Analytical Calculation Methods and Tools for the Design of Unglazed Apertures

Abstract

The first studies by Lambert in geometry and photometry were oriented on practical problems of determining skylight illuminance through rectangular apertures without window frames and glazing. His abstract and simplifying concepts of a fictitious sky hemisphere with unity uniform luminance led him to the importance of the solid angle and its mathematical expressions. However, theoretical photometry also involved the interrelation of basic terms of luminance, luminous flux, and illuminance linking the primary light sources with secondly illuminated surfaces via solid angles.

Appendix 8

8.1.1 Comparison of Graphical Tools for Daylight Prediction and Their Accuracy for Unglazed Apertures Under Uniform Skies

Although in the age of computers the graphical tools seem to be quite obsolete, some traditional assessment routines in practice often still use them. Especially, architects and builders are used to orthogonal projections in the design and drawings of architectural and building objects:

• In urban situations, where the exact orientation of each façade and window to cardinal points is described

• In plans of different storeys with the layout of different rooms as well as the placement of windows and roof lights

• In vertical sections of buildings where the vertical placement of windows and roof lights is given with designed dimensions of window sills, heights, and heads as well as the dimensions of hollow light guides

Also the yearly changes of the sun path and the luminance maps of the sky dome can be reproduced in different projection systems using the representation of the fictitious hemispheres or virtual domes. Many graphical tools such as diagrams, overlays, and protractors were produced and used by students or practitioners to predict and assess the possible sun insolation, to achieve sunlight and skylight illumination of interiors, or to avoid solar overheating gains or control sun glare.

When for the building documentation the orthogonal projection is used either on the horizontal plane (for plans) or on the vertical plane (for sections) these projections and drawings representing the actual design seem to be the most convenient for daylight graphical tools too. Therefore, the most often used design tools should be checked both for their accuracy and simplicity in application for practitioners.

At the Zurich CIE General Session where the CIE (1955) overcast sky standard with 1:3 gradation was adopted, the CIE Scope Committee requested the W-3.2 Daylight Committee “to decide on one or a small number of methods of daylight calculations which the CIE can recommend to the various interested professions.” The aim was:

• To select a method applicable to any position in a room of any size and configuration with any kind of fenestration

• To find a favored method of relatively high accuracy capable of dealing with sky patches of any shape

• To prefer a method of day-to-day use which could be less accurate but readily understood and applicable at a design stage when only scale drawings are available

• To document also methods of lower accuracy but that are usable before scale drawings are decided on.

This proved a difficult task indeed. The CIE W-3.2 British member Peter Petherbridge (1959) prepared a draft and presented an excellent review of currently used graphical tools for calculating SF(1:1) and SF(1:3) in four categories using:

• 1. Graphical charts

• 2. Tabular techniques

• 3. Diagram techniques

• 4. Protractor tools

Although many foreign methods were included in the summary, logically British methods were preferred, i.e., Waldram (1950) diagrams and BRS protractors (Dufton 1946), but unfortunately without the comparison of accuracy and practical advantages. Owing to some opposition and further considerations, a simpler Australian method was proposed and a revised draft was published (CIE 1970) with a summary of 58 other methods briefly annotated and referenced. So the criteria of accuracy were totally abandoned and the calculation basis and assumptions behind the Australian minimum daylight factor diagrams including exterior and interior interreflections were vaguely stated in several correction factors or hidden altogether. So any comparison in terms of ${\hbox{SF}}(1:1)$, ${\rm SF} (1:3)$ for unglazed apertures or accuracy compared with basic formulae to test the CIE (1970) diagram method proved impossible.

