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International Journal of Lighting Research and Technology, 30, 3, pp. 111-118, 1998
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Transparent domed skylights: optical model for predicting transmittance, absorptance and reflectance
Laouadi, A.; Atif, M. R.
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Transparent domed skylights: optical model
for predicting transmittance, absorptance and
reflectance
Laouadi, A.; Atif, M.R.
A version of this paper is published in / Une version de ce document se trouve dans : International Journal of Lighting Research and Technology, v. 30, no. 3, 1998, pp.
111-118
www.nrc.ca/irc/ircpubs
AN OPTICAL MODEL FOR PREDICTING TRANSMITTANCE,
ABSORPTANCE AND REFLECTANCE OF TRANSPARENT DOMED SKYLIGHTS
Abdelaziz Laouadi, and Morad R. Atif Indoor Environment Research Program Institute for Research in Construction National Research Council Canada
Montreal Road, Ottawa, Ontario, Canada, K1A 0R6 Fax: +1 613 954 3733
Tel: +1 613 993 9629 Email: morad.atif@nrc.ca
Short Title: Light Transmittance Model For Transparent Domed Skylights
A version of this paper (NRCC-41926) was originally published in Lighting Research & Technology 30(3) pp. 111-118, 1998
ABSTRACT
Daylighting and thermal loads are very important design issues for skylight design, especially in large spaces such as atria. However, the trade-off between daylighting and thermal performance of skylights has been difficult to solve, due to a lack of daylighting and thermal design tools. A mathematical model was developed to predict the visible/solar transmittance, absorptance and reflectance of multi-glazed domed skylights for both direct and diffuse radiation. The model is based on tracking the beam and diffuse radiation transmission through the dome surface. Since all building-energy simulation and fenestration-rating tools are limited to planar skylights, the model was translated into a simple method where domed skylights were substituted by optically equivalent planar skylights. The results showed that domed skylights yield slightly lower visible/solar transmittance at low sun zenith angles and substantially higher visible/solar transmittance at high sun zenith angles, or near the horizon than planar skylights having the same aperture. Absorptance of domed skylights is higher than that of planar skylights, particularly at high sun zenith angles, or near the horizon. The model was also compared to the IESNA transmittance calculation procedure for domed skylights and to the Wilkinson model. IESNA transmittance calculation procedure overestimates by 19% the transmittance of a dome at low sun zenith angles and significantly underestimates the transmittance of a dome at high sun zenith angles, or near the horizon. However, the Wilkinson model significantly underestimates the transmittance of a dome for both low or high sun zenith angles.
INTRODUCTION
Domed skylights have emerged as new architectural elements in modern building design and in retrofitted buildings. They admit natural light into buildings and can simulate the outdoors in many buildings such as atria and sport arenas. Their potential in reducing electrical lighting and heating/cooling energy costs of buildings is well-recognized(1-4)
. Domed skylights have also been associated with high-energy costs, especially during warm seasons. The trade-off between daylighting design and thermal loads of skylights has always been difficult to solve. The shape of the skylight geometry, among other factors, has an effect on the amount of daylighting contribution and solar heat gains. The optimum design can only be accomplished by a careful understanding of light transmission process though the dome geometry.
Extensive theoretical and experimental investigations have been conducted to predict the daylighting and thermal performance of planar skylights and windows. However, there is a lack of tools to predict the daylighting and thermal performance of domed skylights, mostly because of the difficulty to simulate their geometry. Testing studies of domed skylights in laboratories are very limited(5)
. Testing of domed skylights using physical scale models in artificial skies has been conducted using illuminance measurements to calculate the daylight factor. These studies were restricted to the conditions of the simulation and have not been validated against real data(6)
. Mathematical models to predict the daylighting and thermal performance of domed skylights are also very limited. Wilkinson(7) considered translucent (diffuse transmitter) domed skylights, and developed models to predict daylight factors inside the dome based on horizontal illuminance formulation. Diffuse radiation from isotropic and CIE overcast skies, and beam sun radiation were considered. However, beam solar radiation was treated as diffuse radiation. For daylighting calculations, IESNA(8)
suggested a mathematical procedure to calculate the visible transmittance of single and double glazed domed skylights. The procedure does not account for the dome shape, and it has not been validated against real data. Atif et al(9)
calculated the visible transmittance of an atrium pyramidal skylight based on on-site horizontal illuminance measurements outside and inside the skylight. A significant difference between predicted and measured data was found. ASHRAE(10)
procedure for thermal calculation assumes domed skylights as tilted glazings to calculate the U-value of the structure, and
Fenestration-rating computer programs such as VISION(11) and WINDOW(12)
deal with only planar glazings. Building-energy simulation computer programs accommodate the dome geometry by dividing it into a number of inclined surfaces, and employ a ray-tracing algorithm to predict the transmitted irradiance to the interior space(13,14)
. The accuracy of the results depends on the number of the inclined surfaces composing the dome and the prediction algorithm.
