## 2 Multi-Physics Coupling Mathematical Model
### Mathematical Model of Heat Transfer  
The predominant source of heat generation in DC GIL originates from the Joule heating of the central current-carrying conductor. Neglecting the thermal contributions attributable to insulator leakage currents, the heat generation may be mathematically represented as follows:  
\[P=\frac{l^{2}L}{S\cdot\sigma_{Al}} \tag{1}\]  
Figure 1: Geometric model of the \(\pm\)200 kV DC GIL. where \(P\) is the heating power (unit: W); \(I\) is the load current (unit: A), which is 3150 A\({}^{\prime}\)\(L\) is the length of the conductor (unit: m); \(S\) is the cross-sectional area of the conductor (unit: m\({}^{2}\)); and \(\sigma_{AI}\) is the electric conductivity of the conductor (unit: S/m). \(\sigma_{AI}\) is related to temperature as follows [27]:  
\[\sigma_{AI}=\frac{\sigma_{20}}{1+0.004(T-293.15)} \tag{2}\]  
where \(\sigma_{20}\) is the electric conductivity of the conductor at 20 \({}^{\circ}\)C, which is 3.31 \(\times\) 10\({}^{7}\) S/m. \(T\) is the temperature of the conductor (unit: K). Thermal energy generated by the central conductor dissipates to the surrounding environment via conduction, convection, and radiation. These three heat transfer mechanisms are comprehensively examined herein, with the assumption of optimal contact between the insulator and both the insert and enclosure. In accordance with [PERSON]'s law of heat conduction, the heat transfer within a GIL can be mathematically represented as  
\[\rho C_{p}\frac{\partial T}{\partial t}=\lambda\
abla^{2}T+Q \tag{3}\]  
where \(\rho\) is the density (unit: kg\(\cdot\)m\({}^{3}\)), \(C_{p}\) is the specific heat capacity at constant pressure (unit:)\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)), \(T\) is the absolute temperature (unit: K), \(Q\) represents additional heat sources (unit: W\(\cdot\)m\({}^{-3}\)), and \(\lambda\) is the thermal conductivity (unit: W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)). The density, thermal conductivity, and specific heat capacity of the central conductor and enclosure exhibit minimal variation within the temperature range examined in this study; therefore, they are treated as constants [28]. The relevant parameters are presented in Table 1. The thermophysical parameters of SF\({}_{6}\) gas vary as a function of temperature, with data obtained from the AP1700 physical property database. The specific heat capacity and thermal conductivity of the insulators, as determined experimentally, are presented in Table 2, while the insulator density is 2300 kg/m\({}^{3}\). The Navier-Stokes equations for single-phase fluids can be employed to describe the motion of SF\({}_{6}\) gas in the GIL. The corresponding mass conservation equation, energy conservation equation and momentum conservation equation are as follows:  
\[\frac{\partial\rho}{\partial t}+\rho\
abla\cdot\mathbf{v}=0 \tag{4}\]  
\[\rho C_{p}(\frac{\partial T}{\partial t}+\mathbf{v}\cdot\
abla T)=\lambda\
abla^{ 2}T \tag{5}\]  
\[\rho\bigg{(}\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\
abla\mathbf{v}\bigg{)} =\mu\
abla^{2}\mathbf{v}+\rho\mathbf{g} \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Density kg/m\({}^{3}\)** & **Specific Heat Capacity J/(kg\(\cdot\)K)** & **Thermal Conductivity W/(m\(\cdot\)K)** \\ \hline Conductor & 2690 & 895 & 218 \\ Enclosure & 2660 & 819 & 152 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Center conductor and enclosure thermodynamic parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature K** & **301** & **323.6** & **348.8** & **373.85** & **398.7** \\ \hline Thermal Conductivity W/(m\(\cdot\)K) & 0.946 & 0.967 & 0.934 & 0.917 & 0.984 \\ Specific Heat Capacity J/(kg\(\cdot\)K) & 0.880 & 0.949 & 1.018 & 1.075 & 1.195 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Insulator thermodynamic parameters. where \(\mathbf{v}\) is the velocity vector (unit: m/s), \(\mu\) is the dynamic viscosity (unit: Pa-s), and \(\mathbf{g}\) is the gravity vector (unit: m\(\cdot\)s\({}^{-2}\)).