Daylight Factor Calculation[LINK]
The daylighting factor calculation using the SplitFlux daylighting method is described below.
Variables in Daylighting Calculations
Eh,sky 
Exterior horizontal illuminance due to light from the sky 
lux 
GILSK 
Eh,sun 
Exterior horizontal illuminance due to light from the sun 
lux 
GILSU 
dsky, dsun

Interior illuminance factor due to sky, sun related light 
— 
DFACSK,DFACSU 
wsky, wsun

Window luminance factor due to sky, sun related light 
cd/lm 
SFACSK, SFACSU 
bsky, bsun

Window background luminance factor due to sky, sun related light 
cd/lm 
BFACSK, BFACSU 
N 
Number of exterior windows in a zone 
— 
NWD 
θsky, φsky

Azimuth and altitude angles of a point in the sky 
radians 
THSKY, PHSKY 
Ψcs 
Clear sky luminance distribution 
cd/m2 
— 
Ψts 
Clear turbid sky luminance distribution 
cd/m2 
— 
Ψis 
Intermediate sky luminance distribution 
cd/m2 
— 
Ψos 
Overcast sky luminance distribution 
cd/m2 
— 
φsun 
Altitude angle of the sun 
radians or degrees 
PHSUN 
γ 
Angle between point in the sky and the sun; or angle between vertical and ray from reference point to window element 
radians 
G 
Lz 
Sky zenith luminance 
cd/m2 
ZENL 
m 
Optical air mass of the atmosphere 
m 
AM 
h 
Building altitude 
m 
ELEVATION 
Eh,k 
Exterior horizontal illuminance for sky type k 
lux 
— 
Nθ, Nφ

Number of azimuth, altitude steps for sky integration 
— 
NTH, NPH 
→Rref 
Vector from zone origin to reference point 
m 
RREF 
→Rwin 
Vector from zone origin to window element 
m 
RWIN 
dΩ 
Solid angle subtended by window element 
steradians 
DOMEGA 
Lw 
Luminance of a window element as seen from reference point 
cd/m2 
WLUMSK, WLUMSU 
Lw,shade 
Luminance of window element with shade in place 
cd/m2 
WLUMSK, WLUMSU 
dEh 
Horizontal illuminance at reference point from window element 
lux 
— 
dx, dy

Size of window element 
m 
DWX, DWY 
D 
Distance from reference point to window element 
m 
DIS 
B 
Angle between window element’s outward normal and ray from reference point to window element 
radians 
— 
^Rray 
Unit vector from reference point to window element 
— 
RAY 
^Wn 
Unit vector normal to window element, pointing away from zone 
— 
WNORM 
^W21 
Unit vector along window yaxis 
— 
W21 
^W23 
Unit vector along window xaxis 
— 
W23 
τvis 
Glass visible transmittance 
— 
TVISB 
L 
Luminance of sky or obstruction 
cd/m2 
ELUM, — 
ϕFW 
Downgoing luminous flux from a window 
lm 
FLFW — 
ϕCW 
Upgoing luminous flux from a window 
lm 
FLCW — 
F1 
Firstreflected flux 
lm 
 
ρFW 
Areaweighted reflectance of floor and upper part of walls 
— 
SurfaceWindow%RhoFloorWall 
ρCW 
Areaweighted reflectance of ceiling and upper part of walls 
— 
SurfaceWindow%RhoCeilingWall 
Er 
Average internallyreflected illuminance 
lux 
EINTSK, EINTSU 
A 
Total inside surface area of a zone 
m2 
ATOT 
ρ 
Areaweighted average reflectance of zone interior surfaces 
— 
ZoneDaylight%AveVisDiffREflect 
θ, φ

Azimuth and altitude angle of a sky or ground element 
radians 
TH, PH 
L(θ,φ) 
Luminance of sky or ground element at (θ,φ)

cd/m2 
HitPointLum– 
Aw 
Area of glazed part of window 
m2 
Surface%Area 
β 
Angle of incidence, at center of window, of light from a sky or ground element 
radians 
— 
T(β) 
Glazing visible transmittance at incidence angle β

