## 4 Discussion
### Heating the Porous Media  
The heat accumulated by the porous medium at time \(t\), once the temperature field is known, is very difficult to determine due to the following reasons: it is a non-stationary process, specific heats of both phases are a function of temperature, and the ignorance of the value of the interstitial heat transfer coefficient. However, even if the existence of these factors is accepted, one can ask whether, given the slowness with which the temperature varies, the application of the Fourier law or the methods developed for the non-stationary regime, such as those by [PERSON] (1973) and [PERSON] et al. (2006), can be accepted as a good approximation assuming the local thermal equilibrium condition. This question can be answered by determining the values of the characteristic time of conduction and the value of the Biot number of the porous media being studied. The characteristic time of conduction is defined as \(t_{c}=L_{c}^{2}/\alpha_{f}\), and the Biot number as \(Bi=h_{sf}L_{c}/\lambda_{f}\), where \(L_{c}=\overline{d}_{p}/2\), \(\alpha_{f}\) is the thermal diffusivity, \(\lambda_{f}\) is the thermal conductivity and \(h_{sf}\) is the interstitial heat transfer coefficient, the value of which is calculated by applying the [PERSON] and [PERSON] (1973) correlation for spherical particles, as follows:  
\[\frac{h_{sf}\overline{d}_{p}}{\lambda_{f}}=2+1.1 Re^{0.6}Pr^{1/3} \tag{10}\]  
where \(Re=v_{D}\overline{d}_{p}/\
u=\varepsilon v_{p0}\overline{d}_{p}/\
u\) and \(Pr=\
u/\alpha_{f}\). Once the values of \(t_{c}\) and \(Bi\) are calculated, Tables 5 and 6 show magnitude values and the results corresponding to the condition of maximum velocity of the liquid to the tray, together with the residence time \(t_{r}\). The uncertainty in each of the fluid properties was estimated to be 1%. Figure 11: Power required for the evaporation of the liquid in the tray. From the results, it can be seen that the characteristic time is greater than the residence time in all the porous media studied. Then, at the interface of the solid and boundary layer of the liquid flow, the temperature variation of the solid shows a large delay with respect to the liquid. Therefore, the local thermal equilibrium is far from being reached. Owing to the values of \(t_{c}/t_{r}\) and the Biot number, the calculation of the heating rate of the porous medium over time cannot be carried out with the two-temperature model. Instead, an approximate indirect procedure--the thermal power balance approach--is used:  
\[\dot{W}_{c,mp}(t)=\dot{W}_{dc}(t)-\Big{(}\dot{W}_{c,pl}+\dot{W}_{ev,c}+\dot{W} _{c,l}+\dot{W}_{p}\Big{)}(t) \tag{11}\]  
In this equation, the terms \(\dot{W}_{c,l}\) (increase of the sensible heat of the liquid of the tray and the evaporated liquid in the interface plate-porous medium) and \(\dot{W}_{p}\) (losses due to insulation failure and increase in the temperature of the metallic cover of the porous medium) can be neglected, and then the equation results in:  
\[\dot{W}_{c,mp}(t)\cong\dot{W}_{dc}(t)-\Big{(}\dot{W}_{c,pl}+\dot{W}_{ev,c} \Big{)}(t)=\dot{W}_{tf}(t)-\dot{W}_{ev,c}(t) \tag{12}\]  
By taking into account the previous Equations (1) and (9), the results are shown in Figure 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Porous** & **Re** & **Pr** & \(\mathbf{h_{sf}}\) & \(\mathbf{t_{c}}\) & **Bi** & \(\mathbf{t_{r}}\) \\
**Medium** & & & & **(W-m\({}^{-2}\)K\({}^{-1}\))** & **(s)** & **Bi** & **(s)** \\ \hline B+W & 10.8 \(\pm\) 0.2 & 1.85 \(\pm\) 0.03 & 21,937 \(\pm\) 373 & 0.083 \(\pm\) 0.002 & 3.81 \(\pm\) 0.08 & 0.006 \(\pm\) 0.001 \\ B+DS & 9.6 \(\pm\) 0.1 & 1.85 \(\pm\) 0.03 & 20,833 \(\pm\) 354 & 0.083 \(\pm\) 0.002 & 3.62 \(\pm\) 0.08 & 0.007 \(\pm\) 0.001 \\ S+W & 119 \(\pm\) 2 & 1.85 \(\pm\) 0.03 & 5884 \(\pm\) 100 & 13.12 \(\pm\) 0.29 & 12.86 \(\pm\) 0.28 & 0.090 \(\pm\) 0.001 \\ S+DS & 215 \(\pm\) 3 & 1.85 \(\pm\) 0.03 & 8204 \(\pm\) 139 & 13.12 \(\pm\) 0.29 & 17.93 \(\pm\) 0.39 & 0.050 \(\pm\) 0.001 \\ \hline \end{tabular}
\end{table}