## 3 Thermal Balance and Time Constant of a Solid Slab Containing Phase Change Materials (PCM)
This indicates that thin roof assemblies (i.e., cathedral and sandwiched roofs) containing thin layers of conventional and PCM-enhanced thermal insulation are not effective in time shifting of the thermal peak loads in buildings during the cooling season. For material configurations presented in Figure 5, both the reductions of peak thermal load values and the total load values are insignificant. Assuming that the cost would not be too high, further addition of PCM could still improve this performance. However, PCM would need to be added in a different (concentrated) form with significantly higher thermal storage capacity. This is because the PCM blending with a rate higher than 30% would compromise insulation thermal resistance, also keeping in mind that due to the mentioned earlier simplifications in the numerical representation of external thermal loads, the results discussed above are only approximate projections derived for specific building envelope components, not for the whole building. Similar thermal simulations are repeated for the assembly containing a 30 cm layer of thermal insulation with PCM content of 30% and density of 33.6 kg/m\({}^{3}\). Figure 6 shows a comparison of diurnal heat flux profiles at the internal surface of the 30 cm assembly for different internal and external sol-air temperatures. The comparisons of calculated heat flux values indicate that for cyclic external thermal excitations, the effect of PCM blended with thermal insulation results in time shifts of the heat flux maxima between 3.0 and 6.5 h.  
In addition, an approximate 50% decrease in peak heat flow values was observed. In the case of a 14 cm assembly, the PCM-enhanced thermal insulation did not generate large reductions in total daily heat flow rates, except for the case of the lowest sol-air temperatures during the day, as shown in Figure 4. These results show that an increased thickness of thermal insulation reduces the peak time value of cooling loads but does not generate a significant time shift of the peak cooling load. In the case of PCMs, the factors that impact time delay of the envelope-generated thermal response are PCM enthalpy and the amplitude of external temperature excitations. Table 2 summarizes the results of transient thermal simulations for two thicknesses of envelope assemblies, i.e., 14 and 30 cm, for two configurations of thermal insulation; one containing plain  
Figure 3:0 Comparisons of calculated diurnal heat flow values for 30 cm thick assemblies containing thermal insulation containing 30 wt% of PCM. fibrous insulation layer, and another one containing insulation with 30 wt% PCM. Two internal space temperatures of \(T_{i}\) = 20 \({}^{\circ}\)C and \(T_{i}\) = 25 \({}^{\circ}\)C have been included. A comparison of the heat flux plots and calculated daily heat flow values indicates that for the case of a thicker envelope configuration, the effect of PCM in thermal insulation results in significant time shifting of the heat flux maxima (up to 6.5 h), but not in considerable reductions of the total diurnal heat flow (thermal load). As depicted in Figure 6, roof assemblies without PCM start to rapidly warm up around 10 a.m. and reach daily peak load at about 5 p.m. At the same time, between 10 a.m. and 1 p.m., systems containing PCM still generate the lowest daily thermal loads. It is worthwhile to note that PCM-enhanced roofs containing 30 cm thick assemblies with thermal insulation enhanced with 30 wt% of PCM, reach thermal peaks at significantly later evening hours, i.e., between 9 p.m. and almost midnight. Furthermore, in the case of a thicker envelope assembly containing PCM, simulated reductions of the heat gains maxima are 48%, 51%, 45% for \(T_{i}\) = 20 \({}^{\circ}\)C, and 82%, 65%, 40% for \(T_{i}\) = 25 \({}^{\circ}\)C. Taking this into consideration, thick envelope assemblies containing PCM (i.e., roofs with thick layers of attic insulation) can be utilized as very effective means of whole building peak load management.