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The effect of the O-ring on the end face deformation of mechanical seals based on numerical simulation
Tribology International 97 (2016) 278–287
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Short Communication
The effect of the O-ring on the end face deformation of mechanical seals based on numerical simulation Zhi Chen n, Tinchao Liu, Jianming Li School of Chemical Engineering, Sichuan University, Sichuan 610065, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2015 Received in revised form 4 January 2016 Accepted 23 January 2016 Available online 1 February 2016
The end face deformation of the flexible ring of mechanical seals influenced by the O-ring was analyzed in numerical simulation method based on ANSYS platform. The combined contact model of isolated bodies for the flexible ring and the O-ring was established by transforming nonlinear constraint between them into interface force. The results indicated that the end-face deformation of the flexible ring was directly influenced by the fractional compression of the O-ring and the radial taper of the sealing face increased with increasing fractional compression. The excessive fractional compression had unfavorable effect on sealing performance of mechanical seals in view of face deformation. The elasticity modulus of the material of the flexible ring had a great effect on its deformation. & 2016 Elsevier Ltd. All rights reserved.
Keywords: O-ring Fractional compression Mechanical seal Face deformation
1. Introduction Mechanical seals find wide applications in pumps, compressors and agitators because of their advantages of low leakage rate, long life, and excellent performance. With the development of the mechanical seals, high parameter (high temperature, high pressure, high speed, and large scale) ones have been developed. There are inevitably problems, such as serious wear due to the deformation of seal rings and short life of the seals. A typical mechanical seal is shown in Fig. 1. It is a face sealing device where medium pressure and spring force make the faces of both rotating ring (the flexible ring) and stationary ring contact closely and the former slide relatively along the shaft. As Fig. 1 shows, a mechanical seal has the following elements: a flexible ring, a stationary ring, springs, auxiliary seal rings (Orings), driving parts, anti-rotation pins, spring seat, etc. and there are four leak points in the mechanical seal device. The primary seal point is leak point 3 which prevents the medium from radial leakage through the gap between two end faces and is the key leak point for sealing performance. There exist friction and abrasion between the end faces due to their close contact. There is leak point 2 between the gland and the housing. At the same time, another sealing point between the stationary ring and the gland is leak point 1. Both are static seals with O-rings which work reliably. The auxiliary seal between the rotating ring and the shaft is leak point 4. It is an O-ring seal and belongs to a kind of micro-moving n
Corresponding author. Tel.: þ 86 13880267468; fax: þ 86 028-85407591. E-mail address: [email protected] (Z. Chen).
http://dx.doi.org/10.1016/j.triboint.2016.01.038 0301-679X/& 2016 Elsevier Ltd. All rights reserved.
seal, which could seal process medium and compensate deflection and vibration of the flexible rotating ring simultaneously. Moreover, it is necessary for the O-ring to slide smoothly along the shaft and keep good tracing and floating abilities. The material of an O-ring was frequently chosen as rubber. The O-ring would slide to the lateral wall of the seal groove of a rotating ring under medium pressure. Hence, it is a kind of laterally one side restrained rubber O-ring. There have been many studies on static O-ring seal or elastomeric O-ring seal inserted into a rectangular groove. Through finite element analysis (FEA) and photoelastic experiments, Kim et al. investigated the distribution laws of the primary contact stress and side contact stress, the relationships of the contact width, contact stress and fractional compression for a compressed O-ring [1,2]. Zhang and Zhou obtained the contact stress of an O-ring by studying the deformed shape and stress of an O-ring through finite element method (FEM) with Mooney-Rivlin's linear model [3,4]. Li et al. simulated the Mises stress and the contact stress of sealing contact of an Oring with nonlinear finite element software using the same linear model [5]. They also calculated the friction coefficients from the contact stresses combined with the friction forces measured in experiments. Since the mechanical behavior of a rubber O-ring is very complicated, its constitutive equation is nonlinear. There are some mature constitutive models with the deepening of research of elastomer. Using Neo-Hookean constitutive equation, George et al. studied the changes of stress and displacement under the large deformation of O-rings [6]. Their test data verified that Lindley semi-empirical formula could predict the performance of O-rings well. Green and English established two models according to
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Nomenclature b b´ Cij C01 C10 C20 d E Ii I1 I2 gr gz H1 H2 H3 p1 p0 pM(x) pS(y) r r0 R Rg
Contact width Dimensionless contact width Material constants and C00 ¼0 Material constant Material constant Material constant Cross-section diameter of an O-ring Elasticity modulus Invariants of strain tensor The first invariant of strain tensor The second invariant of strain tensor Radial body force in unit volume Axial body force in unit volume Thickness of a flexible ring Depth of seal shoulder End-face width Medium pressure Atmosphere pressure Contact stress on the main contact Contact stress on the side contact Radial coordinate Local radial coordinate for the end face Shaft radius Seal shoulder radius
Fig. 1. The structure of a mechanical seal: (1) housing; (2) process medium; (3) setting screw; (4) shaft; (5) spring; (6) O-ring; (7) flexible ring; (8) stationary ring and (9) gland.
installation features of O-rings [7]. One is a laterally unrestrained O-ring model and another is a groove-restrained O-ring model. They also studied the relationship between fractional compression and contact stress or peak contact stress. Kim et al. investigated the Young modulus, deformation shape, frictional factor, and extrusion behavior of an O-ring elastomer by means of numerical method and experiments [1,2]. They obtained more precise values of the elasticity modulus through the uni-axial tests and equi-axial tests for a more accurate finite element analysis, detected the deformed shape of the O-ring by a computed tomography and found that the normalized results with respect to the peak stress and contact width followed the typical Hertzian profile. However, majority of research work done in O-rings did not consider the effect of the O-ring on the deformation of the components (e. g. the gland and the housing in Fig. 1) in the mechanical systems. However, the O-ring mounted on the flexible ring of a mechanical
Ri Ro S Sg Su u v w W x y z z0
α δ γzr εr εz εθ θ μ σr σz σθ τzr
279
Inner radius of a flexible ring Outer radius of a flexible ring Boundary condition Boundaries loaded by surface forces Boundaries loaded by displacements Radial displacement Circumferential displacement Axial displacement Strain energy density function Local x coordinate for an O-ring Local y coordinate for an O-ring Axial coordinate Local axial coordinate for the end face Radial taper of end face Fractional compression of an O-ring Shear strain in the z–r plane Radial normal strain Axial normal strain Circumferential normal strain Circumferential coordinate Poisson's ratio Radial normal stress Axial normal stress Circumferential normal stress Shear stress in the z–r plane
seal will lead to the deformation of the flexible ring, which should be studied. In recent years, most studies involving in mechanical seals have focused on sealing and friction characteristics of mechanical seals with sliding surface texture [8–10], materials of end faces [11], temperature field of sealing rings, fluid-thermal or thermal-stress coupled analysis [12,13]. Zhou and Gu built heat transfer model and studied the coupled process of the friction heat of liquid membrane between the sealing faces and thermal deformation of end faces [14]. They analyzed the high nonlinear relationships between thermal flux and thermal deformation of end faces by combining orthogonal design with artificial neural network method. It can be used to forecast the axial thermal deformation of the end faces of sealing rings accurately. Samant et al. presented three basic ideas that are proper for coupled field analysis after thermal-structure coupled analysis of mechanical seal by means of FEM [15]. Tournerie established a corresponding coupled field model using CSTEDY [16]. The model could be used to predict sealing performance, such as deformation and temperature of rings, the thickness of the liquid membrane between end faces and the leakage rate. However, the majority of research work done in face seal assumes that the O-ring seals are rigid or neglects them in the analysis. Meng et al. studied the end faces' deformation influenced by auxiliary seal or O-ring and found that the O-ring played a flexible support role to a rotating ring and made the end faces deform to form a divergent gap [17]. But they did not consider the effects of the varied fractional compression of an O-ring and the changes of contact boundary conditions due to the secondary compression of an O-ring by medium pressure. In addition, contact force was simplified as uniform load or even ignored in the most of previous researches. This simplification could make the computation speed higher but does not accord with the operating conditions of a flexible ring well and the real deformation and temperature field cannot be obtained under the conditions.
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Fig. 2. The schematic diagram of flexible ring's deformation due to the compression of an O-ring.
