## 2 Method  
The simulation of the dynamic behaviour of the considered radiant systems in different kinds of buildings has been carried out with the model DIGITHON [23]. This numerical model performs the detailed simulation of the dynamic behaviour of water-based surface heating and cooling systems. In the model DIGITHON, each surface of the room (containing pipes or not) is divided into elements named tiles (Figure 1); an overall heat balance is carried out for each element. Figure 1: Example of discretization of a room equipped with a radiant system embedded in the floor [24] with the relevant surfaces where the heat balance is solved for structures with or without embedded pipes. To solve the dynamic conduction within the structures the \"response factor\" technique is used [24; 25]. In the present work the heat transfer response factors are calculated by using the commercial software HEAT2 [26], which is based on the Finite Difference Method (FDM). Three simulations must be performed to describe the thermal conduction in a building structure with embedded pipes. In each simulation a triangular impulse of temperature is given on the inner surface, on the outer surface and on the internal surface of the pipes, thus the resulting heat flows on the three surfaces are recorded. After normalization, for a generic trend of temperature on the inner surface of the building structure, on the outer surface of the building structure and on the internal surface of the pipe, the response factor can be used, by superposing of effects, to calculate the specific heat fluxes on the three considered surfaces. More details about this method, included an example showing its accuracy, can be found in [27]. In case of building structures without embedded pipes, two simulations with triangular input have to be performed. The convective heat flux for the \(j\)-th general surface element \(q_{c,j}\) is expressed as:  
\[q_{c,j}=h_{c,j}S_{j}\Big{(}T_{s,j}-T_{f,j}\Big{)} \tag{1}\]  
where \(T_{f,j}\) is the air temperature of the room (for the inner surface elements), the air temperature of the adjacent room (in case of internal walls) or the sol-air temperature (in case of outdoor surface). As for the air temperature, it can be assumed to be uniform when the room is less than 3 m height both in heating [28] and in cooling conditions [29] for radiant systems as well as in a wide range of situations that have been confirmed by other studies [30]. As for convective heat transfer coefficients, a recent review [31] shows that the most reliable analyses are those based on measurements in real size test rooms. In particular, as shown in [23], constant values can be considered for the convective heat exchange coefficients. For this purpose convective heat exchange coefficients have been assumed constant in the calculations. As for the radiative heat exchange, assuming near-black surfaces, in the infrared with small temperature differences the mutual radiation with another surface can be written as:  
\[q_{r,j-k}=F_{j-k}4\sigma T_{m}^{3}S_{j}\big{(}T_{s,j}-T_{s,k}\big{)}=F_{j-k}h _{r,j-k}S_{j}\big{(}T_{s,j}-T_{s,k}\big{)} \tag{2}\]  
Since the surface discretisation is fine enough, in the model view factors \(F_{j,k}\) are calculated in a detail way, as shown in [27]. Shortwave radiation entering from glazing elements has to be summed on the right side of Equation (2) as well as the internal radiant gains. As demonstrated in detail in [27] for cooling conditions and in [32] in heating period, there is no difference in the overall balance when considering solar radiation in detail or when it is assumed to be uniformly distributed. Hence, the solar radiation entering the room in the present work is considered as uniformly diffused.