Static metal sealing mechanism of a subsea pipeline mechanical connector

Abstract

The static metal sealing mechanism is essential for the structural design of subsea pipeline mechanical connector. By analyzing the sealing mechanism and contact pressure on sealing face, the critical condition of the seal is obtained. Based on the superposition theorem of elasticity, the formula of critical mean contact pressure ensuring seal is deduced. The finite element analysis result of critical mean contact pressure agrees with the analysis result very well and the critical mean contact pressure decreases with the increase in pipeline wall thickness and sealing width; however, the radius of pipeline has less influence on it. The critical mean contact pressure of 8-in subsea pipeline mechanical connector is 223.34 MPa and subsea pipeline mechanical connector is designed with ANSYS. The internal pressure experiments indicate that the sealing performance of final designed subsea pipeline mechanical connectors is higher than the design pressure of 4.5 MPa and all the initial designed ones cannot seal up to 4.5 MPa. By observing the sealing indentation on pipeline surface, when the mean contact pressure is higher than the critical mean contact pressure and the contact pressure variance is smaller, a full contact is seen to form a seal. Finally, the vibration experiments prove that the number of vibration is more than 10⁷ and final designed subsea pipeline mechanical connector meets the experiment specification.

Introduction

Because of the rapid growth of offshore oil and gas exploitation, more and more pipelines are applied to deep seabed to transport oil and gas, ¹ and the operation will increase in pipeline laying, merger, and maintenance. ² However, how to accomplish mechanical connections between pipelines reliably as well as the related equipment is the key problem to build a subsea oil and gas production system successfully. ³

For various operation depths and connection requirements, corporations in the Unites States, such as Oil state, Cameron, FMC technologies, Quality Connector Systems, and Oceaneering, have developed a series of mechanical connectors. Two typical connectors are well known in the industry. One is used for the flange-end pipelines and the other is for non-flange pipelines. The first type of the connectors includes common flange connector, articulated clamp connector, ⁴ and collet connector, ⁵ which are mainly used for connection in pipeline laying and merging. The installation of the connectors is achieved by a special tool, and the sealing principle of the connectors is to squeeze the gasket between the flange ends by the axial clamping force. The other type of connectors is applied in the connection of pipeline without flange. A flange can be fitted at the end of the pipeline by mechanical connection or using double-grip mechanical connectors, such as hydraulic double-grip and seal connector, ⁶ smart flange, ⁷ and pipeline connector for connecting a branch pipe to a carrier pipe, ⁸ which are also used for destructed pipeline renewal. During the connecting operation, destructed pipeline is cut off by the subsea cutting machinery, replaced by an intact pipeline prefabricated on board, and connected together by the mechanical connectors. The complicated design of the conventional subsea mechanical connectors results in inefficient operation, and particularly most of the bolts have to be preloaded. Then, a simple mechanical connector is designed for the connection of water supply and drainage pipelines under low or zero pressure on land. ⁹ On the basis of this connector design, Haelok Inc. in Switzerland supplies the connectors connecting pipeline less than 168.3 mm in diameter. To make this kind of connector work well in complicated subsea environment, this article presents an improved design, the subsea pipeline mechanical connector (SPMC), that can be used for a larger diameter and a higher internal pressure.

The seal of SPMC is a static metal seal without a gasket, which is the key technique to realize static metal sealing in the connector. The static metal seal is extensively applied in subsea non-weld connections to prevent the leakage of the transportation medium, because of its advantages in condition with high pressure, corrosion, and complicated external forces. ¹⁰ In the design of a mechanical connector with a gasket, the gasket factor and minimum gasket seating stress must be determined by the design codes^11,12 according to the internal pressure and structural geometry, but these codes are based on a series of assumptions and hence may not calculate the contact pressure and its distribution between flange ends and metal gasket.^13,14 ASME Boiler and Pressure Vessel Code (BPVC) Special Working Group-Bolted Flanged Joints (SWG-BFJ) ¹⁵ improved the design method of conventional bolted flange joints with gasket, and new gasket constants were defined. However, all conclusions were obtained by the assumption that the contact pressure was uniform on the sealing surface. Sawa et al. ¹⁶ had made a research in which the distribution of contact stress which governs the sealing performance is analyzed as a three-body contact problem, using an axisymmetric three-dimensional (3D) theory of elasticity. This work was further simplified by treating the pipe flange and the gasket as hollow cylinders. The formula of gasket contact stress and its variation through the gasket width was deduced on the basis of nonlinear characteristic of the gasket materials, ¹⁷ which was supported by the experiment and finite element analysis (FEA). Nonetheless, this method only predicted that flange rotation had effects on sealing performance. In addition, finite element method was applied to research the distribution of the contact pressure on the sealing surface.^18,19 Their work considered the influence of nonlinear characteristic of gasket and internal pressure; however, the analysis was based on the spiral wound gasket. Our previous article ²⁰ proved that the mean contact pressure and its distribution on sealing surface had an important effect on static metal seal. However, no relevant public research has been done on sealing mechanism of the mechanical connector without a gasket.

In order to obtain reliable sealing performance of SPMC, we analyzed the sealing mechanism of static metal sealing without a gasket and established the critical condition of the seal. Based on the superposition theorem of elasticity, the formula of the critical mean contact pressure is deduced, which can be used to calculate the minimum contact pressure of static metal seal. Meanwhile, the effects of the key structural parameters on the critical mean contact pressure are analyzed. The 8-in SPMC is designed with ANSYS, and the internal pressure experiments, indentation experiments of the sealing face, and vibration experiments are carried out.

Structure and sealing mechanism of SPMC

The structure of SPMC is shown in Figure 1, which is composed of a basic body and two press rings. The external surface of the press ring is a straight cylinder, and the internal surface is a profile that consists of cylindrical and conical surfaces. On the external surfaces of the basic body, two convex rings are produced on each side. The two ends of the basic body are gripper parts. On the internal surface of the basic body, two sealing rings are produced on each side.

Figure 1.

Structure of SPMC.