The sky factor ${\hbox{SF}}(1:1)$ is, by the CIE (1939) definition, a geometrical quantity independent of the sky luminance distribution (assuming the Lambert unit uniform sky hemisphere) or glass transmission losses (assuming a free unglazed opening) and therefore can be calculated with a desired degree of precision. Of course, it has to be noted that applying the Lambert concept of relative terms based on the projection of the sky hemisphere solid angle onto the horizontal plane equal to $\pi$, ${\hbox{SF}}(1:1)$ is inevitably linked as a ratio to the horizontal illuminance from the unobstructed sky outdoors ${E_{\rm{h}}}$. Thus, it expresses also the indoor horizontal illuminance level ${E_{\rm{i}}}$ in lux in a black interior illuminated only by sky luminance equal to 1 cd/m² and

$$ \frac{{{E_{\rm{i}}}}}{{{E_{\rm{h}}}}} = \frac{{{E_{\rm{i}}}}}{\pi } = {\hbox{SF}}(1:1). $$

(A8.1)

Currently the ${\hbox{SF}}(1:1)$ criterion seems to be quite unreal but the uniform sky exists worldwide under certain weather situations and was adopted as sky type 5 in the set of standard skies (ISO 2004). Thus, if an absolute sky luminance ${L_{\rm{vz}}}$ in candelas per square meter is known, then

$$ {E_{\rm{i}}} = {\hbox{SF}}(1:1)\pi {L_{\rm{vz}}}\,({\hbox{lx}}){.} $$

(A8.2)

The relatively simple determination of the ${\hbox{SF}}(1:1)$ value, either in angular or in coordinate form, enabled several tabular or graphical techniques to be developed as practical tools for the prediction of skylight in actual rooms.

After the comparison of the trigonometric basis of the aperture solid angle and its projection for the simple position of a rectangular vertical window in the appendix in Chap. 7, there is a possibility to evaluate also the graphical tools in the same style. It is evident that all graphical methods and techniques are based on the original Lambert ${\hbox{SF}}(1:1)$ formula in angular form after (8.13) or (8.15) or after inserted aperture coordinates are applied in (8.12) or (8.14) results can be considered in the first class of accuracy. Suspect accuracy is associated with techniques and tools based on the simplified (8.17) if the “lune” vanishing perspective of rectangular apertures is not respected or additionally taken into account.

Although the first ${\hbox{SF}}(1:1)$ diagrams after Higbie (1934) formulae were included in the basic illuminating engineering book of Moon (1936) and earlier a protractor method had been suggested (Higbie et al. 1930), even earlier more versatile graphical charts based on equal-area projection by Waldram and Waldram (1923, 1932) and Waldram (1946, 1950) were used in the UK. Later protractors by Dufton (1940, 1946), better known as BRS protractors, became preferred there, whereas in Russia and eastern Europe similar older nomograms were applied by Danilyuk (1931, 1934, 1935, 1941). Therefore, the last three tools will serve for the example and ${\hbox{SF}}(1:1)$ accuracy test at first assuming for all tools the same uniform sky luminance and the same 3 m × 3 m unglazed window opening.

To compare ${\hbox{SF}}(1:1)$ values obtained by different formulae or graphical tools, a simplified example of a window without a sill, in fact assuming the horizontal working plane coincides with the sill height, is used. The illuminated horizontal elements are placed on a perpendicular line from the 3 m × 3 m window corner at a distance from it so that their elevation angles ${\varepsilon_0} = 70^\circ, \;50^\circ, \;40^\circ$, and 30°, respectively. With these assumptions and configuration parameters, two formulae from the appendix in Chap. 7 can be used to calculate ${\hbox{SF}}(1:1)$ values directly, i.e., the exact ones and their very good Lambert approximations are included in Table A8.1. Although angularly defined positions for critical illuminated horizontal elements were chosen, these can be easily determined also by the distances from the 3-m-high window by a simple tangent relation as

Table A8.1 Comparison of ${\hbox{SF}}(1:1)$ values as percentages calculated after different formulae or graphical tools assuming unity sky luminance and a 3 m × 3 m unglazed vertical aperture

$$ x = \frac{z}{{\tan \,{\varepsilon_0}}} = z\,\cot \,{\varepsilon_0}\quad {\hbox{or}}\quad x = \frac{z}{{\tan \,{\varepsilon_0}}} = z\,\cot \,{\varepsilon_0}\,({\hbox{m),}} $$

(A8.3)

where $z$ is the aperture height, i.e., $z = 3\,{\hbox{m}}$ and $x$ is the distance from the window determined by the angular elevation of the window head ${\varepsilon_0}$. This distance is also given in Table A8.1 for each elevation angle.

