## 3 Thermal Balance and Time Constant of a Solid Slab Containing Phase Change Materials (PCM)
When solar radiation absorbed on the exterior surface, _q_\({}_{s}\), is included, the boundary condition takes the form:  
\[q(L)=\frac{1}{R_{se}}[T(L)-T_{e}]-q_{s}=\frac{1}{R_{se}}[T(L)-T_{se}] \tag{24}\]  
\[T_{es}=T_{e}+R_{se}\cdot q_{s} \tag{25}\]  
\(T_{es}\) is the so-called sol-air temperature [24]. After consideration of _R_\({}_{se}\) = 0.04 m\({}^{2}\cdot\)K/W, solar radiation flux of 1000 W/m\({}^{2}\) makes (\(T_{es}-T_{e}\)) by 40 \({}^{\circ}\)C.  
The external (sol-air) temperatures used in the analysis are shown in Figure 4. They are computed as follows:  
\[T_{es}=T_{e0}-A_{1}\sin\!\left[\pi\cdot\frac{t+6}{12}\right]\]  
during the night-time, \(t\) < 6 h or \(t\) > 18 h and  
\[T_{es}=T_{e0}-(A_{1}+A_{2})\cdot\sin\!\left[\pi\cdot\frac{t+6}{12}\right]\]  
during the daytime, 6 h \(\leq t\leq\) 18 h.  
Two series of simulations were performed for _T_\({}_{i}\) = 20 \({}^{\circ}\)C, _T_\({}_{e0}\) = 20 \({}^{\circ}\)C and _T_\({}_{i}\) = 25 \({}^{\circ}\)C, _T_\({}_{e0}\) = 20 \({}^{\circ}\)C, with amplitudes values _A_\({}_{1}\) = 5 \({}^{\circ}\)C and _A_\({}_{2}\) = 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, and 60 \({}^{\circ}\)C. Subsequently, simulation cycles were repeated to minimize the effect of initial conditions. The specific PCM used in the simulations had a melting temperature of 23 \({}^{\circ}\)C and complete solidification at 22 \({}^{\circ}\)C. This selection of PCM phase transition characteristics was intentional because both external and internal air temperatures during the night were below 22 \({}^{\circ}\)C, i.e., the PCM solidification temperature, to ensure that the stored latent heat is fully discharged during the night. This strategy created perfect conditions for PCM heat storage and maximum utilization of phase change enthalpy. Please note that an appropriate design of the PCM-enhanced building envelopes should always include both a detailed temperature analysis in the place where PCM is located, as well as climatic conditions. The goal is to minimize the number of days, when PCM is either not fully melted during the day or solidified during the night. Figures 5 and 6 show the comparisons of diurnal heat flux profiles on internal surfaces for two thicknesses of envelope assemblies containing 14 cm and 30 cm of thermal insulation, and for two configurations of thermal insulation, i.e., one containing plain fibrous insulation layer, and another one containing 30% PCM enhanced insulation. The comparison of the plots and calculated daily heat flow values indicates that, for cyclic processes, the effect of PCM in an insulation layer results in the following: (i) notable time shifting of the heat flux maxima, (ii) lowering the peak heat flux value, and (iii) reduction of the overall heat flow. Heat losses corresponding to minimum external temperatures at midnight are shifted to morning time. This can significantly reduce the morning energy consumption peak during the cooling season. At the same time, the daily heat gain peaks (peak cooling loads) are moved from the midday hours to the afternoon. Figure 4: Internal and external temperature daily profiles used in thermal simulations of envelope assemblies containing thermal insulation and PCM (showed _Ti_ = 20 \({}^{\circ}\)C). Figure 5: Comparisons of diurnal heat flux values for 14 cm thick assemblies containing insulation with 30 wt% PCM. For assemblies with a 14 cm insulation/PCM-insulation layer, comparisons of calculated diurnal heat flow values indicate cyclic external peak load shifts up to 2.5 h and peak heat flow value reductions up to 11% (see Figure 5). However, PCM does not cause any considerable reductions in total daily heat flow rates. At the same time, heat losses corresponding to minimum external temperatures at midnight are shifted to morning hours. As depicted in Figure 5, more noticeable dynamic effects can be observed in the cases of lower sol-air temperatures during the day (\(T_{es}\) (a) and \(T_{es}\) (b) cases on Figure 4).