OBJECTIVES
The specific objectives of this study are:
• To develop a model to predict transmittance, absorptance and reflectance of transparent multi-glazed domed surfaces under natural or artificial light; and
• To translate the model into a simple method where domed glazed surfaces are substituted by optically-equivalent planar glazed surfaces. The prediction of the daylighting and thermal characteristics of domed skylights is important not only for energy calculation but also for solving the trade-off between daylighting and solar heat gains.
MATHEMATICAL FORMULATION
Solar irradiance incident on a surface depends on the curvature, the orientation with respect to the south, and the inclination with respect to the horizontal. Two types of geometrical surfaces are considered: planar and curved. Optical properties transmittance, absorptance and reflectance -of transparent surfaces are available only for planar surfaces(8). Optical properties of curved surfaces can be evaluated based on the optical properties of their counterpart planar surfaces and their geometry. A curved surface can be divided into a number of infinitesimal inclined planar surfaces. Irradiance incident on the curved surface may, thus, be readily calculated by summing up all irradiances incident on the inclined infinitesimal surfaces. Transmitted irradiance through the curved surface may be calculated by summing up all transmitted irradiances through the inclined infinitesimal surfaces that reach the dome base surface. Transmittance of the domed surface may, thus, be readily obtained. Other optical properties may be obtained in a similar manner.
Consider a transparent domed surface with multiple glazed panes. The transmittance, absorptance and reflectance of its counterpart planar surface for beam radiation at a given incidence angle are noted as τ, α and ρ, respectively. The transmittance, absorptance and reflectance for diffuse radiation are noted as τd, αd and ρd, respectively. Solar irradiations on inclined planar surfaces, and on domed surfaces are calculated as follows:
Irradiation on an Inclined Planar Surface
The incident solar irradiance on an inclined planar surface is given by(15)
: (1) radiation diffuse for , radiation; beam for , cos , , , t d d b b I A I I A I ⋅ = ⋅ = β β β θ where:
Ib,β : the beam irradiance incident on an inclined surface (W); Id,β : the diffuse irradiance incident on an inclined surface (W); Ib : the beam solar radiation (W/m2
);
Id : the sky diffuse solar radiation on a horizontal surface (W/m2
); Id,t : the total diffuse radiation on an inclined surface (W/m2
); A : the surface area (m2
);
θβ : the incidence angle on an inclined surface for beam radiation; β : the inclination angle of the surface with respect to the horizontal. The incidence angle θβ is given by(15)
: (2) ) cos( sin sin cos cos cosθβ = β θz + β θz ψs −ψf with:
θz : the sun zenith angle (or the incidence angle on a horizontal planar
ψs : the sun azimuth angle; and
ψf : the surface azimuth angle.
The sun zenith angle θz is expressed in terms of the site latitude L, the sun declination angle δ and the hour angle ω as follows(15):
(3) sin sin cos cos cos cosθz = L δ ω + L δ
The maximum incidence angle on a horizontal planar surface occurs at noontime (ω=0) where the zenith angle θz = L - δ.
Total diffuse radiation on an inclined surface Id,t is composed of the sky diffuse radiation and the ground-reflected radiation. For a general non-isotropic diffuse sky, the total diffuse radiation Id,t may be written as follows(15) : (4) ) cos ( ,t d d b z d r d I c I I c I = + θ +
with cd and cr coefficients for diffuse and ground-reflected radiation, respectively.
Using Perez et al model(15) for the sky diffuse radiation, the coefficients cd and cr are given by:
(5) 2 cos 1 ; sin ) 2 cos 1 )( 1 ( − 1 + β + 1 + 2 β =ρ − β = r g d F c c b F F c where:
ρg : the ground reflectance (albedo);
F1 : circumsolar brightness coefficient; F2 : horizon brightness coefficient; and
b, c : terms that account for angles of incidence of the cone of
circumsolar radiation on the inclined and horizontal surfaces.
For isotropic diffuse skies, the coefficients F1 and F2 are equal to zero. For overcast skies, the coefficient F1 is equal to zero.
Irradiation on a Domed Surface
A domed surface is a hemispheric cap, defined by its truncation angle (σ0) and its radius (R). A dome shape is a representative form for any curved surface. The family of shapes covered range from a fully hemispheric surface to a planar surface. Domed surfaces receive beam solar radiation as well as sky and ground-reflected diffuse solar radiation. The amount of solar irradiance transmitted through, absorbed or reflected by a domed surface is dependent on the dome geometry and the beam and diffuse solar radiation.
