Analytical and experimental study on sealing contact characteristics of subsea collet connectors

Transcription

1 Research Article Analytical and experimental study on sealing contact characteristics of subsea collet connectors Advances in Mechanical Engineering 2017, Vol. 9(4) 1 14 Ó The Author(s) 2017 DOI: / journals.sagepub.com/home/ade Feihong Yun 1, Liquan Wang 1,ShaomingYao 2, Jun Liu 1, Tong Liu 1 and Renhua Wang 1 Abstract The mathematical model of the contact stress is established based on the theory of Hertz contact with the assumption that the contact of a seal gasket and hubs of collet connectors is cylindrical. The mechanical models of the seal are established separately by sealing theory in the preloading and operating modes, and the relationships between the preload, the contact stress, and the sealing width are obtained. The relationships of structural parameters and the preload are analyzed and verified with finite element simulation and experiments. An experimental system has been designed and experiments have been conducted to verify the analytical models. Finite element simulation results and experimental results agree with analytical ones very well and the differences of the preload and sealing width are acceptable. Keywords Sealing contact characteristics, subsea collet connector, sealing width, seal gasket, sealing performance Date received: 26 May 2016; accepted: 5 March 2017 Academic Editor: Yangmin Li Introduction Subsea collet connector is a jumper connection device in subsea pipeline system, as shown in Figure 1. It is handled by remotely operated vehicle (ROV) for connection between wellhead, pipeline terminal, Christmas tree, and pipe manifold. The collet connector works in extremely severe operating environment. In all instances, the connector is subjected to internal high pressure, high temperature, and external low temperature load. 1,2 Its reliability depends on the seal performance; therefore, it is important to analyze the seal contact characteristics of the collet connector. Sealing performance of pipeline connectors has been studied extensively, but most of them focus on other points rather than seal contact characteristics. K Kondo et al., 3 K Tenma et al., 4 and M Abid et al. 5 researched on the contact stress and the sealing performance for the bolted flange connections. K Kondo et al. 3 found that the sealing performance with oval ring gasket was better than that of the connection with octagonal ring gasket. K Tenma et al. 4 found that the stress distribution at the contact surfaces depended on the bolting procedure, and octagonal ring gasket shows a better sealing performance than spiral wound gasket. M Abid et al. 5 found that the sealing performance is related to both the internal pressure and transient thermal load. To investigate the box-shaped bolted flange connection, K Tenma et al. 6 and T Sawa et al. 7 analyzed the gasket contact stress distributions and 1 College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, China 2 Advanced Manufacturing Research Centre with Boeing, The University of Sheffield, Sheffield, UK Corresponding author: Feihong Yun, College of Mechanical and Electrical Engineering, Harbin Engineering University, Room 2062, Building 61, No. 145 Nantong Street, Nangang District, Harbin , China. yunfeihong@hrbeu.edu.cn; yfh88818@163.com Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages ( open-access-at-sage).

2 2 Advances in Mechanical Engineering Figure 1. Installation of the subsea collet connector. evaluated their sealing performance by finite element (FE) method and experiment without analytic analysis. R Madazhy et al. 8 analyzed the contact stress of the tapered and flat seals and found that of the tapered seal is sufficient and that of flat gasket surface is not. H Zhao et al. 9 investigated the subsea X-tree wellhead connector VX gasket and analyzed theoretical relation between seal contact stress and its structural parameters and operation pressure. Z Wang et al. 10 analyzed the seal characteristics of horizontal collet connector with oval metal ring. F Peng et al. 11 optimized the collet connector locking structure and analyzed the friction influence on the contact surface. This article will analyze the seal contact characteristics of the collet connector with a lens seal gasket. The seal structure is for the connection of jumpers with the size of in under the internal pressure of 34.5 MPa. First, based on Hertz theory, the contact between the seal gasket and the hub can be equivalent to the contact of cylinders and the contact stress mathematical model is established. Second, mechanical models of the seal in the preloading and operating modes are established separately by the sealing theory, 12 and the relationship between the preload, the contact stress, and the sealing width is obtained. Third, in order to examine the sealing performance of the collet connector, contact characteristics of seal gasket subjected to preload and operation pressure are analyzed using FE calculations. Finally, the analytical results are compared with the experimental results to validate the mathematic model. The effect of the contact characteristics on the sealing performance of the collet connector is discussed as well. Principle of the collet connector As shown in Figure 2, both of the bottom hub and top hub are dedicated for specific collet connector. Fingers are arranged around the outer surface of the hubs with equal space and work along with the press ring to lock hubs. The bottom hub, top hub, and seal gasket constitute the core of the seal structure in the connector and mate with each other to achieve a seal. In preloading, fingers cooperate with the movement of the press ring and convert the translational motion of the press ring into radial motion of fingers, then Figure 2. Structure of the collet connector. strongly hold the hubs and preload the seal. The axial force on the press ring can regulate the sealing preload. In the operation, the high-pressure fluid goes through the inside of the connector, and the high pressure acts on the gasket and hubs. In this case, the self-locking of the friction between the internal conical surface of the press ring and the back surface of the fingers will guarantee that the collet connector has a good seal performance under the subsea static load. The concept of self-energizing seal is that when the internal pressure increases, the sealing performance of the gasket increases automatically without any extra preload. The sealing performance of the self-energizing gasket is not only related to the shape, size, and material of the hubs and gasket, but also related to the medium pressure and preload. Mechanics of the seal gasket A metal seal gasket was employed in the seal structure analyzed in this article. When the spherical surface on the seal gasket contacts the conical surface on the hub, it is line contact initially. As the deformations of the seal gasket and hubs increase with the increasing axial load, the contact line turns into a strip in the annular region, which is much less than the dimension of the seal gasket, and the width of this strip is called contact width or sealing width. 13 The sealing width and average contact stress of the contact region are the critical parameters for calculating the preload. And the sealing width and the distribution of the contact stress are indispensable when the sealing performance is discussed. 14 Therefore, analysis on metal seal mechanics is quite necessary, and this article aims to establish analytic equations used in the future engineering design. Analysis on sealing contact mechanics It is typically metal-to-metal seal between the seal gasket and hubs. Theoretically, the sealing width has a

