Energy and Buildings, 6 (1984) 229 - 240 229

The Calibration and Error Analysis of an Artificial Mirrored Sky Converted to a Clear Sky Luminance Distribution

ROBERT WHITE Centre for Building Studies, Concordia University, 1455 de Maisonneuve W., Montreal, Quebec H3G 1M8 (Canada)

SUMMARY

An inexpensive conversion o f a mirrored artificial sky from an overcast sky source to a clear sky source allows more extensive model investigations. For a solar altitude angle o f 40 °, the model sees the relative CIE standard clear sky luminance distribution away from the sun caused by a low perimeter source illuminating a cloth drape covering the mir- rored walls. Problems in calibrating the " s ky" are explained and compared to a model in a similar real sky and a computer calculation. The errors due to shifts in angular relationship to the luminance distribution o f a close artifi- cial sky source are calculated for different room locations, window types and model sizes.

INTRODUCTION

Model studies were and are now an essen- tial element of daylight analysis and design, especially for complex spaces. Models were used for museum studies [1] in the 19th Century, and by Pleijel [2] and Buning [3] to show the effects of exterior building obstruc- tions on daylit interior spaces. Evans [4] and Griffith [5] used models under a hemispheri- cal artificial sky to develop the Lumen Method of daylight design currently recom- mended by the Illuminating Engineering Soci- ety of North America. Inexpensive mirror- type artificial skies are in common use [6, 7]. Recently Selkowitz et al. [8] described a 71~-m diameter artificial sky that can be pro- grammed for a variety of overcast and clear sky conditions, a remarkable state-of-the-art instrument.

It would seem useful to have an artificial sky that could provide a capability somewhat less than the Berkeley sky of Selkowitz et al. if it could be built quickly from common ma- terials in a small space. It should provide both overcast and clear sky luminance distribu- tions. These reference skies would be bench marks which are extremely important in day- light studies since daylight is so variable throughout the day and year. While incapable of showing the full range of daylight effects it would demonstrate the very different clear sky and overcast sky luminance distributions in the model and allow accurate measure- ments.

DESCRIPTION OF THE SKY

The mirrored artificial sky that will be con- vetted is a rectangular wood box with a square cross section lined by mirrors 0.38 m high. The ceiling is a 1.21 m X 1.21 m white diffuse translucent plastic panel suspended by wire hangers. Above is a bank of 8 fluorescent lamps. The interreflections in the mirrors produce a diminution of the sky luminance which corresponds to the standard CIE for- mula for an overcast sky

(1 + 2sin ®) Bo = Bz (1)

3

The value of B o is independent of azimuth angle (see Fig. 1). Because the overcast mirror sky is a standard, inexpensive and easily con- structed apparatus, it is the ideal candidate for an apparatus that can be modified to simulate a clear sky as well.

It should be noted that the standard dis- tribution assumes a ground reflectance of

0378-7788/84/$3.00 © Elsevier Sequoia/Printed in The Netherlands

230

a p p r o x i m a t e l y 10%. Thus fo r l ight g round re f lec tances due to sand or snow, the artificial skies ' d i s t r ibu t ion is conserva t ive for vert ical windows , since the luminance values near the hor i zon are lower than t hey would be wi th a

Ceiling

Side Wall

t°t t't Ii )))))) Fig. 1. CIE Overcast Sky and artificial sky -- equi- distant diagram. B® = luminance of the sky at a ver- tical angle O from the horizon (cd/m2). B z = lumi- nance of the sky at the zenith (cd/m2), i.e., O = 90 °.

grea ter g round re f lec tance , i.e., a 3:1 var ia t ion for the f o r m e r versus a 2:1 var ia t ion for the la t ter .

The clear sky luminance d i s t r ibu t ion is m u c h m o r e c o m p l i c a t e d than an overcas t dis- t r i bu t ion since it varies wi th solar a l t i tude and az imu th angles as well as wi th a tmosphe r i c t u r b i d i t y [9, 10] . The wel l -known CIE clear u n p o l l u t e d sky f o r m u l a shown graphical ly in Fig. 2 is:

L(~', ~)

Lz

[ 0 .91 + 10 e x p ( - - 3 7 ) + 0 .45 cos27 ] / I X LO.- Vi x + 0.45 cos'Z01

1 - - e x p ( - - 0 . 3 2 ) sec ~" ]

× 1 - - e x p ( - - 0 . 3 2 ) (2)

90 Bz

WALL

~'---~-~' ~ ~--~/SIDE WALL

NG

6 0 ° _ _ ~ ' • I 70 6 0 ~

O •

Fig. 2. CIE Clear Sky and artificial sky (40 ° solar altitude) -- equidistant diagram.

where

L(~', ~) = luminance of the point considered ( c d / m 2)

Lz = luminance of the zenith ( c d / m 2)

= angle between the zenith and the point considered (radians)

= azimuth angle of the point w/r to the sun at 0 (radians)

3' = angle between the sun and the point considered (radians)

Z0 = angle between the zenith and the sun (radians)

The complications can be reduced if one makes the following assumptions for the arti- ficial clear sky:

(1) Use the hal f hemisphere away f r o m the sun. If one looks at the curves for a clear sky distribution, (Fig. 2), it is clear that an open- ing facing away from the sun receives much less illumination than from the diffuse sky towards the sun. Thus for a conservative design benchmark, an opening will receive at least this illumination during the day even if the sun changes position with respect to it.

