Daily solar heat gains of hemispherical domed skylights are higher than those of horizontal planar skylights in
both winter and summer season. In summer, the solar heat gain of a hemispherical domed
skylights can reach 3% to 9% higher than those of horizontal planar skylights for latitudes
varying between 0° and 55° (north/south). In winter, however, the solar heat gains of hemispherical domed skylights increase significantly with the increase of the site latitude, and can reach
232% higher than those of horizontal planar skylights, particularly for high latitude countries
1
PREDICTING OPTICAL AND THERMAL CHARACTERISTICS OF TRANSPARENT SINGLE-GLAZED DOMED SKYLIGHTS
Abdelaziz Laouadi, Ph.D.; and Morad R. Atif, Ph.D.
A version of this paper (NRCC-42027) was originally published in
ASHRAE Transactions Vol. 105 Pt. 2. pp. 325-333, 1999
ABSTRACT
Optical and thermal characteristics of domed skylights are important to solve the trade-off between
daylighting and thermal design. However, there is a lack of daylighting and thermal design tools for
domed skylights. Optical and thermal characteristics of transparent single-glazed hemispherical domed
skylights under sun and sky light are evaluated based on an optical model for domed skylights. The
optical model is based on tracking the beam and diffuse radiation transmission through the dome surface.
A simple method is proposed to replace single-glazed hemispherical domed skylights by optically and
thermally equivalent single-glazed planar skylights to accommodate limitations of energy computer
programs. Under sunlight, single-glazed hemispherical domed skylights yield slightly lower equivalent
solar transmittance and solar heat gain coefficient (SHGC) at near normal zenith angles than those of
single-glazed planar skylights. However, single-glazed hemispherical domed skylights yield substantially
higher equivalent solar transmittance and SHGC at high zenith angles and around the horizon. Under
isotropic skylight, single-glazed hemispherical domed skylights yield slightly lower equivalent solar
transmittance and SHGC than those of single-glazed planar skylights. Daily solar heat gains of singleglazed
hemispherical domed skylights are higher than those of single-glazed horizontal planar skylights in
both winter and summer season. In summer, the solar heat gain of single-glazed hemispherical domed
skylights can reach 3% to 9% higher than those of horizontal single-glazed planar skylights for latitudes
varying between 0° and 55° (north/south). In winter, however, the solar heat gains of single-glazed
hemispherical domed skylights increase significantly with the increase of the site latitude, and can reach
232% higher than those of horizontal single-glazed planar skylights, particularly for high latitude countries.
2
INTRODUCTION
Domed skylights have increasingly been incorporated into modern building design and in retrofitted
buildings as new elements for daylighting and aesthetic purpose. They admit abundant 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 (AAMA
1987; AAMA 1981; Treado et al 1983; Jensen 1983). Domed skylights have also been associated with
high energy costs, especially during warm seasons. Optical and thermal characteristics of domed
skylights are very important to solve the trade-off between daylighting and thermal design. The shape of
the skylight geometry is a crucial factor for the amount of daylighting contribution, solar heat gains and
thermal heat gains/losses. The optimum design can only be accomplished by an accurate determination
of the optical and thermal characteristics of domed skylights.
Extensive theoretical and experimental investigations have been conducted to predict the optical and
thermal performance of planar skylights and windows. However, there is a lack of tools to predict the
optical and thermal performance of domed skylights, mostly because of the difficulty to simulate their
geometry. Theoretical models to predict the optical and thermal performance of domed skylights are very
limited. Wilkinson (1992) considered translucent (diffuse transmitter) domed skylights, and developed
models to predict the Daylight Factor 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
(1993) 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 measurement.
Recently, Laouadi and Atif (1998) have developed an optical model to predict the transmittance,
absorptance and reflectance of transparent, multi-glazed, hemispherical, domed skylights under sun and
sky light. The model is based on tracking the beam and diffuse radiation transmission through the dome
shape. Experimental testing studies of real domed skylights are also very limited, due to their complex
geometry and size (Enermodal 1994). Atif et al (1997) calculated the visible transmittance of an atrium
pyramidal skylight based on on-site horizontal illuminance measurements outside and inside the skylight.