3 Yun et al. 3 r 1 Body 1 Body 2 O(y) z 1 z 2 x gasket (Body 1) a hub (Body 2) z P O(y) r d x r 2 z Figure 4. Contact principle diagram of the seal gasket. Figure 3. Non-conforming surfaces in contact at O. direct effect on the sealing performance when the sealing width increases, the traveling distance of the medium along the micro path increases, and the leakage will be reduced. For non-conforming surfaces in contact at O, as shown in Figure 3, r 1 and r 2 are the radii of the two bodies, the x-y plane is the common tangent plane of the two surfaces, and the z-axis lies along the common normal line directed positively into Body 2. The undeformed shapes of two surfaces are defined in this coordinates system by the form z 1 = f 1 (x, y) ð1þ z 2 = f 2 (x, y) where z 1 is the undeformed shape of Body 1; z 2 is the undeformed shape of Body 2; f 1 and f 2 are functions. Therefore, the gap, h, between them before loading can be given by h = z 1 + z 2 = f (x, y) ð2þ where f is a function. In the elastic contact with a compression load, each surface is supposed to be topographically smooth on both macro and micro scale. On the macro scale, the profiles of the surfaces are continuous up to their second derivative in the contact region. On the micro scale, it is irregular to disregard small surfaces, which would lead to discontinuous contact or high local variations of contact stress. Thus, the profile of the surface in the contact region of Body 1 close to O can be expressed approximately as z 1 = A 1 x 2 + B 1 y 2 + C 1 xy + ð3þ where higher order terms in x and y are neglected; A 1, B 1, and C 1 are positive constants. In the x-y coordinate system, the term of C 1 xy is zero. Equation (3) can be written as z 1 = 1 2r 0 x r 0 0 y 2 ð4þ 1 where r1 0 and r 00 1 are the principal curvature radii of the surface of Body 1 at O, and they are the maximum and minimum values of the curvature radii of all cross sections of the profile. Similarly, the expression of Body 2 can be written as z 2 = 1 2r 0 x r 0 0 y 2 ð5þ 2 where r2 0 and r 00 2 are the principal radii of curvature of the surface of Body 2 at O. Consider of equation (2) (5), gap, h, can be expressed as h = Ax 2 + By 2 = 1 2r 0 x2 + 1 y2 ð6þ 2r00 where A and B are positive constants; r# and r$ are principal relative curvature radii. Suppose that (1) the deformation of the sealing contact of the collet connector is elastic, (2) the contact area of each point is a circle, and (3) only the vertical load is applied on the contact surfaces. Thus, it satisfies Hertz theory. In the meantime, as the metal seal of the collet connector is a contact of a spherical surface on a conical surface, seal contact can be equivalent to the contact between a cylindrical body and a plane. The cross section can be simplified into a two-dimensional problem, as shown in Figure 4, which is the limit of an elliptical contact, when r$ is infinite. For the contact of two cylinders, 15 r1 0 = r 00 1, r 1 =, r2 0 = r 00 2, r 2 =, A = (1/r 1 +1/r 2 )/2, B = 0. When the surface of Body 2 is a plane, r 2 = N and A = 1/2r 1. Equation (6) can be rewritten as h = z 1 = Ax 2 = x2 2r 1 = x2 2r ð7þ where r is the radius of the spherical surface of the seal gasket, r = r 1. After loading, the elastic displacement (u z ) of the points within the contact area is u z = d h = d x2 ð8þ 2r where d is the displacement of Body 1 after the deformation. Meanwhile, outside the contact area

