## 2 Materials and Methods for Coupled Thermal-Electrical Modeling of the Grounding Electrode
### Electrode-Soil Heat Exchange Model
 \tag{1}\]  
where \(\alpha_{i}\) is the thermal diffusivity of the medium \(i\) (\(i=1\), \(2\)) and \(\hat{\epsilon}_{i}\) is the power density generated by the internal sources of the medium \(i\) (\(i=1\), \(2\)). If the thermal diffusivity of medium \(1\) is large and the radius \(R_{1}\) is small, we can ignore the thermal gradients inside this medium and treat it as a lumped-parameter system, which means that it will only be necessary to solve the heat equation in medium \(2\) and add a boundary condition (BC) at the surface of medium \(1\), which is the interface with medium \(2\). Given that the equation satisfied by medium \(1\) is  
\[-\frac{k_{1}}{\alpha_{1}}\frac{dT_{1}}{dt}V_{1}+\hat{\epsilon}_{1}V_{1}=\phi( S_{1})=-k_{2}\left.\frac{\partial T_{2}}{\partial r}\right|_{r=R_{1}} \cdot S_{1} \tag{2}\]  
where \(\phi(S_{1})\) is the power flux leaving the surface \(S_{1}\) of the buried conductor and \(V_{1}\) is its volume, since this equation must be satisfied at the surface of the conductor where \(T_{1}(t)=T_{2}(R_{1},t)\), Equation (2) becomes a boundary condition for medium \(2\):  
\[-k_{2}\left.\frac{\partial T_{2}}{\partial r}\right|_{r=R_{1}}=-\frac{k_{1}}{ \alpha_{1}}\frac{V_{1}}{S_{1}}\left.\frac{\partial T_{2}}{\partial t}\right|_ {r=R_{1}}+\hat{\epsilon}_{1}\frac{V_{1}}{S_{1}} \tag{3}\]  
and the conduction problem to be solved is for medium \(2\), which is a cylindrical layer with radii \(R_{1}\) and \(R_{2}\), having a thermal diffusivity \(\alpha_{2}\) and thermal conductivity \(k_{2}\):  
\[\begin{array}{l}\frac{1}{\alpha_{2}}\left.\frac{\partial T_{2}(r,t)}{ \partial t}\right.=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial T_ {2}(r,t)}{\partial r})+\frac{\hat{\epsilon}_{2}}{k_{2}}\quad R_{1}<r<R_{2}\\ -k_{2}\left.\frac{\partial T_{2}(r,t)}{\partial r}\right|_{r=R_{1}}=-\frac{k_{ 1}}{\alpha_{1}}\frac{V_{1}}{S_{1}}\left.\frac{\partial T_{2}(r,t)}{\partial t }\right|_{r=R_{1}}+\hat{\epsilon}_{1}\frac{V_{1}}{S_{1}}\\ T_{2}(r,t)\bigEquation (4) contains sources in one of the boundary conditions and also sources within the equation itself. Both types of sources are due to the Joule effect, but they exhibit some essential differences. The sources \(\dot{e}_{1}\) are generated inside the conductor by the current \(I_{c}\) (A) passing through it, while \(\dot{e}_{2}\) are generated by the current \(I_{s}\) that leaves the conductor through its surface, known as the leakage current. The general expression for the Joule power generated by a current density \(\overrightarrow{J}\) (A/m\({}^{2}\)) flowing in a volume region \(V\) with electrical conductivity \(\sigma\), where an electric field \(\overrightarrow{E}\) exists, is given by  
\[W=\int\limits_{V}\overrightarrow{J}\cdot\overrightarrow{E}dV=\int\limits_{V }\overrightarrow{J}\cdot\overrightarrow{\sigma}dV=\int\limits_{V}\rho \overrightarrow{J}dV=\int\limits_{V}\dot{\varepsilon}dV \tag{5}\]  
Therefore, the Joule power density is defined from the current density by the expression \(\dot{\varepsilon}=\rho\overrightarrow{J}^{2}\). For a cylindrical conductor with cross-sectional area \(S\) and length \(L\) carrying a uniform current \(I_{c}\) (A), the current density can be considered uniform if the conductor is thin, and it is \(\overrightarrow{J}=I_{c}/S\cdot\overrightarrow{u}\), where \(\overrightarrow{J}\) is the current density and \(\overrightarrow{u}\) is a unit vector in the direction of the symmetry axis of the cylindrical conductor. In this case, the Joule power density is \(\dot{\varepsilon}=\rho\frac{I_{c}^{2}}{S^{2}}\), and it is easy to obtain that \(W=I_{c}^{2}R_{\Omega}=I_{c}^{2}\rho\frac{L}{S}\). The symbol \(R_{\Omega}\) denotes the ohmic resistance of the conductive medium. A different case occurs when the electric current flows from a cylindrical surface of radius \(R\) and length \(L\), very large, toward the outside.