The installation of SPMC is shown in Figure 2. The press ring is compressed by installation tool and moves toward the middle of basic body along the axial direction. When the press ring moves toward the middle of SPMC, its internal conical surfaces will compress sealing rings and gripper parts to the external surface of the pipeline. The press ring is compressed further, which will increase the contact pressure on the sealing face. When the contact pressure exceeds the critical contact pressure, enough deformation on sealing face will happen, which forms a static metal sealing. Meanwhile, gripper parts are embedded into the external surface of the pipeline and connect the pipelines as a whole piece, which resists the complicated external force, protects sealing face, and improves reliability and lifetime of SPMC in submarine environment. In the installation of SPMC, Remote Operated Vehicle (ROV) can operate other equipments, and operators on board can observe work environment in underwater by ROV.

Figure 2.

Installation of SPMC.

Static metal sealing mechanism

The internal surfaces of the sealing rings are not perfect and its roughness is a little better than that of the pipeline, as shown in Figure 3. If the deformation on the sealing face is not enough, two sealing faces will not fully contact each other, which tends to generate the micro leak path, ²¹ particularly when external forces and internal pressure are applied, as shown in Figure 4. In order to achieve a reliable static metal sealing, the contact pressure on the sealing faces should reach a critical value. The micro leak paths will be blocked by reducing the height of micro-peaks and the depth and diameter of micro-pits or increasing the diameter of micro-peaks, which achieves a reliable static metal sealing. ²² The micro leak paths are formed by linking micro chambers formed by the overlay of micro-peaks and micro-pits on the sealing faces. Because micro-peaks and micro-pits on the sealing surface obey the normal distribution, the increase in the sealing face width will effectively reduce the probability of the micro leak paths generation. The contact pressure and sealing face width are two important parameters in the design of static metal seal.^23–25 In order to generate reliable static metal sealing, which will not be affected by the surface roughness, the contact pressure on the sealing face should be more than the Brinell hardness H_B of the softest material. ²⁶ Taking the design of Badische Anilin-und-Soda-Fabrik (BASF) lenticular gasket of Germany as a basis, the width of the sealing face is not less than 1/16 in. ²⁷

Figure 3.

Micro surface.

Figure 4.

Micro leak path.

Critical condition of seal

By considering subsea environment, the sealing rings and pipeline should be in elastic deformation and have enough elastic spring-back to resist the negative effects of internal pressure, temperature, and external load for seal. ²⁸ Assuming that the deformation of the sealing part is always kept within the range of elastic deformation range in installation and operation condition. L_AD is the width of the sealing face. Because of the intercoordination of elastic deformation between the sealing ring and the pipeline, the contact pressure on the boundary of the sealing face is higher than that of the central plane and the distribution is symmetric about the central plane of the sealing face, as shown in Figure 5. On the basis of static metal sealing mechanism in section “Static metal sealing mechanism,” the critical condition to realize reliable static metal sealing is

$$\{\begin{array}{c}{P}^{\mathrm{B}}=H_{\mathrm{B}}\hfill \\ \multicolumn{1}{c}{P^{\mathrm{C}}=H_{\mathrm{B}}}\\ \multicolumn{1}{c}{L_{\mathrm{AB}}+L_{\mathrm{CD}}=1/16\mathrm{in}\approx 1.6\mathrm{mm}}\end{array}$$

(1)

where P^B is the contact pressure at B, P^C is the contact pressure at C, L_AB is the distance between A and B, and L_CD is the distance between C and D.

Figure 5.

Distribution of contact pressure.

In order to realize the static metal sealing, $\overline{P}$ is the mean contact pressure on the sealing face and should be higher than the critical mean contact pressure

$$\overline{P}\ge {\overline{P}}^{*}=\frac{1}{L_{\mathrm{AD}}}\underset{0}{\overset{L_{\mathrm{AD}}}{\int }}P(z)\mathrm{dz}$$

(2)

where the contact pressure on the sealing face, P(z), meets the critical condition of equation (1) and ${\overline{P}}^{*}$ is the critical mean contact pressure.

Solution of critical mean contact pressure

The contact pressure on the sealing face includes two parts, that is, the contact pressure from installation and that from the internal pressure. Figure 6 shows the distribution of the internal pressure. The internal and external surfaces of the pipeline are under the internal pressure on the left of “W” line and there is no expansion tendency for the pipeline. However, on the right of W line, the internal pressure is applied on the internal surface only and expansion of the pipeline tends to happen, which increases the contact pressure on the sealing face and improves the sealing performance of SPMC. Based on the above analysis, the SPMC is a self-tight seal. As long as the mean contact pressure from installation is more than the critical mean contact pressure, the SPMCs will maintain their high sealing performance. Therefore, the mean contact pressure from installation is analyzed to design SPMC.

Figure 6.

Distribution of internal pressure.

Equivalent model of contact pressure

The static metal seal of SPMC is realized by the squeeze on the sealing surfaces. By the analysis of structural feature and force condition of the pipeline and SPMC, the geometrical profile, constraints, and boundary conditions of pipeline are symmetric about the central line, and the mechanical model of the pipeline is a 3D axial symmetric constitutive model.

The cylindrical coordinate system is established, as shown in Figure 7. z-axis coincides with the central line of the pipeline, r points along the radial direction, θ is along the circumferential direction, and r–θ plane coincides with the low end of band of the contact pressure. In axisymmetric elastic problem, stress, strain, and displacement of pipeline wall only relate to r and z. Because the contact pressure is non-uniform on the sealing face, we use a sectional approximation method to solve the formula of the contact pressure on the sealing face.

Figure 7.

Approximate model of contact pressure.

L _AD is divided uniformly into n sections, and the width of every section is l_i = L_AD/n = l. If n is big enough, we can assume that the contact pressure p_i is uniform with a width of l_i, and the equivalent mechanical model is n bands with uniform contact pressure p_i with a width of l_i applied on the external surface of the pipeline, where i = 1, 2,…, n.

Under band of uniform contact pressure p_i with a width of l_i, the stress in the pipeline can be solved by superposition method in Figure 8. The stress of the outer radius at z = 0 and z = l₁ is p_i/2 or p_i, which will not affect the distribution of stress in the pipeline, because the area is 0.

Figure 8.

Superposition of pressure.

