## 4 Discussion  
Besides several specific results, in this paper we have shown quantitatively two general features: how the radiator performance bfor thermal comfort changes with panel area, and the existence of optimal sizes for the UFH and ceiling heater. These findings are supported by well-tested approaches such as the view factors (see e.g., the standard [25]), which constitute the foundation of our analysis. In this respect, the analytical calculation is necessary for two main reasons: first, IDA ICE calculates the operative temperature in a very specific way that is different from the ISO prescription (As we have shown, the \(t_{op}\) obtained this way is not radically different from the analytical result.). Since we are aiming to contribute to the heat emission code EN 15316-2-1:200, a more standard procedure is preferable. Secondly, and more importantly, we are able to extrapolate and generalize our results to _any_ 10- or 21-type emitter with size included in the studied range, by virtue of the simple interpolation method introduced in [27]. The exact formulas listed in Tables A1-A4 can be of practical use for qualitative assessments of thermal comfort induced by radiators of 10- or 21-type, with dimensions \(0.3m\leq h\leq 0.9m\) and \(1.2m\leq w\leq 3m\) and assuming no back wall losses. In our simulation setup, temperature limits for the water supplied to the emitters are imposed on their systems for a variety of reasons. In theory, any size of emitter in any configuration could be used to offset the heat loss through the external wall, with smaller devices needing hotter surfaces (thus requiring a higher supply water temperature). However, practical reasons bound these temperatures greatly. For example, floor surfaces with UFH in living rooms are limited to 27/29 \({}^{\circ}\)C to avoid thermal discomfort [35]. Moreover, large surface temperature differences in opposing directions can cause local discomfort via a phenomena called \"radiant temperature asymmetry\" [25]. In addition, supply water at low temperature can be generated by ground and air source heat pumps with a higher thermal efficiency. Finally, warmer supply water would also yield higher embedded losses. It should be also noted that if a UFH heated square could follow and track an occupant location, this solution would represent the best performing heater with the highest operative temperature, a result that may have practical implications for personal thermal comfort solutions (Clearly, though the UFH and ceiling smallest squares provided the highest op.t, such cases are not realistic, because a small square heater cannot track the occupants when they move around the room). As the differential between outdoor and indoor air temperature decreases, we expect the corresponding \(\Delta t=t_{air}-t_{op}\) to do the same, since each and every wall surface temperature tends to approach \(t_{air}\). The emitters' surface temperatures will decrease as well, affecting \(\tilde{t}_{r}\) and therefore \(t_{op}\) accordingly. In other words, for a more realistic (higher) outdoor temperature we expect \(\Delta t\) to be generally smaller than what reported in this study. The surface temperatures for our enclosure are listed in Table 2 for the smallest size of each emitter type addressed in this study. The occurrence of a maximum \(t_{op}\) for UFH and ceiling heater strips that do not cover the floor or ceiling entirely implies that the remaining area does not contribute to thermal comfort, which might be interesting for energy saving purposes. More generally, the finding that \(t_{op}\) approaches in some cases \(t_{air}\) is very advantageous for energy saving, as it was shown e.g., in [36] that the energy demand is very sensitive to operative temperature corrections. In particular, a difference of only 0.1 \({}^{\circ}\)C can increase the annual heating need by 1-2% [36]: such effect is found for both the connector and the 21-type panel radiator, which are therefore fairly underperforming.