Beam Radiation Transmission Process
Figure 1 shows a schematic description of the beam radiation transmission process through a horizontal domed surface in a system of coordinates (x, y, z) moving with the sun. Beam irradiance incident on a domed surface is given by:
(6) cos 2 1 ,
∫
+ = A A b dome b I ds I θA portion of the beam irradiance is transmitted directly to the interior space through the surface area A1. The other portion is transmitted through the surface area A2, and then reflected by the dome interior surface to the interior space. The following assumptions are made to calculate the dome transmitted, absorbed and reflected irrandiances:
1. the light transmittance, absorptance and reflectance at any point on the dome surface are equal to those of a planar surface at the same incidence angle; and
2. The amount of light reflection from the interior space under the dome back to the dome interior surface is not accounted for.
The transmitted beam irradiance is given by:
(7) cos ) ( ) ( + cos ) ( 2 1 , =
∫
∫
A b A b dome b I ds I ds IT τ θ θ τ θ ρ θ θ(8) cos ) ( ) ( + cos ) ( 2 2 1 ,
∫
∫
+ = A b A A b dome b I ds I ds IA α θ θ τ θ α θ θ (9) cos ) ( + cos ) ( 2 2 1 2 ,∫
∫
+ = A b A A b dome b I ds I ds IR ρ θ θ τ θ θ where:ds : the area of an elementary surface associated with the point P;
P : a point that moves in a plane perpendicular to the plane of the sun’s rays and inclined with an angle σ with respect to the dome-base’s plane; and
θ : the incidence angle on the elementary surface ds. Figure 2 shows the coordinates of the elementary surface ds.
Substituting the inclination angle of the elementary surface β=π/2-ξ, and the azimuth angle difference ψs - ψf = π/2-ϕ′ in Equation (2), the incidence angle on the elementary surface reads:
(10) sin sin cos cos sin cosθ = ξ θz + ξ θz ϕ′ where:
ξ : the elevation angle of the point P with respect to the dome-base’s plane (varies from 0 to π/2); and
ϕ′ : the relative azimuth angle of the point P (varies from 0 to 2π). The area of the elementary surface ds is given by:
(11) sin ds=R2 ϕdσ dϕ
and the elevation angle ξ is expressed as:
(12) sin sin sinξ = ϕ σ
with ϕ the equivalent angle to ϕ′ in the inclined plane of the point P (varies from 0 to π), given by:
(13) cos cos cosϕ = ξ ϕ′
The surface portions A1 and A2 are defined as follows:
(14) ) ( ) ( : ; ) ( ) ( : 0 0 2 1 2 0 0 1 0 1 − ≤ ≤ ≤ ≤ − ≤ ≤ ≤ ≤ σ ϕ π ϕ σ ϕ σ σ σ σ ϕ π ϕ σ ϕ σ σ σ A A
The angles σ1, σ2 and ϕ0 are given by:
(15) ) sin / (sin sin ) ( ); , -min( ); , 2 -min( 0 1 0 0 z 2 0 z 0 1 σ σ σ ϕ σ π θ π σ σ π θ π σ σ − = = + =
Diffuse Radiation Transmission Process
Figure 3 shows the diffuse radiation transmission process through a domed surface. Total diffuse irradiance incident on a domed surface is expressed as follows: (16) , , =
∫
dome A t d dome d I ds Iwhere Adome is the dome surface area.
The diffuse radiation is spread diffusely after its transmission through the dome surface. A portion of the transmitted diffuse radiation goes directly to the interior space while the other portion experiences multiple reflections and transmissions through the interior surface of the dome. The first reflected radiation from the dome interior surface is diffuse and treated in a similar manner as the first transmitted diffuse radiation, and so on for subsequent reflections. The total transmitted diffuse irradiance is expressed as follows: (17) ...} ) ( ) ( 1 { 3 11 2 11 11 12 + + + + =A IT F F F F ITd,dome dome d,t d d d
Similarly, the absorbed and reflected diffuse irradiances read:
(18) ...} ) ( 1 { 11 11 11 11 2 11 , , =A IT +F + F F + F F + IA d d d d t d dome dome d α τ ρ ρ
(19) ...} ) ( { 11 11 11 11 2 11 , , =A IT + F + F F + F F + IR d d d d d d d t d dome dome d τ τ τ ρ τ ρ ρ where:
F11 : the view factor of the dome interior surface to itself;
F12 : the view factor of the dome interior surface to its base surface; and
ITd,t : the transmitted total diffuse radiation, given by:
(20) cos I 1 dome d, , d d dome t d A IT = θ τ
where θd is the incidence angle for diffuse radiation. Equations (17)-(19) reduce to: (21) 1 11 12 , , − = F F I IT d d dome d dome d ρ τ (22) 1 11 11 , , − + = F F I IA d d d d dome d dome d ρ α τ α (23) 1 ) ( 11 11 2 , , − + = F F I IR d d d dome d dome d ρ τ ρ
The view factors F12 and F11 are expressed as:
(24) )/2 sin (1 /A A and ; -1 12 12 h dome 0 11 = F F = = + σ F
with Ah the area of the dome base surface.
