## 3 Thermal Balance and Time Constant of a Solid Slab Containing Phase Change Materials (PCM)
A curve showing the enthalpy relation with temperature is depicted in Figure 3a for used PCM. Figure 3c contains a comparison of heat fluxes across a 140 mm layer of cellulose fiber and a second specimen of the cellulose insulation with 30 wt% of microencapsulated organic PCM. The heat fluxes are the result of a rapid increase in temperature from 14 to 44 \({}^{\circ}\)C, on one surface. Time constants, \(\tau_{s}\), for the thermal excitation obtained from Equations (16), (19), and (23) are 56 min for insulation alone and 153 min for the PCM-insulation blend. It is worthwhile to note that in the case of the analyzed building envelope containing an insulation layer with a thickness of 0.14 m, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **L** & **k** & \(\rho\) & \(\epsilon_{\text{p}}\) \\  & **(m)** & **(W/(m-K))** & **(kg/m\({}^{3}\))** & **(kJ/(kg\(\cdot\)K))** \\ \hline Gypsum board & 0.013 & 0.16 & 800 & 1.088 \\ Insulation & 0.140 & 0.039 & 25.6 & 1.381 \\ Ins. 30\% PCM & 0.140 & 0.039 & 33.6 & \(\epsilon_{\text{p}}\)(T) \\ Plywood & 0.013 & 0.12 & 544 & 1.244 \\ Envelope finish & 0.013 & 0.07 & 556 & 1.255 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties of analyzed thermal envelope. Figure 3: Illustration of dynamic thermal performance of phase change materials (PCM)-enhanced cellulose insulation. (**a**) Calorimetric data for PCM used in the analysis; (**b**) Microscope image of the cellulose-PCM blend; (**c**) Heat fluxes on the surface of a 14 cm thick layer of thermal insulation after a rapid temperature increase on the opposite surface. the addition of 30% by weight of PCM microppellets to the cellulose fibers tripled the time constant of the resulted material blend. This is illustrated in Figure 3c, where heat flux rose significantly faster in the case of thermal insulation not containing PCM. Dynamic Thermal Response of PCM-Enhanced Insulation under Different Magnitudes of Thermal Excitations: A Theoretical Analysis  
A series of one-dimensional thermal simulations were performed to illustrate the impact of PCM and building envelope configuration on the dynamic thermal behaviour. The major goal was to investigate the transient performance of lightweight building elements with and without PCM-enhanced insulation. Insulation thickness, internal space temperature, and external thermal conditions were varied. The thermodynamic data shown in Figure 3 were used to establish _Ceff_, _Ceff_(_T_) and \(l_{a}\)(_T_) from Equation (6), and in turn were used in the following analysis. Building envelope assembly similar to a typical 165 mm thick cathedral roofing structure was considered. The 140 mm space enclosed by the gypsum board and plywood was either filled with cellulose insulate or with PCM-cellulose blend. In the second case, the 140 mm thick layer of PCM-cellulose was used with density of 33.6 kg/m\({}^{3}\). It contained 30 wt% of microencapsulated PCM. This yields 1.4 kg/m\({}^{2}\) of PCM of the building element, that provides \(\sim\)168 kJ/m\({}^{2}\) of energy storage. For comparison, a 300 mm thick layer would have about 360 kJ/m\({}^{2}\) energy storage capacity. Assuming one-dimensional heat transfer without consideration of thermal bridging due to structural components, the air-to-air thermal resistance of analysed building element is _R_\({}_{u}\) = 4.14 m\({}^{2}\cdot\)K/W (this includes surface film resistances _R_\({}_{si}\) = 0.13 m\({}^{2}\cdot\)K/W and _R_\({}_{se}\) = 0.04 m\({}^{2}\cdot\)K/W). The assembly U-value is 0.24 W/m\({}^{2}\cdot\)K.  
The main goal is to analyse the dynamic responses of the assumed roof assembly, due to external thermal excitations of different magnitudes. In this analysis, the external boundary conditions are represented by a single value for the external air temperature and incident solar radiation.