Solution of pressure band with width l_i

Superposition method

On z positive axis, the stress state in the pipeline under band of p_i with a width of l_i is equivalent to that under band of p_i with a width of l₁ which moves (i − 1)l along the z-axis direction. In Figure 8, according to superposition principle of elasticity, the stress state in orz is equal to the sum of the stress state in o₁r₁z₁ and that in o₂r₂z₂, whose equation is

$${\sigma }_m^1(r,z)={\sigma }_m^{o_1}(r,z)+{\sigma }_m^{o_2}(r,z)$$

(3)

The relational expression is obtained by theory of elasticity

$${\sigma }_m^{o_2}(r,z)=-{\sigma }_m^{o_1}(r,z-l)$$

(4)

where ${\sigma }_m^1(r,z)$ is the stress in orz, ${\sigma }_m^{o_1}(r,z)$ is the stress in o₁r₁z₁, and ${\sigma }_m^{o_2}(r,z)$ is the stress in o₂r₂z₂; m = r, θ, z.

Solution of stress in o₁r₁z₁

When the influence of gravity is negligible, the equilibrium differential equation of stress in o₁r₁z₁ is

$$\{\begin{array}{c}\frac{\partial {\sigma }_r^{o_1}}{\partial r}+\frac{\partial {\tau }_{\mathrm{zr}}^{o_1}}{\partial z}+\frac{{\sigma }_r^{o_1}-{\sigma }_{\theta }^{o_1}}{r}=0\hfill \\ \multicolumn{1}{c}{\frac{\partial {\sigma }_z^{o_1}}{\partial z}+\frac{\partial {\tau }_{\mathrm{zr}}^{o_1}}{\partial r}+\frac{{\tau }_{\mathrm{zr}}^{o_1}}{r}=0}\end{array}$$

(5)

To meet equation (5), the stress function Ψ should be the biharmonic function

$${\nabla }^4\Psi =0$$

(6)

Four stress components ( ${\sigma }_r^{o_1}$ , ${\sigma }_{\theta }^{o_1}$ , ${\sigma }_z^{o_1}$ , ${\tau }_{\mathrm{rz}}^{o_1}$ ) can be expressed as

$$\{\begin{array}{c}{\sigma }_r^{o_1}=\frac{\partial }{\partial z}(\mu {\nabla }^2\Psi -\frac{{\partial }^2\Psi }{\partial r^2})\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta }^{o_1}=\frac{\partial }{\partial z}(\mu {\nabla }^2\Psi -\frac{1}{r}\frac{\partial \Psi }{\partial r})}\\ \multicolumn{1}{c}{{\sigma }_z^{o_1}=\frac{\partial }{\partial z}[(2-\mu ){\nabla }^2\Psi -\frac{{\partial }^2\Psi }{\partial z^2}]}\\ \multicolumn{1}{c}{{\tau }_{\mathrm{rz}}^{o_1}={\tau }_{\mathrm{zr}}^{o_1}=\frac{\partial }{\partial r}[(1-\mu ){\nabla }^2\Psi -\frac{{\partial }^2\Psi }{\partial z^2}]}\end{array}$$

(7)

where µ is the Poisson’s ratio, ∇² is the Laplace operator, and

$${\nabla }^2=\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^2}{\partial z^2}$$

(8)

If ∇²Ψ = 0, the stress function Ψ will meet biharmonic function (6). Substitute Ψ into equation (8), and the result is

$$\frac{{\partial }^2\Psi }{\partial r^2}+\frac{1}{r}\frac{\partial \Psi }{\partial r}+\frac{{\partial }^2\Psi }{\partial z^2}=0$$

(9)

Suppose Ψ = T(r)cos(kz), the equation (9) can be rewritten as

$$\frac{{\partial }^{2}T(r)}{\partial r^2}+\frac{1}{r}\frac{\partial T(r)}{\partial r}-k^{2}T(r)=0$$

(10)

Equation (10) is correctional zero-order Bessel function. Because the pipeline is hollow, the general solution of equation (10) is

$$T(r)=a_0{I}_0(\mathrm{kr})+a_1\mathrm{kr}{I}_1(\mathrm{kr})+b_0{K}_0(\mathrm{kr})+b_1\mathrm{kr}{K}_1(\mathrm{kr})$$

(11)

Substitute equation (11) into Ψ = T(r)cos(kz), the stress function, Ψ, can be expressed as

$$\Psi =\cos (\mathrm{kz})[a_0{I}_0(\mathrm{kr})+a_1\mathrm{kr}{I}_1(\mathrm{kr})+b_0{K}_0(\mathrm{kr})+b_1\mathrm{kr}{K}_1(\mathrm{kr})]$$

(12)

where four coefficients, a₀, a₁, b₀, and b₁, are solved by the boundary conditions of the pipeline, I₀(kr) is the first kind correctional zero-order Bessel function, I₁(kr) is the first kind correctional one-order Bessel function, K₀(kr) is the second kind correctional zero-order Bessel function, and K₁(kr) is the second kind correctional one-order Bessel function; the expressions are

$$\{\begin{array}{c}{I}_v(\mathrm{kr})=\sum _{n=0}^{+\infty }\frac{1}{n!\Gamma (v+n+1)}{\left(\frac{\mathrm{kr}}{2}\right)}^{v+2n}\hfill \\ \multicolumn{1}{c}{K_v(\mathrm{kr})=\underset{m\to v}{lim}\left\{\frac{\pi }{2\sin (m\pi )}[I_{-m}(\mathrm{kr})-I_m(\mathrm{kr})]\right\}}\end{array}$$

(13)

where v is equal to 0 or 1.

Assume a₀ = e₁b₁, a₁ = e₂b₁, b₀ = e₃b₁, and equation (12) can be rewritten as

$$\Psi =b_1\cos (\mathrm{kz})[e_1{I}_0(\mathrm{kr})+e_2\mathrm{kr}{I}_1(\mathrm{kr})+e_3{K}_0(\mathrm{kr})+\mathrm{kr}{K}_1(\mathrm{kr})]$$

(14)

The stress function Ψ meets equation (6) regardless of the value of k. Substitute b₁ = b(k)dk the equation (14) can be rewritten as

$$\mathrm{d}\Psi =\cos (\mathrm{kz})[e_1{I}_0(\mathrm{kr})+e_2\mathrm{kr}{I}_1(\mathrm{kr})+e_3{K}_0(\mathrm{kr})+\mathrm{kr}{K}_1(\mathrm{kr})]b(k)\mathrm{d}k$$

(15)

Equation (15) meets equation (6). Integrate equation (15) within (0, +∞), and the general solution of the stress function Ψ is

$$\Psi =\underset{0}{\overset{+\infty }{\int }}\mathrm{d}\Psi =\underset{0}{\overset{+\infty }{\int }}b(k)\cos (\mathrm{kz})[e_1{I}_0(\mathrm{kr})+e_2\mathrm{kr}{I}_1(\mathrm{kr})+e_3{K}_0(\mathrm{kr})+\mathrm{kr}{K}_1(\mathrm{kr})]\mathrm{d}k$$