There was a significant difference between predicted and measured data. Laboratory 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 fact data (Navvb 1990). ASHRAE’s procedure for
thermal calculation assumes domed skylights as tilted glazings to calculate the U-value of the structure,
and the extent of error of such an assumption has never been tested (ASHRAE 1997). Fenestration
rating computer programs such as VISION (CANMET 1995) and WINDOW (LBL 1992) deal with only
planar glazings. Building energy-simulation computer programs accommodate the dome geometry by
dividing it into a number of inclined surfaces (Clarke 1985; ESRU 1996).
OBJECTIVES
The specific objectives of the paper are:
1. To predict the transmittance, absorptance, reflectance and solar heat gain coefficient of
transparent single-glazed hemispherical domed skylights.
2. To develop a simple method where single-glazed hemispherical domed skylights are replaced by
optically and thermally equivalent single-glazed planar skylights for energy calculations.
3. To compare the thermal characteristics between single-glazed hemispherical domed and
horizontal planar skylights.
The prediction of the optical 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.
3
MATHEMATICAL FORMULATION
A domed surface is 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
hemispherical surface to a planar surface. Solar irradiance incident on a domed surface depends on its
geometry, orientation with respect to the south, and inclination with respect to the horizontal. Thermal
and optical properties of transparent surfaces - U-value, solar heat gain coefficient (SHGC),
transmittance, absorptance, and reflectance - are available only for planar surfaces (ASHRAE 1997;
IESNA 1993). Thermal properties of domed surfaces are more complicated to predict than those of
planar surfaces.
Optical properties of domed surfaces can be evaluated based on the optical properties of their
counterparts planar surfaces and their geometry. A domed surface can be divided into a number of
infinitesimal inclined planar surfaces. Irradiance incident on the domed surface may, thus, be readily
calculated by summing up all irradiances incident on the inclined infinitesimal surfaces. Transmitted
irradiance through the domed 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.
The mathematical model has been developed to predict the optical properties of transparent domed
surfaces (Laouadi and Atif 1998). The underlying assumptions of this model are:
1. The light transmittance, absorptance and reflectance at any point on the dome surface are equal
to those of a flat surface at the same incidence angle.
2. The amount of light reflection from the interior space under the dome back to the dome interior
surface is not accounted for.
Optical Properties of Domed Surfaces
Domed surfaces receive beam solar radiation as well as sky and ground-reflected diffuse solar
radiation. The amount of solar irradiance transmitted, absorbed, or reflected by a domed surface
depends on the dome geometry and the beam and diffuse solar radiation.
Beam Radiation Transmission Process Through a Domed Surface. 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. In tracking the solar radiation transmission, the dome surface is
split into two portions: portion (A1), which corresponds to the directly-transmitted component that reaches
the dome base surface; and portion (A2), which corresponds to the transmitted-reflected component.
Only the first reflected radiation from the dome interior surface is assumed to reach the dome base
surface. The extent of the maximum error of this assumption is less than 5% at an incidence angle 85°
and less than 3% at an incidence angle 80°.
The incident beam irradiance is given by:
∫
+
=
21
cos ,
A A
I
b dome I
b θds (1)
The transmitted, absorbed and reflected beam irradiances are given by:
∫ ∫
=
1 2
)( cos + )()( cos ,
A
b
A
ITb dome I
b
τ θ θds I τ θ ρ θ θds (2)
∫ ∫
+
=
21 2
)( cos + )()( cos ,
A
b
A A
IAb dome I
bα θ θds I τ θ α θ θds (3)
∫ ∫
+
=
21 2
)( cos + )( cos 2
,
A
b
A A
IRb dome I
b ρ θ θds I τ θ θds (4)
where:
A1, A2 : the dome surface portions for the directly-transmitted and transmitted-reflected
4
radiation, respectively;
ds : the area of an elementary surface associated with the point P;
Ib : the beam solar radiation (W/m2
);
P : a point that moves in a plane perpendicular to the sun’s rays plane and inclined with an
angle σ with respect to the dome base surface plane;
τ,α,ρ : the transmittance, absorptance and reflectance of a planar surface, respectively;
θ : the incidence angle on the elementary surface ds.
Figure 2 shows the coordinates of the elementary surface ds.
