## 4 Results  
The software DIGITHON provides, in the case of ideal convective system, the thermal power needed in the time step to get the required air temperature. This means that the air temperature is fixed and heating is needed only if the air temperature could go below 20 \({}^{\circ}\)C as calculated by the thermal balance. If the thermal balance leads to air temperatures greater than 20 \({}^{\circ}\)C the air temperature is calculated as a result of the thermal balance. As results the surface temperatures are provided  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Surface** & \(\mathbf{\alpha_{c}}\) **(W m\({}^{-2}\) K\({}^{-1}\))** \\ \hline Radiant floor & 5.5 \\ \hline Radiant ceiling & 1.0 \\ \hline Non-active surfaces & 2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Convective heat transfer coefficient of each surface of the room.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & \\ \hline Air change rate & 0.3 h\({}^{-1}\) = 90 m\({}^{3}\)/h (infiltrations + ventilation) \\ \hline Internal gains & 374 W, 70\% convective and 30\% radiant \\ \hline \hline \end{tabular}
\end{table}
Table 7: Other parameters of the simulations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Building** & \(\mathbf{T_{o}}\) **(\({}^{\circ}\)C)** & \multicolumn{3}{c}{\(\mathbf{T_{air}}\) **(\({}^{\circ}\)C)**} \\ \cline{3-4}  & & **FW, FD** & **C** \\ \hline N & 19.4 & 19.3 & 18.8 \\ I-M, I-E, I-I & 19.8 & 19.7 & 19.4 \\ WI-E, WI-I & 20.0 & 19.8 & 19.6 \\ WI-X, WI-L & 20.0 & 19.7 & 19.6 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mean value of operative temperature and air temperature setting of the radiant simulations.  
and, hence, the mean radiant temperature, as well as the operative temperature, can be calculated. The heating system has been considered to work constantly 24 h, i.e. no overnight set-back has been assumed. The seasonal energy required by the ideal convective heating system has been named \(Q_{id}\).  
When the simulation regards the radiant system, the inputs are the supply water temperature and the mass flow rate in each circuit, and the thermal balance gives as output the return temperature of the water; in this way the energy delivered/absorbed by the water in the time step can be evaluated, as well as the surface temperatures and, hence, the mean radiant and operative temperature. As already mentioned, the control system has been supposed to be on-off with a set-point of 20 \({}^{\circ}\)C and a dead band of \(\pm\)0.5 \({}^{\circ}\)C. This means that if the temperature of the room in the time step is lower than 19.5 \({}^{\circ}\)C the water circulates in the circuits with a defined supply temperature which could be fixed or variable as a function of external temperature. If the air temperature reaches 20.5 \({}^{\circ}\)C the water stops circulating in the circuits and the temperature fluctuates depending on the thermal balance of the room without any active system working in that time step. As for the circuits in the room either all of them work or they do not operate, hence no zone control has been set since the room has been considered as an open space. From the hydronic point of view no other assumptions have been considered, such as a tank or a limiting power supplied by the generator, except for the water volume and related thermal capacity in the circuits in the room. The seasonal energy required by the water embedded heating system has been named \(Q_{w}\).  
The on-off control between 19.5 \({}^{\circ}\)C and 20.5 \({}^{\circ}\)C has been chosen based on a survey provided to companies producing radiant systems in Italy. All producers declared that in flats on-off regulation is the standard control strategy and that PI or PID control is only installed in single family houses due to the high costs of this latest technology.