## 3 Results  
### Panel radiator  
As discussed in the previous section, in this study we consider a room with a single external wall and adiabatic internal walls, floor and ceiling under steady-state conditions. Thermal layer properties and room dimensions (Figure 2) are chosen according to the European CEN technical committee TC130 working group WG13 specifications, with a U-value for the external wall \(U\) = 0.25 W/m\({}^{2}\) K. A supply temperature of 55 \({}^{\circ}\)C was used. Different types and sizes of heat emitters (radiators, UFH and ceiling heater) are used in IDA ICE simulations to offset the heat loss through the external wall (specific details for each emitter type are presented in their relevant sections). The resulting surface temperatures as calculated by the software are logged and used as input in the analytical calculation. The operative temperatures computed by IDA ICE are also used for comparison with the analytical result. Figures 7 and 8 illustrate first of all that the area by itself is not a good parameter for assessing the performance belated to the operative temperature: given the same area, the efficiency varies with height. Additionally, and more importantly, we observe that one cannot identify a reference ideal convector with 100% convection and 0% radiation as an ideal heater, since it returns the lowest \(t_{op}\). Operative temperatures for fixed heights are plotted in Figures 9 and 10. These hold respectively for a 10-type and a 21-type panel radiator (some values for the 10-type are missing, as it could not reach 134 W of power output). One can see that in general, the numerical and analytical solutions are nearly equivalent. Only for the 10-type we see a slight deviation; moreover, the 10-type reveals to be the most performing radiator, with \(t_{op}\) values always exceeding those of the 21-type by \(\sim\)0.1 \({}^{\circ}\)C. They can even approach the air temperature 20 \({}^{\circ}\)C at \(h=0.9\) m. We can thus conclude that for the study at hand the 10-type can be identified as our \"ideal\" radiator. \begin{table}
\begin{tabular}{l c c} \hline  & **Standing** & **Seated** \\ \hline Up/down & 0.08 & 0.18 \\ Left/right & 0.23 & 0.22 \\ Front/back & 0.35 & 0.30 \\ \hline \end{tabular}
\end{table} 
Table 1: Projected area factors of a person [25]. Figure 8: IDA ICE operative temperatures for a 21-type radiator and convector in function of the panel area. Figure 7: IDA ICE operative temperatures for a 10-type radiator and convector in function of the panel area. Figure 9: Operative temperatures for a 10-type panel radiator, with \(h=0.3\), 0.45, 0.6, 0.9 m.  
Further conclusions can be made rigorous by means of our analytical solution. First of all, the \(t_{op}\) values are linearly distributed along different heights \(h\). By applying a method first introduced in [27], we interpolate the operative temperature versus the height (minimum square method), for a fixed width. The according curves can be generally written as  
\[t_{op}(h,w)=A(w)h+B(w)\,, \tag{8}\]  
returning the operative temperature in the range \(0.3m\leq h\leq 0.9m\), for _any_ desired height \(h\), by using the explicit formulas listed in Tables 1 and 2. In contrast, assuming a given height provides \(t_{op}\) values that are _not_ linearly distributed along different widths \(w\). They rather follow a quadratic law in the form  
\[t_{op}(h,w)=A(h)w^{2}+B(h)w+C(h)\,. \tag{9}\]  
As it is shown in Tables 3 and 4, one finds \(A(h)<0\) for any height. \(t_{op}\) grows instead linearly with increasing height, \(t_{op}(h,w)=A(w)h+B(w)\) with \(A(w)>0\) (Tables 1 and 2): this verifies the physical result that the operative temperature is more dependent on the height than on the width (One can also prove that Equations (8) and (9) are equivalent, namely by substituting one value for \(h\) and \(w\) they return the same \(t_{op}\) (discrepancy of order \(\sim\)0.001, around 0.02%)). The analytical solution allows to make even more specific conclusions. As an example, consider the 21-type radiator.