## 2 Materials and Methods
### Mathematical Model  
The electric radiant panel considered in the experiment is composed of a fiber glass layer on the front face, with embedded copper electrical conductors to provide heating by Joule effect, a polyurethane foam layer in the middle to limit the heat diffusion on the back, and an aluminum sheet on the rear side to guarantee mechanical strength and lightness at the panel. Scheme and dimensions of the panel are reported in Figure 1 and Table 1, respectively. In Table 1, the nominal values of the thermophysical properties of the different materials used in the panel are reported as well. From the thermal point of view, the panel can be modeled as a hot plate of glass fiber with thickness \(L_{x}\), adiabatic on the rear (due to the insulating layer of polyurethane), and extended on the plane \(yz\) with \(-L_{y}<y<L_{y}\) and \(-L_{z}<z<L_{z}\), as shown in Figure 2. The panel is considered adiabatic on its lateral boundaries as well. On its surface (the plane \(x=0,y,z\)), a thin layer of a good electrical conductor is deposited with a negligible thickness. Heat is generated by Joule effect, uniformly distributed, in \(x=0\). The thickness of the glass fiber layer is small enough to consider negligible the characteristic time to propagate heat along the \(x\) direction in comparison with the dynamic of the whole system. Moreover, the temperature \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Layer** & **Material** & **Thickness** & **Density** & **Specific Heat** & **Thermal Conductivity** \\  & **[mm]** & **[kg m\({}^{-3}\)]** & **[ly kg\({}^{-1}\) K\({}^{-1}\)]** & **[W m\({}^{-1}\) K\({}^{-1}\)]** \\ \hline Front & Glass fiber & 1.2 & 2500 & 800 & 1.1 \\ Middle & Polyurethane foam & 22.0 & 40 & 1480 & 0.03 \\ Rear & Aluminum metal sheet & 1.2 & 2700 & 800 & 170 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stratigraphy of the electric radiant panel. Figure 1: Sketch of the radiant panel. Figure 2: Thermal model of the panel. The \(z\) axis is not represented in the image. field is assumed to be uniform on the plane \(x=0\). It allows for the application a lumped formulation to solve the surface temperature variation as a function of time. The ordinary differential equation is [21]:  
\[\rho c_{p}L_{x}\frac{dT}{dt}=q_{E}-(q_{CONV}+q_{RAD}) \tag{1}\]  
in which \(T\equiv T(t)\) is the temperature function of time \(t\), \(\rho\) is density, and \(c_{p}\) is specific heat at constant pressure. Initial conditions are as follows:  
\[T=T_{0}t=0 \tag{2}\]  
in which:  
* the specific heat flux [Wm\({}^{-2}\)] generated on the surface by Joule effect is given by: \[q_{E}=\frac{I^{2}R}{4L_{y}L_{z}}\] (3) being \(I\)[A] the electric current and \(R\)[\(\Omega\)] the electrical resistance;
* the heat flux exchanged with the ambient at \(T_{AMB}=T_{0}\) by convection is: \[q_{CONV}=h_{CONV}(T(x=0,t)-T_{0})\] (4)
* the heat flux exchanged with the ambient at \(T_{AMB}=T_{0}\) by radiation is: \[q_{RAD}=h_{RAD}(T(x=0,t)-T_{0})\] (5)  
The last equation is the linearization of the well-known Stefan-Boltzmann equation according to the following scheme [22]:  
\[ \begin{array}{ll}q_{RAD}&=\varepsilon\sigma(T^{4}-T_{0}^{4})=\varepsilon \sigma(T+T_{0})(T^{2}+T_{0}^{2})(T-T_{0})\\ &\hskip 14.226378 pt=2\varepsilon\sigma\bigg{(}\frac{T^{2}+T_{0}^{2}}{2} \bigg{)}4\bigg{(}\frac{T^{2}+T_{0}^{2}+2 TT_{0}-2 TT_{0}}{4}\bigg{)}(T-T_{0})\\ &\hskip 14.226378 pt=8\varepsilon\sigma\overline{T}\Big{(}T^{2}-\frac{TT_{0}}{ 2}\Big{)}(T-T_{0})\approx 4\varepsilon\sigma\overline{T}^{3}(T-T_{0})=h_{RAD}(T-T_{0}) \end{array}