# Simulation Research on the Effects of Air Gaps and Ambient Temperature on a 27.5 kV Power Cable
## 2 Cable Model Construction
### Boundary Conditions and Formulas  
After the material was set up, the conductor was loaded with current, the solid loaded heat transfer module, and physical field coupling boundary conditions of a joule heat source, heat conduction and thermal expansion were established. The specific process was as follows: In the electric field, material resistivity changes with temperature, the skin effect and conductivity under the influence of pressure were considered, while the proximity effect was ignored. Contact thermal resistance under the influence of a joule heat source and pressure was considered in the thermal field, and eddy current loss under the action of the magnetic field was ignored in the heat source. The electric field affected the joule heat source in the thermal field, and the thermal field affected material resistivity in electrical stress analysis; therefore, the electric field and thermal field showed bidirectional coupling.  
Firstly, the solid heat transfer control equation was constructed as follows:  
\[\rho_{T}C_{P}\frac{\partial T}{\partial t}-\lambda_{T}\
abla^{2}T=Q_{E} \tag{1}\]  
Here, \(t\) is the time; \(\rho_{T}\) is the density of the material at temperature \(T(K)\); \(C_{P}\) is the specific heat capacity of the material at constant pressure; \(\lambda_{T}\) is the thermal conductivity of the material at temperature \(T\); and \(Q_{E}\) is the joule heat source, calculated by the following formula:  
\[Q_{E}=J\cdot E \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure** & **Materials** & **Outside Radius** \\  & & **/(mm)** \\ \hline \hline Cable core & Copper-aluminum alloy & 9.77 \\ Semiconductor layer & Semiconductor polymer & 19.57 \\ stress control tube & Polymer materials & 18.27 \\ Glue & Hot melt glue & / \\ heat shrink tube & PE & 23.77 \\ Sheathing layer & PVC & 23.92 \\ Umbrella skirt & Rubber & 27.12 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cable and terminal material type and geometry.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structure** & **Electric** & **Relative Dielectric** & **Thermal** & **Poisson** \\  & **Conductivity** & **Constant** & **Conductivity** & **Ratio** \\  & **/S/m** & **/(I)** & **/(W/(m-k))** & **/(P)** \\ \hline Cable core & \(5.998\times 10^{7}\) & 1 & 400 & 0.35 \\ Semiconductor layer & 2 & 100 & 10 & 0.25 \\ stress control tube & \(1.3\times 10^{-9}\) & 25.3 & 35.3 & 0.44 \\ Glue & \(7.7\) & \(0.5\times 10^{-8}\) & 10 & 0.38 \\ heat shrink tube & \(0.25\times 10^{-10}\) & 3.34 & 0.25 & 0.22 \\ Sheathing layer & \(1\times 10^{-18}\) & 2.6 & 0.46 & 0.38 \\ Umbrella skirt & \(0.8\times 10^{-18}\) & 0.85 & 2.25 & 0.26 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cable and terminal material parameters.  
Figure 2: Cable and terminal model geometry.  
Here, \(J\) is the current density of the contact source, and \(E\) is the electric field inside the contact source. \(J\) and \(E\) are solved by the electric field control equation satisfying Ohm's law and the current conservation law:  
\[\begin{array}{l}J=\sigma_{T}\cdot E\\ \
abla\cdot J=0\\ E=-\varphi\end{array} \tag{3}\]  
Here, \(\sigma_{T}\) is the conductivity of the contact material at temperature \(T\), and \(\varphi\) is the electric potential. When solving the electric field control equation, linear temperature change characteristics and the skin effect of conductivity should also be considered:  
\[\sigma_{T}(T)=\frac{1}{\rho_{20}[1+\alpha_{20}(T-293.15)]} \tag{4}\]  
This study focused on the transient process. The construction process of the electro-thermal coupling simulation model is shown in Figure 3.  
Figure 3: Construction process of the electric–thermal field cable coupling simulation model.