The curves for the sky opposite the sun from the horizon to approximately 40 ° above the horizon are quite parallel to the circum- ferential altitude line. Thus the luminance values are almost independent of azimuth angle. This reduces modeling errors due to changes of the measured point in the model with respect to the sky source.

Since the artificial sky can produce only relative luminance values, absolute effects such as glare cannot be studied.

231

(2) Use a 40 ° solar angle as the design con- dit ion. No existing artificial sky can simulate the continuously changing clear sky distribu- t ion throughout the day and an inexpensive sky must select a reference condit ion that can be a bench mark for much of the year. This will vary with latitude since the maximum variation in altitude angle over the year is:

c~ = 90 +- L + 23.26 ° (3)

where ~ = solar altitude angle and L = latitude angle.

If the 40 ° sky "represents" solar angles from 35 ° to 45 ° the hours from mid morning and mid af ternoon are well covered for the equinox for all latitudes as well as 12:00 for the mid latitudes from 30 ° to 48 ° . The 40 ° angle also represents early morning and late af ternoon for all latitudes at the summer solstice (see Table 1). When the altitude angle is increased above 40 ° there is less variation from horizon to zenith in the luminance dis- tr ibution, thus daylight for those conditions is expected to fall between the overcast and the 40 ° clear sky model. For lower altitude angles (10 ° to 20 ° ) the horizon luminances are I 'A- 2 times those of the 40 ° sky; thus for windows, the 40 ° sky will be conservative to determine sky factors.

It could be argued that the 30 ° clear sky would be a more appropriate luminance dis- t r ibution since it corresponds to irtitial winter conditions. However, for many locations there are fewer clear skies in winter months (see Table 2). The problem of design bench- marks for clear skies needs more intensive

TABLE 1

Solar a l t i tude angle w/r solar hour for mid- la t i tude locat ions

Lat i tude Winter solstice Dec. 21 Equ inox March 21 Summer solstice June 21

08:00 10:00 12:00 08:00 10:00 12:00 08:00 10:00 and and and and and and 16:00 14:00 16:00 14:00 16:00 14:00

12:00

32 ° 10.39 27.74 34.74* 25.07 47.24 59.97 36.80 62.13 81.26 36 ° 8.02 24.30 30.74 23.84 44.46* 53.97 37.14 61.14 77.26 40 ° 5.63 20.84 26.74 22.50 41.54" 49.97 37.28 59.69 73.26 44 ° 3.22 17.34 22.74 21.08 38.53* 46.00* 37.21 57.84 69.26 48 ° - - 13.82 18.74 19.53 35.39* 41 .97" 36.93 55.68 65.26 52 ° - - 10.29 14.74 17.91 32.20 37.98* 36.44 53.23 61.26

*Times during the year when the 40 ° a l t i tude angle clear sky approx imates an actual clear sky.

232

TABLE 2

Availability of sunshine and clear skies for 3 cities

Location Latitude Number of hours/% possible sunshine hours

June September December

Montreal, Quebec [13] 44.5 ° Knoxville, Tennessee [ 14 ] 33.5 ° Jackson, Mississippi [15 ] 32.3 °

240/54% 170/48% 70/27% 268/62% 212/59% 112/38% 291/69% 212/58% 145/47%

(a)

xj~lD~ WALL ,

:~." + • <: ~: ~-': I: ++:

CLOTH ENCLOSURE o~oo

bJ

,+!

50O i

FLUORESCE NTS, O.C. S K Y ~ ~ / / - - ~ ~e

CABLE TO RAISE 8 ~ . ~ =T, LOWER CEILING P A N E L ~ " • , ~ J I ~ D I E F U S E

CEL,NG PANEL ~ l i ~ ' / Ill - WHITE CLOTH DRAPE ~ t I~ -'-ZENITH / I1'~ -MIRROR ,~=o.,, ~ J 4111 tco~R~

~ O IRROR VERE

I.+ _+ ................... PROJECTOR ~ H I ~ I Z O ~

~°~y%%%~-~/,=0E.+ ~ ~ ~ ~,CKBOROER

( b ) o Ioo soo

Fig. 3. (a) Plan of the artificial sky. (b) Section AA of the artificial sky.

invest igat ion. See for e x a m p l e the discussion o f the sun ' s m o v e m e n t in ref . 11.

