#### 1. Introduction

Due to the depletion of land oil reserves, more attention is being paid to offshore oil, especially deep-sea resources. As a safe and reliable transportation method, the subsea oil pipeline played a very important role in deep-water oil and gas exploitation. Flange connection is a common way to connect the deep-sea oil pipelines [13]. Therefore, it is necessary to study the safety and reliability of flange connection.

In fact, the distribution of bolt loads is very important for the sealing performance of the flange, which cannot be ignored. This paper proposed an analytic and finite element-combined modelling method; taking both temperature change and the elastic interaction into account, we obtained the distribution trend of bolt preload with the temperature change with a given initial bolt preload deviation.

#### 2. The Heat Transfer Model

As shown in Figure 1, the flange of the subsea pipeline is preloaded by eight bolts (B1–B8), and the medium liquid goes through the pipeline inside.

The heat transfer model includes heat conduction, heat radiation, and heat convection. The calculation of heat radiation is highly nonlinear, including the material nonlinearity and geometric nonlinearity. Heat convection is usually temperature-dependent; it means that the heat convection coefficient cannot be directly calculated, so an appropriate heat transfer model is required. The heat transfer model can be simplified as an equivalent coefficient of thermal conductivity with a good calculation accuracy.

##### 2.1. Equivalent Heat Conductivity Coefficient of the Water Layer between the Bolt and Hole

Although the heat transfer in the seawater layer is low, if it is fully ignored, the temperature difference between the bolt and the flange can be exaggerated. If the seawater layer is treated as a liquid thermal material with a conductivity, a big error can be introduced as well. The heat conduction is the easiest to be calculated on comparing with heat convection and heat radiation, and the coefficient of heat conductivity can be set in the material property in simulation. Therefore, the heat transfer can be treated as an equivalent coefficient of heat conductivity. With the setting of the temperature-dependent equivalent coefficient of heat conductivity, the heat radiation, heat convection, and heat conduction can be simulated. So the heat transfer can be idealized as a heat conduction process with a temperature-dependent equivalent heat conductivity coefficient.

As the real surface is not perfect, the nature of the surface can increase or decrease energy exchange. The direct energy flux between the two surfaces is expressed by an angular factor. The determination of the angle factor is to find the energy exchange between differential area elements of two surfaces and after that integrate it on the whole surface. The result relies on the geometric conditions. The angular factor represents only the percentage of the radiant energy that reaches the other surface, which is independent of the absorption capacity of the other surface. It is a parameter that depends on the size and position of the surfaces only. The heat exchange model between the hole and the bolt can be simplified as a heat exchange case between two concentric cylinders. The two concentric cylinders with the same length, L, and the inner surface of the outer cylinder with radius, r2, radiate directly to the outer surface of the inner cylinder with radius, r1. The angle factor of the hole radiation to the bolt can be obtained [1921] as follows:where , , , and .

Considering the effects of heat radiation, the heat emissivity of the bolt and hole can be expressed aswhere ε1 and ε2 denote the blackness of the bolt and hole.

As the materials of the bolt and flange are corrosion-resistant alloy and heat-resistant alloy, the corrosion of seawater also produces an oxidation layer on the surfaces for the long-term operation in deep water, so the blackness of all parts is selected as 0.8.

When the heat convection can be ignored, the heat transfers through the heat radiation and heat conduction of the seawater layer; then, the energy that transfers between the flange and the bolt iswhere Q1-2 is the heat transfer of the seawater layer between the flange and bolt, J, T1 is the bolt surface temperature, K, T2 is the flange bolt hole temperature, K, C0 is the blackbody emissivity, 5.67 W/(m2·K4), and λk2 is the seawater thermal conductivity at T2 [22], W/(m·K); the heat transfer can be expressed with the equivalent heat conductivity between the bolt and hole, λe1:

Substitute equation (4) into (3) and yield

The equivalent heat conductivity can be expressed as

Considering that the heat conductivity is a function of temperature, this equation is a coupling equation, which can be solved by the dichotomy method. The relation of the equivalent heat conductivity coefficient and temperature is plotted in Figure 2.

##### 2.2. Equivalent Heat Conductivity between Flanges

The heat convection between the outer surface of the seal ring and the seawater is much lower than the seawater heat conduction and heat radiation, so it can be ignored in simulation. The heat transfer between the flanges can be treated as the heat conduction and heat radiation, and we yield the following:where λe2 is the equivalent heat conductivity between the flanges, W/(m·K), λk4 is the seawater heat conductivity at T4 [22], W/(m·K), T3 is the seal surface temperature, K, T4 is the seawater temperature, K, r3 is the seal ring radius, and r4 is the flange radius.

The corresponding temperature-dependent equivalent heat conductivity coefficient λe2 is shown in Figure 3.

##### 2.3. Equivalent Heat Transfer Coefficient of the Outer Surface of the Flange

As the flange is exposed to the seawater environment, the main heat transfer is natural convection heat exchange with the seawater, q2, and monomer radiation heat transfer, q3. The convective heat loss is defined as Qp.