4 4 Advances in Mechanical Engineering u z.d x2 2r Differentiate u z with respect of x and yield ð9þ u z x = x ð10þ r As shown in Figure 4, seal half-width is a; the radius of the seal spherical surface is r = mm. The materials of the seal gasket and hubs are stainless steel 316 and 12Cr2Mo1, of which the elastic modulus and Poisson s ratio are E 1 = MPa, E 2 = MPa, m 1 = 0.25, and m 2 = 0.3. Thus, the equivalent elasticity modulus, E *, can be expressed as 1 E = 1 m m2 2 1 = E 1 E 2 1: ð11þ MPa The gradient in the elastic half space due to concentrated normal force, P, is 15 u z x = 2 ð a pe a q(x 0 ) x x 0 dx0 ð12þ where q is the contact stress due to P; q(x) is the distribution of the contact stress; x# is the distance between the point in the contact area and O. Plug equation (10) into equation (12) and yield ð a a q(x 0 ) x x 0 dx0 = pe 2r x ð13þ Equation (13) is a regular integral equation of the first kind, and its general solution 16 is q(x)= p 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 x 2 ð a a pe 2r x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 x 0 2 x x 0 dx 0 + where K is a constant, see Appendix 1. Equation (14) can be expressed as q(x)= pe 2r p 2 I p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi + a 2 x 2 p 2 p 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 x 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 x 2 ð14þ ð15þ where I is a function of x, see Appendix 1. Substituting K and I into equation (15), the distribution of contact stress can be rewritten as q(x)= pe 2r pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 x 2 p 1 + p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi P pe a 2 a 2 x 2 4r ð16þ The boundary condition of the contact stress is q( 6 a) = 0. Solve equation (16) and yield Figure 5. Seal structure. P = pe a 2 ð17þ 4r Plug equation (17) into equation (16) and rewrite the contact stress as 8 < E q(x)= 2r (a2 x 2 ) 1=2, a x a : 0, x\ a and x.a The maximum contact stress is q m = qðþ= 0 E a 2r The mean contact stress is ð18þ ð19þ Ð a a q = q(x)dx = pe a ð20þ 2a 8r Then the sealing half-width can be described as Axial preload analysis a = 8qr pe ð21þ The axial preload has a direct effect on the sealing performance of the collet connector. 17 In order to analyze this effect, the process can be divided into preloading mode and operating mode according to the load on the seal gasket. As shown in Figure 5, D k is the contact diameter, mm; a is the angle of the sealing surface, 20.