(16)

In o₁r₁z₁, the force boundary conditions of the pipeline are

$$\begin{array}{c}\multicolumn{1}{c}{{{\sigma }_r^{o_1}|}_{r=r_{\mathrm{in}}}=0}\\ \multicolumn{1}{c}{{{\tau }_{\mathrm{rz}}^{o_1}|}_{r=r_{\mathrm{out}},r_{\mathrm{in}}}=0}\\ \multicolumn{1}{c}{{{\sigma }_r^{o_1}|}_{r=r_{\mathrm{out}}}=\{\begin{array}{c}{p}_i/2z>0\hfill \\ \multicolumn{1}{c}{0z=0}\\ \multicolumn{1}{c}{-p_i/2z<0}\end{array}}\end{array}$$

(17)

Because

$$\underset{0}{\overset{+\infty }{\int }}\frac{\sin (\mathrm{kz})}{k}\mathrm{d}k=\{\begin{array}{c}\pi /2z>0\hfill \\ \multicolumn{1}{c}{0z=0}\\ \multicolumn{1}{c}{-\pi /2z<0}\end{array}$$

(18)

So

$${{\sigma }_r^{o_1}|}_{r=r_{\mathrm{out}}}=\frac{p_i}{\pi }\underset{0}{\overset{+\infty }{\int }}\frac{\sin (\mathrm{kz})}{k}\mathrm{d}k=\{\begin{array}{c}{p}_i/2z>0\hfill \\ \multicolumn{1}{c}{0z=0}\\ \multicolumn{1}{c}{-p_i/2z<0}\end{array}$$

(19)

where r_out is the outer radius of the pipeline and r_in is the inner radius.

By solving equations (7), (13), and (16) with boundary conditions (17) and (19), the expressions (e₁, e₂, e₃, b(k)) are

$$\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]={\left[\begin{array}{ccc}{G}_1(k{r}_{\mathrm{out}})& G_2(k{r}_{\mathrm{out}})& G_3(k{r}_{\mathrm{out}})\\ G_1(k{r}_{\mathrm{in}})& G_2(k{r}_{\mathrm{in}})& G_3(k{r}_{\mathrm{in}})\\ H_1(k{r}_{\mathrm{in}})& H_2(k{r}_{\mathrm{in}})& H_3(k{r}_{\mathrm{in}})\end{array}\right]}^{-1}\left[\begin{array}{c}Q(k{r}_{\mathrm{out}})\\ Q(k{r}_{\mathrm{in}})\\ P(k{r}_{\mathrm{in}})\end{array}\right]$$

(20)

where

$$\{\begin{array}{c}{G}_1(\mathrm{kr})=I_1(\mathrm{kr})\hfill \\ \multicolumn{1}{c}{G_2(\mathrm{kr})=\mathrm{kr}{I}_0(\mathrm{kr})+2(1-\mu )I_1(\mathrm{kr})}\\ \multicolumn{1}{c}{G_3(\mathrm{kr})=-K_1(\mathrm{kr})}\\ \multicolumn{1}{c}{Q(\mathrm{kr})=\mathrm{kr}{K}_0(\mathrm{kr})-2(1-\mu )K_1(\mathrm{kr})}\\ \multicolumn{1}{c}{H_1(\mathrm{kr})=I_0(\mathrm{kr})-I_1(\mathrm{kr})/(\mathrm{kr})}\\ \multicolumn{1}{c}{H_2(\mathrm{kr})=(1-2\mu )I_0(\mathrm{kr})+\mathrm{kr}{I}_1(\mathrm{kr})}\\ \multicolumn{1}{c}{H_3(\mathrm{kr})=K_0(\mathrm{kr})+K_1(\mathrm{kr})/(\mathrm{kr})}\\ \multicolumn{1}{c}{P(\mathrm{kr})=(1-2\mu )K_0(\mathrm{kr})-\mathrm{kr}{K}_1(\mathrm{kr})}\end{array}$$

(21)

$$b(k)=\frac{p_i}{\pi k^4\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_2(k{r}_{\mathrm{out}})& H_3(k{r}_{\mathrm{out}})\end{array}\right]{\left[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right]}^T-P(k{r}_{\mathrm{out}})}$$

(22)

Substitute equation (22) into equation (16), and the stress function, Ψ, can be rewritten as

$$\Psi =p_i\underset{o}{\overset{+\infty }{\int }}\frac{\cos (\mathrm{kz})}{\pi k^4}\frac{\{\left[\begin{array}{c}{I}_0(\mathrm{kr})\mathrm{kr}{I}_1(\mathrm{kr})K_0(\mathrm{kr})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T+\mathrm{kr}{K}_1(\mathrm{kr})\}}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_2(k{r}_{\mathrm{out}})H_3(k{r}_{\mathrm{out}})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(k{r}_{\mathrm{out}})\}}\mathrm{d}k$$

(23)

By solving equations (3), (4), (7), (8), and (23), ${\sigma }_r^1(r,z)$ , ${\sigma }_{\theta }^1(r,z)$ , and ${\sigma }_z^1(r,z)$ can be expressed as