An O-ring should be usually fractionally compressed when mounted and will endure secondary compression under the operating pressure, which may lead to the deformation of a flexible ring. The same phenomenon may be observed for a stationary ring. The deformation properties of the flexible ring have been mainly studied in this paper. The contact surface between a seal ring and an O-ring cannot be easily determined. It will change with different loads, materials, boundary conditions and some other factors because O-ring is made of hyper-elastic material. It becomes a hot and difficult spot in the field how to find a model in order to simulate the deformation accurately and make it clear that how the mechanical properties of an O-ring influence the deformation of the flexible ring of a mechanical seal. The combined model of a flexible ring, that is, a rotating ring and an O-ring is shown in Fig. 2. A rotating ring will deform when the rubber O-ring is fractionally compressed or/and compressed under the medium pressure. In present work, the Mooney–Rivlin model with a higher order term was adapted and the complicated non-linear constraints on the contact surface between the rotating ring and the O-ring were simplified to interaction surface forces. The effects of both the fractional compression of an O-ring and medium pressure on the deformation of a rotating ring were then analyzed in numerical simulation method. Finally, the influence of the material of a rotating ring on its deformation was also investigated.
Fig. 3. The contact stresses acted by a laterally one side restrained O-ring.
2. Mathematical models
local coordinate system was used in order to illustrate the contact stresses clearly. The coordinate origin is at the center of the cross section of an O-ring, x axis is a local coordinate in the horizontal direction and y axis is a local coordinate in the radial direction. The contact stress between the O-ring and the top surface is defined as main contact stress pM(x) and the interface is called main contact surface. There is the same contact stress between the O-ring and the bottom surface due to the equivalent Cauchy stress effect when friction force on an O-ring is very small [1]. The contact stress between the O-ring and lateral wall of the groove of a flexible ring is defined as side contact stress pS(y) and the interface is called side contact surface. The direction of main contact stress on the top surface is in the y direction and that of side contact stress on the lateral wall is in the opposite direction of x-axis. The p0 is atmospheric pressure and the p1 is the medium pressure. According to the structure characteristics and sealing principle, whether an O-ring can prevent process medium from leakage or not depends on the main contact stress. Leaking is possible when the peak value of main contact stress is smaller than the pressure of sealing medium. NBR is a typical kind of hyper-elastic material, called Green elastomeric material. Its stress state could be described by strain energy density function instead of elastic-plastic response curve like common materials. After a lot of experiments, researchers put forward some constitutive models to describe the stress-strain relationship for this kind of material. Mooney–Rivlin model with high order items could be used to describe the mechanical behavior of the NBR O-ring, which is expressed in following form:
2.1. O-ring
W¼
N X
C ij ðI 1 3Þi ðI 2 3Þj
ð1Þ
i; j ¼ 0
As Fig. 3 shows, the O-ring seated in the groove of a rotating ring is fractionally compressed in the radial direction. It will slide to the side or lateral wall of the groove when under medium pressure and be then compressed in the axial direction. It is a kind of laterally one side restrained O-ring [2]. The O-ring made of nitrile-butadiene rubber (NBR) was studied in the work. NBR is an oil-resistant synthetic rubber with high elasticity, good deformation capacity and low compression set. It has isotropic property. It not only has the elastic property like metal but also absorbs energy like viscous liquid. At the same time, its geometry changes nonlinearly due to its complicated material characteristics. Nonlinear problems with complicated boundary conditions are very common in engineering problems, such as the description of an O-ring mechanical behavior, which requires complex procedures in their analysis. The boundary condition of a laterally one side restrained O-ring is shown in Fig. 3. It is shown that the O-ring is subjected to a vertical compression and lateral pressure. The relative vertical compression of an O-ring is defined as fractional compression. A
Mooney–Rivlin model with second order term can adapt rubber materials well no matter they are filled or not [18]. It is presented as: WðI 1 ; I 2 Þ ¼ C 10 ðI 1 3Þ þ C 01 ðI 2 3Þ þ C 20 ðI 1 3Þ2
ð2Þ
where C10 ¼1.147, C01 ¼0.038 and C20 ¼0.063 [19]. The simulation model was established under the following assumptions. i. Rubber material is isotropic. ii. Rubber is regarded to be incompressible because its Poisson ratio is close to 0.5. iii. Lubrication is quite sufficient so that the friction between the O-ring and the rigid body might be ignored. iv. Both the flexible ring and the O-ring require shaft alignment and operate at room temperature. v. The rigidity of the solid surface contacting with the O-ring is about 1000 times larger than that of the rubber so that its deformation of the contact surface might be negligible. The
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Fig. 4. The gap types between the end faces of mechanical seals: (a) parallel gap; (b) convergent gap and (c) divergent gap.
surface could be defined as the contact surface between a rigid body and a flexible body. In the calculation, the surface of a rotating ring can be defined as non penetrating boundary conditions. vi. The O-ring elasticity modulus is kept constant and its stiffness does not vary with the fractional compression for the fractional compression is smaller than 30% [1]. Lindley studied the plane stress state of an O-ring which was in a no lateral restraint condition and presented a dimensionless semi-empirical formula [1,6]. 0:5 b 6 1:5 6 ¼ ð1:25δ þ 50δ Þ d π
ð3Þ
where b is the width of contact surface between the O-ring and the top surface of a rigid body and it could be computed by means of simulation software; d is the cross-section diameter of an O-ring; δ is the ratio of the diametric compression of an O-ring to its cross-section diameter d (see Eq. (7)). 2.2. Flexible ring In an ideal state, the end faces of friction pair of a mechanical seal fit well, in other words, the end faces are parallel as shown in Fig. 4(a), but it is often encountered that the wear of the end faces is not uniform in use. Some of them are abraded severely close to the inner diameter of the rings while others close to the outer diameter, which would lead to non-parallel gap shape between the friction pairs, as shown in Fig. 4(b) or 4(c), due to the deformation of end faces when being under medium pressure and the contact stress acted by an O-ring as well as in nonuniform temperature distribution of sealing rings. The supports and fractional compression of an O-ring could influence the deformation of end faces of both a rotating ring and stationary ring. In this research, we mainly focus on the deformation of the end face of a rotating ring under the contact stress of an O-ring and medium pressure. Because the geometry, constraints and loads are axisymmetric, the displacement, stress and strain under loads are also axisymmetric. As a result, the numerical simulation for the deformation could be simplified into an axisymmetric problem. A global or a cylindrical coordinate system was set to solve this problem. In cylindrical coordinates (r, θ, and z), the coordinate origin is at the center of end face of the flexible ring and the symmetry axes of the flexible ring and the O-ring are defined as z-axis, as shown in Fig. 6. The displacement, stress and strain are functions of radial coordinate r and axial coordinate z and independent of circumferential coordinate θ. Equilibrium differential equations for space axisymmetric problem can be expressed by Eq. (4). The stress calculating equations for an axisymmetric problem are as follow [20]. ( ∂σ r
σr σθ ∂τzr þ gr ¼ ∂r þ ∂z þ r ∂τrz ∂σ z τrz þ g ¼ 0 þ þ z ∂r ∂z r
0
ð4Þ
Fig. 5. The longitudinal section of the combination body of a flexible ring with an O-ring.
where 8 h i E ð 1 μÞ μ > > > σ r ¼ ð1 þ μÞð1 2μÞ εr þ 1 μ εθ þ εz > > > h i > E ð 1 μÞ > > < σ θ ¼ 1 þ μ 1 2μ εθ þ 1 μ μðεr þ εz Þ ð Þð Þ h i > > σ z ¼ ð1 þEμð1Þð1μÞ2μÞ εz þ 1 μ μ εθ þ εr > > > > > > > : τzr ¼ 2ð1Eþ μÞγ zr The strains can be calculated by Eq. (6). 8 εr ¼ ∂u > ∂r > > > < εθ ¼ ur εz ¼ ∂w > ∂z > > > : γ ¼ ∂u þ ∂w zr
∂z
ð5Þ
ð6Þ
∂r
3. Simulation 3.1. Computational domain The combination of a flexible ring and an O-ring was selected as a research object based on steady state analysis. The longitudinal section of the combination body is shown in Fig. 5. R is the radius of the shaft. Ro is the outer radius of the flexible ring. Ri þH3 is the outer radius of its end face. H1 is the thickness of the flexible ring. Rg is the shoulder radius and H2 is its depth. H3 is the end-face width. Ri is the inner radius of the ring and Ri–R is the gap size between the inner hole of the flexible ring and the shaft. The fractional compression of the O-ring δ is defined by Eq. (7) and changing the Rg will alter the fractional compression for the given R.