5 Yun et al. 5 hub f 0 α Q 0 D k seal gasket ρ α+ρ Figure 6. Force of the seal gasket in preloading mode. Axial preload of the seal gasket in the preloading mode. In the preloading mode, the seal gasket is compressed by top and bottom hubs from both ends. The compression turns the initial contact line into a strip due to deformation to ensure the reliable sealing performance of the collet connector. Since the seal is symmetric against the central plane, the analysis of one side is enough. As shown in Figure 6, not only the normal force (N 0 ) but also the friction (f 0 ) of the hubs acts on the gasket in the preloading mode N 0 = 2pD k aq f 0 = 2pD k aq tan r ð22þ ð23þ The combination of the axial components of N 0 and f 0 is the preload of the seal gasket, Q 0 sin (a + r) Q 0 = 2pD k aq = pd krq 2 m sin (a + r) cos r E ð24þ cos r where r is the frictional angle; r = arctan(m) = with the steel-to-steel friction coefficient of m = Axial preload of the seal gasket in operational mode. In the operating mode, the compression of hubs acts on the seal gasket, as well as the pressure produced by the internal medium. Then the axial force on the seal gasket is composed of two components, Q 0 01 due to the residual preload and Q 0 02 due to the internal pressure. First, the internal pressure acts on the surfaces of the gasket and hubs in the gap communicating with internal medium. It will reduce the preload on the spherical surface of the seal gasket and the conical surfaces of the hubs. The axial force produced by the residual preload 12 is Q 0 01 = 2pD kaq 0 tan (a r)= p2 q 0 m D ka tan (a r) ð25þ 2 where q 0 and q 0 m are the mean contact stress and the maximum contact stress of the contact surface in the operating mode, respectively. The axial force due to the internal pressure on the seal gasket 12 is N 0 Q 0 02 = p 4 D k 2 p ð26þ where p is the internal pressure (MPa). Thus, the axial force in the operating mode, Q 0 0, is the sum of Q0 01 and Q 0 02 Q 0 0 = p 4 D k 2 p + 2pD k aq 0 tan (a r) = p 4 D k 2 p + p2 q 0 m D ka tan (a r) 2 Minimum axial preload ð27þ The contact stress between the sealing surfaces is a good design reference of the connector. If the contact stress is not big enough to produce enough deformation to block the micro path, the sealing performance is not reliable. While if the contact stress is too big, the maximum stress on the contact area of the spherical surface and the conical surface will exceed the admissible limit, the spherical surface on the seal gasket will be damaged, and the sealing will fail. Therefore, the contact stress should be neither so big nor too small. 18 In the preloading mode, it is critical to ensure enough pressure on the surface of the seal gasket to block the leakage. The minimum preloading pressure, n, is the minimum pressure to satisfy sealing requirements, which depends on connector structure and material property. Generally, the harder the material is, the greater the value of n is. In the operating mode, the internal pressure acting on the seal structure tends to separate the sealing surfaces. Then micro path maybe appear as the force between the gasket and hubs decreases, and the internal medium will get through. Therefore, in order to ensure a good sealing effect, the residual preload must be big enough. The ratio of the pressure on the seal gasket and the internal pressure is the gasket factor, m, which depends on geometry and material property of seal gaskets. Generally, m is positively correlated with the hardness of the seal gasket. When the material of the seal gasket is stainless steel 316, the minimum preloading pressure, n, is MPa and the gasket factor, m, is In the preloading mode, the mean contact stress of the gasket and hubs, q, should not be less than n q n ð28þ In the operating mode, q 0, should not be less than mp q 0 mp ð29þ Substituting equations (28) and (29) into equations (24) and (27), respectively, the minimum axial preloads in two modes are

6 6 Advances in Mechanical Engineering maximum contact stress MPa P 1 Analytical results Simulation results axial preloading pressure MPa Figure 8. Curve of the maximum contact stress versus axial preloading pressure in the preloading mode. Figure 7. Boundary conditions of the seal structure. sin (a + r) Q 0 2pD k an cos r Q 0 0 p 4 D k 2 p + 2pD k amp tan (a r) FE analysis of the sealing contact ð30þ ð31þ To investigate the preloading effect of collets, the axial preload is applied on the end of the top hub directly in the FE analysis. There is just one loading step in this case and the seal structure model is axisymmetric about the centerline in the meantime. Two-dimensional (2D) axisymmetric FE model is applied, which will efficiently save computational resources and increase the computing speed. 19 The contacts between seal gasket and hubs are frictional contacts with friction coefficient of The boundary conditions are shown in Figure 7. A is the fixed support of the bottom hub to constrain the axial displacement of the bottom hub; B is the preloading pressure; C is the internal pressure. The mean contact stress is not available in the FE analysis; thus, the maximum contact stress is investigated and compared with analytical q m. Sealing contact characteristics in the preloading mode Relationship between the preload and the contact stress. In the preloading mode, q n = 179:3 MPa, plug it into equations (19), (21), and (30), then a = mm, q m MPa and Q kn. The area of the end of the top hub is S = mm 2 and then the minimum preloading pressure is Q 0 /S = MPa. Set the preloading pressure varying from 0 to 3 MPa with step of 0.02 MPa to investigate the relationship between the preload and the maximum contact stress. Figure 8 shows the curve of the maximum contact stress versus axial preloading pressure. The maximum contact stress increases with increasing the axial preloading pressure while the increment tends to decrease for both the analytical and simulation results. It is because that the initial contact is line contact, and the deformation on the contact spherical surface due to the increasing preloading pressure turns the line contact into a strip contact. At point P 1 shown in Figure 8, when preloading pressure is MPa, the maximum contact stress is MPa, which is a little higher than MPa and indicates that the preloading pressure fits the sealing requirements. The corresponding axial preload of the top hub is 66.5 kn. Comparing with the analytical result, the difference is 4.23%. Further increasing the preload (.3 MPa) tends to introduce more errors. The model should be carefully used when in a higher preload (.3 MPa). Simulation of sealing contact characteristics. Figure 9 shows the x-axis and y-axis deformation of the seal gasket in the preloading mode. The deformation of the seal gasket is negative along the two axes indicating that the seal gasket is compressed, as shown in Figure 9(a) and (b). In the direction of x-axis, the deformation of the seal gasket is symmetrical against the central plane. The minimum deformation is in the central area about mm. The maximum deformation appears on two ends about mm. The deformation on y-axis of the seal gasket is shown in Figure 9(c) and (d). The seal gasket moves downward with the top hub pressed and the bottom fixed, and the deformation on