This clear sky f ac to r is n o t s t r ic t ly the same kind as for an overcas t day since the same w i n d o w will have a var iable sky f ac to r as the sun moves across the sky relat ive to a f ixed window. Al te rna t ive ly one could re la te the in ternal i l luminance in a m o d e l to the to t a l p r o d u c e d by the art if icial sky. However , this is a q u a n t i t y so numer ica l ly and concep tua l ly d i f f e ren t f r o m the overcas t sky f ac to r t h a t re la t ing the two values would be di f f icul t for the designer. I f the clear sky day l igh t f ac to r is

Fig. 4. View to the artificial clear sky from the model opening.

k n o w n for one solar a l t i tude angle the varia- t ion for o the r hours o f the day and for the yea r can be rough ly es t ima ted .

Artificial clear sky conversion Since the clear sky is br ight at the ho r i zon

and da rk at the zeni th , the mi r ro r skies ' over- cas t sky d i s t r ibu t ion {dark at the ho r i zon and br ight at the zeni th) should be reversed. To do this it was necessary to da rken the ceiling and p rov ide a br ight ho r i zon source .

(1) Wall surface (Figs. 3, 4). The mi r ro r walls were covered wi th a whi te 50 /50 c o t t o n / p o l y e s t e r c lo th d rape s t r e t ched t au t ly at the corners , t o p and b o t t o m to e l imina te folds or wrinkles . A re ference grid was m a r k e d on the c lo th wi th light ink marks for ca l ibra t ion purposes . The d rape had a t op seam wi th an inser ted rod to be secured to the channe l above the mi r ro r surface. The b o t t o m c lo th seam was filled wi th a heavy me ta l rod. The c lo th surface was f o u n d to be subs tant ia l ly L a m b e r t i a n when washed and i roned and had a re f l ec tance of 0.71.

(2) The ceiling surface (Figs. 4, 5). The ceil- ing surface was m a d e of a mosaic of grey

233

) 5o '~

160

4

_1 170

SUN

~ ' ~ / T THEORETICAL

0 . 9

18°* o ~oo 200 I I I

Fig. 5. Ceiling panel luminance distribution for a 40 ° altitude angle artificial clear sky.

papers glued to a s t y ro foam foam core panel. The choice o f ref lec tances and thei r loca t ion was made by trial and er ror and was designed to be reproduc ib le by simple measurements . A m or e compl i ca ted mosaic which would more accura te ly r ep roduce the theore t ica l re- qui red luminances was no t believed to be w or th the added complex i ty .

(3) The pe r ime te r source. Two f luorescent lamps each 1.21 m (4 f t) with ref lec tors were suppor t ed below the hor i zon line on the side, and one f luorescen t lamp o f 0.91 m (3 f t) was at the rear.

To change the sky f rom the overcast to the clear sky cond i t ion the ceiling panel was raised by wires at the fou r comers . The c loth drape was also raised and h o o k e d at the top edge to a pe r ime te r channel . The convers ion can take 15 - 20 minutes including some spot check re fe rence luminance cal ibrat ions. The ceiling source is t u rned o f f and the b o t t o m per ime te r tu rned on and stabil ized (15 min- utes) . The to ta l convers ion takes 15 - 30 min- utes.

Calibration To cal ibrate the sky, the vertical and hori-

zonta l angles were drawn on elevations and re f lec ted ceiling plans o f the sky (Figs. 5 - 7) along with luminance con tou r s o f the 40 ° clear sky.

A grid o f re fe rence poin ts was t ransfer red to the c loth drape and to an initial middle grey ca rdboard ceiling panel . Using a sensitive

4'.

4c

3c

2c .~ I=

01

MEASURED Be/B z ~ THEORETICAL Be/B~X "

aa -z \ 0.9

~ / . "

. . . .

"22 ....... ~

: . . - ~ , ~ _ : 2 ~ --..-...J ~ ,~_____~.o~

1.0

90 I00 I10 120 130 140 150

0 I00 200 L l I

Fig. 6. Side wail luminance distribution for a 40 ° altitude angle artificial clear sky.

HORIZ.

"'1 . . . . . . ' " . . . . . "'l

- - iO __2.0_

5 ' . .........

160 171:

~ J

J

J 1.2

. . . . . . . 1.5

~ 2 . 0

0 I00 2 0 0 | | I

Fig. 7. Rear wall luminance distribution for a 40 ° altitude angle artificial clear sky.

luminance mete r , the luminance d is t r ibut ion was surveyed, and compared to the theore t i - cal values. It was modi f i ed by moving the pe r ime te r source and the ceiling mosaic.