δ¼
Rg R 100% d
ð7Þ
Because of axial symmetry of the geometry and the load, an axisymmetric FEM model was built, as shown in Fig. 6. The deformation behavior of a flexible ring and the mechanical behavior of its O-ring were simulated by ANSYS software. The domain was meshed by the PLANE182 element that is used for 2-D modeling of solid structures and can be used either as a plane element (plane stress or plane strain) or as an axisymmetric element. The
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element is defined by four nodes having two degrees of freedom at each node: translations in the nodal z and r directions. The element has plasticity, stress stiffening, large deflection, and large strain capabilities [21]. The contact analysis is based on the Mooney–Rivlin second order model. For better description of the characteristics of the deformation of a flexible ring, another local coordinate system was used, the coordinate origin is in the end-face plane at the point of the inner radius before ring deformation, r0 axis is in the radial direction and z0 axis is in the axial direction, as shown in Fig. 7(a). The sign (positive or negative) of the relative displacement of end face along z0 axis was set and the radial taper α of end face due to its axial displacement was defined to depict the deformation characteristic of the end face. The α being ‘ þ’ implies that the end face is convex at inner radius and the gap is convergent. The α being ‘–’ means that the end face is convex at outer radius and the gap is divergent, shown in Fig. 7(b). 3.2. Boundary conditions When some of the boundaries are given as surface forces and others are given as displacements, the mixed boundary condition is used. Sg indicates the boundaries loaded by surface forces, and Su does the boundaries loaded by displacements. Then the global boundary is as follows. S ¼ Sg þ Su
ð8Þ
The boundaries of forces, acted on the outer surface of the assembly of the O-ring and the flexible ring, are shown in Fig. 8(a). The thick lines and curves indicate the surface was acted by medium pressure p1 but the film-pressure on the end face was assumed to be the linear distribution along the radial direction. The pressure at outer diameter of the end face was medium pressure p1 and that at the inner diameter of the end face was atmospheric pressure p0 providing the leakage stream was an inward flow along the seal faces. In addition, the back of flexible ring was also exerted by the pressure resulted from spring force. O-ring was constrained by a shaft in radial direction as shown in Fig. 8(b). The flexible ring could rotate around the triangle vertex as shown in Fig. 8(c). The force of the O-ring acting on flexible ring was treated as interface force as shown in Fig. 8(d). 3.3. Influence of materials of flexible ring Four materials of flexible ring which are frequently used were selected. They were 316L matrix (spraying or surfacing Satellite alloy on surface), carbon-graphite, tungsten carbide (Co-based) and reaction bonded silicon carbide, respectively. As a friction pair, the material of stationary ring is silicon carbide in practice. Material characteristics related to stress analysis are shown in Table 1.
4. Results and discussion 4.1. Model verification
r z Fig. 6. Computational domain of a flexible ring with an O-ring.
The physical properties of a NBR O-ring and the geometry of the O-ring assembly studied are shown in Table 2. Hertz contact theory can describe the contact stress between the seal plate and the O-ring and the contact stresses must present a parabolic pattern [2]. Lindley's semi-empirical formula may predict the dimensionless contact width of O-ring accurately [6]. Wendt studied NBR (E ¼7.9 MPa) O-ring's peak contact stresses at different fractional compressions through the experiments [23,24]. Our previous simulation results [25] showed that the simulated profile of contact stress of the O-ring without the action of lateral pressure was in good accordance with Hertz theory, as shown in Fig. 9 and the simulated dimensionless contact widths (b´¼ b/d) agreed well with those calculated by Lindley semi-empirical formula, as shown in Fig. 10 except the simulated peak stresses that were smaller than Wendt's results due to the lubrication. In that research, 1131 finite elements were used for the same O-ring as used in this research and the detailed simulation method and conditions were introduced in Ref. [25]. So the Mooney–Rivlin
Fig. 7. The local coordinate system of the end face and rule of sign: (a) the local coordinates and (b) conical angle of the end face and its sign.
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283
Fig. 8. Force load boundary and constrain: (a) force boundary; (b) displacement boundary; (c) axial constrain and (d) contact surface force boundary. Table 1 Main properties of seal ring materials [22]. Material
E (MPa)
μ
316L Carbon-graphite WC (Co-based) SiC
200,000 25,000 600,000 400,000
0.30 0.15 0.26 0.15
Table 2 The physical properties of a NBR O-ring [4] and the and geometry of the O-ring and the flexible ring NBR O-ring Rigidity (IRHD) Elasticity modulus E (MPa) Poisson ratio μ Inner diameter (mm) Cross-section diameter d (mm)
Flexible ring 85 7.8 0.49 28 5.3
Shaft radius R (mm) Outer radius Ro (mm) Inner radius Ri (mm) Thickness H1 (mm) Shoulder depth H2 (mm) End-face width H3 (mm)
14 29 14.2 15 6 10
Fig. 10. Comparison of the dimensionless contact width with Lindley's results.
Fig. 11. The contact stresses on main contact surface at the medium pressure of 1.0 MPa. Fig. 9. Comparison of the contact stress simulated with Hertz's profile.
formula with second-order term adopted in this research might properly describe the mechanical behavior of the NBR O-ring. Properly reducing mesh size could make the calculation more accurate and consume more computating time at the same time. The mesh size of the O-ring was determined according to the comparison with Lindley's results of dimensionless contact width in Ref. [25]. The mesh size for the flexible ring was 0.5 mm in order to achieve higher accurate and consume shorter simulation time. 1767 finite elements were used in the domain of the O-ring and the flexible ring.
4.2. Mechanical behavior of a laterally one side restrained rubber Oring 4.2.1. Influence of fractional compression How the fractional compression of a rubber O-ring influenced the main contact stress and side contact stress at the lateral medium pressure p1 of 1.0 MPa was given in Figs. 11 and 12, respectively. It was shown in Fig. 11 that contact stress distributions at different fractional compressions did not present a parabolic pattern on the main contact and the stresses at the lateral pressure boundary were a bit larger than corresponding lateral pressures, which was similar to the experimental result provided by Kim et al. [2]. The state of internal stress of an O-ring is very different from that of an O-ring compressed only in the normal direction. The O-ring will slide to the lateral wall under the medium pressure resulting in the side contact stress. The contact
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Fig. 12. The contact stresses on the side contact surface at the medium pressure of 1.0 MPa.
Fig. 14. The profiles of the contact stresses on the main contact surface.
Fig. 15. The profiles of the contact stresses on the side contact surface. Fig. 13. The contact stress profiles at the medium pressure of 0.2 MPa and the fractional compression of 4%.
stresses at the atmosphere boundaries will be near zero according to the Hertz theory, but the internal stress of an O-ring will increase, which makes the peak stress become larger. Moreover, the part of an O-ring near the medium would be compressed much greatly leading to larger contact stress on the contact area near the lateral pressure boundary, which was verified by the experimental results from Ref. [2]. Main contact stress, contact width and peak stress increased as the fractional compression increased. Peak stress was always larger than medium pressure, which could prevent leakage of medium. The higher the fractional compression was, the larger the deformation of O-ring was and the larger the main contact stress was. The relatively large main contact stress of an O-ring could be obtained in a small fractional compression. The peak stress was 5 times larger than the medium pressure when fractional compression was 8%. The peak stress on the main contact increased linearly as the fractional compression of an O-ring increased. It can be figured out from Fig. 12 that the side contact stress on the lateral wall had just a parabolic distribution, and the peak stress hardly changed and kept about 7 MPa. So the peak stress on the side contact might have weak relationship with the change of fractional compression. It meant that the peak stress on the side contact would generally keep constant for a laterally one side restrained O-ring under the given medium pressure. The profiles of side contact stress were similar to the results given by Ref. [1] when friction force was very small.
The contact stress profiles were given in Fig. 13 at the medium pressure p1 of 0.2 MPa and the fractional compression of 4% for the O-ring. In the figure, the peak stress on the main contact was about 12 times larger than the medium pressure. Moreover, the Oring on the flexible ring is a kind of micro-moving seal. It is demanded that the O-ring not only has the function of sealing but also can move along a shaft without large friction. Thus, the fractional compression cannot be too large [26] and its range from 4% to 12% is more appropriate. 4.2.2. The influence of medium pressure The influences of medium pressure on the contact stresses on both main contact and side contact surfaces at fractional compression of 12% are given in Figs. 14 and 15, respectively. The profile of the contact stress on main contact surface was approximately a parabolic distribution as shown in Fig. 14. The peak stress was always at the midpoint of contact width and the curve was not symmetrical. The peak stress would increase from 5.7MPa to 9 MPa when the medium pressure rose from 0.5MPa to 2MPa. This is because that increasing medium pressure could make the O-ring be under the larger secondary compression resulting in increasing contact stress, which may cause the peak stress to increase more greatly than the medium pressure. Moreover, the contact width on main contact increased with the increasing medium pressure at the same time. Fig. 15 reveals the distribution of the side contact stress on the side contact. It was a good quadratic parabolic curve. The stress decreased from the peak point to both contact sides and increased
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with the medium pressure. The contact width on the side contact also increased with the increase of the medium pressure. Therefore, the secondary compression of O-ring due to the medium pressure was the main reason for generation of the stress on the side contact surface. The peak stress on main contact increased with the increasing medium pressure and was larger than medium pressure. As a result, O-ring could also prevent the medium from leakage when medium pressure increases abruptly. Due to the ‘self-tight sealing’ and ideal parabolic distribution pattern of the contact stress, an Oring has the ability to adapt to the changes of the medium pressure. This good sealing performance makes the O-ring find wide application in mechanical seals.