7 Yun et al. 7 Figure 9. Deformation of the seal gasket when the preloading pressure is MPa: (a) contour plot of radial deformation, (b) radial deformation in section of MM, (c) contour plot of axial deformation, and (d) axial deformation in sections of N up N up and N down N down. cross section N up N up is greater than that on cross section N down N down. Figure 10 shows that the contact stress on the seal spherical surface decreases from the middle to the both sides in the preloading mode. The maximum contact stress is MPa, which is greater than q m = MPa and fits the sealing requirements. The trend of the ANSYS simulation results is consistent with which of the analytical results very well. Sealing width of the analytical result is the distance between C and D, mm. Sealing width of simulation result is the distance between A and B, mm, which is 21.8% bigger than the analytical result. This error is because of the assumptions in the analytic method and tends to make the design a little bit safe. Sealing contact characteristics in the operating mode Relationship between the preload and the contact stress in the operating mode. In the operating mode, p = 34.5 MPa, q 0 mp = 224:25 MPa, then a = mm, q 0 m 285:52 MPa, Q :6 kn by equations (22), (24), and (34). Minimum preloading pressure is Q 0 0 =S = 17:91 MPa. In order to get a quality relationship between the internal and the preloading pressure, set the preloading pressure varying from 17 to 20 MPa with step of 0.04 MPa. Figure 11 shows the curve of the maximum contact stress versus axial preloading pressure in the operating mode. The tends of the maximum contact stress of the analytical and simulation results are similarly. And at point P 2 in Figure 8, preloading pressure is MPa and the maximum contact stress is MPa, which is a little higher than MPa and satisfies the sealing requirements. The corresponding axial preload is kn, which is 1.69% higher than that of analytical result. Simulation of sealing contact characteristics in the operating mode. The simulation of sealing contact characteristics is to get the contact stress and sealing width in the operating mode with internal pressure of 34.5 MPa and compare with the analytical result. As shown in Figure 12(a) and (b), preloading pressure is MPa, and the maximum deformations of x-axis are and

8 8 Advances in Mechanical Engineering Figure 10. Contact stress of seal spherical surface in the preloading mode: (a) distribution of contact stress and (b) comparison of the contact stress of analytical and simulation results. maximum contact stress MPa P 2 Analytical results Simulation results axial preloading pressure MPa Figure 11. Curve of the maximum contact stress versus axial preloading pressure in the operating mode mm, respectively, when the internal pressures are 0 and 34.5 MPa. In Figure 12(c), the curve under 34.5 MPa is above the curve under 0 MPa, and the maximum deformation increment is mm, which occurs on the ends of the seal gasket. As shown in Figure 13(a) and (b), on y-axis, comparing with the internal pressure of 0 MPa, internal pressure of 34.5 MPa makes the seal gasket tend to separate from top and bottom hubs, and the preload decreases, and the maximum deformation in y-axis changes from to mm. Meanwhile, the bottom of the seal gasket is fixed; thus, the maximum deformation on cross section N down N down is lower than that on cross section N up N up (see Figure 13(c)). Contact stress of seal spherical surface is shown in Figure 14(a). Contact stress in the central area is the maximum and decreases on the both sides. The maximum contact stress is MPa, a little higher than the analytical result, MPa, and fits the sealing requirements. As shown in Figure 14(b), the contact stresses of analytical and simulation results are parabolic distribution in the contact area. The differences of the contact stresses are on the acceptable level. Sealing width of the analytical result is the distance between C and D, mm. Sealing width of simulation results is the distance between A and B, mm, which is 15.9% higher than the analytical result. This error is introduced by the assumptions of the analytical method, as mentioned above, and tends to make the design a little bit safe. Sealing contact characteristics with 1.5 times of the rated pressure Relationship between the preload and the contact stress with 1. 5 times of the rated pressure. To ensure the sealing reliability, based on ISO 21329, 20 the seal structure must be able to seal 1.5 times of the rated internal pressure, MPa. The corresponding parameters are p = MPa, q 0 mp = 336:375 MPa, and a = mm, q 0 m 428:29 MPa, Q :3 kn. Minimum preloading pressure is Q 0 0 =S = 27:53 MPa. In the simulation, set the preloading pressure varying from 25 to 32 MPa with step of 0.06 MPa and the maximum contact stress is shown in Figure 15. At point P 3, preloading pressure is MPa and the maximum contact stress is MPa, which is a little higher than MPa and fits the sealing requirements. The corresponding axial preload is kn. The difference is 0.30% comparing with the analytical result. Simulation of sealing contact characteristics with 1. 5 times of the rated internal pressure. When the preloading pressure is MPa, the deformations with internal pressure of 0 MPa and 1.5 times of the rated internal pressure in x-axis are shown in Figure 16(a) and (b). The maximum deformations in the cross section MM are

