# Thermal Time Constant of PV Roof Tiles Working under Different Conditions
## 3 Thermal Resistance and Thermal Capacity of the Roof and PV Roof Tiles  
By knowing the construction and material parameters of PV roof tile layers as well as the roof structure, one can identify the thermal resistance and thermal capacity of each layer and the entire PV roof tile based on current legal regulations and standards concerning building materials. PN-EN ISO 6946:2017 \"Building components and building elements. Thermal resistance and thermal transmittance. Calculation methods\" is the standard valid currently in Poland [36]. The thermal resistance and thermal capacity of a homogeneous layer can be identified based on the following equations [18; 36; 37]:  
\[R_{th}=\frac{d_{m}}{k}, \tag{1}\]  
\[C_{th}=\rho_{n}c_{p,n}\cdot d_{m}, \tag{2}\]  
where \(R_{th}\) is the material (layer) thermal resistance (m\({}^{2}\)K/W), \(C_{th}\) is the material (layer) thermal capacity (J/m\({}^{2}\)K), \(d_{m}\) is the layer thickness (m), \(k\) is the thermal conductivity coefficient of the material (W/mK), \(\rho_{n}\) is the material density (kg/m\({}^{3}\)), and \(c_{p,n}\) is the material specific heat (J/kgK). The values of thermal resistance and thermal capacity of PV roof tile layers and roof structure, identified according to Equations (1) and (2), are collected in Table 3. Individual thermal resistance values \(R_{th}\) describe the material resistance to transmit heat, while thermal capacity values \(C_{th}\) refer to the ability to absorb and transmit heat. Heat transmittance (conduction) between the material layers (building component layers) is a process involving inert movement energy transmittance by adjacent particles; it is one of three kinds of heat transfer, with convection and radiation being the other two. The process strictly depends on the material parameters of the layers. The total value of thermal resistance \(R_{TH}\) and thermal capacity \(C_{TH}\) of a PV roof tile or a roof is the total of thermal resistance and thermal capacity values of all layers in the analysed structure, which is expressed by the following equations [18; 36]:  
\[R_{TH}=\sum_{i}R_{th\ i}, \tag{3}\]  
\[C_{TH}=\sum_{i}C_{th\ i}. \tag{4}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Layer** & **Parameter** & \(R_{th}\) & \(C_{th}\) \\  & **(m\({}^{2}\)K/W)** & **(J/m\({}^{2}\)K)** \\ \hline
**ETFE** & \(2.08\times 10^{-3}\) & 900 \\
**PV cell** & \(10\times 10^{-9}\) & 2.17 \\
**Grid connection** & \(0.1\times 10^{-9}\) & \(3.6\times 10^{-3}\) \\
**PVDF** & \(16.66\times 10^{-3}\) & 4029.78 \\
**Pine board** & \(71.43\times 10^{-3}\) & 18,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated \(R_{th}\) and \(C_{th}\) values of PV PVL68 materials (layers) and roof components. The abovementioned Equations (3) and (4) were used to calculate the values of thermal resistance and thermal capacity of the reference PVL68 PV roof tile, which amounted to \(18.74\times 10^{-3}\) m\({}^{2}\)K/W and 4931.95 J/m\({}^{2}\)K respectively. The roof structure in the deliberations was treated as a homogeneous layer composed only of pine boards, without taking other components (e.g., rafters) into account, which would not greatly affect the identified RC time constant values due to the small size of the components compared to the roof slope area. Details of the method used to identify the total resistance of a roof composed of different layers, taking all components into account, are presented in [38]. When Equation (1) is transformed into one which enables the determination of the coefficient of thermal conductivity (thermal conductance) \(k\) of a PV roof tile, the following relationship is obtained:  
\[k=\frac{d_{m}}{R_{th}}. \tag{5}\]  
The coefficient of thermal conductivity \(k\) of the studied PV PVL68 roof tile was 0.13 W/mK, that is, much lower than the value of \(k\) for standard ceramic tiles (i.e., 1 W/mK, according to standard [36]) and slightly lower than the coefficient value for the top layer of asphalt roofing felt (for which it amounts to 0.18 W/mK).