— 
TVISBR 
dϕinc 
Luminous flux incident on window from sky or ground element 
lm 
— 
dϕ 
Luminous flux from sky or ground element transmitted through window 
lm 
— 
dϕFW, dϕCW

Luminous flux from sky or ground element transmitted through window and going downward, upward 
lm 
 
θmin, θmax

Azimuth angle integration limits 
radians 
THMIN, THMAX 
φw 
Window normal altitude angle 
radians 
 
ϕsh, ϕunsh

Transmitted flux through window and shade, without shade 
lm 
— 
ϕCW,sh, ϕFW,sh

Upgoing and downgoing portions of transmitted flux through window with shade 
lm 
— 
ϕCW,unsh, ϕFW,unsh

Upgoing and downgoing portions of transmitted flux through window without shade 
lm 
— 
f 
Fraction of hemisphere seen by the inside of window that lies above the window midplane 
— 
SurfaceWindow%Fraction UpGoing 
ϕinc 
Flux incident on glazing from direct sun 
lm 
— 
fsunlit 
Fraction of glazing that is sunlit 
— 
SunLitFrac 
ϕ 
Transmitted flux from direct sun 
— 
— 
Lsh 
Luminance of window with shade 
cd/m2 
— 
Lb 
Window background luminance 
cd/m2 
BLUM 
G 
Discomfort glare constant 
— 
GTOT 
Gi 
Discomfort glare constant from window i

— 
— 
ω 
Solid angle subtended by window with respect to reference point 
steradians 
SolidAngAtRefPt 
Ω 
Solid angle subtended by window with respect to reference point, modified to take direction of occupant view into account 
steradians 
SolidAngAtRefPtWtd 
Nx, Ny

Number of elements in x and y direction that window is divided into for glare calculation 
— 
NWX, NWY 
p(xR,yR) 
Position factor for horizontal and vertical displacement ratios xR and yR