Fig. 16. The influences of different materials on the axial displacement of flexible rings.
285
4.3. End face deformation of a flexible ring 4.3.1. Influence of material of end face Fig. 16 shows the axial displacements of the end faces of the flexible rings made of 316L, carbon-graphite, SiC and WC increased with the increase of radius r0 under the conditions that medium pressure p1 ¼1.0 MPa, p0 ¼0 MPa (gauge pressure), rotation speed n ¼0 rpm, the fractional compression of the O-ring was of 12% and the radial leakage flow was inward, which illustrates the axial displacements of flexible rings under the static condition. The end face made of carbon-graphite had 2.4 μm axial displacement at inner diameter and 5.6 μm axial displacement at outer diameter. The radial taper of the end face was formed. It is possible that there was a divergent gap between contact pairs and axial deformation angle of 32.5 10 5 rad while the axial displacement difference between the inner and outer diameters of the end face made of 316L, SiC or WC was only about 0.4 μm and the axial
Fig. 18. The influence of fractional compression on axial displacement for a 316L matrix flexible ring.
Fig. 17. The deformation of flexible rings (scale up 100 times): (a) carbon-graphite; (b) 316L; (c) SiC and (d) WC (Co-based).
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Fig. 19. The influence of fractional compression on axial displacement for a carbongraphite flexible ring.
Table 3 Effects of O-ring's fractional compression on axial deformation of the sealing face for p1 ¼ 1.0 MPa (G). Fractional compression of an O-ring
0% 8% 12% 16% 20% 24%
Axial displacement at inner diameter (μm)
Axial displacement at outer diameter (μm)
α 105 (rad)
316L
Carbongraphite
316L
Carbongraphite
316L
Carbongraphite
0.31 0.31 0.31 0.30 0.30 0.30
2.40 2.40 2.38 2.35 2.32 2.31
0.67 0.71 0.74 0.76 0.80 0.84
5.11 5.45 5.63 5.85 6.09 6.46
3.58 4.03 4.29 4.62 4.98 5.49
27.1 30.5 32.5 35.0 37.7 41.5
deformation angle was about 3.0 10 5 rad. So it suggested that material's elasticity modulus E had a remarkable influence on the force deformation of flexible ring under the same operating and geometry parameters. The seal ring with a larger elasticity modulus had a greater ability to resist force deformation under static condition and then the deformation of the end face would be smaller. Fig. 17 illustrates the influences of the material properties of the flexible ring on their deformation under the static condition. It was apparent that the deformation of a carbon-graphite ring was the largest and that of a WC ring was the smallest for the given simulation conditions. When the flexible ring is rotated at high speed, there are additional effects of the centrifugal force and frictional heat on its deformation. The total deformation is the sum of the deformation produced by the loads, such as the medium pressure, contact stresses due to the O-ring, centrifugal force and frictional heat, on the basis of the superposition principle under small deformation. The deformation of the flexible ring is then different from Fig. 17. However, the results of this work show that there will be some wear of the end face of the ring at the moment it starts up or shuts down if it deforms like Fig. 17(a). Therefore, too large deformation of the end face should be avoided as far as possible during the design. 4.3.2. Influence of fractional compression of an O-ring Figs. 18 and 19 show the end-face deformation curves of seal rings made of 316L matrix and carbon-graphite at different fractional compressions of an O-ring, respectively. The other computation conditions were the same as those for Fig. 16. It was shown from the curves that with same structure and force load, the deformation of the sealing ring made of carbon-graphite was greater than that of the ring made of 316L. With the increasing of
fractional compression, the radial taper increased for both materials. So it might be disadvantageous that the fractional compression of an O-ring was too large in view of the end face deformation of a sealing ring. Excess fractional compression of an O-ring would lead to larger displacement of the end face of a flexible ring resulting in more serious wear to the end face at the moment it starts up or shuts down. It could also be avoided by choosing smaller fractional compression of the O-ring. Table 3 shows that the axial deformation at the inner diameter of the end face decreased with increasing fractional compression of an O-ring and that at outer diameter of the sealing face increased as the fractional compression increased. The reason for this is that different fractional compressions lead to different deformations of an O-ring, the contact area between the O-ring and the sealing ring is then different and at last, the relative distances of acting points of total contact force to the ring centroid are different. Therefore, this kind of axial deformation is induced [17].
5. Conclusions Aiming at understanding of the influence of an elastomeric Oring on the end-face sealing performance of a mechanical seal, the combined contact model of isolated bodies for a flexible ring and an O-ring was established. The mechanical deformation of a flexible ring was studied with different fractional compression of an elastomeric O-ring and different ring materials under the given operating conditions. Based on ANSYS platform, the mechanical behavior of the Oring under the laterally one side restrained condition in the seal groove was simulated by transforming the complicated nonlinear constraint between a flexible ring and an O-ring into an interface force so that the problem resulted from simplifying the contact boundary due to the unknowing contact state might be completely solved. Fractional compression of an O-ring was one of main factors influencing sealing ring’s deformation. The laterally one side restrained O-ring in the seal groove had an important property that the peak contact stress on the main contact was always larger than the medium pressure for the given fractional compression, e.g., ‘self-tight sealing’. The main contact stress profile was approximately parabolic. Higher peak stress on main contact surface could prevent working medium from leaking when medium pressure increased rapidly. The fractional compression of an Oring could not be too large and its range from 4% to 12% was more appropriate. The elasticity modulus of the material of the flexible ring had a great effect on deformation of its end face. The larger the elasticity modulus of the material was, the better the ability to resist force deformation was and then, the smaller the deformation of end face was. Thus, material elasticity modulus is one of the most important parameters that influence the deformation of the flexible sealing ring. The fractional compression of an O-ring had much larger effect on the end face deformation of a carbon-graphite ring than that of a 316L ring. Increasing the fractional compression would vary the supporting boundary condition of the flexible ring with the end-face deformation increasing. Therefore, it is inappropriate to make the fractional compression too large from the view point of deformation of the end face.
References [1] Kim HK, Park SH, Lee HG, Kim DR, Lee YH. Approximation of contact stress for a compressed and laterally one side restrained O-ring. Eng Fail Anal 2007;14:1680–92.
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[2] Kim HK, Nam JH, Hawong JS, Lee YH. Evaluation of O-ring stresses subjected to vertical and one side lateral pressure by theoretical approximation comparing with photoelastic experimental results. Eng Fail Anal 2009;16(6):1876–82. [3] Zhang J, Jin G. Finite element analysis of contact pressure on O-ring. Lubr Eng 2010;35(2):80–3 (Chinese). [4] Zhou ZH, Zhang KL, Li J, Xu TL. Finite element analysis of stress and contact pressure on the rubber sealing O-ring. Lubr Eng 2006;4:86–9 (Chinese). [5] Li SX, Cai JN, Zhang QX, Jie L, Gao JJ. Performance analysis of o-ring used in compensatory configuration of mechanical seal. Tribology 2010;30(3):308–13 (Chinese). [6] George AF, Strozzit A, Rich JI. Stress fields in a compressed unconstrained elastomeric O-ring seal and a comparison of computer predictions and experimental results. Tribol Int 1987;11:237–47. [7] Green, I., English, C. Stresses and deformation of compressed elastomeric Oring seals. In: Proceedings of the 14th international conference on fluid sealing; 1994, pp. 83 95. [8] Nian X, Khonsari MM. Thermal performance of mechanical seals with textured side-wall. Tribol Int 2012;45(1):1–7. [9] Antoszewski B. Mechanical seals with sliding surface texture-model fluid flow and some aspects of the laser forming of the texture. Procedia Eng 2012;39:51–62. [10] Brunetière N, Tournerie B. Numerical analysis of a surface-textured mechanical seal operating in mixed lubrication regime. Tribol Int 2012;49:80–9. [11] Wang JL, Jia Q, Yuan XY, Wang SP. Experimental study on friction and wear behaviour of amorphous carbon coatings for mechanical seals in cryogenic environment. Appl Surf Sci 2012;258(24):9531–5. [12] Brunetière N, Modolo B. Heat transfer in a mechanical face seal. Int J Therm Sci 2009;48:781–94. [13] Luan ZG, Khonsari MM. Analysis of conjugate heat transfer and turbulent flow in mechanical seals. Tribol Int 2009;42:762–9. [14] Zhou JF, Gu BQ. Analysis of thermal deformation of end face of mechanical seal ring and forecast based on BP ANN. J Chem Industry Eng 2006;57(12):2902–7 (Chinese).
