## 3 Results
### Underfloor Heating  
Regarding underfloor heating (UFH), we considered two different cases: a square heater in the centre of the floor, with varying side length (Figure 13), and a strip setup with fixed width as the floor and varying depth from the external wall (Figure 14). A nominal heat output of 50 W/m\({}^{2}\) at a water-side temperature drop of \(\Delta T=7\) K was used as input for the IDA ICE model, with the piping placed at 25 mm depth in screed. A 30 W/m\({}^{2}\) K fluid-to-slab heat transfer coefficient and a supply temperature of 35 \({}^{\circ}\)C were used.  
Figure 11: First derivative of the analytical operative temperature for a 21-type radiator, \(h=0.3\) m.  
Figure 12: First derivative of the analytical operative temperature for a 21-type radiator, \(h=0.9\) m.  
The operative temperature for square and strip UFH is plotted in Figures 15 and 16 respectively. Here we compare IDA ICE (dots) with an analogous analytical calculation with no projection on perpendicular surfaces (crosses) and with the full analytical calculation (all the 6 directions with perpendicular surfaces), diamonds.  
In both cases the analytical model agrees with the numerical computation with an excellent precision. Notice however how the full solution deviates by \(\sim\)0.1 \({}^{\circ}\)C from IDA ICE unless the square width is around 2 m: considering the horizontal and vertical walls as a whole, this case corresponds to the most symmetric configuration indeed. As it can be seen by investigating the view factors of each surface, for smaller squares \(w<2\) m IDA ICE does not account for the heat dissipation to the vertical walls (These lower values hold also if one uses the IDA ICE data directly, therefore they are not due to errors related to the interpolations of the surface temperatures). On the other hand, for \(w>2\) m these contribute to increasing \(t_{op}\) at the calculation point (2 m from each wall, at 0.6m from the floor). The main result in any case is that there is no evident optimal size for the UFH with this configuration.  
In the case of UFH as a strip running between the side walls, starting from the cold wall, we find instead something more interesting. Neglecting the vertical walls we get again an excellent cross-check with IDA ICE; furthermore, the extension to the full enclosure shows a systematic difference of nearly 0.1 \({}^{\circ}\)C, accounting for the effect of vertical walls. While qualitatively there is basically no deviation from the numerical solution, this is interesting when considering precision calculations.  
Figure 14: Underfloor heating—strip setup.  
Figure 13: Underfloor heating—square setup.  
More importantly, we find a very distinct maximum for \(t_{op}\) between 3 m and 3.1 m, Figure 16. By means of the analytical form of the solution, we can compute its location precisely at \(w=3.0372\) m, see Figure 17. Notice also that, qualitatively, the operative temperature difference between radiators and a fully covering UFH is comparable to that obtained in the experimental paper [8].  
Figure 16: Operative temperature for a strip UFH.  
Figure 17: Operative temperature maximum for a strip heated area underfloor.  
Figure 15: Operative temperature for a square UFH.