The above analysis shows that all the parameters needed to calculate the dome transmittance, absorptance and reflectance are now available. The dome transmittance is the ratio of the transmitted irradiance to the incident irradiance on the dome surface. Similarly, the dome absorptance or reflectance is the ratio of the absorbed or reflected irradiance to the incident irradiance. These are expressed as:
For beam radiation, (25) ; ; , , , , , , dome b dome b dome dome b dome b dome dome b dome b dome I IR I IA I IT = = = α ρ τ
For diffuse radiation,
(26) 1 ; 1 ; 1 11 2 11 , 11 11 , 11 12 , F F F F F F d d d dome d d d d d dome d d d dome d ρ τ ρ ρ ρ α τ α α τ ρ τ − + = − + = − =
For diffuse radiation, the dome transmittance is lower than that of a planar surface. The dome absorptance and reflectance are, however, higher than those of a planar surface.
Equations (25) and (26) are general for any source of light (natural or artificial) with specular or diffuse radiation. For energy and daylighting calculations, Equations (25) and (26) may be used to compute the solar or visible transmittance of transparent domed skylights.
Computing the dome transmittance, absorptance, and reflectance and, consequently, the transmitted, absorbed and reflected irradiances is not straightforward before the evaluation of the double integral in Equations (7), (8), (9) and (16) for each time of the day. An alternative approach is to compute the optical properties of a planar surface that is optically equivalent to a domed surface. This is particularly important for building-energy simulation and fenestration-rating computer programs. This approach would have the following benefits:
1. It eliminates the need to input complex geometrical data of domed skylights; the user only needs to input the optical properties of the dome-equivalent planar surface;
2. It allows daylighting performance comparison between domed and planar skylights;
3. It can handle inclined domed skylights by simply reducing them to inclined dome-equivalent planar skylights; and
Horizontal illuminance measurement inside and outside the dome may be used to measure the equivalent transmittance.
DOME-EQUIVALENT PLANAR SURFACE
A simple method is proposed to calculate the transmittance, absorptance, and reflectance of a planar surface that is optically equivalent to a domed surface. The dome-equivalent planar surface would have the same aperture, the same construction materials, the same orientation and inclination angles, and would produce the same transmitted, absorbed and reflected irradiances as the domed surface. However, incident irradiance on the dome-equivalent planar surface can not be equal to that incident on the dome surface. Hence, transmittance, absorptance, and reflectance of a dome-equivalent planar surface may be greater than unity. Transmittance, absorptance and reflectance of a dome-equivalent planar surface may be defined as follows:
(27) ) ( ) ( ) ( ); ( ) ( ) ( ); ( ) ( ) (θ θ τ θ α θ θ α θ ρ θ θ ρ θ τeq =t eq =a eq =r
where (t), (a) and (r) are transmittance, absorptance and reflectance coefficients to be determined. The equivalent transmittance τeq may indicate the solar heat gains of the interior space under the dome, or may be interpreted as a daylight factor, which indicates the amount of daylight entering the space through the dome(8). The absorptance αeq indicates the amount of solar radiation stored in the dome surface, which is then convected and radiated to the interior and the exterior spaces.