The incidence angle on the elementary surface (θ) is given by:
cosθ = sinξ cosθ + cosξ sinθ sinϕ′
z z
(5)
where:
ξ : the elevation angle of the point P with respect to the dome base surface plane;
ϕ′ : the relative azimuth angle of the point P;
θz : the sun zenith angle
The sun zenith angle θz and the elevation angle ξ are expressed as:
cosθ z = cosLcosδ cosω + sinLsinδ (6)
sinξ = sinϕ sinσ (7)
with:
L : the site latitude angle;
δ : the sun declination angle;
ω : the hour angle;
σ : the inclination angle of the plane of the point P with respect to the dome base surface
plane;
ϕ : the equivalent angle to ϕ′ in the inclined plane of the point P, given by:
cosϕ = cosξ cosϕ′ (8)
Diffuse Radiation Transmission Process Through a domed Surface. The total diffuse irradiance inc ident
on a domed surface is expressed as follows:
∫
=
Adome
I
d dome I
td ds , ,
(9)
The transmitted, absorbed and reflected diffuse irradiances are calculated as follows (Laouadi and Atif
1998):
−
=
11
12
, ,
1 F
F
IT I
d
d
d dome d dome
ρ
τ
(10)
−
= +
11
11
, ,
1 F
F
IA I
d
d d
d dome d dome d
ρ
τ α
α (11)
−
= +
11
2
11
, ,
1 F
F
IR I
d
d
d dome d dome d
ρ
τ
ρ (12)
where:
Adome : the dome surface area;
5
F11 : the view factor of the dome interior surface to itself;
F12 : the view factor of the dome interior surface to its base surface;
Id,t : the total diffuse sky and ground-reflected radiation on an inclined surface (W/m2
);
τd,αd,ρd : the transmittance, absorptance and reflectance of a planar surface for diffuse radiation;
The view factors F11 and F12 are expressed as:
F11 = -1 F12 ; and F12 = Ah
/Adome = (1+ sinσ 0
)/2 (13)
with Ah the area of the dome base surface.
By definition, the dome transmittance is the ratio of the transmitted irradiance to the incident irradiance.
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,
b dome
b dome
dome
b dome
b dome
dome
b dome
b dome
dome I
IR
I
IA
I
IT
,
,
,
,
,
,
τ = ; α = ; ρ = (14)
For diffuse radiation,
11
2
11
,
11
11
,
11
12
,
1
;
1
;
1 F
F
F
F
F
F
d
d
d dome d
d
d d
d dome d
d
d
d dome
ρ
τ
ρ ρ
ρ
τ α
α α
ρ
τ
τ
−
= +
−
= +
−
= (15)
Solar Heat Gain Coefficient And U-Value
By definition, the solar heat gain coefficient (SHGC) is the fraction of the incident irradiance that enters
the glazing and becomes heat gain. It includes both the directly-transmitted portion and the fraction of the
absorbed portion, which is redirected to the indoor space by convection and radiation. The SHGC for
diffuse radiation is calculated assuming a uniform radiance distribution (ASHRAE 1997). The U-value of
the dome structure is calculated based on one-dimensional heat flow. These parameters are expressed
as follows:
For beam radiation,
SHGCdome dome Ni α dome = τ + (16)
For diffuse radiation,
SHGCd,dome d,dome Ni α d,dome = τ + (17)
/1 /1
1
0 i c
dome h h R
U
+ +
= (18)
where:
h0 : the outdoor combined convection and radiation coefficient;
hi
: the indoor combined convection and radiation coefficient;
Rc : the glazing conduction resistance.
Ni
: the fraction of the absorbed irradiance that is inwardly redirected to the indoor space
under the dome, and is expressed as follows (ASHRAE 1997):
0 Ni = Udome / h (19)
Computing the optical and thermal properties of a dome is not straightforward before the evaluation of the
double integral in Equations (2), (3), (4) and (9) for each time of the day. An alternative approach is to
compute the optical and thermal properties of a planar surface that is optically and thermally equivalent to
a domed surface. This is particularly important to building energy-simulation and fenestration-rating
6
computer programs. This approach has the following advantages:
1. It eliminates the need for the input of complex geometrical data of domed surfaces. The user
only needs to input the optical properties of the dome-equivalent planar surface.
