## 7 Results and Discussion  
The model was used to investigate some cases, whose dynamic behavior in transient condition was considered to be of relevant interest. The first case took into consideration a period of one week, made up of seven different daily profiles of temperature with circadian oscillations. In practice, the simulation was extended to a 10-day period, in which the first day was repeated three times at the beginning, to get to a stabilized quasi-periodic condition and to eliminate a possible bias, which might be have been induced by an initial unphysical distribution of the temperature of each layer. For this reason, the significant results were gathered for one week long, from the third day onwards. A thermostat controlled the heater by means of a simple on-off logic, with a \(\pm 1.0\)\({}^{\circ}\)C activation band and a set point at 20 \({}^{\circ}\)C. Similar to a relay, the heater's switch-on point was at 19 \({}^{\circ}\)C and the switch-off point was at 21 \({}^{\circ}\)C, during the day. Analogously, at night, the set point was at 17 \({}^{\circ}\)C.  
Figure 18 illustrates the temperature variation of each layer of the external walls for ten days, together with the external and internal temperatures. The results show that the Simulink model controlled the internal temperature of the considered building between 19 to 21 \({}^{\circ}\)C, whereas the external temperature fluctuated between 2 and 17 \({}^{\circ}\)C. The temperature of the inner layers gradually increased and then decreased, following the fluctuations of the external temperature in a smooth way,  
Figure 16: Weather temperature. Figure 17: Energy consumption of a single-zone building with and without PCMs. between the fourth and seventh days. The thermal variation of the external layer was close to the external temperature, and the amplitude of fluctuation within this layer was the highest between the layers. In fact, each layer decreased the rate of fluctuation, from the outer to the inner layer. In the presence of the night time set point, fixed at 17 \({}^{\circ}\)C, the building exploited the high thermal capacity and the heater remained switched off, all night long, until the daytime set point was restored. Besides, it can be seen precisely that the fluctuations of the nodes near the outer layer were greater than the internal nodes, and the variation of node two slightly increased when the outside temperature increased. Figure 19 shows the temperature variation of an internal wall. The profile of node three, red in Figure 19, represents the mean radiant temperature, including the heater surface. On the other hand, while the outdoor temperature was increasing, the minimum temperature of node three and four gradually increased. The second case describes the thermal behavior of a country house, located in the countryside or in the mountains, used for leisure or at weekends, in a non-continuous way, for pleasure. It was assumed that the apartment was empty at the beginning of the simulation, as well as the surrounding ones. The initial internal temperature was fixed at 6 \({}^{\circ}\)C; the internal initial temperature corresponds to the set-point common no-icing controls, which maintain the internal temperature of the house always higher than 5 \({}^{\circ}\)C, in order to prevent the formation of ice in the pipes and their consequent rupture. The simulations were accomplished in case of three different heater powers and a set of outdoor temperatures: 0, 5, 10, 15, and 20 \({}^{\circ}\)C.  
The graph in Figure 20 shows the starting transient condition for the internal air temperature of the building, which was calculated for a set of different cases. The cases, which were considered, differed one from another for the maximum thermal power of the heater and for the temperature of the external environment. At first glance, it was easy to appreciate that the time, which was needed to achieve thermal comfort, sharply decreased as the power and the outdoor temperature increased. Figure 18: Inlet air temperature variation.