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[15] Samant RN, Plelan PE, Ullah MR. Finite element analysis of residual stress induced flatness deviation in banded carbon seals. Finite Elem Anal Des 2002;38:785–801. [16] Tournerie B. 2D numerical modelling of the TEHD transient behaviour of mechanical face seals. Seal Technol 2003;6:10–3. [17] Meng XK, Wu DZ, Wang LQ. Study on deformation behavior of mechanical seal faces influenced by secondary seals. Lubr Eng 2008;33(1):101–3 (Chinese). [18] Tschoegl NW. Constitutive equations for elastomers. J Polym Sci 1971;12:1959–70. [19] Yamabe J, Nishimura S. Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas. Int J Hydrog Energy 2009;31:1979–89. [20] Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed.. Beijing: Tsinghua University Press; 2004. [21] ANSYS online manuals, Release 5.5. 〈http://mostreal.sk/html/elem_55/chap ter4/ES4-182.htm〉. [22] Mayer E. Mechanical seals (Chinese edition). Beijing: Chemical industry press; 1981. [23] Wendt, G. Investigation of rubber O-ring and X-rings, 1. Stress Distributions, Service and Groove Design. BHRA Fluid Engineering; 1971. Internal Report T1115. [24] Dragoni E, Strozzi A. Theoretical analysis of an unpressurized elastomeric Oring seal inserted into a rectangular groove. Wear 1989;130:41–51. [25] Chen Z, Gao Y, Dong R, Wu B, Li JM. Finite element analysis of sealing characteristics of the rubber O-ring for a mechanical seal. J Sichuan Univ 2011;43 (5):234–9 (Engineering Science Edition, Chinese). [26] Yu HG, Zhang QX, Cai JL, Li SX. Experimental investigation on friction performance of O-ring used in compensatory configuration of mechanical seal. Fluid Mach 2012;40(6):1–4 (Chinese).
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Short Communication
The effect of the O-ring on the end face deformation of mechanical seals based on numerical simulation Zhi Chen n, Tinchao Liu, Jianming Li School of Chemical Engineering, Sichuan University, Sichuan 610065, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2015 Received in revised form 4 January 2016 Accepted 23 January 2016 Available online 1 February 2016
The end face deformation of the flexible ring of mechanical seals influenced by the O-ring was analyzed in numerical simulation method based on ANSYS platform. The combined contact model of isolated bodies for the flexible ring and the O-ring was established by transforming nonlinear constraint between them into interface force. The results indicated that the end-face deformation of the flexible ring was directly influenced by the fractional compression of the O-ring and the radial taper of the sealing face increased with increasing fractional compression. The excessive fractional compression had unfavorable effect on sealing performance of mechanical seals in view of face deformation. The elasticity modulus of the material of the flexible ring had a great effect on its deformation. & 2016 Elsevier Ltd. All rights reserved.
Keywords: O-ring Fractional compression Mechanical seal Face deformation
1. Introduction Mechanical seals find wide applications in pumps, compressors and agitators because of their advantages of low leakage rate, long life, and excellent performance. With the development of the mechanical seals, high parameter (high temperature, high pressure, high speed, and large scale) ones have been developed. There are inevitably problems, such as serious wear due to the deformation of seal rings and short life of the seals. A typical mechanical seal is shown in Fig. 1. It is a face sealing device where medium pressure and spring force make the faces of both rotating ring (the flexible ring) and stationary ring contact closely and the former slide relatively along the shaft. As Fig. 1 shows, a mechanical seal has the following elements: a flexible ring, a stationary ring, springs, auxiliary seal rings (Orings), driving parts, anti-rotation pins, spring seat, etc. and there are four leak points in the mechanical seal device. The primary seal point is leak point 3 which prevents the medium from radial leakage through the gap between two end faces and is the key leak point for sealing performance. There exist friction and abrasion between the end faces due to their close contact. There is leak point 2 between the gland and the housing. At the same time, another sealing point between the stationary ring and the gland is leak point 1. Both are static seals with O-rings which work reliably. The auxiliary seal between the rotating ring and the shaft is leak point 4. It is an O-ring seal and belongs to a kind of micro-moving n
Corresponding author. Tel.: þ 86 13880267468; fax: þ 86 028-85407591. E-mail address: [email protected] (Z. Chen).
http://dx.doi.org/10.1016/j.triboint.2016.01.038 0301-679X/& 2016 Elsevier Ltd. All rights reserved.
seal, which could seal process medium and compensate deflection and vibration of the flexible rotating ring simultaneously. Moreover, it is necessary for the O-ring to slide smoothly along the shaft and keep good tracing and floating abilities. The material of an O-ring was frequently chosen as rubber. The O-ring would slide to the lateral wall of the seal groove of a rotating ring under medium pressure. Hence, it is a kind of laterally one side restrained rubber O-ring. There have been many studies on static O-ring seal or elastomeric O-ring seal inserted into a rectangular groove. Through finite element analysis (FEA) and photoelastic experiments, Kim et al. investigated the distribution laws of the primary contact stress and side contact stress, the relationships of the contact width, contact stress and fractional compression for a compressed O-ring [1,2]. Zhang and Zhou obtained the contact stress of an O-ring by studying the deformed shape and stress of an O-ring through finite element method (FEM) with Mooney-Rivlin's linear model [3,4]. Li et al. simulated the Mises stress and the contact stress of sealing contact of an Oring with nonlinear finite element software using the same linear model [5]. They also calculated the friction coefficients from the contact stresses combined with the friction forces measured in experiments. Since the mechanical behavior of a rubber O-ring is very complicated, its constitutive equation is nonlinear. There are some mature constitutive models with the deepening of research of elastomer. Using Neo-Hookean constitutive equation, George et al. studied the changes of stress and displacement under the large deformation of O-rings [6]. Their test data verified that Lindley semi-empirical formula could predict the performance of O-rings well. Green and English established two models according to
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Nomenclature b b´ Cij C01 C10 C20 d E Ii I1 I2 gr gz H1 H2 H3 p1 p0 pM(x) pS(y) r r0 R Rg
Contact width Dimensionless contact width Material constants and C00 ¼0 Material constant Material constant Material constant Cross-section diameter of an O-ring Elasticity modulus Invariants of strain tensor The first invariant of strain tensor The second invariant of strain tensor Radial body force in unit volume Axial body force in unit volume Thickness of a flexible ring Depth of seal shoulder End-face width Medium pressure Atmosphere pressure Contact stress on the main contact Contact stress on the side contact Radial coordinate Local radial coordinate for the end face Shaft radius Seal shoulder radius
Fig. 1. The structure of a mechanical seal: (1) housing; (2) process medium; (3) setting screw; (4) shaft; (5) spring; (6) O-ring; (7) flexible ring; (8) stationary ring and (9) gland.