2. It allows daylighting and thermal performance to be compared as between domed and planar
surfaces.
3. It can treat inclined domed surfaces by simply reducing them to inclined dome-equivalent planar
surfaces.
4. It facilitates the prediction of the optical properties of domed surfaces on the basis of
measurements conducted for the dome-equivalent planar surface. Horizontal illuminance
measurement inside and outside the dome may be used to measure the equivalent
transmittance.
OPTICAL AND THERMAL CHARACTERISTICS OF A DOME-EQUIVALENT PLANAR SURFACE
A simple method is proposed to calculate the optical and thermal characteristics of a planar surface that
is optically and thermally 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 similar amount of transmitted, absorbed and reflected irradiances, and similar amount of
thermal heat losses/gains as the domed surface.
The optical and thermal characteristics of the dome-equivalent planar surface are calculated as follows:
Equivalent Optical Properties
The equivalent transmittance (τeq), absoptance (αeq) and reflectance (ρeq) are expressed as:
For beam radiation,
eq dome dome dome τ = τ ⋅ α = α ⋅ ρ = ρ ⋅
eq eq ; ; (20)
For diffuse radiation,
d
d
d
d eqd d
d
d d
d eqd d
d
d
eqd
F
F
F
F
F
F
ε
ρ
τ
ε ρ ρ
ρ
τ α
ε α α
ρ
τ
τ ⋅
−
⋅ = +
−
⋅ = +
−
=
11
2
11
,
11
11
,
11
12
,
1
( )
;
1
;
1
(21)
Conservation of solar radiative heat flux on the dome surface yields the following relationship:
τ +α + ρ = ε eq eq eq (22)
where ε is the ratio of the incident irradiance on the domed surface to that incident on the domeequivalent
planar surface. εd is for diffuse radiation. For horizontal domed surfaces, ε and εd are given
by:
;
1
;
cos
1 cos ,
21
∫ ∫
= =
+ Adome d
td
h
d
h A A z
ds
I
I
A
ds
A
ε
θ
θ
ε (23)
Equation (23) for beam radiation can be expressed as:
{ } Fs Fc θ z
π σ
ε tan
cos
1
4
3
0
2
= + + (24)
with:
7
( -/2 ( ))sin sin cos ( )
cos sin
(1 sin )sin 2sin
cos sin
2
1
( ( -) /2)cos
0 2 2 0 0 2
2
2
0
2
0
2
2
2
0
2
1-
0
2
0 2 2
π ϕ σ σ σ ϕ σ
σ σ
σ σ σ
ϕ σ π σ σ
= −
+ −
= −
c
s
F
F
(25)
where σ2 is the angle that delimits the surface A2 , and ϕ0 is given by:
( ) sin (sin / sin )
0 2
1 ϕ 0 σ 2 σ σ
−
= (26)
For isotopic diffuse overcast skies, Equation (23) for diffuse radiation reduces to:
{ } 12
12
1 1( )
2/1
F
F
d ρ g ρ g
ε = + + − (27)
where ρg is the ground reflectance (albedo).
Equivalent Solar Heat Gain Coefficient and U-Value
SHGC and U-value for the dome-equivalent planar surface are given as follows:
For beam radiation,
SHGCeq eq Ni α eq = τ + (28)
For diffuse radiation,
SHGC ,eqd ,eqd Ni α ,eqd = τ + (29)
12 Ueq = Udome /F (30)
The daily solar heat gain of a horizontal domed surface (or, the dome-equivalent horizontal planar
surface) is calculated as follows:
∫
= ⋅ + ⋅
day
SHGdome {SHGCeq I
b
cos z SHGC eqd
I
d
cos d
}Ahdt θ , θ (31)
Similarly, the daily solar heat gain of a horizontal planar surface, having the same surface area as the
dome base surface, is given by:
∫
= ⋅ + ⋅
day
SHGh
{SHGCh
I
b
cos z SHGC hd
I
d
cos d
}Ahdt θ , θ (32)
where Id the diffuse radiation intensity on a horizontal surface, and θd the incidence angle for diffuse
radiation.