The sky's luminance d is t r ibut ion is a func- t ion o f in te r re f lec ted f lux f rom the c lo th drapes and ceiling emit ters as well as the initial f lux f rom the f luorescen t sources. Thus changing the ceiling ref lec tances also changes the wall luminance . The actual luminances were sensitive to the source locat ions and thei r r e f l ec to r angles with respect to the hor izon . The pe r ime te r side lamps and reflec- tors were u l t imate ly s loped vert ically and dis- placed towards the sky's cen te r at the opening to the mode l and were located below the hor i zon (Fig. 3).

The re fe rence hor izon ta l i l luminance for an u n o b s t r u c t e d clear artificial sky is a func t ion o f h o w m u c h o f the sky can be seen f rom any r o o m pos i t ion th rough a window/sky l igh t opening. The to ta l con t r i bu t ion o f the artifi- cial sky is measured by a cell covered by a

234

0 .

~l~R, WALL I¢ Avg = 190Lu l REFERENCE ILLUMINANCE: 190X3 ,768 :715 Lull

Fig. 8. Unobstructed horizontal illuminance from the artificial clear sky facing away from the sun (40 ° altitude angle).

quarter sphere mask which represents 7r/2 steradians. For a 40 ° solar altitude angle, the ratio of the diffuse illuminance for the half of the sky away from the sun to the total diffuse illuminance for an unpolluted clear sky is 1:2.768, therefore the total artificial sky con- tr ibution to the horizon at the model opening is multiplied by 3.768 to get the unobstructed horizontal diffuse illuminance for the clear sky daylight factor. Note that other solar altitude angles and atmospheric pollution will change this ratio. For this modeling condi- tion, however, it will remain fixed. The refer- ence unobstructed illumination was (190 lux X 3.768) or 715 lux (see Fig. 8).

The opening to the artificial sky was sur- veyed for horizontal illuminance with a quarter sphere mask as shown in Fig. 8. The decrease with height was due to the reduced sky source visible at any point. Thus a high clerestory faces a reduced source compared to a low window at the horizon. This problem also plagues overcast artificial skies, especially small mirrored skies.

COMPARISON TESTS

Several models were tested under a real clear sky around the 40 ° solar altitude angle. Test A had a high clerestory window 0.9 m high, 0.10 m below a ceiling 3 m high. Test B used the same room (7.0 m X 7.0 m X 3.0 m high) with a low window whose sill is at

0.8 m, and is 1.2 m high. Measurements of horizontal illuminance were made on the 4 X 6 grid shown in Figs. 9 and 10. These daylight factors of the direct component are compared to each other and the values calcu- lated by Quicklite [12] (see Figs. 11, 12). Correspondence is reasonable for daylight measurements (-+20%) except at point A where the calculated values are larger than the measured ones. For line 3 near the wall there was a greater difference between measured and calculated values than between the mea- sured values alone. It thus appears that the artificial sky can give direct daylight factor values that are close to a real sky or calculated values for certain window configurations.

There were more difficult correlation prob- lems however with the interreflected compo- nent (IRC). The high window model (test A) consistently showed a lower IRC from the artificial sky than the real sky values. Con- versely, the low window model (test B) showed higher IRC from the artificial sky than from the real sky. These variations may be due to the difficult problems in measuring the interreflected component both in real spaces and in models. Ideally, the cell should be shielded only from the direct flux of the room openings and thus every measuring position would require a different shield. It would, in fact, be possible to make such a shield although it would be tedious to use. The quarter sphere shield is very accurate for positions near the window wall since the cell sees all room surfaces and reads the illumi- nance coming from them, however, the same cell at the rear position of the room sees only the rear wall and small portions of the floor, wall and ceiling surfaces. It does not record the interreflected flux from room surfaces near the f ront of the room and thus under- estimates the reflected component .

GEOMETRICAL ERRORS AND WALDRAM ANALYSIS

There are some interesting geometrical modeling problems with artificial skies since the distance to any point on the sky is of the same order as the model size. This may vary from 1 5 - 2 0 model dimensions for large hemispherical skies to 2 - 3 model dimensions for small mirror skies. If the sky surfaces are

Lambertian, then their luminance will be constant from any angle of view.

If the sky areas are close to the observation points then the same area on the sky will be at different angular coordinates throughout the room. The same sky will thus appear to have different distributions to each model position.