Conservation of radiative heat flux on the dome surface yields the following relationship: (28) ε ρ α τeq + eq + eq =
where ε is the ratio of the incident irradiance on the domed surface to that incident on the dome-equivalent planar surface. For horizontal domed surfaces, ε is given by:
(29) radiation diffuse for ; 1 radiation; beam for ; cos cos 1 ) ( , 2 1
∫
∫
= = + dome d t d h d A A z h z ds I I A ds A ε θθ θ εEquation (29) for bean radiation can be expressed as:
{
tan}
(30) cos 1 4 3 ) ( 0 2 s c z z F F θ σ π θ ε = + + with: (31) ) ( cos sin ))sin ( -/2 ( sin cos sin 2 )sin sin (1 sin cos 2 1 /2)cos -) ( ( 2 0 0 2 2 0 2 2 0 2 0 2 2 2 0 2 1 -0 2 2 2 0 σ ϕ σ σ σ ϕ π σ σ σ σ σ σ σ π σ ϕ − = + − − = c s F FFor isotopic diffuse skies, F1 = F2 = 0, Equations (29) for diffuse radiation reduce to:
{
1 (1 )}
(32) 2 / 1 12 12 F F g g d ρ ρ ε = + + −By equating the transmitted, absorbed and reflected irradiances of a horizontal domed surface to that of a dome-equivalent horizontal planar surface, one obtains:
For beam radiation,
(33) ) r( ; ) a( ; ) t( z z z ) ) ) ( ) ) ) ( ) ) ( ) ( z z z dome z z z dome z z z dome = = = τ τ
For diffuse radiation,
(34) ) 1 ( ) ( 1 r ; 1 1 a ; 1 1 11 2 11 11 11 11 11 d d d d d d d d d d d d F F F F F F t ε ρ ρ τ ε ρ τ ε ρ − + = − + = − − =
For beam radiation, the optical properties τeq, αeq and ρeq depend on the optical properties τ, α and ρ, on the sun zenith angle θz (which itself depends on the site latitude and the day of the year), and on the dome geometry. For diffuse radiation, the optical properties (τeq)d, (αeq)d and (ρeq)d depend on the optical properties τ d, α d and ρd, on the beam and diffuse radiation intensities, on the ground reflectance ρg, and on the dome geometry. The dome geometry is described by only one dimensionless parameter, the dome truncation angle σ0 (the radius has no effect). The dome truncation angle parameter is very important to domed skylight designs. The dome shape may be chosen to yield high/low visible equivalent transmittance for daylighting purpose, or to yield high/low solar heat gains to reduce heating/cooling costs in winter/summer, depending on the site latitude and the prevailing climate.
COMPARISON WITH EXISTING OPTICAL MODELS FOR DOMED SKYLIGHTS
The model previously developed is compared to that of two widely known models. These models are: (1) IESNA(8)
calculation procedure to predict daylight transmittance of domed skylights, and (2) Wilkinson(7) model to predict the daylight factor of translucent domed skylights.
Using the IESNA transmittance calculation procedure, daylight transmittance coefficient (t) of single glazed domed skylights is calculated using the following relation:
(35) ) 416 . 0 18 . 1 ( 25 . 1 − τ = IES t
IESNA Equation (35) does not account for the dome shape.
The equivalent transmittance for double-glazed domed skylights is calculated in a similar manner of calculating transmittance of planar double glazed skylights. The equivalent transmittance and reflectance for each dome layer are used instead of transmittance and reflectance of each layer of a planar skylight. However, IESNA does not give any procedure to calculate the equivalent reflectance for domed skylights. Furthermore, the IESNA procedure for double-glazed domed skylights may not be correct since equivalent transmittance, absorptance and reflectance may be greater than unity (see Equation 28).
Wilkinson(7) developed theoretical relations for daylight factor calculation based on horizontal illuminance formulation inside and outside a dome. Three types of radiation were considered: diffuse radiation from isotropic and CIE overcast skies, and beam sun radiation. The dome surface was assumed diffuse transmitter and reflector (translucent). Beam solar radiation was treated as diffuse radiation. The following relations were developed (with no internal space light reflection):
(36) 1 11 12 F QF t d W ρ − =
where Q takes on the following values:
(37) radiation diffuse isotropic for ; ) 1 1 ( 2 1 12 F Qu = + (38) radiation beam for ; cos cos cos sin ) cos sin ( 1 0 2σ θ π η η η χ χ χ π π z s Q = − + + −
with η and χ angles given by:
(39) for ; 0 for ; ) tan / (tan cos and ), sin / (sin cos 0 z 0 z 0 1 0 1 σ θ χ η σ θ θ σ χ θ σ η < = = ≥ = = − − z z
Equation (36) was developed for the sky diffuse radiation and, therefore, does not account for the ground-reflected radiation. Under isotropic diffuse sky conditions, Equation (36) is equivalent to Equation (34) for
ρg=0.
RESULTS AND DISCUSSION
The comparison between the developed model in this paper and the two existing models is based on a single glazed dome with 6mm clear float glass. The solar transmittance and absorptance of the clear float glass at normal incidence angle are(16)
: τ = 0.78 and α = 0.15. The solar transmittance and absorptance at other incidence angles are calculated and then fitted using five order polynomial series with argument cos(θz), similar to those used in ASHRAE(10)
(8), (9) and (16) is evaluated using Simpson’s rule for numerical integration.