The daily solar heat gain of horizontal domed surfaces is an important parameter to compare the thermal
performance of different dome shapes. Likewise, the ratio of the daily solar heat gain of a horizontal
domed surface to that of a horizontal planar surface - SHGdome/SHGh - is an important parameter to
compare the monthly/seasonal solar heat gains of horizontal domed surfaces with those of horizontal
planar surfaces.
RESULTS AND DISCUSSION
The models developed above are applied to predict the optical and thermal performance of a singleglazed
hemispherical domed skylight. A comparison is made between the optical and thermal
characteristics of a dome-equivalent planar skylight and those of a horizontal planar skylight. A clear float
8
glass is used as the dome glazing. The solar transmittance and absorptance of the clear float glass at
normal incidence angle are τ = 0.78 and α = 0.15 (Pilkington 1988). 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 by ASHRAE (1997). The U-value of the skylight is 5.84
W/m2
K in summer and 6.19 W/m2
K in winter, based on the ASHRAE design conditions (ASHRAE 1997).
The double integral in Equations (2), (3), (4) and (9) is evaluated using Simpson’s rule for numerical
integration.
Figure 3 shows the profile of the equivalent solar transmittance of a single-glazed dome as a function of
the sun zenith angle for a number of dome shapes. Contrary to planar surfaces, the equivalent solar
transmittance of single-glazed domes increases with the increase in the sun zenith angle, especially for
domes with truncation angles up to 45°. The equivalent solar transmittance of single-glazed domes with
truncation angles σ0 < 30° is slightly lower at near normal zenith angles (θz < 40°), and significantly higher
at high zenith angles and around the horizon than that of single-glazed planar surfaces. The equivalent
solar transmittance of single-glazed domes with truncation angles greater than 30° is approximately the
same as that of single-glazed planar surfaces for zenith angles up to 55°, and much higher at high zenith
angles and around the horizon. The equivalent solar transmittance increases slightly with the dome
truncation angle at near normal zenith angles. However, at high zenith angles and around the horizon, the
equivalent solar transmittance decreases with the increase of the dome truncation angle. Single-glazed
fully hemispherical domes have the lowest equivalent solar transmittance at near normal zenith angles
(about 10% lower than that of single-glazed planar surfaces) and the highest equivalent solar
transmittance at high zenith angles and around the horizon. As a result, depending on the site latitude
and the day of the year, single-glazed domes may yield lower or higher equivalent solar transmittance
than that of single-glazed planar surfaces. For tropical regions (latitude lower than 24°), single-glazed
domes yield lower equivalent solar transmittance at noontime than that of single-glazed planar surfaces in
both winter and summer season. For higher latitudes, single-glazed domes yield lower equivalent solar
transmittance at noontime in summer, and higher equivalent solar transmittance in winter than that of
single-glazed planar surfaces. This is a very important feature of domed surfaces, particularly for
daylighting in summer and winter season.
Figure 4 shows the profile of the equivalent solar heat gain coefficient (SHGC) of a single-glazed dome
for beam radiation as a function of the sun zenith angle for a number of dome shapes. The equivalent
SHGC is calculated based on summer design conditions (Ni=0.2573). The equivalent SHGC follows the
same trend as the equivalent solar transmittance (Figure 3). At near normal zenith angles, the equivalent
SHGC is slightly higher than the equivalent solar transmittance. However, at high zenith angles and
around the horizon, the equivalent SHGC is much higher than the equivalent solar transmittance. This is
because the equivalent solar absorptance of single-glazed domes is much higher at high zenith angles
and around the horizon than that at near normal zenith angles (θz < 40°). At near normal zenith angles,
the equivalent SHGC of single-glazed domes is slightly lower than that of single-glazed planar surfaces.
However, at high zenith angles and around the horizon, the equivalent SHGC of single-glazed domes is
substantially higher than that of single-glazed planar surfaces. The equivalent SHGC increases slightly at
near normal zenith angles, and decreases significantly at high zenith angles and around the horizon with
the increase of the dome truncation angle. Fully hemispherical single-glazed domes have the lowest
equivalent SHGC at near normal zenith angle (about 9% lower than that of single-glazed planar surfaces)
and the highest equivalent SHGC at high zenith angles and around the horizon. For tropical regions,
single-glazed domes yield lower equivalent SHGC at noontime than that of single-glazed planar surfaces
in both winter and summer season. For high latitude countries, single-glazed domes yield lower
equivalent SHGC at noontime in summer, and higher equivalent SHGC in winter than that of singleglazed
planar surfaces.