A Waldram diagram for the 40 ° clear sky is shown in Table 3 for a real clear sky. This is the same as seen by point 0A in the artifi- cial sky and thus the daylight factor from the model may be compared to that calculated from this diagram for a real sky. Because there are apparent geometrical errors in luminance values from different model room positions, it is useful to compare how much error there is for different window configura- tions in different model sizes. These are errors apart from calibration differences shown in

235

Figs. 5 - 7. A compute r program was writ ten which produces a Waldram diagram for any room position with the sky source being the artificial skies' luminance distribution. Thus one can compare the effect of the constant sky luminance source appearing to have a different angular distribution due to the shift in room position. Table 4 shows such a diagram for a room position 200 mm along the X axis. It can be seen that the distribution is now asymmetrical. The error for any win- dow can be computed by comparing the values measured from Table 3 to those mea- sured from Table 4. For windows in the centre of the diagram the error is negligible. For windows comprising most of the diagram the maximum error is +5%. Window areas in the extreme lower right corner will produce more significant local errors, however the maximum error from any area in the Waldram

A B C O E F i 7 t ' ~ i I 0 0 0 • 1 7 0 0 0 AC , T U A L ROOM ,

~..~ o.')7 '3,14 ' / , 90 I .o 4

~ . ~ }3,~9 .75

~.~.D '~..oo ' / . /Z. ~ SUN

Fig. 9. Clear sky dayl igh t fac tors f rom tes t " A " for a 40 ° solar a l t i tude angle: D.C. art if icial sky/D.C, real sky.

236

A B C D E F .i ~7(_'~ I I 0 0 0 • 1 7 0 0 0 AC ,TUALROOM

f i ! I I

~-~,/6 '5'5 I~,'Zl ~ ~ ,1,1£. ' 3,, l + ¢_ ,'Z.'~

'~e> '~,~o ' ~ . 7 I

SUN

Fig. 10 . Clear s k y d a y l i g h t f a c t o r s f r o m t e s t " B " fo r a 40 ° so la r a l t i t u d e ang l e : D.C. a r t i f ic ia l s k y / D . C , real s k y .

9- o ~o I.D

ul 7 - -

A o

o - i

Q

~WlNDOW

L,oo Io oo Io,oo I oo ' DISTANCE FROM MODEL WINDOW WALL (min i

Fig. 11 . D a y l i g h t f a c t o r c o m p a r i s o n s fo r a rea l c lear s k y , an a r t i f i c ia l c lear s k y a n d c a l c u l a t e d v a l u e s - - t e s t " A " . L ine 3 n e a r t h e wal l ; 40 ° so la r a l t i t u d e a n g l e ; w i n d o w wal l f a ce s 180 ° f r o m t h e s u n ; w i n d o w sill is a t 200 c m , win- d o w h e a d is a t 2 9 0 c m a n d t h e w i n d o w w i d t h is 7 0 0 c m ; a n d r o o m d i m e n s i o n s a re 7 0 0 c m × 7 0 0 c m × 300 c m H.

237

II

. - - ~ ~ Qu ick l i te I0

/ J = LINE A , - / /

- - ~ - - A r t i f i c i a l Sky

s S i ~ Rea~ S~v

~ 6

~ s

~ 4

~ _ _ . m ~ . _ , . - - . . _ ~ ~ ---@-----_._~ ~ _ . ~ ~ _ _ ~ _ _ _ _ _ _ _ . ~ Real Sky 3 A r t i f i c i a l Sky LINE C

. / ~ Qu ick l i te

~x-- - - - - ~ w Ar t i f i c ia l Sky I ~ ~ _ _ . - - ~ . - - - a Real Sky LINE E

~ " Quicklite

t I I I o (3) (2) (~) (o)

WALL ROOM POSITION q. ROOM

Fig. 12. Daylight factor comparisons for a real clear sky, an artificial clear sky and calculated values - - t e s t "B". Clear sky opposite sun at 40 ° altitude angle; 7.0 m x 7.0 m x 3.0 m room; window sill at 0.80 m, head at 2.0 m; window width is 7.0 m.

TABLE 3

Waldram diagram for a clear sky (40 ° Solar altitude angle)

Angle of Azimuth angles Total elevation

90 110 130 150 170 --170 --150 --130 --110 --90

9O 1.19 1.11 1.05 1.02 1.02 1.02 1.05 1.11 1.19 9.76

74 1.14 1.02 0.95 0.90 0.88 0.90 0.95 1.02 1.14 8.90

68 1.12 0.98 0.90 0.86 0.84 0.86 0.90 0.98 1.12 8.56

62 1.11 0.97 0.88 0.84 0.84 0.84 0.88 0.97 1.11 8.44

58 1.11 0.95 0,88 0.84 0.84 0.84 0.88 0.95 1.11 8.40

53 1.11 0.96 0.89 0.86 0.84 0.86 0.89 0.96 1.11 8.48 49 1.13 0.97 0.91 0.88 0.88 0.88 0.91 0.97 1.13 8.66 45 1.15 1.00 0.94 0.92 0.92 0.92 0.94 1.00 1.15 8.94 41 1.17 1.04 0.99 0.98 0.98 0,98 0.99 1.04 1.17 9.34