In the original Waldram diagram, the droop lines for 70°, 50°, 40°, and 30° vertically and horizontally were chosen to calculate and sum up their area in each case to divide it by the area of the whole diagram. Similarly, after the summation of all relevant areas in the droop-line diagram, also corrections for ${\varepsilon_0} = 70^\circ, \;{ }50^\circ, \;{\hbox{ and }}40^\circ$ were found and these were divided by the area of the whole diagram. Thus, the final ${\hbox{SF}}(1:1)$ results as a percentage were calculated and are included in Table A8.1.

Of course, the details and calculation procedures using the original angular charts of Danilyuk (1931, 1934, 1935, 1941) are now out of date, so only the better known BRS (1944) protractor is shown in Fig. A8.1 as an example indicating how to predict the value of ${\hbox{SF}}(1:1)$ from the vertical apertures at point B. Similarly, such overlays can be used also at points A, C, or D, respectively, or further protractors for different glazed and sloped apertures. Both Danilyuk and BRS protractors were later approximately corrected for ${\hbox{SF}}(1:3)$ calculations, and Kittler’s protractors (Kittler and Kittlerova 1968, 1975) were determined for ${\hbox{SF}}(1:3)({\hbox{TG}})$ and ${\hbox{SF}}(1:2)({\hbox{TG}})$ as well as for ${\hbox{SF}}(1:3)({\hbox{WG}})$ and ${\hbox{SF}}(1:2)({\hbox{WG}})$ for glazed apertures, in Chapter 9. An example of a protractor for a vertical window overlaid on the room section is shown in Fig. A8.2.

Fig. A8.1

Building Research Station protractor overlaid on sectional and plan drawing of a room

Fig. A8.2

Kittler’s protractor overlaid on the section

From Table A8.1 it is evident that all the results follow the same tendency of decreasing ${\hbox{SF}}(1:1)$ values from the window opening to the rear places with only minor errors. Whereas Danilyuk and BRS protractors show a positive error, the Waldram diagram indicates a rather conservative minor decrease compared with the exact values or Lambert approximations except for the place furthest from the window. However, all the above-mentioned graphical tools take into account only dense overcast skies. The vertical windows are the most common rectangular apertures and correspond to the sky luminance raster of meridians; thus, a method of arbitrary meridians published by Kittler and Darula (2006) enabled a user-friendly computer program to be developed (Roy et al. 2007). This computer tool enables one to use all 15 sky types adopted by the ISO (2004) standard. Of course, this program can also calculate ${\hbox{SF}}(1:3)$ values for unglazed vertical apertures if the input of normal transmittance ${\tau_{\rm{n}}} = 1$ or any other transmittance, but no directional transmittance except the normal transmittance, is taken into account.

Most existing graphical tools even in their corrected versions for ${\hbox{SF}}(1:3)$ are relatively out of date. Some recent computer programs e.g., (Kota and Haberl 2009) are available for calculations. However, often their source information, assumptions, and software structures are unpublished. Some are restricted only for the determination of DF (1:3) values under overcast conditions as they include interreflections in rooms. Unfortunately, many user-friendly computer tools are oriented to prove that some prescribed criteria or standards were obeyed in building designs. So, users usually forget or do not care on what basis, formulae, or their simplifications the computer tool was developed, but they trust it because it gives apparently favorable results. However, not even the comparison of results produced by few computer software programs can help (Ubbelohde and Humann 1998). Recently the basic ${\hbox{DF}}(1:3)$ criterion has been criticized because of the unrepresentative ratio system disregarding the lifetime performance of buildings in annual daylight climate as well as because of the rigidity respecting only overcast winter conditions in temperate zones while ignoring regions where extensive sun-shading and air-conditioning is of prime importance.