# Thermal-Structural Coupling Analysis of Subsea Connector Sealing Contact
## 3 Research on the Heat Transfer Model of the Subsea Connector
### Heat Transfer Model of the seawater Layer between the Hub and Collet  
Figure 3 shows a simplified cross-sectional view of the position relationship between the hub and the collet. The seawater layer between the hub and the collet is very thin, but not negligible, otherwise, the temperature difference between the hub and the collet will increase. Since the hub outer surface and the collet inner surface are not flat, the introduced shape factor represents the energy flux between the two surfaces. Firstly, the energy exchange between the area elements of the two surfaces is obtained, and then the integral of the two surfaces is determined. Therefore, the shape factor is mainly related to the geometric state of the two surfaces. In addition, the shape factor only represents the percentage of radiative transfer on one surface and has nothing to do with the energy absorption capacity of the other surface. The heat transfer model between the hub and the collet can be simplified as the heat transfer problem between two concentric cylinders. As shown in Figure 3, \(L_{1}\) is the height of the seawater layer. \(\varepsilon\) is the emissivity, \(0\leq\varepsilon\leq 1\); \(\sigma\) is the Stephen-Boltzmann constant; \(A_{1}\) is the area of the smaller surface, set as surface \(1\); \(T_{1}\) is the temperature of surface \(1\); \(T_{2}\) is the larger surface (set as the temperature of surface \(2\)); \(\varphi_{1-2}\) is the shape factor from surface \(1\) to surface \(2\). The radiation shape factor [18; 19] of the hub and collet surface is  
\[\varphi_{1-2}=R_{1}\bigg{[}1-\frac{1}{\pi}\cos^{-1}\bigg{(}\frac{\chi_{1}}{ \chi_{2}}\bigg{)}\bigg{]}+\frac{\sqrt{\big{(}\chi_{1}+2\big{)}^{2}-4R_{1}^{2}} }{2\pi R_{2}}\cos^{-1}\bigg{(}R_{1}\frac{\chi_{1}}{\chi_{2}}\bigg{)}+\frac{ \chi_{1}}{2\pi R_{2}}\sin^{-1}R_{1}-\frac{\chi_{2}}{4R_{2}} \tag{1}\]  
Figure 2: Heat transfer model of the subsea collet connector. where \(r_{1}\) is the outer diameter of the flange; \(r_{2}\) is the inner diameter of the collett; \(R_{1}=\frac{r_{1}}{r_{2}}\); \(R_{2}=\frac{L_{1}}{r_{2}}\); \(\chi_{1}=R_{2}^{2}+R_{1}^{2}-1\); \(\chi_{2}=R_{2}^{2}+R_{1}^{2}+1\). Considering the influence of radiation in the system on heat transfer, the emissivity of the system can be obtained, \(\varepsilon_{s}\)  
\[\varepsilon_{s}=\frac{1}{1+\varphi_{1-2}\Big{(}\frac{1}{\varepsilon_{1}}-1 \Big{)}+\varphi_{2-1}\Big{(}\frac{1}{\varepsilon_{2}}-1\Big{)}} \tag{2}\]  
where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the emissivity of the hub and collett. Since the materials selected for each part of the subsea connector are corrosion-resistant alloy steel and heat-resistant alloy steel, during the long-term operation in deep water, they will be corroded by seawater to produce an oxide layer, so the emissivity of each part is taken as 0.8. The flow of seawater is very slow because the seawater layer is wrapped in the press ring and the wall tube. Therefore, convective heat transfer can be ignored. Accordingly, only two heat transfer methods, thermal conduction and thermal radiation, are considered between the hub and the collett and can be obtained from the energy conservation on the hub surface  
\[Q_{1-2}=2\pi L_{1}\cdot\lambda_{k2}\cdot\frac{T_{1}-T_{2}}{\ln(r_{2}/r_{1})}+ \varepsilon_{s}2\pi r_{1}L_{1}\varphi_{1-2}C_{0}\Bigg{[}\bigg{(}\frac{T_{1}}{1 00}\bigg{)}^{4}-\bigg{(}\frac{T_{2}}{100}\bigg{)}^{4}\Bigg{]} \tag{3}\]  
where \(Q_{1-2}\) is the total heat transferred from the seawater layer between the hub and the collett; \(\lambda_{k2}\) is the thermal conductivity of seawater at temperature \(T_{2}\)[18]; \(T_{1}\) is the temperature of the hub outer surface (as shown in Figure 3); \(T_{2}\) is the temperature of the collett inner surface (as shown in Figure 3); \(C_{0}\) is the black-body radiation coefficient. In the finite element analysis, compared with convective heat transfer and thermal radiation, the calculation and regulation of thermal conduction are easier.