9 Yun et al. 9 Figure 12. Deformation of x-axis: (a) internal pressure is 0 MPa, (b) internal pressure is 34.5 MPa, and (c) deformation comparison on cross section MM. Figure 13. Deformation of y-axis: (a) internal pressure is 0 MPa, (b) internal pressure is 34.5 MPa, and (c) deformation comparison on cross section N up N up and N down N down and mm. The maximum difference of the deformation is in the central area of the gasket, as shown in Figure 16(c), and it increases mm when internal pressure increases from 0 to MPa. The deformations on y-axis are shown in Figure 17. The maximum deformations with internal pressures of 0 and MPa are and mm, respectively. The biggest deformation increases mm along the positive direction of y- axis with the increase of the internal pressure from 0 to MPa. As shown in Figure 18(a), the distribution of the contact stress with 1.5 times of the rated internal pressure is similar to that in the operating mode. The maximum contact stress is MPa, a little higher than MPa, and fits the sealing requirements. Figure 18(b) shows that the sealing width of analytical result is CD = mm, and that of simulation result is AB = mm, which is 30.0% bigger than analytical result. This error is because of the assumption in the analytical method, as mentioned previously.

10 10 Advances in Mechanical Engineering Figure 14. Contact stress of seal spherical surface in the operating mode: (a) distribution of contact stress and (b) contact stress comparison of analytical and simulation results. maximum contact stress MPa P3 Analytical results Simulation results axial preloading pressure MPa Figure 15. The maximum contact stress versus axial preloading pressure with the 1.5 times of the rated internal pressure. Experiment and analysis In order to validate the sealing performance of the collet connector and investigate the effect of the preload, an equivalent sealing experiment system was designed. The contact stress of the seal spherical surface cannot be measured directly; thus, the sealing width and the preload were used to measure the sealing performance. The experiment results have been compared with the analytical results to validate on the design of sealing contact mechanics. Experiment system The experiment system is shown in Figure 19, which is composed of a 200 tons (2000 kn) compression-testing Figure 16. Deformation on x-axis: (a) internal pressure is 0 MPa, (b) internal pressure is MPa, and (c) deformation comparison on cross section MM.

11 Yun et al. 11 Figure 17. Deformation on y-axis: (a) internal press is 0 MPa, (b) internal pressure is MPa, and (c) deformation comparison on cross sections N up N up and N down N down. Figure 18. Contact stress of seal spherical surface with 1.5 times of the rated internal pressure: (a) distribution of contact stress and (b) contact stress comparison of analytical and simulation results. Figure 19. Experiment system of the collet connector: (a) structure diagram of the experimental connector and (b) photo of the experiment system.

12 12 Advances in Mechanical Engineering Table 1. Results of sealing experiment with the internal pressure. Preload (kn) Initial pressure (MPa) Packing pressure (MPa) Table 2. Comparison of the experimental, analytical, and simulation results with the internal pressure. Internal pressure (MPa) Preload (kn) Sealing width (mm) Experiment Analysis FE simulation Experiment Analysis FE simulation FE: finite element. machine, a top hub, a bottom hub, a seal gasket, and a hydraulic pump. The material of the seal gasket is stainless steel 316, and the roughness of the seal spherical surface is 0.4 mm. The hubs are blind-hubs with one side open. The hydraulic pump pressure is 0 80 MPa. In the experiment, it pumps water into the experimental connector and the pressure gauge indicates the actual pressure in the experimental connector. Sealing experiment in the preloading mode In the preloading mode, the compression-testing machine applies a force upon the hubs until a given load. Then unload and measure the sealing width on the seal spherical surface with a vernier caliper. Indentations on the seal spherical surface before and after the experiment are shown in Figure 20, and the experimental, analytical, and simulation results are shown in Figure 21. It can be noted that the experimental results are a little smaller than the analytical and simulation ones, particularly when the preload is low. When the preload is 10 kn, the deformations are very small, even the analytical sealing width is only 0.54 mm. Meanwhile, the deformation on the edge is too small to be observed, which leads to an inaccurate measurement. The lower the preload is, the bigger the difference will be. Thus, the main cause of the differences is that the deformations are inconspicuous, and the measurement error is also a reason. The simulation results are greater than the analytical ones because of assumptions in the analytic method, and the difference between Figure 20. Comparison of the indentations of the seal spherical surface before and after the experiment. them has been discussed as mentioned above. In Figure 21, it can also be seen that when preload is 65 kn, which reaches the minimum preload required, analyzed in section Sealing contact characteristics in the preloading mode, the difference between the experimental sealing width (1.34 mm) and the analytical result (1.38 mm) is 2.90%, which meets the simulation result too. Sealing experiment with the internal pressure To test the relationship between the preload and the maximum packing pressure, the compression-testing machine applies a force upon the hubs and the hydraulic pump supplies water into the interior of the hubs until the pressure does not increase, maintain the pressure for 1 h, and record the pressure value. Then this pressure is the maximum packing pressure with the given preload. Increase the force and repeat the above