installation features of O-rings [7]. One is a laterally unrestrained O-ring model and another is a groove-restrained O-ring model. They also studied the relationship between fractional compression and contact stress or peak contact stress. Kim et al. investigated the Young modulus, deformation shape, frictional factor, and extrusion behavior of an O-ring elastomer by means of numerical method and experiments [1,2]. They obtained more precise values of the elasticity modulus through the uni-axial tests and equi-axial tests for a more accurate finite element analysis, detected the deformed shape of the O-ring by a computed tomography and found that the normalized results with respect to the peak stress and contact width followed the typical Hertzian profile. However, majority of research work done in O-rings did not consider the effect of the O-ring on the deformation of the components (e. g. the gland and the housing in Fig. 1) in the mechanical systems. However, the O-ring mounted on the flexible ring of a mechanical
Ri Ro S Sg Su u v w W x y z z0
α δ γzr εr εz εθ θ μ σr σz σθ τzr
279
Inner radius of a flexible ring Outer radius of a flexible ring Boundary condition Boundaries loaded by surface forces Boundaries loaded by displacements Radial displacement Circumferential displacement Axial displacement Strain energy density function Local x coordinate for an O-ring Local y coordinate for an O-ring Axial coordinate Local axial coordinate for the end face Radial taper of end face Fractional compression of an O-ring Shear strain in the z–r plane Radial normal strain Axial normal strain Circumferential normal strain Circumferential coordinate Poisson's ratio Radial normal stress Axial normal stress Circumferential normal stress Shear stress in the z–r plane
seal will lead to the deformation of the flexible ring, which should be studied. In recent years, most studies involving in mechanical seals have focused on sealing and friction characteristics of mechanical seals with sliding surface texture [8–10], materials of end faces [11], temperature field of sealing rings, fluid-thermal or thermal-stress coupled analysis [12,13]. Zhou and Gu built heat transfer model and studied the coupled process of the friction heat of liquid membrane between the sealing faces and thermal deformation of end faces [14]. They analyzed the high nonlinear relationships between thermal flux and thermal deformation of end faces by combining orthogonal design with artificial neural network method. It can be used to forecast the axial thermal deformation of the end faces of sealing rings accurately. Samant et al. presented three basic ideas that are proper for coupled field analysis after thermal-structure coupled analysis of mechanical seal by means of FEM [15]. Tournerie established a corresponding coupled field model using CSTEDY [16]. The model could be used to predict sealing performance, such as deformation and temperature of rings, the thickness of the liquid membrane between end faces and the leakage rate. However, the majority of research work done in face seal assumes that the O-ring seals are rigid or neglects them in the analysis. Meng et al. studied the end faces' deformation influenced by auxiliary seal or O-ring and found that the O-ring played a flexible support role to a rotating ring and made the end faces deform to form a divergent gap [17]. But they did not consider the effects of the varied fractional compression of an O-ring and the changes of contact boundary conditions due to the secondary compression of an O-ring by medium pressure. In addition, contact force was simplified as uniform load or even ignored in the most of previous researches. This simplification could make the computation speed higher but does not accord with the operating conditions of a flexible ring well and the real deformation and temperature field cannot be obtained under the conditions.
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Fig. 2. The schematic diagram of flexible ring's deformation due to the compression of an O-ring.
An O-ring should be usually fractionally compressed when mounted and will endure secondary compression under the operating pressure, which may lead to the deformation of a flexible ring. The same phenomenon may be observed for a stationary ring. The deformation properties of the flexible ring have been mainly studied in this paper. The contact surface between a seal ring and an O-ring cannot be easily determined. It will change with different loads, materials, boundary conditions and some other factors because O-ring is made of hyper-elastic material. It becomes a hot and difficult spot in the field how to find a model in order to simulate the deformation accurately and make it clear that how the mechanical properties of an O-ring influence the deformation of the flexible ring of a mechanical seal. The combined model of a flexible ring, that is, a rotating ring and an O-ring is shown in Fig. 2. A rotating ring will deform when the rubber O-ring is fractionally compressed or/and compressed under the medium pressure. In present work, the Mooney–Rivlin model with a higher order term was adapted and the complicated non-linear constraints on the contact surface between the rotating ring and the O-ring were simplified to interaction surface forces. The effects of both the fractional compression of an O-ring and medium pressure on the deformation of a rotating ring were then analyzed in numerical simulation method. Finally, the influence of the material of a rotating ring on its deformation was also investigated.
Fig. 3. The contact stresses acted by a laterally one side restrained O-ring.
2. Mathematical models
local coordinate system was used in order to illustrate the contact stresses clearly. The coordinate origin is at the center of the cross section of an O-ring, x axis is a local coordinate in the horizontal direction and y axis is a local coordinate in the radial direction. The contact stress between the O-ring and the top surface is defined as main contact stress pM(x) and the interface is called main contact surface. There is the same contact stress between the O-ring and the bottom surface due to the equivalent Cauchy stress effect when friction force on an O-ring is very small [1]. The contact stress between the O-ring and lateral wall of the groove of a flexible ring is defined as side contact stress pS(y) and the interface is called side contact surface. The direction of main contact stress on the top surface is in the y direction and that of side contact stress on the lateral wall is in the opposite direction of x-axis. The p0 is atmospheric pressure and the p1 is the medium pressure. According to the structure characteristics and sealing principle, whether an O-ring can prevent process medium from leakage or not depends on the main contact stress. Leaking is possible when the peak value of main contact stress is smaller than the pressure of sealing medium. NBR is a typical kind of hyper-elastic material, called Green elastomeric material. Its stress state could be described by strain energy density function instead of elastic-plastic response curve like common materials. After a lot of experiments, researchers put forward some constitutive models to describe the stress-strain relationship for this kind of material. Mooney–Rivlin model with high order items could be used to describe the mechanical behavior of the NBR O-ring, which is expressed in following form:
2.1. O-ring
W¼
N X
C ij ðI 1 3Þi ðI 2 3Þj
ð1Þ
i; j ¼ 0
As Fig. 3 shows, the O-ring seated in the groove of a rotating ring is fractionally compressed in the radial direction. It will slide to the side or lateral wall of the groove when under medium pressure and be then compressed in the axial direction. It is a kind of laterally one side restrained O-ring [2]. The O-ring made of nitrile-butadiene rubber (NBR) was studied in the work. NBR is an oil-resistant synthetic rubber with high elasticity, good deformation capacity and low compression set. It has isotropic property. It not only has the elastic property like metal but also absorbs energy like viscous liquid. At the same time, its geometry changes nonlinearly due to its complicated material characteristics. Nonlinear problems with complicated boundary conditions are very common in engineering problems, such as the description of an O-ring mechanical behavior, which requires complex procedures in their analysis. The boundary condition of a laterally one side restrained O-ring is shown in Fig. 3. It is shown that the O-ring is subjected to a vertical compression and lateral pressure. The relative vertical compression of an O-ring is defined as fractional compression. A
Mooney–Rivlin model with second order term can adapt rubber materials well no matter they are filled or not [18]. It is presented as: WðI 1 ; I 2 Þ ¼ C 10 ðI 1 3Þ þ C 01 ðI 2 3Þ þ C 20 ðI 1 3Þ2
ð2Þ
where C10 ¼1.147, C01 ¼0.038 and C20 ¼0.063 [19]. The simulation model was established under the following assumptions. i. Rubber material is isotropic. ii. Rubber is regarded to be incompressible because its Poisson ratio is close to 0.5. iii. Lubrication is quite sufficient so that the friction between the O-ring and the rigid body might be ignored. iv. Both the flexible ring and the O-ring require shaft alignment and operate at room temperature. v. The rigidity of the solid surface contacting with the O-ring is about 1000 times larger than that of the rubber so that its deformation of the contact surface might be negligible. The
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281
Fig. 4. The gap types between the end faces of mechanical seals: (a) parallel gap; (b) convergent gap and (c) divergent gap.
surface could be defined as the contact surface between a rigid body and a flexible body. In the calculation, the surface of a rotating ring can be defined as non penetrating boundary conditions. vi. The O-ring elasticity modulus is kept constant and its stiffness does not vary with the fractional compression for the fractional compression is smaller than 30% [1]. Lindley studied the plane stress state of an O-ring which was in a no lateral restraint condition and presented a dimensionless semi-empirical formula [1,6]. 0:5 b 6 1:5 6 ¼ ð1:25δ þ 50δ Þ d π
ð3Þ
where b is the width of contact surface between the O-ring and the top surface of a rigid body and it could be computed by means of simulation software; d is the cross-section diameter of an O-ring; δ is the ratio of the diametric compression of an O-ring to its cross-section diameter d (see Eq. (7)). 2.2. Flexible ring In an ideal state, the end faces of friction pair of a mechanical seal fit well, in other words, the end faces are parallel as shown in Fig. 4(a), but it is often encountered that the wear of the end faces is not uniform in use. Some of them are abraded severely close to the inner diameter of the rings while others close to the outer diameter, which would lead to non-parallel gap shape between the friction pairs, as shown in Fig. 4(b) or 4(c), due to the deformation of end faces when being under medium pressure and the contact stress acted by an O-ring as well as in nonuniform temperature distribution of sealing rings. The supports and fractional compression of an O-ring could influence the deformation of end faces of both a rotating ring and stationary ring. In this research, we mainly focus on the deformation of the end face of a rotating ring under the contact stress of an O-ring and medium pressure. Because the geometry, constraints and loads are axisymmetric, the displacement, stress and strain under loads are also axisymmetric. As a result, the numerical simulation for the deformation could be simplified into an axisymmetric problem. A global or a cylindrical coordinate system was set to solve this problem. In cylindrical coordinates (r, θ, and z), the coordinate origin is at the center of end face of the flexible ring and the symmetry axes of the flexible ring and the O-ring are defined as z-axis, as shown in Fig. 6. The displacement, stress and strain are functions of radial coordinate r and axial coordinate z and independent of circumferential coordinate θ. Equilibrium differential equations for space axisymmetric problem can be expressed by Eq. (4). The stress calculating equations for an axisymmetric problem are as follow [20]. ( ∂σ r
σr σθ ∂τzr þ gr ¼ ∂r þ ∂z þ r ∂τrz ∂σ z τrz þ g ¼ 0 þ þ z ∂r ∂z r
0
ð4Þ
Fig. 5. The longitudinal section of the combination body of a flexible ring with an O-ring.