37 1.23 1.10 1.06 1,06 1.06 1.06 1.06 1.10 1.23 9.96

32 1.30 1.18 1.16 1.16 1.18 1.16 1.16 1.18 1.30 10.78

28 1.42 1.31 1.30 1.33 1.34 1.33 1.30 1.31 1.42 12.06

22 1.62 1.54 1.56 1.61 1.64 1.61 1.56 1.54 1.62 14.30 16 2.07 2.05 2.13 2.25 2.30 2.25 2.13 2.05 2.07 19.30 0

Total 17.87 16.18 15.60 15.51 15.56 15.51 15.60 16.18 17.87 145.88

D i a g r a m is n o m o r e t h a n 0 . 1 6 lux . F o r p o i n t s f u r t h e r f r o m t h e i dea l c e n t e r t h e r e l a t i v e

e r r o r s i n c r e a s e (see a lso Fig . 13) .

T a b l e 5 s h o w s f o u r cases f o r b o t h c l ea r a n d

p o l l u t e d a t m o s p h e r e s f o r a r e f e r e n c e w i n d o w

d e f i n e d b y a z i m u t h angles o f l e f t 160 ° t o +

r i gh t 1 6 0 ° and a l t i t u d e angles o f 16 ° t o 45 ° .

( A c t u a l l y a r a t h e r o d d - l o o k i n g w i n d o w ! )

D r o o p l i n e s a re o m i t t e d f o r s i m p l i c i t y . N o t e

t h a t i t d o e s n o t m a k e sense t o c o m p a r e t h e

t o t a l e r r o r f o r o n e sky p o s i t i o n t o a n o t h e r

s ince n o w i n d o w p o s i t i o n e x c e p t a t t h e glass

238

TABLE 4

Waldram diagram for a model point on the window wall 0.2 m from the centerline of the artificial clear sky opening

Angle of Azimuth angles Total elevation

90 110 130 150 170 --170 --150 --130 --110 --90

9O 1.19 1.11 1.05 1.03 1.02 1.03 1.05 1.11 1.18 9.76 74 1.15 1.01 0.94 0.90 0.90 0.92 0,97 1.04 1.15 8.98 68 1.13 0.97 0.89 0.86 0.86 0.88 0.93 1.01 1.14 8.67 62 1.10 0.93 0.86 0.84 0.84 0.87 0.92 1.01 1.14 8.51 58 1.08 0.92 0.85 0.82 0.84 0.87 0.92 ] .03 1.16 8.49 53 1.08 0.92 0.85 0.84 0.85 0.89 0.95 1.05 1.20 8.63 49 1.08 0.92 0.87 0.86 0.88 0.91 0.98 1.09 1.24 8.83 45 1.08 0.94 0.90 0.90 0.92 0.95 1.03 1.15 1.29 9.16 41 1.10 0.96 0,94 0.95 0.98 1.01 1.09 1.21 1.36 9.60 37 1.12 1.01 1,01 1.03 1.07 1.09 1.17 1.39 1.44 10.33 32 1.17 1.07 1.10 1.15 1.18 1.21 1.29 1.41 1.56 11.14 28 1.25 1.18 1.24 1.31 1.35 1.30 1.45 1.57 1,90 12.55 22 1.42 1.38 1.49 1.58 1.65 1.66 1.71 1.80 1.92 14.70 16 1.89 1.94 2.10 2.25 2.30 2.26 2.21 2.21 2.29 19.45 0

Total 16.84 15.24 15.09 15.21 15.64 15.94 16.67 18.08 19.97 148.68

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I I l - - ] - I - r - J - 9 0

- ,6o - ,~o -14o -13o-,zo-,o.4oo ( b ) A Z I M U T H F R O M THE. S U N

Fig. 13. Apparent luminance distribution for a model point on the window wall 0.20 m from the centerline of the artificial clear sky opening. (a) Left side wall; (b) right side wall.

sees t h e e n t i r e h a l f h e m i s p h e r e sky . T a b l e 5 s h o w s c o m p a r a b l e e r ro r s f o r al l m o d e l p o s i - t i o n s o f less t h a n 1.1 l u x / 1 0 0 c d / m 2 z e n i t h l u m i n a n c e .

