## 4 Results
### Radiant Systems Efficiency  
The hourly values of the energy provided as results of the simulations performed by DIGITRON permit the evaluation of the efficiency of the radiant system. Considering the seasonal energy \(Q_{id}\) of an ideal convective system and the seasonal energy provided to the radiant system \(Q_{w}\), the efficiency of the radiant system can be defined as:  
\[\eta_{emb+ctrl}=\frac{Q_{id}}{Q_{w}}=\eta_{emb}\cdot\eta_{ctrl} \tag{5}\]  
This is a joint embedded and control efficiency since it takes into account the losses of the radiant system behind the pipes as well as the effects of the control system. In Figure 6 the results obtained for the combination of eight building structures, three radiant systems, and two supply water temperature control modalities are represented. As already explained, these efficiency values are calculated along the whole heating season and hence they include the thermal inertia of the building structures, influence of solar and internal gains, the way the water circulating in the circuit is controlled as well as the heat losses in the room behind the radiant system. Since it is a combination of all these factors, it is not possible to split among a control efficiency and an embedded efficiency but they have to be considered together. In the same figure, the steady-state embedded efficiency calculated through HEAT2 is represented in green colour. These efficiency values take into account only the geometry and the thermal properties of the building structure where the radiant system is placed and proper boundary conditions. Considering the useful thermal power \(\Phi_{u}\) towards the heated space and the thermal losses \(\phi_{l}\) towards the adjacent space, the embedded efficiency in steady-state conditions of the radiant system can be calculated as:  
\[\eta_{emb,H2}=\frac{\phi_{u}}{\phi_{u}+\phi_{l}} \tag{6}\]  
The following boundary conditions were fixed: room temperatures equal to 20 \({}^{\circ}\)C, convective heat transfer coefficient on the surfaces according to Table 6, adding the radiant heat transfer coefficient on the surfaces equal to 5.5 W m\({}^{-2}\) K\({}^{-1}\), water temperature according to the specific design heat load for each case (Table 4). In order to maintain the same comfort conditions (i.e. same operative temperature), the radiant simulations were repeated reducing the air temperature set-point compared to the convective ideal simulation. As already mentioned, the set-point was chosen in such a way that the resulting mean value of the operative temperature in the point P in the period from December to February was the same of the mean value of the ideal convective simulations with set-point of the air \(T_{air}\) = 20 \({}^{\circ}\)C. This way an iterative process has been carried out for each single case in order to match the same operative temperature. The period from December to February has been chosen to evaluate the mean value of the operative temperature, because in the insulated building the average temperature in mild months is quite high, the heating system works rarely and it is difficult to compare buildings under the same operating conditions. In fact the heating energy need in the period December-February is, depending on the insulation level, from 75% to 90% of the total heating energy need of the buildings. The embedded and control efficiency values for the radiant systems have been calculated via Equation (5). The results are shown for the period December-February (Figure 6), as well as for the whole season (Figure 7). As a matter of fact, the first value shows the results under the same indoor conditions (operative temperature), the second one gives an idea on the embedded and control efficiency over the whole season. In Figures 6 and 7 also the embedded values which can be calculated in steady state conditions by using Equation (6) with results of FDM are represented.