The use of shading masks makes it quite easy to calculate the instantaneous incident solar radiation on a surface. As incident solar radiation levels varies quite significantly due to cloudiness and the movement of the Sun through sky, a more important value is usually a cumulative exposure value measured over some specific time period. This is where shading masks come into their own as they make the calculation of cumulative solar exposure even easier.

Fast Solar Exposure Calculations

Whilst it is possible to calculate the position of the Sun, the incidence angle it makes with the surface and the diffuse radiation at each time-step over the measurement period, this can be time consuming especially when it involves many surfaces in a complex model.

However, if we instead sum the available direct and diffuse radiation from the sky in each sky segment over the measurement period, we could simply iterate through each sky segment adding the direct and diffuse values and multiplying the result by that segment's shading mask value. We would only have to sum the sky values once as the sky conditions are common to all objects in the model over that period. Thus we have turned a complex calculation with many thousands of trigonometric operations into a series of simple additions and multiplications.

Cumulative Sky

The distribution of direct solar radiation is obviously concentrated in those areas of the sky through which the Sun actually passes over the year. Also, because of factors such as cloudiness in Winter and monsoonal activity in the afternoons, there will likely be both seasonal and diurnal variation in its distribution. Thus, in order to determine the relative (or even absolute) contribution of each sky segment, it is possible to map the occurrence of direct and diffuse radiation values across the entire sky dome.

Cumulative Direct Radiation

This requires that the position of the Sun as it crosses the sky each day of the year be calculated and measured direct beam solar radiation values in the weather file at each time apportioned to the sky segment through which it passes.

To avoid concentration of solar radiation in specific segments that correspond to the exact hourly values, ECOTECT uses 5 minute time steps (hour/12) and linearly interpolates between hourly recorded radiation values. Where consecutive time steps result in the same sky segment, a segment-averaged value is apportioned.

A radiation map is then formed by summing data for each segment, to give a cumulative sky showing the annual radiant distribution. If this is divided by the number of times the Sun was within each segment, an annual average hourly radiant distribution can be displayed.

Cumulative Diffuse Radiation

The calculation of a cumulative diffuse sky is actually more complex than one might think as incidence angle effects are significant due to the dynamic and non-uniform radiant distribution of solar energy over the sky dome. For a clear sky, the corona around the Sun obviously dominates the overall distribution with the rest of the sky being much less radiant. In this case the corona moves through the sky each hour with the Sun. Similarly, as clouds gather and disperse, pretty much any area of the sky can be at times bright or dark, as shown in Figure 3 below. This makes for very dynamic conditions which can change significantly within just a few minutes.

Whilst Task 21 (Daylighting in Buildings) of the IEA Solar Heating and Cooling Programme included a number of research projects in which real-time recordings of actual sky conditions were taken (Roy, et. al, 1995), such detailed data is not available from most weather stations on which to base calculations. Even if it were, with conditions so variable, even hourly data would need to be an average of the minute-by-minute state of the sky.

Thus, as it is not possible to accurately model over any period exactly where in the sky the majority of diffuse radiation is coming from, it make sense that calculations are based on long-term average conditions. For this purpose, CIE have developed standard mathematical models of sky illuminance under different sky conditions (CIE, 1973). Figure 4 below illustrates the CIE Standard Clear, Intermediate and Overcast Sky models. A further Uniform Sky model can be used, but is not illustrated below as all areas of the sky are taken to emit the same level of radiation.

The CIE Standard Overcast Sky distribution, taken as representative of long-term average cloudy sky conditions, assigns sky luminance values at the zenith three time greater than those at the horizon. Assuming a close to linear relationship between luminance and radiance, this means that under overcast conditions the zenith has a much greater overall contribution to diffuse solar gains than the horizon. Thus, for a vertical surface that is exposed to half the sky, incidence angles mean that the zenith contribution is lost – making the maximum possible diffuse radiation available on an overcast day only 40% of that available to a horizontal surface (this is further explained in the Vertical Sky Component topic).

To illustrate this, the image to the right shows an example CIE Overcast Sky distribution. To obtain the total unobstructed diffuse radiation incident on a surface, each segment in the distribution map is multiplied by the incidence angle factor in the corresponding segment of the incidence mask. This clearly shows why vertical surfaces receive significantly less than half the radiation available to the horizontal.

ECOTECT assumes a linear relationship between the luminous distribution of the sky and its radiant distribution, allowing the user to choose between either the CIE Overcast or CIE Uniform Sky to govern the distribution of diffuse radiation. The resulting map can be generated for any hour of the year based on the corresponding measured diffuse solar radiation component in the weather file.

To represent conditions more accurately in ECOTECT, there is ongoing research looking at dynamically interpolating between cloudy and clear sky distributions based on direct-to-diffuse solar ratios, absolute direct solar values and even humidity ratios. Research by Linacre (Linacre, 1992) suggests that a method based on a relationship between average station pressure and the average partial pressure of water vapour can be used for this purpose. However, as no conclusive approach has yet been developed, this will not be covered here.

The Result

The total cumulative sky is calculated by simply summing the total direct and diffuse values in each segment. Figure 6 shows and example.

All Square One calculations maintain the direct and diffuse cumulative sky data as separate layers in order to facilitate other forms of analysis that consider them differently. Also, by storing the cumulative sky data separate from each object's shading mask, many different time periods can be analysed very quickly by simply recalculating the cumulative sky data. Alternatively, it is possible to test different orientations by simply rotating the sky data and the overshadowing/reflection layers within each mask.

Total Solar Exposure

The task of calculating the total cumulative solar radiation incident on any surface is simply a matter of adding up the direct and diffuse availability at each segment and then multiplying by both the surface incidence layer and the overshadowing and reflections layer. The total solar exposure is then calculated by summing up the results for each segment over the entire sky dome. Based on the solar reflections example used in the Shading Mask Calculations topic, Figure 7 illustrates this calculation sequence.

The resulting incident radiation map shows not only where in the sky the majority of incident solar radiation is coming from, but when because it is mapped onto a sun-path diagram.

This allows you to decide on the most appropriate shading design strategies or to better consider trade-offs with other design considerations.

REFERENCES

1. Linacre, E., 1992, Climate Data and Resources. New York, Routledge.
2. Roy, G.G., et al, 1995, The Development of Modelling Strategies for Whole Sky Spectrums under Real Conditions for International Use, University of Sydney and Murdoch University, (view as PDF).

Useful Links

• Irradiation Modelling with Cumulative Skies (PDF)
  http://www.radiance-online.org/radiance-workshop3/cd/Robinson_talk.pdf