O t h e r w i n d o w s see ing a s y m m e t r i c a l p o r - t i o n s o f t h e s k y c o u l d c o n c e i v a b l y have l a rge r e r r o r s a n d t h u s t h e i n v e s t i g a t o r s h o u l d be a w a r e o f t h e p r o b l e m a n d c h e c k i f g e o m e t r i - ca l m o d e l i n g e r r o r s wi l l be a p r o b l e m f o r a p a r t i c u l a r case . I t a p p e a r s f o r t h i s s k y (1 .2 m X 1 .2 m n o m i n a l s ize) a m o d e l d i m e n - s ion o f 0 .3 m f r o m t h e c e n t e r l ine in t h e " x " ax i s a n d 0 ,6 m a w a y f r o m t h e s k y o p e n i n g a re p o s s i b l e f o r n o r m a l r o o m d i m e n s i o n s . I t is t h e o r e t i c a l l y p o s s i b l e t o p r o j e c t a m o d e l i n t o t h e s k y u p t o 0 .4 m s ince t h e e r r o r s i t u a t i o n is t h e s a m e f o r case I I a n d case I I I in T a b l e 5. T h e m o d e l i t s e l f wi l l c h a n g e t h e l u m i n a n c e d i s t r i b u t i o n o n t h e c l o t h d r a p e a n d ce i l i ng p a n e l a n d t h u s t h e p r a c t i c a l p e n e t r a t i o n i n t o t h e s k y is s o m e w h a t less , s ay 0 .3 m.

As seen in F ig . 1, even t h e m i r r o r e d ove r - c a s t s k y wi l l p r o d u c e g e o m e t r i c m o d e l i n g e r r o r s . W h i l e t h e n e a r i n f i n i t e r e f l e c t i o n s a t t h e h o r i z o n p u t t h o s e a r eas o f l u m i n a n c e ap- p a r e n t l y v e r y far a w a y , t h e r e f l e c t i o n s n e a r t h e t o p o f t h e m i r r o r s a n d t h e ce i l ing l u m i - n a n c e is c l o s e t o t h e m o d e l a n d t h u s d i f f e r e n t p o s i t i o n s in t h e m o d e l wi l l see t h e f i x e d l u m i n a n c e d i s t r i b u t i o n f r o m d i f f e r e n t a n g u l a r p o s i t i o n s g iv ing d i f f e r e n t a p p a r e n t l u m i n a n c e d i s t r i b u t i o n s .

I t is a l so c l e a r f r o m F ig . 1 t h a t t h e ce i l ing p a n e l o f t h e m i r r o r s k y s h o u l d n o t have a u n i f o r m l u m i n a n c e b u t s h o u l d be g r e a t e s t a t

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TABLE 5

Illumination on a horizontal plane for a window facing an artificial clear sky away from the sun*

Case Coordinates of position in the model

x y (metres) (metres)

Clear atmosphere Polluted atmosphere

Illum. Error Illum. Error (lux) (lux) (lux) (lux)

0 0 0 14.20 11.42 I 0.2 0 14.26 +0.06 11.52 +0.06 II 0 --0.2 13.10 --1.03 10.68 --0.58 III 0.2 +0.2 13.19 --0.98 9.86 --0.65

*Solar altitude 40 ° . Window azimuth --20 ° to +20 ° around a horizontal line opposite the sun. Window altitude angles 16 ° to 45 ° illuminance on a horizontal plane for a zenith luminance of 100 cd/m 2. Derived from a Waldram diagram analysis.

the open ing to the m o d e l and should shif t if a m o d e l is inser ted into the sky.

MEASURING VALUES FROM THE ARTIFICIAL SKY

S o m e care m u s t be t aken in m ak i ng m o d e l m e a s u r e m e n t s and cal ibra t ing the luminance d i s t r ibu t ion f r o m the artificial clear sky. T w o sources o f p r o b l e m s are the low luminance and i l luminance values in the m o d e l and the co r r ec t r e fe rence u n o b s t r u c t e d d i f fuse artifi- cial c lear sky i l lumina t ion values.

The l imits set b y the sky size and the re f l ec tance losses m a k e the abso lu te values o f sky l uminance and m o d e l i l luminance ( 1 8 0 - 200 lux) r a the r low at the w i n d o w plane and on the mode l walls. These could be boos t ed wi th high o u t p u t l amps . Sensit ive instru- m e n t s are requi red to accura te ly de t ec t the O.1Bo/Bz steps b e t w e e n the dark spo t (0 .86) , the zeni th (1.0) and the ho r i zon (2.0}. Even if the abso lu te l uminance levels on sky surfaces are increased, the p r o b l e m will still arise; t h e r e f o r e to ca l ibra te the sky, a sensit ive l uminance m e t e r is required. N o t e t ha t on ly accura te re la t ive l uminance values are re- quired, thus abso lu te ca l ib ra t ion to s e c o n d a r y sources is n o t cri t ical .

The i l luminance m e t e r ' s r e q u i r e m e n t s for the m o d e l m e a s u r e m e n t s have been well de- scr ibed e lsewhere [16 ] . Li t t le a t t e n t i o n has been paid to l uminance surveys and since wha t we see is the subject ive evaluat ion o f l u m i n a n c e (br ightness) , l u m i n a n c e surveys

should be an integral pa r t o f the m o d e l s tudy . I f m o d e l surfaces are l amber t i an , the lumi- nance survey fo r m o s t o f the r o o m can be m a d e f r o m one obse rva t ion pa r t in the mode l . Using mi r ro rs on p robes , the inaccessible r o o m surfaces can be read ( accoun t ing fo r mi r ro r losses). One m u s t be careful wi th specular surfaces a n d reading at near glancing angles (85 ° - 90 ° f r o m the no rma l ) since m a t t e sur faces ' character is t ics m a y change.