$$\{\begin{array}{c}{\sigma }_r^1(r,z)=p_i\underset{0}{\overset{+\infty }{\int }}\frac{1}{\pi }\frac{\sin (\mathrm{kz})-\sin [k(z-l)]}{k}\frac{\{\left[\begin{array}{c}{H}_1(\mathrm{kr})H_2(\mathrm{kr})H_3(\mathrm{kr})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(\mathrm{kr})\}}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_{2}k{r}_{\mathrm{out}}{H}_{3}k{r}_{\mathrm{out}}\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-\mathrm{Pk}{r}_{\mathrm{out}}\}}\mathrm{d}k\hfill \\ \multicolumn{1}{c}{{\sigma }_{\theta }^1(r,z)=p_i\underset{0}{\overset{+\infty }{\int }}\frac{1}{\pi }\frac{\sin (\mathrm{kz})-\sin [k(z-l)]}{k}\frac{\left[\begin{array}{c}{I}_1(\mathrm{kr})/(\mathrm{kr})(1-2\mu )I_0(\mathrm{kr})-K_1(\mathrm{kr})/(\mathrm{kr})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_2(k{r}_{\mathrm{out}})H_3(k{r}_{\mathrm{out}})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(k{r}_{\mathrm{out}})\}}\mathrm{d}k}\\ \multicolumn{1}{c}{-p_i\underset{0}{\overset{+\infty }{\int }}\frac{1}{\pi }\frac{\sin (\mathrm{kz})-\sin [k(z-l)]}{k}\frac{(1-2\mu )K_0(\mathrm{kr})}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_2(k{r}_{\mathrm{out}})H_3(k{r}_{\mathrm{out}})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(k{r}_{\mathrm{out}})\}}\mathrm{d}k}\\ \multicolumn{1}{c}{{\sigma }_z^1(r,z)=p_i\underset{0}{\overset{+\infty }{\int }}\frac{1}{\pi }\frac{\sin (\mathrm{kz})-\sin [k(z-l)]}{k}\frac{\left[\begin{array}{c}-I_0(\mathrm{kr})-2(2-\mu )I_0(\mathrm{kr})-\mathrm{kr}{I}_1(\mathrm{kr})-K_0(\mathrm{kr})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_2(k{r}_{\mathrm{out}})H_3(k{r}_{\mathrm{out}})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(k{r}_{\mathrm{out}})\}}\mathrm{d}k}\\ \multicolumn{1}{c}{+p_i\underset{0}{\overset{+\infty }{\int }}\frac{1}{\pi }\frac{\sin (\mathrm{kz})-\sin [k(z-l)]}{k}\frac{2(2-\mu )K_0(\mathrm{kr})-\mathrm{kr}{K}_1(\mathrm{kr})}{\{\left[\begin{array}{c}{H}_1(k{r}_{\mathrm{out}})H_2(k{r}_{\mathrm{out}})H_3(k{r}_{\mathrm{out}})\hfill \end{array}\right]{[\begin{array}{c}{e}_1{e}_2{e}_3\hfill \end{array}]}^T-P(k{r}_{\mathrm{out}})\}}\mathrm{d}k}\end{array}$$

(24)

Based on elasticity method, physical equation and geometric equation of 3D axial symmetric model are

$${\epsilon }_{\theta }^1=\frac{1}{E}[{\sigma }_{\theta }^1-\mu ({\sigma }_z^1+{\sigma }_r^1)]$$

(25)

$${\epsilon }_{\theta }^1=\frac{u_r^1}{r}$$

(26)

Using equations (24)–(26), the radial displacement of the pipeline can be obtained as

$$\begin{array}{c}{u}_r^1(r,z)=p_i{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r}{E\pi }\frac{\sin (kz)-\sin \left[k(z-l)\right]}{k}\frac{\left[\begin{array}{ccc}{I}_1(kr)/(kr)& I_0(kr)& -K_1(kr)/(kr)\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T}{\left\{\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_2(k{r}_{\mathrm{out}})& H_3(k{r}_{\mathrm{out}})\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-P(k{r}_{\mathrm{out}})\right\}}\mathrm{d}k}\\ -p_i{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r}{E\pi }\frac{\sin (kz)-\sin \left[k(z-l)\right]}{k}\frac{K_0(kr)}{\left\{\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_2(k{r}_{\mathrm{out}})& H_3(k{r}_{\mathrm{out}})\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-P(k{r}_{\mathrm{out}})\right\}}\mathrm{d}k}\end{array}$$

(27)

Verification of stress and displacement

In order to verify equations (24) and (27), stress components ( ${\sigma }_r^1$ , ${\sigma }_z^1$ , ${\sigma }_{\theta }^1$ ) and radial displacement ( $u_r^1$ ) are compared with the results of FEA. Define λ = (r − r_in)/(r_out − r_in), when r_out =  is 109.5 mm, r_in = 100 mm, l =10mm and p_i =–1 MPa, the distribution of stress ( ${\sigma }_r^1$ , ${\sigma }_z^1$ , ${\sigma }_{\theta }^1$ ) and displacement ( $u_r^1$ ) along the z-axis are shown in Figure 9.

Figure 9.

Stress and displacement: (a) distribution of ${\sigma }_r^1$ along z-axis, (b) distribution of ${\sigma }_z^1$ along z-axis, (c) distribution of ${\sigma }_{\theta }^1$ along z-axis, and (d) distribution of $u_r^1$ along z-axis.

Figure 9(a) shows the distribution of stress ${\sigma }_r^1$ along z-axis, Figure 9(b) shows the distribution of stress ${\sigma }_z^1$ , Figure 9(c) shows the distribution of stress ${\sigma }_{\theta }^1$ , and Figure 9(d) shows the distribution of displacement $u_r^1$ . The stress components ( ${\sigma }_r^1$ , ${\sigma }_z^1$ , ${\sigma }_{\theta }^1$ ) and displacement component ( $u_r^1$ ) are symmetrical about the central plane and the absolute values decrease progressively when λ decreases. When λ is 0.4 or 1, the theoretical results ( ${\sigma }_r^1$ , ${\sigma }_z^1$ , ${\sigma }_{\theta }^1$ , $u_r^1$ ) agree with FEA results very well. The minor error is from the assumption of uniform pressure in a band as well as computation error of finite element method. Figure 9 also indicates that the expressions of stress and displacement are deduced correctly by elasticity.

Displacement

The radial displacement $u_r^i$ is equal to the radial displacement $u_r^1$ in the pipeline under band of contact pressure p_i (width is l₁) moved (i − 1)l along z-axis. Solving equation (27), the radial displacement, $u_r^i$ , is

$$\begin{array}{c}{u}_r^i(r,z)=p_i{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r}{E\pi }\frac{\sin \left\{k\left[z-(i-1)l\right]\right\}-\sin \left[k(z-il)\right]}{k}\frac{\left[\begin{array}{ccc}{I}_1(kr)/(kr)& I_0(kr)& -K_1(kr)/(kr)\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T}{\left\{\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_{2}k{r}_{\mathrm{out}}& H_{3}k{r}_{\mathrm{out}}\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-Pk{r}_{\mathrm{out}}\right\}}\mathrm{d}k}\\ -p_i{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r}{E\pi }\frac{\sin \left\{k\left[z-(i-1)l\right]\right\}-\sin \left[k(z-il)\right]}{k}\frac{K_0(kr)}{\left\{\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_2(k{r}_{\mathrm{out}})& H_3(k{r}_{\mathrm{out}})\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-P(k{r}_{\mathrm{out}})\right\}}\mathrm{d}k}\end{array}$$

