and a surface wave having a height of between 0.6 and 1.0 times the minimum film thickness. For a given seal design the minimum film thickness will usually be a compromise between heat dissipation and leakage requirements. The face loading necessary to achieve the selected film thickness may be calculated from hydrodynamic lubrication theory as outlined in the Paper. It is shown that the theory can be applied to the design of a mechanical seal for any practical duty provided that the working conditions can be closely controlled and the faces adequately cooled; it is suggested that for effective dissipation of the heat generated in the fluid film the seal faces should be made rather wider than is present practice. ACKNOWLEDGEMENT

The work was carried out at the Central Electricity Research Laboratories is published by permission of the Central Electricity Generating Board.

and

Wear, 18 (1971) 5149

66

E. I-1. Ih\r

NOMENCLATURE

A b

F G H h ho h, h, hm J M N : PS

Q RI R2 ; W TV

X Y lEl 82 A P 4 cu

f(4)

Area of seal face, width of seal face, viscous drag of fluid film, dimensionless flow coefficient defined in eqn. (4), heat generated between faces, local film thickness, minimum separation of aligned seal faces, amplitude of relative face run-out, height of surface waviness, mean film thickness, Joule’s mechanical equivalent of heat, dimensionless flow coefficient defined in eqn. (4), rotational speed of shaft (rev./min), number of surface undulations, film pressure, sealed fluid pressure, radial flow, inner radius of seal face, outer radius of seal face, radial co-ordinate, dimensionless stiffness factor defined in eqn. (9), load applied to seal faces, dimensionless load capacity obtained by numerically integrating all positive values of film pressure, dimensionless load capacity obtained by integrating the tilm pressure where f(6) ’ 0, dimensionless load-capacity defined in eqn. (3), dimensionless friction coefficient defined in eqn. (7), geometry term defined in eqn. (4), dimensionless amplitude h,/ho, dimensionless amplitude h2/ho, dimensionless duty parameter defined in eqn. (4), absolute viscosity of sealed fluid, angular displacement, angular velocity of seal face, dimensionless pressure coefficient defined in eqn. (2).

REFERENCES 1 B. A. BATCHAND

E. H. INY,

Pressure generation in mechanical seals, Proc. 2nd Intern. Corsf. Fluid

Seuling, Cranfield, England, 1964, Paper F4. 2 B. A. BATCHANDE. H.

INY. Face lubrication and the sealing mechanism of the radial-face mechanical seal, submitted for publication in J. Lubrication Technol., Tram ASME, Ser. F. 3 G. B. DUBOISANIJF. W. OCVIKK,Analytical derivation and experimental evaluation of short-bearing approximation for full journal bearings, Natl. Advisory Comm. Aeron., Rept. 1157, 1953. 4 J. THEWLISet al., Encyclopaedic Dictionary of Physics, Vol. 2, Pergamon, Oxford, 1961, pp. 74-81.

Weur, I8 (1971) 51-69

THEORY OF SEALING WITH RADIAL FACE SEALS 5

67

D. F. DENNY,

Some measurements of fluid pressure between plane parallel thrust surfaces with special reference to radial-face seals, Wear, 4 (1961) 64-83. 6 A. NAHAVANDIAND F. OSTERLE,The effect of vibration on the load-carrying capacity of parallel surface ‘thrust bearings, Am. Sot. Mech. Engrs., Paper 60-LUBS-3, 1960. I A. CAMERON,Principles of Lubrication, Longmans, London, 1966, p. 306. 8 E. H. INY, The design of hydrodynamically lubricated seals with predictable operating characteristics, 5th Intern. Conf. Fluid Sealing, Coventry, England, 1971, Paper Hl.

APPENDIX: RANGE OF APPLICATION OF THE THEORY

In the fundamental analysis of the radial-face seal on which this treatment is based two simplifying assumptions were made in order to obtain an analytical solution of the Reynolds’ differential equation of hydrodynamic pressure generation : the seal face was assumed narrow compared with its circumference and the film pressure was assumed positive where its dynamic component was positive and zero where the dynamic component had theoretically negative values, i.e. the sealed fluid pressure is assumed to have no effect on the boundaries of the cavitated region’. Both these assumptions will impose limits on the application of the hydrodynamic theory to seal design, and it is the purpose here to define these limits. I. The narrow face assumption