The radiative heat exchange between the external surface of the flange and the seawater can be replaced by the equivalent air convection heat exchanger in simulation and calculated aswhich iswhere Qp is the heat loss due to convection heat transfer, J, hp is the seawater natural heat transfer coefficient [23], 200 W·m−2·K−1, Sp is the outer surface area of the flange, m2, T5 is the flange outer surface temperature, K, and T6 is the seawater temperature, K.

In addition, the heat loss due to heat radiation QR iswhere QR is the heat loss caused by heat radiation from the outer surface of the pressure ring, J, and ε is the blackness, 0.8.

Introducing a temperature-dependent equivalent heat transfer coefficient of convection and heat radiation, he, is

The relation of the equivalent heat transfer coefficient and temperature is plotted in Figure 4.

#### 3. Calculation Results and Analysis

The temperature-dependent conductivity coefficients and heat transfer coefficient acquired in Section 2 can be applied in the flange assembly in the simulation of the thermal process.

##### 3.1. Simulation Settings

The material of the seal ring is 12Cr1MoV, and the material of flange and bolts is 060A35 (British Standards Institution). The material properties are listed in Table 1. In the step module, create 2 steps: the first step was static and general to compute the bolt loads, and the second step was coupled temp-displacement to compute the heat transfer process. In the load module, set the boundary conditions: the one flange end was set as fixed, as shown in Figure 5, and the translational degrees of freedom of the XYZ axis of surface A is on the flange set as 0, and the translational degrees of freedom of the XZ axis of surface B on all the bolts set as 0, and the translational degree of freedom of the Y axis of surface C which is on all the bolts were 0; then, set the temperature boundary of the inner pipeline as 373 K, 423 K, 473 K, and 523K, respectively, in Step 2; in the final, set the bolt loads. In the interaction module, create interaction property Inprop-1, Inprop-2, and Inprop-3 and list in Table 2, and the Inprop-1 was used for the bolts and the hole, the Inprop-2 was used for the flanges, and the Inprop-3 was used for the outer surface of the flange and the surface of the bolts which contact the water.

##### 3.2. Temperature Field Analysis

The temperature field of the flange is shown in Figure 6. It can be seen that the temperature of the flange gradually decreases from inside to outside, but the temperature of the bolt and hole was not evenly distributed. The temperature at the flange joint decreases significantly from the sealing ring, but the total decrease is lower than that of the flange external surface.

The temperature distribution of the bolt is shown in Figure 7. The temperature in the middle of the bolt in the flange hole is the highest and gradually decreases to both ends. The distributions of the temperature from the inside to the outside along the axial direction of the bolt are similar, but the outside is slightly different due to the influence of heat dissipation from the two end faces of the bolt, and the radial direction is nonlinear and gradually decreases due to the influence of the seawater layer.

##### 3.3. The Change of Bolt Loads

Because of the elastic interaction, the bolt preloads are not exactly the same. If the temperature of the flange changes, the bolt loads change as well. When the bolt loads change, the elastic interaction between the bolts further affects the uniformity of the bolt loads which affects the sealing of the flange in the final [24]. Therefore, the temperature change affects the sealing of the flange as a result. In order to study the influence of the temperature change on the uniformity of bolt preloads’ distribution, this study investigates the bolt load change under different temperatures with the proposed finite element method.

The simulation shows that when the bolts’ preload is uniform and the medium liquid temperature is 373 K (Condition 1), the bolt loads change are shown in Figure 8.

When the bolt preloads are uniform, the medium liquid temperature is 473 K (Condition 2), and the bolt loads’ change is shown in Figure 9.

When the bolts’ preload is uniform, the medium liquid temperature is 523 K (Condition 3), and the bolt loads’ change is shown in Figure 10.

When the bolts’ preload variation is 1% and the temperature of the medium liquid is 523 K (Condition 4), the bolt loads’ change is shown in Figure 11.

When the bolts’ preload variation is 2% and the temperature of the medium liquid is 523 K (Condition 5), the bolt loads’ change is shown in Figure 12.

When the bolts’ preload variation is 5% and the temperature of the medium liquid is 523 K (Condition 6), the bolt loads’ change is shown in Figure 13.

The bolt loads increase quickly from 0 to 2000 s and after that gently until stable.

Then, we can see that the high temperature of the medium liquid leads to a high increase of the bolt load, and the bolt loads are more scattered; meanwhile, a big bolt preload deviation leads to more scattered bolt loads; in order to study the effects of the distribution of the bolts’ preload on the bolt loads with the same deviation, define the standard deviation aswhere S is the standard deviation, Fi is the bolt load of bolt i, i = 1–8, and μF is the average of the bolt loads.

When the bolt preload deviation is 5%, the temperature of the medium liquid is 523 K, and the bolts’ preload was axial symmetry (Condition 7), the standard deviation is shown in Figure 14.

When the bolt preload deviation is 5%, the temperature of the medium liquid is 523 K, and the bolts’ preload was decreasing (Condition 8), the standard deviation is shown in Figure 15.

When the bolt preload deviation is 5%, the temperature of the medium liquid is 523 K, and the bolt preloads are staggered (Condition 9), the standard deviation is shown in Figure 16.

The bolt load peak deviations are around 6–6.4 kN, happen around 1000 s, and tend to be stable after 2000 s.