(28)

Contact pressure on sealing surface

According to the analysis of section “Equivalent model of contact pressure,” the radial displacement in a pipeline under n bands of contact pressure meets the superposition principle, and based on equation (28), the radial displacement is

$$u_r(r,z)=\sum _{i=1}^n{u}_r^i(r,z)$$

(29)

If we choose n points at different locations of the pipeline, the radial displacement u_r at point j is

$$u_r(r_j,z_j)=\sum _{i=1}^n{u}_r^i(r_j,z_j)$$

(30)

Solve equation (30) with n points radial displacements as the input, and contact pressure p_i can be expressed as

$$\left[\begin{array}{c}{p}_1\hfill \\ \multicolumn{1}{c}{p_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{p_n}\end{array}\right]={\left[\begin{array}{cccc}{a}_{11}\hfill & a_{12}\hfill & \cdots \hfill & a_{1n}\hfill \\ \multicolumn{1}{c}{a_{21}}& a_{22}\hfill & \cdots \hfill & a_{2n}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{a_{n1}}& a_{n2}\hfill & \cdots \hfill & a_{\mathrm{nn}}\hfill \end{array}\right]}^{-1}\left[\begin{array}{c}{u}_r(r_1,z_1)\hfill \\ \multicolumn{1}{c}{u_r(r_2,z_2)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{u_r(r_n,z_n)}\end{array}\right]$$

(31)

where

$$\begin{array}{c}{a}_{ji}={\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r_j}{E\pi }\frac{\sin \left\{k\left[z_j-(i-1)l\right]\right\}-\sin \left[k(z_j-il)\right]}{k}\frac{\left[\begin{array}{ccc}{I}_1(k{r}_j)/(k{r}_j)& I_0(k{r}_j)& -K_1(k{r}_j)/(k{r}_j)\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T}{\left\{\left[\begin{array}{ccc}{H}_1\left(k{r}_{\mathrm{out}}\right)& H_2\left(k{r}_{\mathrm{out}}\right)& H_3\left(k{r}_{\mathrm{out}}\right)\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-P\left(k{r}_{\mathrm{out}}\right)\right\}}\mathrm{d}k}\\ -{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{(1+\mu )r_j}{E\pi }\frac{\sin \left\{k\left[z_j-(i-1)l\right]\right\}-\sin \left[k(z_j-il)\right]}{k}\frac{K_0(k{r}_j)}{\left\{\left[\begin{array}{ccc}{H}_1(k{r}_{\mathrm{out}})& H_2(k{r}_{\mathrm{out}})& H_3(k{r}_{\mathrm{out}})\end{array}\right]{[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}]}^T-P(k{r}_{\mathrm{out}})\right\}}\mathrm{d}k}\end{array}$$

(32)

When r_j equals r_out, equation (31) shows the relation between radial displacement on pipeline surface and contact pressure on sealing surface

$$\left[\begin{array}{c}{p}_1\hfill \\ \multicolumn{1}{c}{p_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{p_n}\end{array}\right]={\left[\begin{array}{cccc}{a}_{11}^{*}\hfill & a_{12}^{*}\hfill & \cdots \hfill & a_{1n}^{*}\hfill \\ \multicolumn{1}{c}{a_{21}^{*}}& a_{22}^{*}\hfill & \cdots \hfill & a_{2n}^{*}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{a_{n1}^{*}}& a_{n2}^{*}\hfill & \cdots \hfill & a_{\mathrm{nn}}^{*}\hfill \end{array}\right]}^{-1}\left[\begin{array}{c}{u}_r^{*}(r_{\mathrm{out}},z_1)\hfill \\ \multicolumn{1}{c}{u_r^{*}(r_{\mathrm{out}},z_2)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{u_r^{*}(r_{\mathrm{out}},z_n)}\end{array}\right]$$

(33)

When P  = [p₁, p₂,…, p_n] meets the critical condition of equation (1), the critical mean contact pressure on the sealing face is

$${\overline{P}}^{*}=\frac{1}{L_{\mathrm{AD}}}\underset{0}{\overset{L_{\mathrm{AD}}}{\int }}{P}^{*}(z)\mathrm{d}z\approx \frac{1}{n}\sum _{i=1}^n{p}_i$$

(34)

Analysis of critical mean contact pressure

As the critical mean contact pressure, ${\overline{P}}^{*}$ , is the determinant of SPMC’s sealing performance, the influence of the structure parameters of the pipeline and sealing ring (pipeline wall thickness T and sealing face width L_AD) on the critical mean contact pressure can be analyzed.

When L_AD is equal to 3 mm and r_out is equal to 109.5 or 84.15 mm, the relation of the critical mean contact pressure and pipeline wall thickness is shown in Figure 10. The result indicates that the FEA result agrees with that of analysis and both of them decrease nonlinearly with the increase in pipeline wall thickness T. When r_out = 109.5 mm, the critical mean contact pressure ${\overline{P}}^{*}$ is 252.27 MPa at T = 5.5 mm and is 216.39 MPa at T = 14.5 mm. When r_out = 84.15 mm, the critical mean contact pressure ${\overline{P}}^{*}$ is 245.90 MPa at T = 5.5 mm and is 215.33 MPa at T = 14.5 mm. The radius of the pipeline is increased by 30.1%, but the critical mean contact pressure is increased by 2.6%, which shows the radius of the pipeline influences on the critical mean contact pressure on a low level. Therefore, for different pipeline radius, the pipeline wall thickness T is an important parameter for the critical mean contact pressure.

Figure 10.

Relation of T and ${\overline{P}}^{*}$ .

When T is equal to 9.5 mm and r_out is equal to 109.5 or 84.15 mm, the relation of the critical mean contact pressure and the width of sealing face is shown in Figure 11. The result indicates that the FEA result agrees with that of analysis very well and both of them decrease almost linearly with the increase in the sealing face width L_AD. Under the same width of the sealing face, the critical mean contact pressures of different pipeline radial are nearly the same. When r_out = 109.5 mm, T = 9.5 mm (r_in = 100 mm) and L_AD = 10 mm, the critical mean contact pressure ${\overline{P}}^{*}$ is 117.82 MPa. When r_out = 109.5 mm, the critical mean contact pressure ${\overline{P}}^{*}$ with L_AD = 2 mm is 48.5% higher than that with L_AD = 6.5 mm. When r_out = 84.15 mm, the critical mean contact pressure ${\overline{P}}^{*}$ with L_AD = 2 mm is 48.3% higher than that with L_AD = 6.5 mm. Therefore, for different pipeline radius, the width of sealing face L_AD is an important parameter for the critical mean contact pressure.

