I n d . E n g . Chem. Res. 1987,26, 2180-2185
2180
Technology and presented at 74th National Meeting, American Institute of Chemical Engineers, March 11-15,1973, New Orleans, LA. Wasfi, A. K.; Mathur, G. P.; St. Pierre, C. C.; Gnyp, A. W. Atmos. Enuiron. 1978, 12, 2389-2398. Wei, J. Advances in Catalysis; Academic: New York, 1975; Vol. 24, p 57.
Weldon, J.; Senkan, S. M. Combust. Sci. Technol. 1986,47,229-237. Yao, Y. F. Y. J. Catal. 1973,28, 139-149. Yao, Y. F. Y. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 60-67. Yao, Y. F. Y.; Kummer, J. T. J. Catal. 1973, 28, 124-138.
Received for review J u n e 1, 1987 Accepted August 11, 1987
ARTICLES
Hydrodynamics of Rubber Seals for Reciprocating Motion, Lubricating Film Thickness, and Out-Leakage of 0-Seals Artur Karaszkiewicz The Technical University of Warsaw, 02-524 Warsaw, Poland
This paper gives the fundamentals of the hydrodynamics of rubber seals for reciprocating motion, introducing certain modifications that are aimed at availability of engineering practice and conformity with experiment. The inverse problem in the theory of hydrodynamic lubrication of a rubber seal has been related with the contemporary achievements in elastohydrodynamic lubrication of the contact with a roll made of a low elasticity modulus material. The experimental verification of the relationship which conditions height of the slot (lubricating film thickness) between the sealing ring and surface of the sealed element on material, geometrical, and exploitation parameters of the O-ring is a significant result of this research work. This dependence solves the problem of lubrication and leakage in the case of O-rings during the out stroke of the sealed element. 1. Introduction The sealing technique deals, among others, with resolving the problems in exploitation and design of seals. The task of sealing design is to work out seals of defined, required exploitation features, which of course are leakage, frictional drag, and durability. There are indigences of knowledge about relationships between material properties, geometry of seals, and exploitation features and also between these features and operating conditions, which are determined by sealed pressure, sliding velocity, temperature, and properties of sealed liquid. The sealing technique seems to be at the beginning of the scientific approach to problems of design of seals for reciprocating motion. Hydrodynamics of rubber seals tries to solve the problems of exploitation and engineering design of seals. Theoretical work and experimental research based upon the theory of hyrodynamics attempt to provide fundamental designing knowledge through determination of the relationships between leakage (lubrication), seals parameters (material, geometrical), and operating conditions. This paper refers to previous works published by me (Karaszkiewicz, 1985, 1986).
2. Hydrodynamics of Rubber Seals The base of the inverse problem in the theory of hydrodynamic lubrication of a rubber seal as an axial bearing (Figure 1) is the Reynolds equation dP h - h, - = 67~(1) dx h3
where p = pressure in the slot (Figure 2), x = direction of motion, 17 = dynamic viscosity, u = velocity, h = running height of the slot, and h, = height of the slot where dp/dx = 0. In the classical problem of hydrodynamic lubrication of a bearing, to calculate, e.g., its load bearing capacity, the distribution of pressure, p ( x ) ,is determined analytically for a given geometry of the slot, h ( x ) . The problem is inverse in the case of a seal: a pressure distribution in the slot between the rubber ring and sealed element is assumed, whereas geometry of the slot, and in principle only the height, h,, is searched for. With these assumptions, eq 1 becomes algebraic since dpldx is replaced by the function p ’ ( x ) :
It may easily be proven that the expression [ h ( x )- h,]/ h3(x)attains a maximum at h ( x ) = 1.5hC,which also points out that the maximum positive value of p’(x), called the p’ gradient, is attained at the point of inflection of the pressure distribution curve, p ( x ) : 1.5hC- h, p’ = 617u (3) ( 1.5hJ After transformation, h, = ( 0 . 8 9 ~ p / p ’ ) ~ . ~ (4) Equation 4 has been derived by using quite a complex
0888-5885/87/2626-2180$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2181
b‘
Q
Figure 1. Rubber seal for reciprocating motion: (a) sealing of a cylindrical hole; (b) sealing of a shaft.
Figure 4. Distribution of contact pressure and geometry of the O-ring mounted in the seal groove and subjected to sealed pressure: (a) O-ring; (b) at a pressure p = 0; (c) at p = pl; (d) at p = p z .
Figure 2. Distribution of pressure and velocity of flow in the thrust bearing slot. Figure 5. Fragment of the contact pressure and pressure distribution at the inlet side of the slot under the O-ring.
Figure 3. Distribution of pressure and velocity of flow in the slot between the O-ring and surface of the sealed element (for the cross section where dp/dx = 0, only): (a) for the out-stroke; (b) for the in-stroke.
formalism and has been proposed for the fiist time by Blok (1963). Today, it is regarded as the basic equation of the problem. Hydrodynamic representation of the rate of flow (leakage) in the slot gives
where D = diameter of the sealed element. Thus, when h(x) = h, and dpfdx = 0, the rate of leakage is given by
Q = O.SrDuh,
(6)
pointing to a linear distribution of the velocity of flow over crow section h, (Figure 2). At the same time, eq 5, together with the continuity of flow criterion (Q = constant), results in a fundamental property of rubber seals-the rate of leakage is caused by motion of the sealed element with a defined velocity only. Rubber seals characterized by a pressure distribution where dpldx = 0 show no presence of flow caused by the differences in pressures in spaces separated by the seal. This principle is illustrated in Figure 3, where flow caused by the pressure difference is as if blocked by a barrier of pressure higher than in the sealed space. Equations 6 and 4 give a formula defining the leakage, q = volume of liquid flowing in or out of the sealed space during time t corresponding to the stroke length, L, of the sealed element moving at a velocity u: q = Qt
0.5rDuhJ/u = 0.51rDLhc
q = 0.5lrDL(O.89q~/p’)~~
(7) (74
The volume of the liquid flowing out of the sealed space is called the out-leakage, ql, and that flowing in is called in-leakage, q2.
In practice, we most often use the term net-leakage, Q, understood as the difference between the out-leakage, q l , and in-leakage, q2: = 41 - q 2 = 0.5rDL(hCl- hc2) (8)
a
Q = 0.5rDL(0.89q)o.5[(u1/pl’)o.5 - (V~/P~’)O.~]
(8a)
Net-leakage, Q, is popular because the majority of the investigations carried out so far enabled the measurement of only Q, instead of q1 and q2. If q1 7 q2, ij denotes the amount of liquid removed from the sealed space, while the q1
