## 7 Results and Discussion
### Average Total Efficiency (ATE)  
The efficiency parameters discussed so far reflect only the heliostat field efficiency as a result of the geometric parameters of the field layout. However, a realistic approach should take into account that the sun radiation flux collected by the heliostats depends decisively upon the altitude of the sun for each hour. That is why this flux is more intense at midday than at 08:00, in the summer than in the winter, and close to the equator than at higher latitudes. A simple, elegant way to take the sun's height into account is to multiply HHE by sin(altitude), assuming that absorption of the sun's radiation by the lower layers of atmosphere is directly proportional to 1--sin(altitude). For example, assuming a maximum flux radiation of 1000 W/m\({}^{2}\), the product 1000 \(\times\) sin(altitude) results in an effective radiation flux that is maximum at midday, in summer, and closer to the equator. To make this total efficiency more realistic, it can be multiplied by some loss factor, say 0.85, to account for the heliostat reflectivity (including dust, moisture, etc.) plus atmospheric attenuation effects. Using these assumptions, an hourly total efficiency (HTE) for each heliostat can now be defined:  
\[\mathrm{HTE}=(1-S-B)\times\mathrm{sin(altitude)}\times 0.85 \tag{15}\]  
As before, a daily total efficiency (DTE) and a yearly total efficiency (YTE) can now be defined:  
\[\mathrm{DTE}=\frac{0.5\mathrm{HTE_{S}+\sum_{i=9}^{15}\mathrm{HTE_{i}}+0.5 \mathrm{HTE_{16}}}}{8} \tag{16}\]\[\mathrm{YTE}=\frac{\sum_{i=1}^{366}\mathrm{DTE}_{i}}{366} \tag{17}\]  
DTE curves for the eight configurations can be plotted as before, although the sin(altitude) factor tends to predominate, thus hiding other particularities of these curves, making them relatively similar.  
Finally, the average of YTE for the six representative heliostats give us a new figure of merit, the average total efficiency (ATE), which should be representative of the whole heliostat field, considering both the geometric layout parameters and the sun's position in the sky:  
\[\mathrm{ATE}=\frac{9\times\frac{\mathrm{YTE}_{\mathrm{SF}}+\mathrm{YTE}_{ \mathrm{SF}}+\mathrm{YTE}_{\mathrm{SF}}}{3}+(66-9)\times\frac{\mathrm{YTE}_{ \mathrm{NF}}+\mathrm{YTE}_{\mathrm{NF}}+\mathrm{YTE}_{\mathrm{NF}}}{3}}{66} \tag{18}\]  
Like AHE, this single ATE number permits an easy comparison of the different layout configurations, but with the advantage of also taking into account the sun's radiation intensity. The YTE for the six representative heliostats, plus the corresponding ATE, is reported in Table 3 for the eight configurations studied.  
As expected, ATE values for Julich (Germany) are much lower (\(\sim\)0.13) than for Protaras (Cyprus), as the sun trajectories are much lower in the first location. These ATE values are much smaller than the AHE values seen before, because the low sun height strongly disadvantages the 08:00-10:00 and 14:00-16:00 periods, particularly at Julich. The maximum total efficiencies achieved are above 0.5, corresponding to an average radiation flux of 500 W/m\({}^{2}\) for the entire 2024 year, for all days from 8 h to 16 h, assuming that all days are sunny.  
Of course, the weather will be cloudy during many of these hours, so no sun radiation (specularly reflected) will reach the target. Extensive research has been reported [28; 29] to estimate the distribution of these cloudy days, gathering statistical information--solar radiation data--that plays an important economic role. Assuming that half of the 366 days in 2024 were cloudy (a fairly conservative assumption, to keep this discussion as simple as possible), an average DNI of 250 W/m\({}^{2}\) can be expected for the whole year, from 08:00 to 16:00 LCT, which is still a significant amount of energy. The actual value should be higher, as the cloudy hours are expected to occur mostly in winter, early in the morning, and late in the afternoon, when the sun height is lower and the contribution for ATE is smaller, thus having less of an effect on the whole average.