Figure 11.

Relation of L_AD and ${\overline{P}}^{*}$ .

Design of 8-in SPMC

Critical mean contact pressure

We take 8-in pipelines with API 5L standards (specification for line pipe) as an example to demonstrate the design of SPMC. The outer radius of the pipeline, r_out, is 109.5 mm, the inner radius, r_in, is 100 mm, and the yield strength, σ_s, is 235 MPa. According to the design rules for static metal seal, the yield strength of sealing ring should be higher than that of the pipeline, the material of press rings and the basic body is Q345, and its yield strength is 345 MPa. The Brinell hardness of pipeline, H_B, is HB158. With regard to comprehensive factors such as sealing mechanism, lifetime, and loading force in installation, the width of sealing ring L_AD is determined as 3 mm, so the value of z at B point is 0.8 mm and the value of z at C point is 2.2 mm. The L_AD is divided into 20 sections with equal spaces and n is 20. Taking the above parameters into equation (34), the critical mean contact pressure for 8-in SPMC, ${\overline{P}}^{*}$ , is 223.34 MPa.

Finite element design

Structural parameters analysis

The contact pressure and contact uniformity on sealing face affect sealing performance of SPMC directly. In the design of SPMC, the contact pressure is replaced by the mean contact pressure and contact uniformity is expressed by contact pressure variance. If the mean contact pressure is higher than ${\overline{P}}^{*}$  = 223.34 MPa, the smaller the contact pressure variance, the better the contact uniformity and the higher the sealing performance of SPMC. By analyzing SPMC’s structure, the interference D₁ and offset D₂ affect the mean contact pressure and contact pressure variance on major sealing face, respectively. The interference D₃ and offset D₄ affect the mean contact pressure and contact pressure variance on minor sealing face, respectively, as shown in Figure 12. The variables (D₁, D₂, D₃, D₄) are designed with ANSYS.

Figure 12.

Key variables of SPMC.

Finite element model

The structure of SPMC is axisymmetric and two-dimensional (2D) finite element model is established to save computing time. Because the structure of SPMC is complicated, in order to obtaining high-quality elements, the basic body and press ring are divided into some quadrilateral planes which can be meshed by a mapping method. The elements on the sealing faces are refined to improve the accuracy. The basic element is PLANE182, the contact element is CONTA172, and the target element is TARGE169. The finite element model is shown in Figure 13.

Figure 13.

Finite element model.

Results of FEA

Figure 14(a) shows the distribution of the contact pressure on the major sealing face in the initial design. As the contact pressure variance is 40,316 MPa², the contact of the sealing face is non-uniform, which explains the sealing face is line contact and tends to generate sealing failure. The mean contact pressure is 171.9 MPa and is less than the critical mean contact pressure of 223.34 MPa, which indicates the deformation of the sealing face is not enough and tends to form micro leak path. The distribution of the contact pressure on the major sealing face in the final design is shown in Figure 14(b). The contact pressure variance is 17795 MPa² and decreases by 55.86%. The mean contact pressure is 245.2 MPa and is higher than the critical mean contact pressure of 223.34 MPa. Compared with initial design, the contact pressure increases and contact uniformity is improved in the final design, which can form a reliable seal.

Figure 14.

Contact pressure on major sealing face: (a) initial design and (b) final design.

Figure 15(a) and (b) shows the contact pressure of the initial design and that of the final design on the minor sealing face, respectively. The mean contact pressure of initial design is 148.8 MPa and the contact pressure variance is 20029 MPa². The mean contact pressure of the final design is 262.6 MPa and higher than the critical mean contact pressure of 223.34 MPa. The contact pressure variance is 16912 MPa² and decreases by 15.56%. The final design improves the sealing performance of the minor sealing face.

Figure 15.

Contact pressure on minor sealing face: (a) initial design and (b) final design.

From Figure 16(a) and (b), the Mises stress of the pipeline approximately equals 197.95 MPa, which only makes elastic deformation on the pipeline. When SPMC is under complicated external load, there is enough elastic spring-back to keep the sealing. The Mises stress of the sealing rings and press ring is less than the yield strength, which will keep SPMC safe in operation. After the design of SPMC, the values of the variables are listed in Table 1.

Figure 16.

Distribution of Mises stress of final designed SPMC: (a) major sealing ring and (b) minor sealing ring.

Table 1.

Value of variables.

D 1 (mm)	D 2 (mm)	D 3 (mm)	D 4 (mm)
2.1	2.2	2.65	6

Experiment

The design pressure of SPMC is 4.5 MPa. In order to validate that the critical mean contact pressure can form reliable sealing face for SPMC, the internal pressure experiments, indentation experiments of sealing faces, and the vibration experiments are carried out. These experiments have followed the experiment specification of the mechanical connector. ²⁹ The experimental devices include two initial designed SPMCs, four final designed SPMCs, pressure gauge, shutoff valve, water hydraulic pump, pipelines, Instron8801 (vibration source), and so on. The medium is water and the temperature is 20°C.

The measurement range of the pressure gauge used in the experiments is 0–16 MPa and the minimum scale value is 0.5 MPa. The accuracy grade of the pressure gauge is 1.6 and full-scale error is 0.256 MPa.

Internal pressure experiments

The schematic diagram of the internal pressure experiments is shown in Figure 17. The internal pressure of 2 MPa is increased every 5 min by the water hydraulic pump. When the internal pressure reaches the given value, the shutoff valve is turned off to keep the pressure. When the internal pressure is increased up to a certain value, the value of pressure gauge starts to drop or leakage occurs, and this critical value is the sealing performance of SPMC. Figure 18 shows experiment facility of the internal pressure experiments.

Figure 17.

Schematic diagram of internal pressure experiment.

Figure 18.

Facility of internal pressure experiment.