13 Yun et al. 13 sealing width (mm) Experimental results 0.5 Analytical results Simulation results preload (kn) Figure 21. Comparison of sealing width of the experimental, analytical, and simulation results. processes until the internal pressure is 1.5 times of the rated internal pressure (51.75MPa). The experimental results are listed in Table 1. When preload is 820 kn, the maximum packing pressure is 34.5 MPa; when preload is 1215 kn, the maximum packing pressure is 52 MPa, which is a little higher than MPa. Meanwhile, in the range of the preload, the maximum packing pressure increases with increasing the preload, which indicates that increasing the preload appropriately is helpful to improve the sealing performance of the connector. This experiment uses the leakage of the sealing structure as the criterion, while the analytical calculation and simulation use no leakage as the criterion; thus, the preload in this experiment is lower than that of the analytical calculation and simulation. As shown in Table 2, when the internal pressure is 34.5 MPa, comparing with the analytical results, the difference of the preload is 2.68% and the difference of sealing width is 4.09%; when the internal pressure is MPa, the difference of the preload is 6.20% and the difference of sealing width is 4.69%. The simulation results are also the biggest ones and the difference between the analytical and simulation results has been discussed as mentioned above, which meets the experimental results well. Conclusion This article proposes an analytical calculation method for the contact stress of the lens seal gasket, which has guiding significance in subsea collet connector design. The analytical calculation, FE simulation method, and experiment design of contact stress proposed in this article not only applies to the in subsea collet connectors and seal gaskets, but also contact characteristics of other seals with spherical surface and conical surface in ocean oil and gas industry. The specific conclusions are as follows: 1. According to Hertz contact theory, the contact between the seal gasket and hubs is equivalent to the contact of cylinders, and the contact stress mathematical model is established and then the parameters of contact, including sealing width, mean contact, stress, and maximum contact stress can be calculated. 2. Based on the mechanical analysis of the collet connector under various operating conditions, mechanical models of the seal structure in the preloading and operating modes are established to analyze the relationship between contact characteristics and preload. Then, calculation formulas of minimum preload of the seal gasket in the preloading and operating modes are deduced for the sealing mechanism of the metal seal To simulate the clamping effect of collets, 2D axisymmetric FE model is created with the axial preload applied on the end of the top hub. The sealing contact characteristics in the preloading and operating modes are simulated and compared with the analytical results. In the preloading mode, the difference between the analytical and simulation results of the axial preload is only 4.23%, and that of the sealing width is 21.8%; in the operating mode, the difference between the analytical and simulation results of the axial preload is only 1.69%, and that of the sealing width is 15.9%; in the mode with 1.5 times of the rated pressure, the difference between the analytical and simulation results of the axial preload is only 0.30%, and that of the sealing width is 30%. This error is because of the assumptions in the analytic method and tends to make the design a little bit safe. The differences of the simulation and the analytical results are acceptable. 4. An equivalent sealing experiment has been designed to test the sealing performance of the collet connector and investigate the effect of the preload. In the preloading mode, the experimental result of the sealing width (1.34 mm) is lower than the analytical result (1.38 mm), and the difference is 2.90%. In the experiment with internal pressure, when the internal pressure is 34.5 MPa, comparing with the analytical results, the difference of the preload is 2.68% and the difference of sealing width is 4.09%; when the internal pressure is 34.5 MPa, the difference of the preload is 6.20% and the difference of sealing width is 4.69%. 5. The analytical, the simulation, and experimental results of the sealing contact characteristics agree with each other very well. The accuracy of the analytical results is good enough to obtain the contact force, the sealing width, and the axial preload in the engineering application. Analytic equations obtained in this article can be used in the future engineering design and the simulation is not needed anymore.

