## 4 Results
### Simulation Results
#### 4.2.2 Simulation Results for All Cases, Semi-Infinite Thickness  
The periodic penetration depth Equation (30) is of interest. It must be much smaller than the thickness of the material in order to satisfy the general assumption of semi-infinite material. For gypsum and a time period of 24 h:  
\[\sqrt{\frac{\tau_{p}}{\pi}}=0.0077\ \mathrm{m}=7.7\ \mathrm{mm} \tag{36}\]  
For a six days time period the penetration depth is 18.8 mm. The corresponding values for clay are 7.5 and 18.5 mm for the 12 h and 72 h test, respectively. It means that for diurnal variations, both the thickness 20 mm and 40 mm should be applicable for the semi-infinity analysis. However, it is \begin{table}
\begin{tabular}{l c c} \hline \hline \(Z\) [s/m] & \(f_{A}\) & \(\frac{f_{A}}{f_{A}(Z=0)}\) \\ \hline
0 & 2.2745 & 1 \\
60 & 2.2203 & 0.98 \\
360 & 1.9737 & 0.87 \\
10\({}^{3}\) & 1.5708 & 0.69 \\
10\({}^{4}\) & 0.3684 & 0.16 \\
10\({}^{5}\) & 0.0416 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The amplitude parameter\(f_{A}\) for the determination of the total moisture uptake during a half cycle, case triangular 12/12h. The third column shows the reduction in moisture buffering capacity due to the surface resistance. questionable if the thickness of 20 mm (gypsum case) would work for the six days variation. For the thickness 40 mm (clay case) it would be acceptable. Tables 8 and 9 show the simulated vales for the moisture buffering uptake and the time lag. The first value for the time lag represents the time between the RH-peak and the moisture uptake peak. The second one represents the time between the lowest RH and the lowest moisture uptake value. The difference in these values clearly show the asymmetry and non-linearity of the problem. As also shown in the experimental test (Table 4), the longer lag during the de-humidification in the 8/16 h tests, compared to the humidification can be also explained by analysing Equation (39). However, the time lags in Table 9 also justify the models assumption that the surface temperature follows the room temperature. The time delay for the change in temperature in the boards, due to a room temperature change, is around 1200 s for the 20 mm thick gypsum board and 1600 s for the 40 mm thick clay board. This is roughly 1/3 of an hour. For diurnal variations the ratio between this time delay and the time period is of the order \(10^{-2}\). For the six days period the ratio is almost one order less and can further justify the approximation. Figures 8 and 9 show the moisture uptake function \(m\) as a function of \(t/t_{p}\). Time zero corresponds to the time of the RH peak, and the start time of integration, i.e., we assume the uptake of moisture being zero at time zero, which is an arbitrary choice of level. Consequently, it is possible to follow the accumulation and the drying out of moisture of the specimens during the time period. The two curves with the largest amplitude represent the case with a six days variation. It is clear that the diurnal variations do not significantly differ between 12/12 h and 8/16 h cases. As also shown in the experimental test (Table 4), the longer lag during the de-humidification in the 8/16 h tests, compared to the humidification can be explained by analysing Equation (39). If a 8/8 h period was analysed, instead of the 8/16, the time \(t_{p}\) period would correspond to 16 hours. For the 16/16 h test would be i.e., 32 h. The difference in uptake \(m_{A}\) would then differ between the 8/8 and 16/16 h approximately of \(\sqrt{32/16}=\sqrt{2}\), which correspond to 40% difference. For the time delay, a pure sinusoidal variation suggests that the lag is directly proportional to the time period, which corresponds to a doubled time delay. The time delay ratio in the mixed case (8/16) in Figure 9 is 1.17-1.62. Therefore, the asymmetry is reasonable as the delay ratio is in between 1 and 2, in line with the 8/8 and 16/16 ratio.