The assumption of a narrow seal face allows the use of a simplified form of Reynolds’ equation in which flow induced by a circumferential pressure gradient is neglected in comparison with that induced by radial pressure gradients. Whilst the adoption of this simplification enables the Reynolds’ equation to be solved analytically the method tends to over-estimate the magnitude of the generated pressures at the larger face widths. Limits on the face width will be determined by reference to published data comparing the approximate solution with a numerical solution of the complete Reynolds’ equation. Nahavandi and Osterle6 have treated an annular thrust bearing arrangement in which the film pressure is generated by misalignment of the faces, i.e. the film thickness varies sinusoidally at one cycle per revolution, and found excellent agreement, within 23 % of load carrying capacity, between the narrow face approximation and a numerical solution over a range of face widths up to b/R, = 1. For the two-humped configuration assumed in the seal theory this corresponds to b/R, =& and for greater than two humps the seal aspect ratio is proportionally reduced. The work of Nahavandi and Osterle was by no means comprehensive and needs confirmation. Fortunately, since the pressure generating mechanism is identical, work relating to journal bearings is immediately applicable to this problem. Cameron’ has given a range of data comparing the narrow bearing approximation with a numerical solution of the complete equation. This information is given in terms of the bearing length/diameter ratio L/D and for application to seal faces this must be converted into the seal aspect ratio b/R,. In this conversion L= b and D= 2R, + b whence b 2 LID __=----------n-L/D RI Wear, I8 (1971) 5169

68

E. H. IN)

where n is the number of surface undulations. Using this relationship Cameron’s data have been plotted in Fig. Al as a percentage error against aspect ratio for various values of n. Cameron’s information suggests a rather greater difference between the approximate and exact solutions than that of Nahavandi and Osterle, for instance at b/R, =OS Cameron suggests 8 “() compared with 2$% from Nahavandi and Osterle. It would therefore be prudent to

0

0.2 ASPECT

Fig. Al. Over-estimation E~=O, s,=O.8.

03 RATIO.

0.4

0.5

0.6

b/R,

of dynamic

film pressure

by the narrow-face

approximation

(after

Cameron),

use the former to determine the limiting face width. As would be expected the face width for a given error in film pressure decreases as the number of undulations on the seal face is increased. With two humps the narrow face simplification can be applied with little error to all practical aspect ratios, and at the optimum aspect ratio off derived from heat dissipation considerations the number of humps can be increased to four with only a 15 % over-estimation of load capacity and a consequent 5 “/, overestimation of the working film thickness. If a larger number of undulations is required, for example to reduce the face distortion, and the face width cannot be reduced the load capacity could be corrected using Fig. Al, but the effect on the leakage equations would need to be evaluated (cf Iny*). II. The simplified

boundary

conditions

To facilitate the integration of the film pressure to give the load carried the pressure was assumed to be zero when the hydrodynamically generated pressure term f(b) was negative. Whilst this assumption is correct when the pressure on either side of the annulus is ambient, a positive fluid pressure on one side produces a small area of positive pressure which is neglected. This simplification under-estimates the load carrying capacity of the fluid film, but the contribution of the neglected pressure to the load is small if the sealed fluid pressure is small compared with the hydrodynamic pressures. For a given seal the ratio of hydrodynamic to hydrostatic pressure is characterized by the dimensionless duty parameter LI,and the limiting value of n can be determined by comparing the dimensionless load capacity obtained by numerically integrating all the positive values of film pressure from eqn. (2a) below with that obtained by integrating this expression using the assumption that P =0 when f(4) = 0, eqn. (3a) Wear,18 (1971)

51-69

THEORY

OF SEALING

WITH

RADIAL

69

FACE SEALS

Dimensionless pressure P = g In (RJR,) s

and dimensionless load W = Wln(R21R,) PAR: -R:) R;-R$--2Rf ln(R,,‘R,) R;--R;

+ XA 20~2 [(Rz+R?f

1

ln(R~/R~)-R~~R~l

-I_ Figure 10 shows how the ratio of dimensionless load capacities Wb/Wp varies with the duty parameter A, and it is clear that the analytical solution gives a satisfactory approximation to the load capacity for A 2 lo3 irrespective of the size and con~guration of the seal. For values of A much less than lo3 the analytical solution seriously under-estimates the load required for a given film thickness. This could be corrected using Fig. 10, but at these low values of A the ratio of hydrodynamic to hydrostatic components of the film pressure is becoming sufficiently small to seriously reduce the film stiffness, which becomes zero when the load ratio is reduced to 0.5. It would not be prudent, therefore, to design seals with A 6 103, and the equations derived from the approximate integration are thus applicable to all practical seal designs. Wear, 18 (1971) 51-69