## 2 Materials and Methods
### Parameters of Interest  
#### 2.2.1 Thermal Effectiveness of Vertilated Roof (\(\epsilon\))  
The ventilated roof can be studied as a heat exchanger [18] in which the air exchanges heat with the mass of the slab. Therefore, air exchanges can be calculated by specific heat exchanger methods. Among them, the effectiveness-number of transfer units (\(\epsilon\)-NTU) method is promising because it has some advantages, such as easy calculation of heat transfer without knowing the air outlet temperature and the option to characterize the element in terms of effectiveness (\(\epsilon\)), which is a physically meaningful parameter. This method has been previously used to calculate ventilated structures such as slabs [4; 7], and ventilated walls using the following Equation (1) [19; 20]:  
\[\dot{Q}_{H}=\epsilon\dot{Q}_{max} \tag{1}\]  
where \(\dot{Q}_{H}\) is the heat actually exchanged between the air and the slab surface and \(\dot{Q}_{max}\) is the maximum heat that could theoretically be transferred, which would be the case if the air exited at the slab surface temperature; therefore, the term \(\epsilon\) is the ratio between these two heat fluxes and represents the effectiveness of the heat exchange.  
Moreover, since the slab mass temperature is constant along the air path, the heat exchange between the air chamber and the slab is similar to a heat exchanger with one fluid condensing or evaporating and is therefore of infinite thermal capacity [21]. According to [6], the ventilated slab can be seen as the case of a heat exchanger in which a fluid is at a constant temperature (as it could be in phase change). In these cases, their effectiveness can be related to the so-called \"number of transfer units\" (NTU) by  
Figure 2: Geometry of ventilated roof: air chamber (6.75 \(\times\) 6.90 \(\times\) 0.10 m\({}^{3}\)); air inlet and outlet ducts (0.30 \(\times\) 0.30 \(\times\) 2.00 m\({}^{3}\)).  
\[\epsilon=1-e^{NTU} \tag{2}\]  
where the NTU is a parameter or number widely known and used, by the academic, technical, and scientific community, for the calculation of heat exchangers and is defined with Equation (3) [22].  
\[NTU=\frac{UIA}{C_{min}}=\frac{hA}{C_{Pair}\dot{m}_{air}} \tag{3}\]  
The overall heat transfer coefficient, U, is equal to the mean convection heat transfer coefficient because this is the only heat transfer mechanism between the slab surface and the air.  
Therefore, operating with the equations, the effectiveness can be calculated as  
\[\epsilon=1-e^{\left(-\frac{hA}{C_{Pair}\dot{m}_{air}}\right)} \tag{4}\]  
On the other hand, as a consequence of the second law of thermodynamics, the maximum theoretical heat transfer flux achievable is  
\[\dot{Q}_{max}=C_{Pair}\dot{m}_{air}(T_{slab}-T_{inlet}) \tag{5}\]  
The maximum heat flow (\(\dot{Q}_{max}\)) can easily be calculated using Equation (5) given that all the terms on the right are known, namely, the properties of the air, its mass flow rate, and the inlet temperature of the air and the temperature of the surface of the slab.  
On the other hand, the flow of heat that is really exchanged (\(\dot{Q}\)) is defined in Equation (1), and the NTU value is defined in Equation (2).