# Thermal Time Constant of PV Roof Tiles Working under Different Conditions
## 5 Heat Balance of the Roof and Photovoltaic Roof Tiles  
It was assumed that the PV roof tile temperature was uniformly distributed on its different layers and that the temperature of all PV cells was the same. In the case of photovoltaic roof tiles, it suffices to take conductive and convective heat exchange into account, while neglecting radiation. The value of electrical energy generated by PV roof tiles and the quantity of solar energy reaching the PV module surface were also considered. The thermal balance equation (PET model, photo-electro-thermal) can be formulated as follows [17]:  
\[S\_C_{TH}\cdot\frac{dT_{pv}}{dt}=Q_{solar}-P_{pv}-Q_{conv}, \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Heat Transfer** & \multicolumn{3}{c}{**Heat Stream Direction**} \\ \cline{2-4}
**Resistance (m\({}^{2}\)K/W)** & **Up** & **Horizontal \({}^{*}\)** & **Down** \\ \hline \(R_{si}\) & 0.10 & 0.13 & 0.17 \\ \(R_{se}\) & 0.04 & 0.04 & 0.04 \\ \hline \hline \end{tabular} \({}^{*}\) Values applying to the horizontal direction were used in case the direction of a heat stream was deflected by \(\pm 30^{\circ}\) from the horizontal plane and when it was possible to change the heat stream direction.  
\end{table} 
Table 4: Values of the heat transfer resistance on the surface and air layers, depending on the heat stream flow direction [36]. Figure 4: Equivalent electrical circuit for the resistance–capacity (RC) model for: (**a**) a single PVL68 photovoltaic roof tile; (**b**) the entire roof composed of pine board and PVL68 photovoltaic roof tiles. where \(S\) is the PV module surface (m\({}^{2}\)), \(C_{TH}\) is the PV roof tile thermal capacity (J/m\({}^{2}\)K), \(T_{pv}\) is the PV cell temperature (K), \(Q_{solar}\) is the solar energy reaching the PV module surface (W), \(P_{pv}\) is the electrical power generated by the PV module (W) and \(Q_{conv}\) represents the convective heat losses (W). The power density value of solar radiation reaching the PV module surface can be identified from Equation (8) [17]:  
\[Q_{solar}=\alpha\cdot S\cdot E, \tag{8}\]  
where \(\alpha\) is the PV module coefficient of solar radiation absorption, whose value ranged from 70% to 90%, and \(E\) is the solar radiation power density (W/m\({}^{2}\)). The total value of convective heat losses is the total of forced convection heat on the PV module front surface and free convection heat of the module rear part, whereas free convection is minor compared to forced convection and can be neglected. Convective heat losses can be identified from Equation (9) [17]:  
\[Q_{conv}=S\cdot h_{conv}\cdot\left(T_{pv}-T_{amb}\right), \tag{9}\]  
where \(h_{conv}\) is the coefficient of convective heat exchange (W/m\({}^{2}\)K) and \(T_{amb}\) is the ambient temperature (K). The coefficient of convective heat exchange \(h_{conv}\) depends on the wind speed \(v_{wind}\) (m/s), which is expressed by Equation (10) [18]:  
\[h_{conv}=8.55+2.56\cdot v_{wind}. \tag{10}\]  
The \(h_{conv}\) coefficient was determined experimentally by the authors in [18], who also reviewed the other coefficients of Equation (10) depending on the determination methods--in a wind tunnel or during real measurements--and obtained on the basis of the fundamental theory of heat exchange and criterion numbers. It was assumed that in natural conditions, the change in the ambient temperature at irradiance changes was slow enough to be considered as a constant value. Upon including Equations (8)-(10), a relationship for the momentary temperature value of PV cells was obtained in Equation (7) as an analytical solution of the first-order PET differential equation:  
\[T_{pv}(t)=\left[1-\exp\left(-\frac{h_{conv}}{C_{TH}}.t\right)\right]\cdot \frac{\alpha\cdot S\cdot E-P_{pv}}{S\cdot h_{conv}}+T_{amb}. \tag{11}\]  
An assumption was made that under natural conditions, ambient temperature change at irradiance change occurs slowly enough to be considered a fixed value.