## 3 Methods: Characterisation Model
### Model  
The model developed is based on the theoretical foundations described in Section 4.1. Its parameters have been obtained from experimentation with an experimental prototype of the system described in Section 4.2. It was necessary to formulate a model to predict the absorbed or dissipated heat flux (cooling mode or heating mode). The main challenge was to characterise radiant heat flux. The total heat flux of the radiant system is shown in Equation (2):  
\[P=P_{cv}+P_{rad} \tag{2}\]  
where \(P\) is the total absorbed or dissipated heat flux of the active roof system \(\left(\frac{W}{m^{2}}\right)\), \(P_{cv}\) is the convective heat flux between the surface and the surrounding air, and \(P_{rad}\) is the radiant heat flux between the surface and the rest of surface. The convective flux is modelled according to Equation (3). \[Q_{cv}\left(\frac{W}{m^{2}}\right)=h_{cv}\cdot(LMTD_{cv}) \tag{3}\]  
where \(LMTD_{cv}\) (Equation (4)) corresponds to the logarithmic mean temperature and \(h_{cv}\) is the convective heat transfer. \[LMTD_{cv}(^{\circ}\mathrm{C})=\frac{\Delta T_{in}-\Delta T_{out}}{Ln\left( \frac{\Delta T_{in}}{\Delta T_{out}}\right)} \tag{4}\]  
where \(\Delta T_{in}\) refers to the temperature difference between the leaving water temperature and the ambient temperature and \(\Delta T_{out}\) refers to the temperature difference between the return water temperature and the ambient temperature. Therefore, the use of the \(LMTD\) implies the assumption that the temperature variation between both fluids is proportional to the temperature difference. This assumption has been validated by the data obtained from the experimentation. The convective heat coefficient is modelled based on the Morgan correlation for a long horizontal cylinder [36] according to Equation (5). \[h_{cvi}=K_{i}\cdot LMTD_{cv}{}^{n_{i}} \tag{5}\]  
The coefficients \(K\) and \(n\) are determined based on the data obtained from the experimentation of the experimental prototype. These coefficients vary depending on the regime (heating or cooling), so the value of \(i=1\) is assigned for the cooling mode and \(i=2\) for the heating mode. This is so because the cooling mode cools the surrounding air favouring its circulation (stratification), so a higher convective film coefficient is expected, while in heating mode, this convective film coefficient is much lower since the heating of the surrounding air does not generate air mass movement. The absorbed or dissipated heat flux referred to the radiant thermal gains is linearised and is modelled according to Equation (6). The hypothesis is that the representative temperature difference is less than 100 K and, therefore, the error made when linearising the flux by radiation [37]. \[Q_{rad}=h_{rad}\cdot LMTD_{rad} \tag{6}\]where \(LMTD_{rad}\) corresponds to the mean logarithmic radiant temperature and is defined by Equation (7). \[LMTD_{rad}(^{\circ}\mathrm{C})=\frac{\Delta T_{in}-\Delta T_{out}}{Ln\Big{(} \frac{\Delta T_{in}}{\Delta T_{out}}\Big{)}} \tag{7}\]  
where \(\Delta T_{in}\) refers to the temperature difference between the leaving water temperature and the mean radiant temperature and \(\Delta T_{out}\) refers to the temperature difference between the return water temperature and the mean radiant temperature. The average radiant temperature is deduced based on the temperature of each of the cell surfaces by its corresponding form factor obtained using the EES software [38] and determined by Equation (8). \[T_{rad}(^{\circ}\mathrm{C})=\sum_{i=1}^{8}f_{i}\cdot STI+f_{floor}\cdot STfloor \tag{8}\]  
where \(STi\) is the surface temperature of each one of the walls (see Figure 7) and \(f_{i}\) is the corresponding view factor concerning the radiant cover. \(f_{floor}\cdot STfloor\) represents the product of the form factor corresponding to the soil with the cover and the surface temperature of the soil (See Figure 8). Figure 7: Experimental test: surface temperatures \(ST\) of different surfaces.