Costs The cost o f the convers ion i tself was very

low (CDN $100 - 130) since all mater ia l s were c o m m o n l y avai lable and inexpensive . The mi r ro r sky i tself could be bui l t fo r $ 5 0 0 - $600 in mater ia ls . The m a j o r expense is the cos t o f the mir rors .

Instrumentation All i l luminance and l u m i n a n c e measure -

m e n t s were m a d e with a Hagner p h o t o m e t e r m o d e l S-2. Quar te r - sphere m a t t e b lack masks were p laced over the cell fo r in te r re f lec ted m e a s u r e m e n t s and to ta l i l luminance f r o m the sky. A Texas I n s t r u m e n t s TI -59 ca lcu la to r was used fo r the Quickl i te p rogram.

GENERAL CONCLUSIONS

(1) I t is possible to c o n s t r u c t an inexpen- sive f lexible art if icial sky t ha t can s imula te b e n c h m a r k 2 dayl igh t sources wi th a 15-min changeover .

(2) Both useful qual i ta t ive (pho tog raphs , views) and quan t i t a t ive ( i l luminance and

240

luminance) values can be made that correlate with real sky measurements.

(3) The limitations of the artificial sky can be compensated by calculations and by a geometric error analysis.

(4) There are interesting and important geometric modeling errors with artificial skies that this simple device shares with more com- plicated skies due to the close proximity of the sky source to the model.

(5) The size of the model facing a 1.2 m × 1 . 2 m sky should not exceed 0 . 4 m X 0 . 7 m for normal window glazing.

(6) Accurately measuring the interreflected component is difficult in all model measure- ments.

ACKNOWLEDGEMENTS

This work was supported by an operating grant from the National Science and Engineer- ing Council of Canada. M. Richard Moffett was responsible for much of the model con- struction and computer analysis.

REFERENCES

1 J. Summerson, Union of the arts, Lotus Inter- national, 35 (II) (1982) 69.

2 G. Pleijel, Daylight in central urban areas, Proc. Twelfth Meeting Commission Internationale de l'Eclairage, Stockholm, June 1951.

3 W. Buning, Augemessenes Tageslicht im Woh- nungsbau, Franckusche Verlagshandlung, Stutt- gart, 1953.

4 B. Evans, Daylight in Architecture, McGraw-Hill, 1981.

5 J. W. Griffith, W. J. Arner and E. W. Conover, A modified lumen method for daylighting design, Illuminating Eng., (March) (1955).

6 R. G. Hopkinson, P. Petherbridge and d. Long- more, Daylighting, Heinemann, London, 1966.

7 E. L. Harkness, The Design of a Mirror Chamber Artificial Sky, Report MR1 1970, Dept. of Arch. Sci., Univ. of Sydney, Australia.

8 S. Selkowitz, D. MaeGowan, B. McSwain and M. Navvab, A hemispherical sky simulator for day- lighting studies, Proc. 1981 Annual Illuminating Engineering Society Technical Conference, Toronto, Ontario, Paper #41.

9 Standardization of Luminance Distribution on Clear Skies, CIE Publication No. 22 (TC ~ 4.2), Commission Internationale de I'Eelairage, Paris, 1973.

10 F. C. Hooper and A. P. Brunger, A model for the angular distribution of sky radiance, Proc. Joint ASME/AICHE 18th National Heat Transfer Con- ference, San Diego, California, August 6 - 8 , 1979.

11 R. W. White, Sunpath graphical design aids, Proc. Solar Energy Society o f Canada, Prince Edward Island, August 19 - 21, 1979.

12 H. J. Bryan and R. D. Clear, Calculating interior daylight illumination with programmable hand calculators, J. o f LE.S., 10 (4) (July) (1981) 219 - 227.

13 N. N. Powe, The Climate o f Montreal, Climato- logical Study #15, Department of Transport, Canada, 1969.

14 P. Fisen, Knoxville, T e n n e s s e e - Climatic Data Base, Tennessee Valley Authority, 1980.

15 P. Fisen, Jackson, MS -- Climatic Data Base, Ten- nessee Valley Authority, 1980.

16 M. Spitzglas, State of the Art in Scale Model Photometry for the Evaluation of Daylighting in Buildings, abstract in General Proc., 1983 Inter- national Daylighting Conference, Phoenix, Arizo- na, February 16 - 18, 1983.