Figure 4 shows the predicted transmittance and absorptance of a single-glazed domed surface as a function of the incidence angle on a horizontal planar surface for a number of dome shapes. The figure shows that the transmittance of domed surfaces decreases as the incidence angle increases. The nearly hemispheric domes (σ0 < 30°) have lower transmittance than that of planar surfaces for incidence angles up to 87°. Domes with truncation angles 30° ≤ σ0 ≤ 75° have a slightly lower transmittance than that of planar surfaces for incidence angles up to 60° (up to 4% lower that that of planar surfaces). However, at higher
incidence angles (above 60°), these domes have a much higher
transmittance than that of planar surfaces, reaching a maximum of about 26%. The dome transmittance increases with the dome truncation angle, particularly for domes with σ0 lower than 75°. Hemispheric domes have the lowest transmittance. The absorptance of domes is generally higher than that of planar surfaces. At near normal incidence angle (up to 50°), the absorptance of domes is about 7% higher than that of planar surfaces. At very high incidence angles (near the horizon), the absorptance of domes can be 28% higher than that of planar surfaces. For a given incidence angle, the dome absorptance decreases with the increase of truncation angles. Hemispheric domes have the highest absorptance. This is because a hemispheric dome has the largest exposure area to absorb and reflect solar radiation. Unlike planar surfaces, transmittance and absorptance of domed surfaces are not equal to zero at 90° incidence angle, due to the fact that domes still collect solar radiation when the sun is at the horizon.
Figure 5 shows prediction comparison between the proposed model and both IESNA(8) transmittance calculation procedure and the Wilkinson(7) model. The beam transmittance coefficient (t) is plotted as a function of the incidence angle on a horizontal planar surface for a number of dome shapes. The IESNA Equation (35) yields higher values of the transmittance coefficient at incidence angles up to 55° (about 19% higher than that predicted by the proposed model) and lower values of the transmittance coefficient at incidence angles higher than 55°. However, Wilkinson Equation (36) produces lower values of the transmittance coefficient at both high and low incidence angles, particularly for nearly hemispheric domes. For fully hemispheric domes, the difference between
the Wilkinson model and the proposed model is about 40% at normal incidence angle and 27% at an incidence angle of 75°. This difference is due to the fact that the beam radiation in the Wilkinson model is treated as diffuse radiation. All the three prediction models show that a domed surface has the ability to gather less daylight at near normal incidence angles and to gather more daylight at high incidence angles.
The predictions of the proposed model show that, at normal incidence angles, the equivalent transmittance increases with the dome truncation angle. However, near the horizon, the equivalent transmittance decreases with the increase of the dome truncation angle. Fully hemispheric skylights have the lowest equivalent transmittance at normal incidence angles (about 10% lower than that of planar skylights) and the highest equivalent transmittance around the horizon. Hence, depending on the site latitude and the day of the year, domed skylights may yield lower/higher visible equivalent transmittance or solar heat gains than planar skylights.
Figure 6 shows the beam absorptance coefficient (a) as a function of the incidence angle on a horizontal planar surface for a number of dome shapes. The beam absorptance coefficient for nearly hemispheric domes (σ0 < 30°) increases significantly with the incidence angle. However, the beam absorptance coefficient for domes with truncation angles 30° ≤σ0 ≤ 75° is almost equal to a unity for incidence angles up to 50° while at higher incidence angles it takes on higher values. The beam absorptance coefficient decreases with the dome truncation angle. Fully hemispheric domes have the highest absorptance coefficient. This is because fully hemispheric domes have the largest exposed surfaces with respect to other dome shapes.
Figure 7 shows the diffuse transmittance and absorptance coefficients (td and ad) as a function of the dome shape under isotropic diffuse skies. Some ground coverings are considered, with a reflectance varying from 0 to 0.7. Under isotropic skies, the diffuse transmittance coefficient increases with the increase of the dome truncation angle while the diffuse absorptance coefficient decreases. Fully hemispheric domes have the lowest diffuse transmittance coefficient and the highest diffuse absorptance coefficient. In addition, increasing the ground reflectance results in an increase of the dome diffuse transmittance and absorptance coefficients. For snow-covered ground (ρ=0.7), the increase in the
transmittance and absorptance coefficients is about 16% compared to those for a ground with ρg=0.2.
Figure 8 illustrates the diffuse transmittance and absorptance coefficients of a fully hemispheric dome under non-isotropic skies (Perez et al model(15)
) for the 21st
of June. The selected latitude is 45° with a longitude difference of 0° (such as Ottawa, Canada). The beam (Ib) and diffuse (Id) radiations are calculated using the ASHRAE(10)
standard method for solar radiation calculation. The ratio of diffuse-to-beam radiation is 0.134. Unlike diffuse isotropic skies, the diffuse transmittance and absorptance coefficients under non-isotropic skies increase significantly with the decrease of the sun zenith angle. At low zenith angles (around noontime), the diffuse transmittance and absorptance coefficients under non-isotropic skies are 30% higher than those under isotropic skies. At high zenith angles (at horizon), the diffuse transmittance and absorptance coefficients under non-isotropic skies are 13% lower than those under isotropic skies.
