## 2 Heat Transfer Theory of Linear Heat Sources in Water Media
### Analysis of Heat Conduction Laws during the Heating Process  
As commonly used core materials, metal materials represented by iron and plastic materials represented by polyvinyl chloride (PVC) have good thermal conductivity and heat resistance [11], making them alternative materials for the development of linear heat sources required for equipment research and development [12; 13; 14]. To verify the transient heat transfer analytical solution of linear heat sources in a water medium, this section calculates the heating processes of iron-based linear heat sources and PVC-based linear heat sources in seawater. In this case, the seawater temperature \(T_{\infty}=25\)\({}^{\circ}\)C, density \(\rho_{water}=1020\) kg/m\({}^{3}\), and specific heat capacity \(c_{water}=820\) J/(kg.\({}^{\circ}\)C) are used. Since the heat transfer mode of the heat source in seawater is convective heat transfer [15], the convective heat transfer coefficient \(h\) between the heat source and seawater is related to the physical properties of seawater, the surface characteristics of the linear heat source, the size of the heat source, and the flow velocity of seawater near the heat source. It is difficult to derive the convective heat transfer coefficient for natural convection and forced convection of linear heat sources in seawater [16; 17; 18]. Therefore, the heating and cooling processes of linear heat sources in seawater are calculated for convective heat transfer coefficients \(h=50\) W/(m\({}^{2.}\)\({}^{\circ}\)C), 100 W/(m\({}^{2.}\)\({}^{\circ}\)C), 200 W/(m\({}^{2.}\)\({}^{\circ}\)C), 300 W/(m\({}^{2.}\)\({}^{\circ}\)C), 500 W/(m\({}^{2.}\)\({}^{\circ}\)C), 1000 W/(m\({}^{2.}\)\({}^{\circ}\)C), 1500 W/(m\({}^{2.}\)\({}^{\circ}\)C), 2000 W/(m\({}^{2.}\)\({}^{\circ}\)C) and 2500 W/(m\({}^{2.}\)\({}^{\circ}\)C) to investigate the laws of heat conduction [19]. The parameters of the linear heat sources are shown in Table 1.  
According to Equation (12), we can calculate the temperature rise process for a linear iron heat source and a PVC linear heat source heating up in seawater at a constant power of 1000 W. The results are shown in Figure 3.  
As can be seen from Figure 3, due to the high heating power, both the iron linear heat source and the PVC linear heat source reach a stable temperature in a short period of time [20]. The temperature at which they stabilize gradually increases as the convective heat transfer coefficient \(h\) decreases. The relationship between the steady-state temperature reached and the convective heat transfer coefficient is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{MQ}\) & \(\mathbf{T_{0}}\) (\({}^{\circ}\)C) & \(\mathbf{r}\) (m) & \(\mathbf{\rho}\) (kg/m\({}^{3}\)) & \(\mathbf{\epsilon}\) [J/(kg\({}^{\circ}\)C)] \\ \hline Iron & 25 & 0.05 & 7850 & 460 \\ PVC & 25 & 0.05 & 1380 & 1200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of linear heat source parameters.  
Figure 4: The relationship between the final steady-state temperature of a linear heat source in seawater and the convective heat transfer coefficient \(h\).  
Figure 3: Temperature rise curve of linear heat sources in seawater.  
As shown in Figure 4, the cumulative heat transferred from the linear heat source to seawater gradually increases under the action of ocean currents. The final steady-state temperature of the heat source in seawater gradually decreases, and there is a logarithmic relationship between the magnitude of the final steady-state temperature and the convective heat transfer coefficient \(h\)[21; 22].