## 4 Numerical Analyzes  
In order to determine the heat flux \(Q\), numerical calculations were made in the PSI THERM program based on the finite element method. The same assumptions were made as when calculating in accordance with the guidelines of EN ISO 6946:2017 [28]. The upward flow and the downward flow were modelled. The presence of reinforcement, air voids and materials of different densities and different heat transfer coefficients were taken into account. The heat transfer coefficient in air voids was selected based on the air void database available in the computer program. The value of the heat transfer coefficient of the void is selected depending on its thickness (in the program, the thickness is graded every 5 mm-a range of values for voids with a thickness of 5 mm to 300 mm), depending on the direction of heat flow (horizontally, upwards, downwards) and the type of void (nonventilated or poorly ventilated). Therefore, alternative resistance was used. Ceiling models were divided into finite elements with the dimension of the side equal to 20 mm. The obtained results are shown in Figures 6-12. On the basis of heat flux \(Q\) of the obtained numerical models, the heat transfer coefficient \(U\), thermal resistance \(R_{T,}\) and thermal resistance of the ceiling \(R_{U}\) were calculated. The results of these calculations are summarized in Tables 13 and 14. Table 13 presents the results obtained for individual variants of the Teriva Panel ceiling, whereas Table 14 contains the results obtained for the Smart system and three Teriva systems: Base, Plus and Termo. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(U\) & **Flow** & **Teriva Panel 160** & **Teriva Panel 180** & **Teriva Panel 200** \\ \hline \multirow{2}{*}{0.25 W/(m\({}^{2}\cdot\)K)} & upwards & 155.2 mm & 154.4 mm & 153.6 mm \\  & downwards & 149.2 mm & 148.4 mm & 147.6 mm \\ \multirow{2}{*}{0.45 W/(m\({}^{2}\cdot\)K)} & upwards & 78.7 mm & 77.9 mm & 77.1 mm \\  & downwards & 72.7 mm & 71.9 mm & 71.1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Thicknesses of the polystyrene layer necessary for the analyzed ceilings to obtain the required \(U\) values. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(U\) **Coefficient** & **Flow** & **Smart** & **Teriva Base** & **Teriva Plus** & **Teriva Termo** \\ \hline \multirow{2}{*}{0.25 W/(m\({}^{2}\cdot\)K)} & upwards & 158.3 & 151.4 & 147.5 & 140.5 \\  & downwards & 152.3 & 145.3 & 141.5 & 134.4 \\ \multirow{2}{*}{0.45 W/(m\({}^{2}\cdot\)K)} & upwards & 81.7 & 74.8 & 70.9 & 63.9 \\  & downwards & 75.7 & 68.8 & 64.9 & 57.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Thicknesses of the polystyrene layer necessary for the analyzed ceilings to obtain the required \(U\) values. Figure 8: Results of numerical calculations of the Teriva Panel ceiling with a height of 180 mm: (**a**) flow upwards, (**b**) flow downwards. Figure 6: Results of numerical calculations of the Smart ceiling: (**a**) flow upwards, (**b**) flow downwards. Figure 7: Results of numerical calculations of the Teriva Panel ceiling with a height of 160 mm: (**a**) flow upwards, (**b**) flow downwards. Figure 10: Results of Teriva Base ceiling calculations: (**a**) flow upwards, (**b**) flow downward. Figure 9: Results of numerical calculations of the Teriva Panel ceiling with a height of 200 mm: (**a**) flow upwards, (**b**) flow downwards. Figure 11: Results of Teriva Plus ceiling calculations: (**a**) flow upwards, (**b**) flow downwards.