CONCLUSION
An optical model was developed to predict the transmittance, absorptance and reflectance of multi-glazed domed skylights for both direct and diffuse radiation. A simple method was proposed to substitute domed skylights by optically equivalent planar skylights. The model is compared with the IESNA(8)
transmittance calculation procedure and the Wilkinson(7)
model for domed skylights.
The results of the comparison showed that the IESNA transmittance calculation procedure predicts the equivalent transmittance for domed skylights, which is 19% higher than that predicted by the proposed model for sun zenith angles up to 55°. At low sun zenith angles, or near the horizon, the IESNA procedure significantly underestimates the equivalent transmittance with respect to the proposed model. However, the Wilkinson model substantially underestimates the equivalent transmittance with respect to the proposed model for both low and high sun zenith angles, particularly for fully hemispheric domes. At normal sun zenith angles, the equivalent transmittance predicted by the Wilkinson model is 40% lower than that predicted by the proposed model for fully hemispheric domes. At a sun zenith angle of 75°, the equivalent transmittance predicted by the Wilkinson model is 27% lower than that predicted by the proposed model for fully hemispheric domes.
Predictions of the proposed model showed that domed skylights have the ability to gather less daylight at low sun zenith angles and more daylight at high sun zenith angles. Nearly hemispheric domed skylights (σ0 < 30°) have lower visible/solar equivalent transmittance at near normal sun zenith angles and higher visible/solar equivalent transmittance near the horizon than that of planar skylights. The equivalent absorptance for nearly hemispheric domed skylights is higher than that of planar skylights, particularly near the horizon. Skylights with truncation angles between 30° and 75° yield approximately the same visible/solar equivalent transmittance and absorptance as planar skylights for sun zenith angle up to 55°, and yield much higher visible/solar equivalent transmittance and absorptance near the horizon.
The significance of the above work is important to provide designers with tools to solve the trade-off between daylighting and thermal performance of skylights. Future work will address thermal characteristics of domed skylights. Since the visible equivalent transmittance depends on the sun zenith angle (which itself depends on the site latitude and the day of the year), skylight shapes for daylighting applications should be chosen according to the site latitude. Fully hemispheric domes may be better candidates than other domes for skylight design, especially for latitudes where the sun altitude can not reach the normal during the day. However, for thermal design, fully hemispheric domes may have the highest solar heat gain coefficient. Furthermore, fully hemispheric domes have the largest surface area, and, thus, the highest thermal loads (high dome-equivalent planar surface U-value). Therefore, domed skylight shapes should be chosen according to the site latitude and the prevailing climate. In cold climates, high solar heat gain coefficient and low equivalent U-value are important to reduce the heating costs. In warm/hot climates, low solar heat gain coefficient and low equivalent U-value are important to reduce the cooling costs.
ACKNOWLDGEMENTS
This work has been funded by the Institute for Research in Construction, National Research Council Canada; Natural Resources Canada; Public Works and Government Services Canada; Sociéte Immobilière du Québec; and Hydro-Québec. The authors are very thankful for their contribution.
REFERENCES
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2 Architectural Aluminum Manufacturers Association (AAMA), Chicago, Illinois, Design for energy conservation with skylights, (AAMA TIR-A6) (1981).
3 Treado S, Gillette G, and Kusuda T, Evaluation of the daylighting and energy performance of windows, skylights, and clerestories, (U.S. Department of commerce, Report NBSIR 83-2726) (1983).
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5 Enermodel Engineering Ltd., Thermal performance of complex fenestration systems: skylights, greenhouse windows, and curtainwalls, (Natural Resources Canada, Report 23440-92-9615) (1994).
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7 Wilkinson M A, Natural lighting under translucent domes, Lighting Research Technology 24(3) 117-126 (1992).
8 IESNA lighting handbook, reference and application volume, (New York: Illuminating Engineering Society of North America) (1993).
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10 ASHRAE, ASHRAE handbook – Fundamentals (1997).