where 8 h i E ð 1 μÞ μ > > > σ r ¼ ð1 þ μÞð1 2μÞ εr þ 1 μ εθ þ εz > > > h i > E ð 1 μÞ > > < σ θ ¼ 1 þ μ 1 2μ εθ þ 1 μ μðεr þ εz Þ ð Þð Þ h i > > σ z ¼ ð1 þEμð1Þð1μÞ2μÞ εz þ 1 μ μ εθ þ εr > > > > > > > : τzr ¼ 2ð1Eþ μÞγ zr The strains can be calculated by Eq. (6). 8 εr ¼ ∂u > ∂r > > > < εθ ¼ ur εz ¼ ∂w > ∂z > > > : γ ¼ ∂u þ ∂w zr
∂z
ð5Þ
ð6Þ
∂r
3. Simulation 3.1. Computational domain The combination of a flexible ring and an O-ring was selected as a research object based on steady state analysis. The longitudinal section of the combination body is shown in Fig. 5. R is the radius of the shaft. Ro is the outer radius of the flexible ring. Ri þH3 is the outer radius of its end face. H1 is the thickness of the flexible ring. Rg is the shoulder radius and H2 is its depth. H3 is the end-face width. Ri is the inner radius of the ring and Ri–R is the gap size between the inner hole of the flexible ring and the shaft. The fractional compression of the O-ring δ is defined by Eq. (7) and changing the Rg will alter the fractional compression for the given R.
δ¼
Rg R 100% d
ð7Þ
Because of axial symmetry of the geometry and the load, an axisymmetric FEM model was built, as shown in Fig. 6. The deformation behavior of a flexible ring and the mechanical behavior of its O-ring were simulated by ANSYS software. The domain was meshed by the PLANE182 element that is used for 2-D modeling of solid structures and can be used either as a plane element (plane stress or plane strain) or as an axisymmetric element. The
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element is defined by four nodes having two degrees of freedom at each node: translations in the nodal z and r directions. The element has plasticity, stress stiffening, large deflection, and large strain capabilities [21]. The contact analysis is based on the Mooney–Rivlin second order model. For better description of the characteristics of the deformation of a flexible ring, another local coordinate system was used, the coordinate origin is in the end-face plane at the point of the inner radius before ring deformation, r0 axis is in the radial direction and z0 axis is in the axial direction, as shown in Fig. 7(a). The sign (positive or negative) of the relative displacement of end face along z0 axis was set and the radial taper α of end face due to its axial displacement was defined to depict the deformation characteristic of the end face. The α being ‘ þ’ implies that the end face is convex at inner radius and the gap is convergent. The α being ‘–’ means that the end face is convex at outer radius and the gap is divergent, shown in Fig. 7(b). 3.2. Boundary conditions When some of the boundaries are given as surface forces and others are given as displacements, the mixed boundary condition is used. Sg indicates the boundaries loaded by surface forces, and Su does the boundaries loaded by displacements. Then the global boundary is as follows. S ¼ Sg þ Su
ð8Þ
The boundaries of forces, acted on the outer surface of the assembly of the O-ring and the flexible ring, are shown in Fig. 8(a). The thick lines and curves indicate the surface was acted by medium pressure p1 but the film-pressure on the end face was assumed to be the linear distribution along the radial direction. The pressure at outer diameter of the end face was medium pressure p1 and that at the inner diameter of the end face was atmospheric pressure p0 providing the leakage stream was an inward flow along the seal faces. In addition, the back of flexible ring was also exerted by the pressure resulted from spring force. O-ring was constrained by a shaft in radial direction as shown in Fig. 8(b). The flexible ring could rotate around the triangle vertex as shown in Fig. 8(c). The force of the O-ring acting on flexible ring was treated as interface force as shown in Fig. 8(d). 3.3. Influence of materials of flexible ring Four materials of flexible ring which are frequently used were selected. They were 316L matrix (spraying or surfacing Satellite alloy on surface), carbon-graphite, tungsten carbide (Co-based) and reaction bonded silicon carbide, respectively. As a friction pair, the material of stationary ring is silicon carbide in practice. Material characteristics related to stress analysis are shown in Table 1.
4. Results and discussion 4.1. Model verification
r z Fig. 6. Computational domain of a flexible ring with an O-ring.
The physical properties of a NBR O-ring and the geometry of the O-ring assembly studied are shown in Table 2. Hertz contact theory can describe the contact stress between the seal plate and the O-ring and the contact stresses must present a parabolic pattern [2]. Lindley's semi-empirical formula may predict the dimensionless contact width of O-ring accurately [6]. Wendt studied NBR (E ¼7.9 MPa) O-ring's peak contact stresses at different fractional compressions through the experiments [23,24]. Our previous simulation results [25] showed that the simulated profile of contact stress of the O-ring without the action of lateral pressure was in good accordance with Hertz theory, as shown in Fig. 9 and the simulated dimensionless contact widths (b´¼ b/d) agreed well with those calculated by Lindley semi-empirical formula, as shown in Fig. 10 except the simulated peak stresses that were smaller than Wendt's results due to the lubrication. In that research, 1131 finite elements were used for the same O-ring as used in this research and the detailed simulation method and conditions were introduced in Ref. [25]. So the Mooney–Rivlin
Fig. 7. The local coordinate system of the end face and rule of sign: (a) the local coordinates and (b) conical angle of the end face and its sign.
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283
Fig. 8. Force load boundary and constrain: (a) force boundary; (b) displacement boundary; (c) axial constrain and (d) contact surface force boundary. Table 1 Main properties of seal ring materials [22]. Material
E (MPa)
μ
316L Carbon-graphite WC (Co-based) SiC
200,000 25,000 600,000 400,000
0.30 0.15 0.26 0.15
Table 2 The physical properties of a NBR O-ring [4] and the and geometry of the O-ring and the flexible ring NBR O-ring Rigidity (IRHD) Elasticity modulus E (MPa) Poisson ratio μ Inner diameter (mm) Cross-section diameter d (mm)
Flexible ring 85 7.8 0.49 28 5.3
Shaft radius R (mm) Outer radius Ro (mm) Inner radius Ri (mm) Thickness H1 (mm) Shoulder depth H2 (mm) End-face width H3 (mm)
14 29 14.2 15 6 10
Fig. 10. Comparison of the dimensionless contact width with Lindley's results.
Fig. 11. The contact stresses on main contact surface at the medium pressure of 1.0 MPa. Fig. 9. Comparison of the contact stress simulated with Hertz's profile.
formula with second-order term adopted in this research might properly describe the mechanical behavior of the NBR O-ring. Properly reducing mesh size could make the calculation more accurate and consume more computating time at the same time. The mesh size of the O-ring was determined according to the comparison with Lindley's results of dimensionless contact width in Ref. [25]. The mesh size for the flexible ring was 0.5 mm in order to achieve higher accurate and consume shorter simulation time. 1767 finite elements were used in the domain of the O-ring and the flexible ring.
4.2. Mechanical behavior of a laterally one side restrained rubber Oring 4.2.1. Influence of fractional compression How the fractional compression of a rubber O-ring influenced the main contact stress and side contact stress at the lateral medium pressure p1 of 1.0 MPa was given in Figs. 11 and 12, respectively. It was shown in Fig. 11 that contact stress distributions at different fractional compressions did not present a parabolic pattern on the main contact and the stresses at the lateral pressure boundary were a bit larger than corresponding lateral pressures, which was similar to the experimental result provided by Kim et al. [2]. The state of internal stress of an O-ring is very different from that of an O-ring compressed only in the normal direction. The O-ring will slide to the lateral wall under the medium pressure resulting in the side contact stress. The contact
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Fig. 12. The contact stresses on the side contact surface at the medium pressure of 1.0 MPa.
Fig. 14. The profiles of the contact stresses on the main contact surface.