Figure 5 shows the profiles of the equivalent solar transmittance, SHGC and U-value of a single-glazed
dome as a function of the dome shape under isotropic diffuse skies. The equivalent SHGC and U-value
are calculated based on summer design conditions (Ni
= 0.2573). Single-glazed domes under isotopic
diffuse skies have lower equivalent solar transmittance and SHGC than those of single-glazed planar
surfaces. Fully hemispherical single-glazed domes have the lowest equivalent solar transmittance and
9
SHGC. The equivalent solar transmittance of a fully hemispherical single-glazed dome is 14% lower than
that of a single-glazed planar surface while the equivalent SHGC is 7% lower than that of a single-glazed
planar surface. However, the equivalent U-value of single-glazed domes is significantly higher than that
of single-glazed planar surfaces. Fully hemispherical single-glazed domes have the highest equivalent Uvalue,
due to the fact that fully hemispherical domes have the largest surface area. As a result, under
isotropic skies, single-glazed domes are susceptible to lower solar heat gains and higher thermal heat
losses/gains than those of single-glazed planar surfaces.
Figure 6 shows the profile of the ratio of the daily solar heat gain of a single-glazed dome to that of a
horizontal single-glazed planar surface as a function of the dome shape for the 21st of December and
June. Several site latitudes with zero longitude difference are covered. The beam (Ib) and horizontal sky
diffuse (Id) radiation are calculated using the ASHRAE standard method for solar radiation calculation
(ASHRAE 1997). The ratio of beam-to-diffuse radiation is 0.134 and 0.057 for December and June,
respectively. Perez et al. model (Duffie and Beckman 1991) is used to estimate the total diffuse radiation
on a sloped surface for the non-isotropic sky diffuse radiation.
The Figure shows that single-glazed domes result in larger daily solar heat gains than those of horizontal
single-glazed planar surfaces during winter and summer season. Fully hemispherical single-glazed
domes have the largest solar heat gains. In summer, the solar heat gain ratio for fully hemispherical
single-glazed domes is between 1.03 and 1.09 for latitudes varying between 0° and 55° (north/south).
However, in winter, the solar heat gain ratio increases significantly with the site latitude. In tropical
countries (latitudes lower than 24°), the solar heat gain ratio can reach 1.19. In subtropical countries
(latitudes between 24° and 35°), the solar heat gain ratio can reach 1.33. In midlatitude countries
(latitudes between 35° and 55°), the solar heat gain ratio can reach 2.32.
CONCLUSIONS
The optical and thermal characteristics of a clear single-glazed hemispherical domed skylight are
evaluated under sun and sky light. Since energy computer programs deal with planar surfaces, a simple
method was proposed to replace single-glazed hemispherical domed skylights by optically and thermally
equivalent single-glazed planar skylights. Single-glazed hemispherical domed skylights have the
following important features:
1. Under sunlight (direct beam), single-glazed domed skylights have low equivalent solar
transmittance and SHGC at near normal zenith angles (noontime) and substantially high
equivalent solar transmittance and SHGC at high zenith angles and around the horizon. Nearly
fully-hemispherical single-glazed domed skylights (σ0 < 30°) have slightly lower equivalent solar
transmittance and SHGC at near normal zenith angles, and significantly higher equivalent solar
transmittance and SHGC at high zenith angles and around the horizon than those of singleglazed
planar skylights. Single-glazed domed skylights with truncation angles greater than 30°
yield approximately the same equivalent solar transmittance and SHGC as single-glazed planar
skylights for zenith angles up to 55°, and yield much higher equivalent solar transmittance and
SHGC at high zenith angles and around the horizon.
2. Under isotropic diffuse overcast skies, single-glazed domed skylights have slightly lower
equivalent solar transmittance and SHGC than single-glazed planar skylights. Fully
hemispherical single-glazed domed skylights have the lowest equivalent solar transmittance
(about 14% lower than that of single-glazed planar skylights) and SHGC (about 7% lower than
that of single-glazed planar skylights).
