## 2 Materials and Method
### Methodology
#### 2.2.2 Surface Thermal Behavior  
On the surface, all three possible heat transmission mechanisms (conduction, convection and radiation) occur. * Convection  
Experience shows that the convection heat transfer highly depends on certain properties of the fluid considered in the system but also on the geometric configuration and the roughness of the solid surface, and the conditions of the fluid flow (laminar or turbulent). According to [PERSON]'s cooling law, the ratio of heat transferred by this mechanism is proportional to the temperature difference and is conveniently expressed by [35; 36]:  
\[\dot{q}_{conv}=\sum\,h_{c}\cdot(T_{s}-T_{\infty}) \tag{5}\]  
where \(\dot{q}_{conv}\) is the convection heat rate (W/m\({}^{2}\)), \(h_{c}\) is the convection heat transfer coefficient (W/m\({}^{2. }\)\({}^{\circ}\)C) and \(T_{s}\) and \(T_{\infty}\) are the temperature of the surface and the ambient temperature, respectively (\({}^{\circ}\)C). For applying Equation (5), it is necessary to define the convection cell and the convection heat transfer coefficient that operates in the conditions of the present experimental test. With that aim, Equation (6) is used to determine the aforementioned convection heat transfer coefficient:  
\[h_{c}\ =\ \frac{Nu\cdot k_{a}}{Xc} \tag{6}\]  
where \(Nu\) is the Nusselt number, \(k_{a}\) is the thermal conductivity of the air (W/m \({}^{\circ}\)C) and \(Xc\) is the length of the boundary layer (m). At the same time, the \(Nu\) and \(Xc\) are calculated following the expression of Equations (7) and (8):  
\[Nu\ =0.664\cdot\sqrt{Re}\cdot Pr^{\gamma/3} \tag{7}\]  
\[Xc\ =\ \frac{\
u\cdot Re}{u_{F}} \tag{8}\]  
where \(Re\) is the Reynolds number, \(Pr\) is the number of Prandtl, \(\
u\) is the kinematic viscosity (m\({}^{2}\)/seg) and \(u_{F}\) is the atmospheric air velocity (m/s). The properties of the air required in the previous equations must be defined for a reference temperature (Equation (9)):  
\[T_{R}=\frac{T_{s}-T_{\infty}}{2} \tag{9}\]  
Once the air properties [37] are determined for the reference temperature (\(T_{R}\)), the values of \(Nu\) and \(Xc\) are calculated using the corresponding Equations (7) and (8), for then obtaining the convection heat transfer coefficient and the convection heat rate. * Radiation  
The total amount of energy being radiated by an object can be estimated by applying the Stefan-Boltzmann law, which states that for an object of a certain temperature, the radiated heat will be [36]:  
\[\dot{q}_{rad}=\sum\varepsilon\cdot\sigma\cdot T_{s}^{4} \tag{10}\]  
where \(\dot{q}_{rad}\) is the radiation heat rate (W/m\({}^{2}\)), \(\varepsilon\) is the emissivity constant that depends entirely on the material (sand) and is established at 1 for an ideal blackbody, \(\sigma\) is the Stefan-Boltzmann constant and \(T_{s}\) is the temperature of the sand (K). * Conduction  
The quantification of the heat exchanged by conduction at the surface is not as simple as in the above conduction and radiation effects. In the conditions of the experimental setup, according to the line source model, the heat transfer rate does not depend on the depth considered, in such a way that it remains constant in the horizontal slides of the model (\(Q_{surface}=Q_{depth}\)). From that equality, and given that heat is only transmitted in depth by conduction, the heat transferred by conduction at the surface can be obtained from the following expression [38]:  
\[Q_{cond}=\ Q_{surface}-\ Q_{conv}-\ Q_{rad} \tag{11}\]  
where \(Q_{cond}\) is the heat transferred by conduction at the surface (W), \(Q_{surface}\) is the total heat transferred at the surface (W) and \(Q_{conv}\) and \(Q_{rad}\) are, respectively, the heat transmitted by convection and radiation at the surface (W). The estimation of each heat transfer mechanism was made with the help of the data collected by the thermal camera. While thermal cameras do not directly derive temperature values, they record the radiance emitted by surfaces that is directly related to the object surface temperature.