Fig. 15. The profiles of the contact stresses on the side contact surface. Fig. 13. The contact stress profiles at the medium pressure of 0.2 MPa and the fractional compression of 4%.
stresses at the atmosphere boundaries will be near zero according to the Hertz theory, but the internal stress of an O-ring will increase, which makes the peak stress become larger. Moreover, the part of an O-ring near the medium would be compressed much greatly leading to larger contact stress on the contact area near the lateral pressure boundary, which was verified by the experimental results from Ref. [2]. Main contact stress, contact width and peak stress increased as the fractional compression increased. Peak stress was always larger than medium pressure, which could prevent leakage of medium. The higher the fractional compression was, the larger the deformation of O-ring was and the larger the main contact stress was. The relatively large main contact stress of an O-ring could be obtained in a small fractional compression. The peak stress was 5 times larger than the medium pressure when fractional compression was 8%. The peak stress on the main contact increased linearly as the fractional compression of an O-ring increased. It can be figured out from Fig. 12 that the side contact stress on the lateral wall had just a parabolic distribution, and the peak stress hardly changed and kept about 7 MPa. So the peak stress on the side contact might have weak relationship with the change of fractional compression. It meant that the peak stress on the side contact would generally keep constant for a laterally one side restrained O-ring under the given medium pressure. The profiles of side contact stress were similar to the results given by Ref. [1] when friction force was very small.
The contact stress profiles were given in Fig. 13 at the medium pressure p1 of 0.2 MPa and the fractional compression of 4% for the O-ring. In the figure, the peak stress on the main contact was about 12 times larger than the medium pressure. Moreover, the Oring on the flexible ring is a kind of micro-moving seal. It is demanded that the O-ring not only has the function of sealing but also can move along a shaft without large friction. Thus, the fractional compression cannot be too large [26] and its range from 4% to 12% is more appropriate. 4.2.2. The influence of medium pressure The influences of medium pressure on the contact stresses on both main contact and side contact surfaces at fractional compression of 12% are given in Figs. 14 and 15, respectively. The profile of the contact stress on main contact surface was approximately a parabolic distribution as shown in Fig. 14. The peak stress was always at the midpoint of contact width and the curve was not symmetrical. The peak stress would increase from 5.7MPa to 9 MPa when the medium pressure rose from 0.5MPa to 2MPa. This is because that increasing medium pressure could make the O-ring be under the larger secondary compression resulting in increasing contact stress, which may cause the peak stress to increase more greatly than the medium pressure. Moreover, the contact width on main contact increased with the increasing medium pressure at the same time. Fig. 15 reveals the distribution of the side contact stress on the side contact. It was a good quadratic parabolic curve. The stress decreased from the peak point to both contact sides and increased
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with the medium pressure. The contact width on the side contact also increased with the increase of the medium pressure. Therefore, the secondary compression of O-ring due to the medium pressure was the main reason for generation of the stress on the side contact surface. The peak stress on main contact increased with the increasing medium pressure and was larger than medium pressure. As a result, O-ring could also prevent the medium from leakage when medium pressure increases abruptly. Due to the ‘self-tight sealing’ and ideal parabolic distribution pattern of the contact stress, an Oring has the ability to adapt to the changes of the medium pressure. This good sealing performance makes the O-ring find wide application in mechanical seals.
Fig. 16. The influences of different materials on the axial displacement of flexible rings.
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4.3. End face deformation of a flexible ring 4.3.1. Influence of material of end face Fig. 16 shows the axial displacements of the end faces of the flexible rings made of 316L, carbon-graphite, SiC and WC increased with the increase of radius r0 under the conditions that medium pressure p1 ¼1.0 MPa, p0 ¼0 MPa (gauge pressure), rotation speed n ¼0 rpm, the fractional compression of the O-ring was of 12% and the radial leakage flow was inward, which illustrates the axial displacements of flexible rings under the static condition. The end face made of carbon-graphite had 2.4 μm axial displacement at inner diameter and 5.6 μm axial displacement at outer diameter. The radial taper of the end face was formed. It is possible that there was a divergent gap between contact pairs and axial deformation angle of 32.5 10 5 rad while the axial displacement difference between the inner and outer diameters of the end face made of 316L, SiC or WC was only about 0.4 μm and the axial
Fig. 18. The influence of fractional compression on axial displacement for a 316L matrix flexible ring.
Fig. 17. The deformation of flexible rings (scale up 100 times): (a) carbon-graphite; (b) 316L; (c) SiC and (d) WC (Co-based).
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Fig. 19. The influence of fractional compression on axial displacement for a carbongraphite flexible ring.
Table 3 Effects of O-ring's fractional compression on axial deformation of the sealing face for p1 ¼ 1.0 MPa (G). Fractional compression of an O-ring
0% 8% 12% 16% 20% 24%
Axial displacement at inner diameter (μm)
Axial displacement at outer diameter (μm)
α 105 (rad)
316L
Carbongraphite
316L
Carbongraphite
316L
Carbongraphite
0.31 0.31 0.31 0.30 0.30 0.30
2.40 2.40 2.38 2.35 2.32 2.31
0.67 0.71 0.74 0.76 0.80 0.84
5.11 5.45 5.63 5.85 6.09 6.46
3.58 4.03 4.29 4.62 4.98 5.49
27.1 30.5 32.5 35.0 37.7 41.5
deformation angle was about 3.0 10 5 rad. So it suggested that material's elasticity modulus E had a remarkable influence on the force deformation of flexible ring under the same operating and geometry parameters. The seal ring with a larger elasticity modulus had a greater ability to resist force deformation under static condition and then the deformation of the end face would be smaller. Fig. 17 illustrates the influences of the material properties of the flexible ring on their deformation under the static condition. It was apparent that the deformation of a carbon-graphite ring was the largest and that of a WC ring was the smallest for the given simulation conditions. When the flexible ring is rotated at high speed, there are additional effects of the centrifugal force and frictional heat on its deformation. The total deformation is the sum of the deformation produced by the loads, such as the medium pressure, contact stresses due to the O-ring, centrifugal force and frictional heat, on the basis of the superposition principle under small deformation. The deformation of the flexible ring is then different from Fig. 17. However, the results of this work show that there will be some wear of the end face of the ring at the moment it starts up or shuts down if it deforms like Fig. 17(a). Therefore, too large deformation of the end face should be avoided as far as possible during the design. 4.3.2. Influence of fractional compression of an O-ring Figs. 18 and 19 show the end-face deformation curves of seal rings made of 316L matrix and carbon-graphite at different fractional compressions of an O-ring, respectively. The other computation conditions were the same as those for Fig. 16. It was shown from the curves that with same structure and force load, the deformation of the sealing ring made of carbon-graphite was greater than that of the ring made of 316L. With the increasing of
fractional compression, the radial taper increased for both materials. So it might be disadvantageous that the fractional compression of an O-ring was too large in view of the end face deformation of a sealing ring. Excess fractional compression of an O-ring would lead to larger displacement of the end face of a flexible ring resulting in more serious wear to the end face at the moment it starts up or shuts down. It could also be avoided by choosing smaller fractional compression of the O-ring. Table 3 shows that the axial deformation at the inner diameter of the end face decreased with increasing fractional compression of an O-ring and that at outer diameter of the sealing face increased as the fractional compression increased. The reason for this is that different fractional compressions lead to different deformations of an O-ring, the contact area between the O-ring and the sealing ring is then different and at last, the relative distances of acting points of total contact force to the ring centroid are different. Therefore, this kind of axial deformation is induced [17].
5. Conclusions Aiming at understanding of the influence of an elastomeric Oring on the end-face sealing performance of a mechanical seal, the combined contact model of isolated bodies for a flexible ring and an O-ring was established. The mechanical deformation of a flexible ring was studied with different fractional compression of an elastomeric O-ring and different ring materials under the given operating conditions. Based on ANSYS platform, the mechanical behavior of the Oring under the laterally one side restrained condition in the seal groove was simulated by transforming the complicated nonlinear constraint between a flexible ring and an O-ring into an interface force so that the problem resulted from simplifying the contact boundary due to the unknowing contact state might be completely solved. Fractional compression of an O-ring was one of main factors influencing sealing ring’s deformation. The laterally one side restrained O-ring in the seal groove had an important property that the peak contact stress on the main contact was always larger than the medium pressure for the given fractional compression, e.g., ‘self-tight sealing’. The main contact stress profile was approximately parabolic. Higher peak stress on main contact surface could prevent working medium from leaking when medium pressure increased rapidly. The fractional compression of an Oring could not be too large and its range from 4% to 12% was more appropriate. The elasticity modulus of the material of the flexible ring had a great effect on deformation of its end face. The larger the elasticity modulus of the material was, the better the ability to resist force deformation was and then, the smaller the deformation of end face was. Thus, material elasticity modulus is one of the most important parameters that influence the deformation of the flexible sealing ring. The fractional compression of an O-ring had much larger effect on the end face deformation of a carbon-graphite ring than that of a 316L ring. Increasing the fractional compression would vary the supporting boundary condition of the flexible ring with the end-face deformation increasing. Therefore, it is inappropriate to make the fractional compression too large from the view point of deformation of the end face.
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