Table 2 is the results of the internal pressure experiments of the initial designed SPMCs and final designed ones. The results indicate that the sealing performance of the initial designed SPMCs is less than the design pressure of 4.5 MPa and that of the final designed ones is more than the design pressure. A pressure drop of 2.0 MPa can be seen in the experiment of the first one of the initial designed SPMCs and the reason is that there are some microdefects on pipeline surface and the micro leakage paths are formed under a low mean contact pressure. A pressure drop of 4.0 MPa is witnessed in the experiment of the second one of the initial designed SPMCs and the reason is that the sealing faces cannot contact fully and is linear contact with a big contact pressure variance, which tends to leak under internal pressure. The final designed SPMCs are designed with finite element method and its mean contact pressure is higher than the critical mean contact pressure of 223.34 MPa, which is enough to form a static sealing. The contact pressure variance is very small and the sealing faces fully contact. Larger mean contact pressure and smaller contact pressure variance will block leakage paths entirely. In the beginning of section “Solution of critical mean contact pressure,” when the internal pressure is higher, the sealing performance of SPMC is higher. Therefore, the SPMC is a self-tight seal and can resist a high internal pressure. Meanwhile, the result shows that the design method based on the critical mean contact pressure is valid for SPMC.

Table 2.

Result for internal pressure experiments.

Class	Group	Keeping time (min)	Pressure (MPa)	Pressure gauge
Initial design	First	5	2.0	Drop
	Second	5	4.0	Drop
Final design	First	5	14.0	No change
	Second	5	10.0	No change

Indentation experiments of sealing faces

To analyze the reasons to generate leakage for the initial designed SPMCs, the initial and final designed SPMCs are divided by wire-electrode cutting method and the indentation on pipeline surface is observed. One half of SPMC is shown in Figure 19.

Figure 19.

Cutting part of SPMC.

Figure 20 shows the indentation on pipeline surface for the initial designed SPMC. The indentation locating in “A” point of the major sealing face is not continuous and the leakage occurs. When the internal pressure is increased to 2.0 MPa, a pressure drop occurs. That is because the mean contact pressure is less than the critical mean contact pressure, which cannot make microstructure on the sealing face generate enough deformation and micro leakage paths be blocked entirely. Although the indentation at “B” location is continuous, it is not uniform. The indentation depth locating at the bottom of the sealing face is more than that locating on the top of the sealing face and the real contact width is less than the critical width of 1.6 in. When the SPMC is under internal pressure, the seal fails easily. The mean contact pressure and contact pressure variance ensure the sealing performance of SPMC.

Figure 20.

Indentation for initial designed SPMC.

Figure 21 shows the indentation on pipeline surface for the final designed SPMC. The mean contact pressure is higher than the critical mean contact pressure and the indentation on pipeline surface is continuous. On the partial enlarged observation, the sealing face has a similar indentation depth, the contact face is uniform, and the width is 3 mm. The sealing performance of SPMC is improved observably.

Figure 21.

Indentation for final designed SPMC.

Vibration experiments

Because the initial designed SPMCs show a poor sealing ability in the internal pressure experiments, the vibration experiments are carried out only for the final designed SPMCs. The schematic diagram of vibration is shown in Figure 22. Based on experiment specification of the mechanical connector, ²⁹ the vibration source (Instron 8801) provides the sine displacement wave, S = A sin(2πft), where the amplitude A is 0.097 mm and the frequency f is 20 Hz. When the cycle number of vibration is on the level of 10⁷, leakage does not happen, which indicates the final designed SPMC meets the experiment specification of the mechanical connector. Before the experiments, the design pressure of 4.5 MPa is applied and the shutoff valve is turned off. Figure 23 shows experiment facility of the vibration experiments.

Figure 22.

Schematic diagram of vibration.

Figure 23.

Experimental facility of vibration.

Table 3 is the results of the vibration experiments of the final designed SPMCs. The cycle numbers of vibration are 10,029,670 and 10,004,652, respectively, which indicates the two final designed SPMCs meet the experiment specification. Therefore, not only does the SPMC-based critical mean contact pressure have a high sealing performance but also it has high ability in resisting vibration.

Table 3.

Result for vibration experiments.

Class	Group	Keeping time (h)	Number of vibration	Pressure gauge
Final design	First	139.3	10,029,670	No change
	Second	139.0	10,004,652	No change

Conclusion

The article investigated sealing mechanism of SPMC and established the critical condition of seal. On the basis of the superposition principle of elasticity, the formula of the critical mean contact pressure was deduced. The SPMC was designed with ANSYS and the sealing performance was verified by the internal pressure experiments, indentation experiments, and the vibration experiments.

According to the analytical analysis, finite element simulation, internal pressure experiments, indentation experiments, and vibration experiments, the following conclusions can be drawn:

1. Based on static metal sealing mechanism, the critical condition realizing reliable static metal seal for SPMC was established and the formula of the critical mean contact pressure was deduced.

2. By comparison of the analysis and FEA result of the critical mean contact pressure, the critical mean contact pressure decreases nonlinearly with the increase in pipeline wall thickness and decreases almost linearly with the increase in sealing face width. The radius of the pipeline has less influence on the critical mean contact pressure. The FEA result agrees with that of analysis very well.

3. The critical mean contact pressure of 8-in SPMC is 223.34 MPa which is used to design SPMC. The internal pressure experiments were carried out for both the initial and final designed SPMCs. The results indicate that the sealing performance of the final designed SPMC is higher than design pressure of 4.5 MPa and all the initial designed SPMCs failed in the experiment. The sealing performance of the final designed SPMCs is better than that of the initial designed ones.

4. By the observation on the indentation on the sealing face, if the mean contact pressure is less than the critical mean contact pressure, it produces discontinuous indentation and generates micro leakage paths. Meanwhile, the large contact pressure variance makes contact of sealing face non-uniform and leakage will occur under high internal pressure. The mean contact pressure and variance of final designed SPMCs meet design requirements, which forms uniform and continuous sealing face. The SPMCs designed by the critical mean contact pressure have good sealing performance.

5. The vibration experiments show that the cycle number of vibration of the final designed SPMCs is more than 10⁷. The final designed SPMCs have good ability to resist both internal pressure and vibration.

In the next work in the soon future, we will develop auxiliary devices and further measure the contact pressure on the sealing surface by ultrasonic technology. By comparison of analysis, simulation, measurement, and experiment, the analysis and design method can be improved further. Once the SPMC is used to connect the pipelines in seabed successfully, it will strongly support offshore oil and gas exploitation.