The error we are interested in is the error in the maximum power absorbed. Figure 7 shows that for a two pane window where the outer pane is a grey 4 mm float glass and the inner pane is a clear 4 mm float glass, the largest values for absorbed power is in the interval 20-30°. In that interval, we have shown that the Angular Variation Model is more accurate than the other two models.

As with the maximum absorbed power analysis (Figure 2), a selection of three panes serves as examples. The three figures 7-9 show that the largest absolute errors are found in the interval 60-80°, depending on what model is used and what pane is examined.

Figure 7 shows the error in the absorptance in the outer pane of a double pane window where the outer pane has a thin silver layer (Ag+) on the inner side of the outer pane.

Figure 7 Absolute error in absorbed power versus angle of incidence for the outer pane in a double pane window where the outer pane has a thin silver layer (Ag+) on the inner side of the outer pane. The errors are shown for all three models and for four orientations for each model.

Figure 8 shows the error in the absorptance in the outer pane of a double pane window where the outer pane has a thin stainless steel and nitride layer (SsTiN) on the inner side of the outer pane.

Figure 8 Absolute error in absorbed power in the outer pane of a double pane window where the outer pane has a thin stainless steel and nitride layer (SsTiN) on the inner side of the outer pane versus angle of incidence. The errors are shown for all three models and for four orientations for each model. Figure 8 shows the error in the absorptance.

Figure 9 shows the error in the absorptance in the inner pane of a triple pane window where both the outer and the inner pane have a thin silver layer (Ag) on the inner and outer side, respectively.

Figure 9 Absolute error in absorbed power absorptance in the inner pane of a triple pane window where both the outer and the inner pane has a thin silver layer (Ag) on the inner and outer side respectively versus angle of incidence. The errors are shown for all three models and for four orientations for each model. Figure 8 shows the error in the absorptance.

Figure 9 shows a pane in a window suited for cold climates, like Stockholm. Most of the power absorbed by the inner pane in a window eventually continues into the room and helps heating it. Figure 9 shows the absolute error in maximum power for different incidence angles in the inner pane of a three pane window where the outer and inner panes have a thin film silver layer and the middle pane is clear. Therefore the absorptance in the inner pane is more interesting for energy balance calculations of the room. Figures 7 and 8 show windows suited for warmer climates, solar control windows. Their coatings are "tuned” so that they not only reflect the radiant heat from the surrounding, but also the near infrared heat in order to minimize the need to use air conditioning [2].

When we went through the 62 tested panes, we noticed that the largest errors in maximum power absorbed generally occurred in the outer glass, a pattern that is also present in Figures 7-9 in this report. When studying the energy balance of the room, though, the amount of irradiation absorbed in the inner pane is more relevant.

All extreme values shown in table 2 occur when the window is facing the west. In eight cases the incidence angle was 30° and in one case 20°.

Conclusions

From the 62 panes in the 27 windows we studied, it can be seen that the largest errors typically occur in the outer panes.

The Angular Variation Model gives smaller errors in absorbed power than with the other models. This is not least the case for the ten worst approximations, as can be seen in Table 1, where the errors with the Angular Variation Model are smaller and more evenly distributed. The errors with the Angular Variation Model is on average 0 for the ten worst cases and the standard deviation of the error for these ten cases is 2 W/m2, whereas the average error for the ten worst cases with the Constant Value Model is -7±4 W/m2 and for the Clear Pane Model 11±3 W/m2.

The main disadvantage with the Angular Variation Model compared to the other two models is that one has to know to which of the 27 window categories the window belongs. When that is known, however, this model is better than the other two presented alternatives.

[1] A. Werner, A. Roos, Trying to find an approximation of the angle dependence of the solar absorptance of a window pane, in proceeding from the conference CIB, Toronto, May 2-5 2004.