3. Under combined sunlight and non-isotropic sky light, single-glazed domed skylights yield higher
daily solar heat gains than those of single-glazed horizontal planar skylights. Fully hemispherical
domed skylights yield the largest daily solar heat gains in both winter and summer season. In
summer, the solar heat gain of fully hemispherical single-glazed domed skylights is 3% to 9%
higher than that of horizontal single-glazed planar skylights for latitudes varying between 0° and
55° (north/south). In winter, however, the solar heat gain of single-glazed domed skylights
increases significantly with the site latitude. In tropical countries (latitudes lower than 24°), the
10
solar heat gain of fully hemispherical single-glazed domed skylights can reach 19% higher than
that of horizontal single-glazed planar skylights. In subtropical countries (latitudes between 24°
and 35°), the solar heat gain of fully hemispherical single-glazed domed skylights can reach 33%
higher than that of horizontal single-glazed planar skylights. In midlatitude countries (latitudes
between 35° and 55°), the solar heat gain of fully hemispherical single-glazed domed skylights
can reach 232% higher than that of horizontal single-glazed planar skylights.
4. Single-glazed domed skylights have higher equivalent U-value than that of single-glazed planar
skylights. Fully hemispherical single-glazed domed skylights have the largest equivalent U-value
(twice higher than that of single-glazed planar skylights). Single-glazed domed skylights are,
thus, susceptible to higher thermal heat gains/losses than those of horizontal single-glazed planar
skylights. Therefore, the shape of the skylight dome should be chosen according to the site
latitude and the prevailing climate to compromise solar heat gains with thermal losses.
NOMENCLATURE
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.
Fc, Fs : functions, Equation (25).
F11 : view factor of the dome interior surface to itself.
F12 : view factor of the dome interior surface to its base surface.
hi
: combined indoor radiation and convection coefficient.
h0 : combined outdoor radiation and convection coefficient.
I : incident irradiance on a surface (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
).
IT : transmitted irradiance (W).
IA : absorbed irradiance (W).
IR : reflected irradiance (W).
L : site latitude.
R : dome radius.
Rc : glazing conductive resistance (m2
K/W).
SHG : daily solar heat gain (J).
SHGC : solar heat gain coefficient.
t : time.
Udome : U-value of the dome structure (W/m2
K).
Ueq : U-value of the dome-equivalent planar surface (W/m2
K).
Geek Symbols
α, ρ, τ : absorptance, reflectance and transmittance of a flat glazing, respectively.
δ : sun declination angle.
ε : ratio of dome irradiance to planar surface irradiance, Equation (23)
ϕ : equivalent angle to ϕ′ in the plane of the point P.
ϕ′ : the relative azimuth angle of the elementary surface (ds).
θ : incidence angle on the elementary surface (ds).
θd : incidence angle for diffuse radiation.
θz : sun zenith angle.
ρg : ground reflectance (albedo).
σ : inclination angle of a plane perpendicular to the plane of the sun’s rays.
σ0 : dome truncation angle.
σ1, σ2 : angles that delimit the surfaces A1 (σ0 ≤ σ ≤ σ1) and A2 (σ1 ≤ σ ≤ σ2), respectively.
ω : hour angle.
ξ : elevation angle of the point P on the dome surface with respect to the dome base surface plane.
11
Subscripts
b : beam radiation.
d : diffuse radiation.
dome : dome surface.
eq : dome-equivalent planar surface.
h : horizontal surface.
I : indoor.
0 : outdoor.
REFERENCES
AAMA . 1987. Skylight Handbook, Design Guidelines. Des Plaines: American Architectural
Manufacturers Association.
AAMA . 1981. Design for Energy Conservation with Skylights. Chicago: Architectural Aluminum
Manufacturers Association.
ASHRAE. 1997. ASHRAE Handbook – 1997 Fundamentals. Atlanta: American Society of Heating,
Refrigerating, and Air-Conditioning Engineers, Inc.
Atif R. M., Galasiu A., Macdonald R. A., And Laouadi A. 1997. “On-Site Monitoring Of An Atrium Skylight
Transmittance: Performance And Validity Of The IES Transmittance Calculation Procedure For
Daylighting”. IESNA Transactions.