11 The advanced glazing system laboratory, University of Waterloo, Ontario, Canada, VISION 4 Reference manual (1995).
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LIST OF FIGURE CAPTIONS
Figure 1 Beam radiation transmission process through a horizontal domed surface
Figure 2 Coordinates of the elementary surface ds
Figure 3 Diffuse radiation transmission process through a horizontal domed surface
Figure 4 Transmittance and absorptance of a domed surface Figure 5 Beam transmittance coefficient of a domed surface Figure 6 Beam absorptance coefficient of a domed surface
Figure 7 Diffuse transmittance and absorptance coefficients of a domed surface under isotropic sky
Figure 8 Diffuse transmittance and absorptance coefficients of a domed surface under non-isotropic sky
NOMENCLATURE
A : surface area
A1 : surface for the directly-transmitted beam radiation through a domed surface.
A2 : surface for the transmitted-reflected beam radiation through a domed surface
Adome : area of the dome surface. Ah : area of the dome-base surface.
a, r, t : absorptance, reflectance and transmittance coefficients,
respectively.
b, c : terms that account for angles of incidence of the cone of circumsolar radiation on inclined and horizontal planar surfaces. cd, cr : coefficients for diffuse and ground-reflected radiation (Equation 5) Fc, Fs : functions, Equation (31)
F1, F2 : circumsolar and horizon brightness coefficients F11 : view factor of the dome interior surface to itself.
F12 : view factor of the dome interior surface to its base surface. I : incident irradiance (W).
Ib : beam solar radiation (W/m2
).
Id : sky diffuse solar radiation on a horizontal surface (W/m2
). Id,t : total diffuse radiation on an inclined surface (W/m2
Id,β : diffuse irradiance incident on an inclined planar surface (W). IT : transmitted irradiance (W). IA : absorbed irradiance (W). IR : reflected irradiance (W). L : site latitude. R : dome radius. Geek Symbols
α, ρ, τ: absorptance, reflectance and transmittance of a planar surface, respectively.
β : inclination angle of a planar surface with respect to the horizontal
δ : sun declination angle
ε :ratio of dome irradiance to planar surface irradiance, Equation (29)
θ : incidence angle on the elementary surface (ds)
θβ : incidence angle on an inclined planar surface for beam radiation θd : incidence angle for diffuse radiation
θz : sun zenith angle
ξ : elevation angle of the point P on the dome surface with respect to the dome-base surface plane
ρg : ground reflectance (albedo)
σ : inclination angle of the plane of the point P with respect to the plane of the dome-base surface
σ0 : dome truncation angle
ϕ : equivalent angle to ϕ′ in the plane of the point P
ϕ0 : value of ϕ at σ = σ0, Equation (15)
ϕ′ : relative azimuth angle of the elementary surface (ds)
ψs : sun azimuth angle
ψf : surface azimuth angle
ω : hour angle
Subscripts
b : beam radiation
d : diffuse radiation dome : dome surface
x y z ξ ϕ´ σ0 σ σ1 σ2 P θz A1 A2 ϕ0´ ϕ R Sun’s rays Figure 1
ϕ ϕ´ ξ σ dσ dϕ´ x y z dϕ Figure 2
Id,t F12ITd,t IRd,t F 11IT d,t ρdF12F11ITd,t ρdρd F12F11F11ITd,t ρdF11F11 ITd,t τ dF 11IT d,t τdρdF11 F11ITd,t Sky Figure 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90
Incidence angle on a horizontal planar surface (deg.)
T ransmittance, τdom e 0 0.1 0.2 0.3 0.4 0.5 Absor p tance, αdom e σ0 = 0ο σ0 = 30ο σ0 = 45ο σ0 = 75ο σ0 = 60ο σ0 = 90ο Figure 4
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 90
Incidence angle on a horizontal planar surface(deg.)
B eam t ransm it tance coef fi cient (t ) σ0 = 0o σ0 = 30 o σ0 = 45 o σ0 = 60 o σ0 = 75o IESNA(8) Wilkinson(7) Figure 5
0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 70 80 90
Incidence angle on a horizontal planar surface (deg.)
B e am absor p tance coef fi cient (a) σ0= 0o σ0= 30o σ0= 45o σ0= 60o σ0= 75o Figure 6
0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 40 50 60 70 80 90
Dome truncation angle, σ0 (deg.)
Di ffuse tran smi ttan c e coeffi ci en t ( td ) 1.0 1.5 2.0 2.5 3.0 Diffuse absor ptance coefficient ( ad ) ρg = 0.0 ρg = 0.2 ρg = 0.5 ρg = 0.7 Isotropic diffuse sky
June 21 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 20 30 40 50 60 70 80 90
Sun zenith angle, θz (deg.)
Diffuse tr ansmittance coefficient ( td ) 1 1.5 2 2.5 3 Diffuse absor ptance coefficient ( ad )
Isotropic diffuse sky Non-isotropic diffuse sky
ρg = 0.2 ; L = 45 o
; σ0 = 0o