14 14 Advances in Mechanical Engineering Acknowledgements The authors also would like to express their gratitude to all the friends in our scientific team, who have always been helping and supporting us without a word of complaint. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by a grant from the National Natural Science Foundation of China (grant no ) and the Royal Society International Exchange (grant no. IE141319). References 1. Yong B and Qiang B. Subsea pipelines and risers. 1st ed. Kidlington: Elsevier, 2013, pp Omiya Y and Sawa T. Thermal stress analysis and the sealing performance evaluation of bolted flange connection at elevated temperature. In: Pressure vessels and piping conference of the American-society-of-mechanicalengineers, Prague, June 2009, paper no. PVP , pp New York: ASME. 3. Kondo K, Sato K, Sawa T, et al. Sealing performance evaluation in bolted flange connections with ring joint gasket subjected to internal pressure. In: ASME pressure vessels and piping conference (PVP 2013), Paris, July 2013, paper no. PVP New York: ASME. 4. Tenma K, Kikuchi T, Sawa T, et al. Evaluation of sealing performance and fem calculations in bolted flange connections with ring joint gasket subjected to internal pressure. In: ASME pressure vessels and piping conference, Baltimore, MD, July 2011, paper no. PVP , pp New York: ASME. 5. Abid M, Chattha J and Khan K. Finite element analysis of a gasketed flange joint under combined internal pressure and thermal transient loading. In: Joint conference of the ASME pressure vessels and piping division/8th international conference on creep and fatigue at elevated temperatures, San Antonio, TX, July 2007, paper no. PVP , pp New York: ASME. 6. Tenma K, Kurosawa R and Sawa T. FEM stress analysis and evaluation of the sealing performance in flexible boxshape bolted flange gasketed joints subjected to internal pressure. In: Proceedings of the ASME international mechanical engineering congress and exposition (IMECE 2009), Lake Buena Vista, FL, November 2009, paper no. IMECE , pp New York: ASME. 7. Sawa T, Kurosawa R and Maezaki W. Stress analysis and sealing performance evaluation in rectangular boxshape bolted flange connection with gasket subjected to internal pressure. J Press Vess: T ASME 2011; 133: Madazhy R, Mathews S and Howard E. Analysis of contact pressure distribution on 3-bolt self-energized connector seals. In: ASME pressure vessels and piping conference, Prague, June 2009, paper no. PVP , pp New York: ASME. 9. Zhao H, Chen R, Luo X, et al. Metal sealing performance of subsea X-tree wellhead connector sealer. Chin J Mech Eng 2015; 28: Wang Z, Li Y, Zhang B, et al. Analysis of flanges mechanics and sealing on horizontal pipes connector in deep sea. Appl Mech Mater 2012; 164: Peng F, Duan M, Wang J, et al. Optimisation method for mathematical model of deepwater collet connector locking mechanism. Sh Offshore Struc 2016; 11: Boqin G, Xinhua L and Zheng T. Static seal design technology. 1st ed. Beijing, China: Standards Press of China, 2004, pp Li Z, Yun F, Jiang Y, et al. Analysis and experimental study of subsea connector s seal performance. J Harbin Eng Univ 2015; 36: Arghavani J, Derenne A and Marchand L. Prediction of gasket leakage rate and sealing performance through fuzzy logic. Int J Adv Manuf Tech 2002; 20: Johnson K. Contact mechanics. 1st ed. Cambridge: University of Cambridge, 1987, pp.1 3, 20, 30 32, 84 88, Mikhlin SG. Singular integral equations. Uspekhi Mat Nauk 1948; 3: Gong H, An S, Liu Y, et al. Seal characteristics and leakage model of metallic large-diameter lenticular gasket. Adv Mater Res 2013; 868: Perez-Rafols F, Larsson R and Almqvist A. Modelling of leakage on metal-to-metal seals. Tribol Int 2016; 94: Aleshin V, Kobyakov V and Seleznev V. A simulation technology for a full cycle of steel line pipe manufacturing operations. Adv Mech Eng 2011; 3: ISO 21329:2004. Petroleum and natural gas industries pipeline transportation systems: test procedures for mechanical connectors. Appendix 1 1. K is a constant and determined by concentrated normal force, P K = p ð a a 2. I is a function of x I = P V q(x)dx = pp ð 1 X 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X 0 2 X X 0 dx 0 where X = x/a and X# = x#/a, and I = px 2 (p=2)=p((x 2 =a 2 ) (1=2)).