CANMET. 1995. VISION 4 Reference Manual. Ottawa: Canadian Centre for Mineral and Energy
Technology.
Clarke J. A. 1985. Energy Simulation In Building Design. Bristol: Adam Hilger Ltd.
Duffie J. A. And Beckman W. A. 1991. Solar Engineering Of Thermal Processes. New York: John
Wiley&Sons, Inc.
Enermodal. 1994. “Thermal Performance Of Complex Fenestration Systems: Skylights, Greenhouse
Windows, And Curtainwalls”. Report 23440-92-9615. Ottawa: Natural Resources Canada.
ESRU. 1996. ESP-r User Manual. Glasgow: Energy Systems Research Unit, University Of Strathclyde.
IESNA. 1993. Lighting Handbook, Reference And Application Volume. New York: Illuminating
Engineering Society Of North America.
Jensen T. 1983. Skylights. Pennsylvania: Running Press.
Laouadi A., and Atif M.R. 1998. “Transparent Domed Skylights: Optical Model For Predicting
Transmittance, Absorptance And Reflectance.” International Journal of Lighting Research and
Technology, in-press.
LBL. 1992. WINDOWS 4.0 User Manual. Berkeley: Lawrence Berkeley Laboratory.
Navvab M. 1990. “Outdoors Indoors, Daylighting within Atrium Spaces”. LD+A. Vol. 20, No. 5, pp. 6-31.
Pilkington. 1988. Glass And Transmission Properties Of Windows, 7th
edition. England: Harwills.
Treado S., Gillette G., and Kusuda T. 1983. “Evaluation of the Daylighting and Energy Performance of
Windows, Skylights, and Clerestories”. Report NBSIR 83-2726. U.S. Department of Commerce.
Wilkinson M. A. 1992. “Natural Lighting Under Translucent Domes”. Lighting Research Technology, Vol.
24, No. 3, pp. 117-126.
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.
12
FIGURE CAPTIONS
Figure 1 Beam radiation transmission process through a horizontal domed surface.
Figure 2 Coordinates of the elementary surface (ds) on the dome.
Figure 3 Profile of the equivalent solar transmittance of a single-glazed dome as a function of the sun
zenith angle.
Figure 4 Profile of the equivalent solar heat gain coefficient (SHGC) of a single-glazed dome as a
function of the sun zenith angle.
Figure 5 Profiles of the diffuse equivalent solar transmittance τd,eq, SHGCd,eq and Ueq of a single-glazed
dome as a function of the dome shape under isotropic diffuse skies.
Figure 6 Profile of the ratio of the daily solar heat gain of a single-glazed dome to that of a horizontal
single-glazed planar surface as a function of the dome shape.
13
x
y
z
ξ
ϕ´
σ0
σ
σ1
σ2
P
θz
A1
A2
ϕ0
´
ϕ
R
Sun’s rays
Figure 1
14
ϕ
ϕ´
ξ
σ
dσ
dϕ´
x
y
z
dϕ
ds = R2sinϕ dϕ dσ
Figure 2
15
0
0.5
1
1.5
0 10 20 30 40 50 60 70 80
Sun zenith angle θz (deg.)
Equivalent solar transmittance
τeq
σ0
= 0o σ0
= 30o
σ0
= 45o σ0
= 60o
σ0
= 75o
σ0
= 90o
Figure 3
16
0
0.5
1
1.5
0 10 20 30 40 50 60 70 80
Sun zenith angle θz (deg.)
SHGCeq
σ0
= 0o σ0
= 30o
σ0
= 45o σ0
= 60o
σ0
= 75o
σ0
= 90o
Figure 4
17
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 50 60 70 80 90
Dome truncation angle, σ0
(deg.)
Diffuse transmittance, or SHGC
5
6
7
8
9
10
11
12
13
U-Value
Ueq
SHGCd, eq
τd, eq
Isotropic diffuse sky
Figure 5
18
Dec. 21st
0.9
1.4
1.9
2.4
0 10 20 30 40 50 60 70 80 90
Dome truncation angle, σ0
(deg.)
Solar heat gain ratio (SHGdome/SHGh)
L=0
L=24
L=35
L=45
L=50
L=55
L=24
L=55
June 21st
Figure 6
