## 2 Method
In the resulting floor coil model, the heat transfer between fluid and active plane is computed via their logarithmic temperature difference; the according heat transfer coefficient includes convection between medium and tube wall, heat conduction through the tube walls and \"fin efficiency\" given by the distance between immersed tubes or actual fins. In steady state, this approach corresponds to the resistance method of the EN 15377-1 standard [29]. For a selection of IDA ICE model and software validations, see, e.g., [30; 31; 32; 33; 34]. The operative temperature \(t_{op}\) is computed analytically, according to the prescriptions of the ISO 7726 standard, as we explain in the following. Considering the contributions of all the six surfaces in the enclosure, we obtain an expression for \(t_{op}=t_{op}(a,b)\) that is a function of the radiator height \(a\) and width \(b\). The eventual global maxima of this function in the \((a,b)\) plane would then correspond to the optimal configuration for that specific heater. Such full analytical solution is then numerically validated by the finite difference method software IDA ICE [26] in the same CEN TC130 test room, in the limit when only the contribution of the surfaces that are parallel to the principal calculation surface is accounted for. The operative temperature at the above location is not uniquely defined. In IDA ICE this is evaluated as the simple arithmetic average of air temperature \(t_{air}\) and mean radiant temp. \(\bar{t}_{r}\)[26],  
\[t_{op}=\frac{t_{air}+\bar{t}_{r}}{2}\,, \tag{3}\]  
(throughout this paper, \([t_{i}]=[^{\circ}\mathrm{C}]\) and \([T_{i}]=[\mathrm{K}]\)). This differs from the exact definition given in the ISO 7726 [25],  
\[t_{op}=\frac{h_{c}t_{air}+h_{r}\bar{t}_{r}}{h_{c}+h_{r}}\equiv At_{air}+(1-A) \bar{t}_{r}\,, \tag{4}\]  
where the average is weighted by the radiation and convection heat transfer coefficients \(h_{r}\) and \(h_{c}\) at the calculation point (As we demonstrate in Appendix A, in reality the different operative temperature values which are obtained with either method show no sizeable difference). The explicit formula for the coefficient \(A\), which is itself a function of \(h_{c}\) and \(h_{r}\), is given in Appendix A; for our setup, it lies within the range \(A\sim 0.5-0.6\). Another difference between the ISO standard and IDA ICE is that the numerical software has a peculiar way of computing the mean radiant temperature \(\bar{t}_{r}\). It considers only the surfaces that are _parallel_ to the principal calculation surface, therefore the sum of view factors in a principal direction is \(<\)1 (Figure 5). Moreover, \(\bar{t}_{r}\) is obtained as the average of mean radiant temperatures from the six principal directions, weighted by the respective view factors,\[\overline{T}_{mrt}=\sqrt{\frac{\sum_{i=1}^{6}\sum_{j=1}^{n}F_{i\to j}T_{j}^{4}}{\sum_{i =1}^{6}\sum_{j=1}^{n}F_{i\to j}}}, \tag{5}\]  
where the \(F_{i\to j}\) are computed for a small area (the observer) that is only parallel to the radiating surface. In contrast, the ISO 7726 prescribes that for each direction one considers both parallel _and_ perpendicular surfaces (see Figure 6), obtaining the plane radiant temperature [14; 25],  
\[\overline{T}_{pr}^{(i)}=\sqrt[4]{\sum_{j=1}^{6}F_{p-A_{j}}T_{sj}^{(i)\,4}}\,, \tag{6}\]  
with the angle factors \(F_{p-A_{j}}\) reported in Appendix A. Now the sum of view factors in each direction is accordingly =1, and the mean radiant temperature is given by [25],  
\[\bar{t}_{r}(a,b,c)=\sqrt[4]{\frac{\sum_{i=1}^{6}\beta_{i}\overline{T}_{pr}^{( i)}(a,b,c)}{\sum_{i=1}^{6}\beta_{i}}}-273.15}, \tag{7}\]  
namely by a weighted average over the projected area factors \(\beta_{i}\) of a person, listed in Table 1.  
\[P=kldT^{n}\,, \tag{1}\]0 Calculation of the mean radiant temperature from IDA ICE [26]. \[P=kldT^{n}\,, \tag{1}\]1 Calculation of the mean radiant temperature from ISO 7726 [25].