## 3 Mathematical Model  
The partial differential equation for transient heat conduction of a two-dimensional solid is [18]:  
\[\frac{\partial(\rho c_{p}T)}{\partial t}=\frac{\partial}{\partial x}\bigg{(} \lambda\frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(} \lambda\frac{\partial T}{\partial y}\bigg{)} \tag{1}\]  
where \(\lambda\) is the thermal conductivity, \(\rho\) is the density, and \(c_{p}\) is the specific heat. The boundary conditions used to solve in the physical model were as follows: The exterior surface (\(y=y_{1}\)) is in contact with the outdoor environment or environmental conditions of different cities of Mexico. Therefore, the roof has convective and radiative exchange with the surroundings. The lateral surfaces are thermally insulated (\(x=0\) and \(x=W\)), and the interior surface (\(y=0\)) also exchanges heat by convection and radiation with the indoor environment. Mathematically, the boundary conditions can be expressed as:  
\[-\lambda\frac{\partial T}{\partial y} = \alpha G+h_{out}(T-T_{out})+\sigma\varepsilon(T^{4}-T_{sky}^{4}) \ \ \text{for}\ \ \ y=y_{1},\ \ \ 0\leq x\leq W \tag{2}\] \[-\lambda\frac{\partial T}{\partial y} = h_{in}(T-T_{in})\ \ \ \text{for}\ \ \ y=0,\ \ \ 0\leq x\leq W\] (3) \[\frac{\partial T}{\partial x} = 0\ \ \ \text{for}\ \ \ x=0,\ \ \ 0<y<y_{1}\] (4) \[\frac{\partial T}{\partial x} = 0\ \ \ \text{for}\ \ \ x=W,\ \ \ 0<y<y_{1} \tag{5}\]  
In Equation (2), \(G\) is the solar radiation received by the roof, \(\alpha\) is the solar absorptance of the coating, \(\varepsilon\) is the thermal emittance, and \(\sigma\) is the Stefan-Boltzmann constant. The beam-solar radiation received by roofs has different angles during the day. The solar radiation data used for the simulations were obtained from measurements performed using weather stations, and these stations used pyranometers for measuring total (beam plus diffuse) radiation over a horizontal surface. The detectors of the pyranometers had a response that did not depend on radiation wavelength over the solar spectrum. Further, these devices had a response independent of the angle of incidence of solar radiation. Thus, it can be supposed that the roofs received total radiation in a perpendicular direction. To calculate the outdoor convective heat transfer coefficient (\(h_{out}\)) in Equation (2), the following empirical correlation was used [19]:  
\[h_{out}=2.8+3.0v \tag{6}\]  
where \(v\) is the wind speed in m/s and \(h_{out}\) is the outdoor convective heat transfer coefficient in W/m\({}^{2}\)K. The value of \(T_{sky}\) is calculated with the following expression [19]:  
\[T_{sky}=0.0552T_{out}^{1.5} \tag{7}\]  
In Equation (7), \(T_{out}\) is the outdoor air temperature, and both temperatures \(T_{sky}\) and \(T_{out}\) are expressed in K. The heat transfer coefficient in the interior environment \(h_{in}\) in Equation (3), which considers both convection and thermal radiation, is equal to 6.13 W/m\({}^{2}\) when the heat flux heat goes to the indoor air and 9.26 W/m\({}^{2}\)K when the heat flux goes from the indoor air to the interior surface [20]. For the roof configuration (b) of Figure 1, due to the different materials involved and the thermophysical properties' changes across the interface between concrete and thermal insulation, the harmonic mean was used to calculate these properties in the interface. Furthermore, perfect contact between both materials was considered, such that heat flux through the interface was the same for the materials involved. The properties of the concrete used for the simulations were \(\rho=2400\) kg/m\({}^{3}\), \(c_{p}=1080\) J/(kg\(\cdot\)K), and \(\lambda=1.8\) W/(m\(\cdot\)K). On the other hand, the properties of the insulation were \(\rho=28\) kg/m\({}^{3}\), \(c_{p}=1800\) J/(kg\(\cdot\)K), and \(\lambda=0.033\) W/(m\(\cdot\)K).