— 
DayltgGlarePositionFactor 
pH 
Hopkinson position factor 
— 
DayltgGlarePositionFactor 
Lb 
Window background luminance 
cd/m2 
BLUM 
Eb 
Illuminance on window background 
lm 
— 
Er 
Total internallyreflected component of daylight illuminance 
lm 
— 
Es 
Illuminance setpoint 
lm 
IllumSetPoint 
GI 
Glare Index 
— 
GLINDX 
There are three types of daylight factors: interior illuminance factors, window luminance factors, and window background luminance factors. To calculate these factors the following steps are carried out for each hourly sun position on the sun paths for the design days and for representative days during the simulation run period:
1. Calculate exterior horizontal daylight illuminance from sky and sun for standard (CIE) clear and overcast skies.
2. Calculate interior illuminance, window luminance and window background luminance for each window/referencepoint combination, for bare and for shaded window conditions (if a shading device has been specified), for overcast sky and for standard clear sky.
3. Divide by exterior horizontal illuminance to obtain daylight factors.
Interior Illuminance Components[LINK]
To calculate daylight factors, daylight incident on a window is separated into two components: (1) light that originates from the sky and reaches the window directly or by reflection from exterior surfaces; and (2) light that originates from the sun and reaches the window directly or by reflection from exterior surfaces. Light from the window reaches the workplane directly or via reflection from the interior surfaces of the room.
For fixed sun position, sky condition (clear or overcast) and room geometry, the skyrelated interior daylight will be proportional to the exterior horizontal illuminance, Eh,sky, due to light from the sky. Similarly, the sunrelated interior daylight will be proportional to the exterior horizontal solar illuminance, Eh,sun.
Daylight Factors[LINK]
The following daylight factors are calculated:
dsky=Illuminance at reference point due to sky−related lightEh,sky
dsun=Illuminance at reference point due to sun−related lightEh,sun
wsky=Average window luminance due to sky−related lightEh,sky
wsun=Average window luminance due to sun−related lightEh,sun
bsky=Window background luminance due to sky−related lightEh,sky
bsun=Window background luminance due to sun−related lightEh,sun
For a daylit zone with N windows these six daylight factors are calculated for each of the following combinations of reference point, window, skycondition/sunposition and shading device:
[Ref pt 1Ref pt 2]⎡⎢
⎢
⎢⎣Window 1Window 2...Window N⎤⎥
⎥
⎥⎦⎡⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣Clear sky, first sun−up hourClear/turbid sky, first sun−up hourIntermediate sky, first sun−up hourOvercast sky, first sun−up hour...Clear sky, last sun−up hourClear/turbid sky, last sun−up hourIntermediate sky, last sun−up hourOvercase sky, last sun−up hour⎤⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦⎡⎢⎣Unshaded windowShaded window(if shade assigned)⎤⎥⎦
Sky Luminance Distributions[LINK]
The luminance distribution of the sky is represented as a superposition of four standard CIE skies using the approach described in (Perez et al. 1990). The standard skies are as follows.
The clear sky luminance distribution has the form (Kittler, 1965; CIE, 1973)
ψcs(θsky,ϕsky)=Lz(0.91+10e−3γ+0.45cos2γ)(1−e−0.32cosecϕsky)0.27385(0.91+10e−3(π2−ϕsun)+0.45sin2ϕsun)
Here, Lz is the zenith luminance (i.e., the luminance of the sky at a point directly overhead). In the calculation of daylight factors, which are ratios of interior and exterior illumination quantities that are both proportional to Lz, the zenith luminance cancels out. For this reason we will use Lz = 1.0 for all sky luminance distributions.
The various angles, which are defined in the building coordinate system, are shown in Figure 1. The angle, γ, between sun and sky element is given by:
γ=cos−1[sinϕskysinϕsun+cosϕskycosϕsuncos(θsky−θsun)]
The general characteristics of the clearsky luminance distribution are a large peak near the sun; a minimum at a point on the other side of the zenith from the sun, in the vertical plane containing the sun; and an increase in luminance as the horizon is approached.
Clear Turbid Sky[LINK]
The clear turbid sky luminance distribution has the form [Matsuura, 1987]:
ψts(θsky,ϕsky)=Lz(0.856+16e−3γ+0.3cos2γ)(1−e−0.32cosecϕsky)0.27385(0.856+10e−3(π2−ϕsun)+0.3sin2ϕsun)
The intermediate sky luminance distribution has the form [Matsuura, 1987]:
ψis(θsky,ϕsky)=LzZ1Z2/(Z3Z4)
where
Z1=[1.35(sin(3.59ϕsky−0.009)+2.31)sin(2.6ϕsun+0.316)+ϕsky+4.799]/2.326
Z2=exp[−0.563γ{(ϕsun−0.008)(ϕsky+1.059)+0.812}]
Z3=0.99224sin(2.6ϕsun+0.316)+2.73852
Z4=exp[−0.563(π2−ϕsun){2.6298(ϕsun−0.008)+0.812}]
The overcast sky luminance distribution has the form [Moon & Spencer, 1942]:
ψos(ϕsky)=Lz1+2sinϕsky3
Unlike the clear sky case, the overcast sky distribution does not depend on the solar azimuth or the sky azimuth. Note that at fixed solar altitude the zenith (ϕsky=π/2 ) is three times brighter than the horizon (ϕsky=0 ).
Direct Normal Solar Illuminance[LINK]
For purposes of calculating daylight factors associated with beam solar illuminance, the direct normal solar illuminance is taken to be 1.0 W/m2. The actual direct normal solar illuminance, determined from direct normal solar irradiance from the weather file and empiricallydetermined luminious efficacy, is used in the timestep calculation.
Exterior Horizontal Illuminance[LINK]
The illuminance on an unobstructed horizontal plane due to diffuse radiation from the sky is calculated for each of the four sky types by integrating over the appropriate sky luminance distribution:
Eh,k=2π∫0π/2∫0ψk(θsky,ϕsky)sinϕskycosϕskydθskydϕsky
where k = cs, ts, is or os. The integral is evaluated as a double summation:
Eh,k=Nθ∑i=1Nϕ∑j=1ψk(θsky(i),ϕsky(j))sinϕsky(j)cosϕsky(j)ΔθskyΔϕsky
where
θsky(i)=(i−1/2)Δθskyϕsky(j)=(j−1/2)ΔϕskyΔθsky=2π/NθΔϕsky=π/2Nϕ
Nθ=18 and Nϕ=8 were found to give a ±1% accuracy in the calculation of Eh,k.
Direct Component of Interior Daylight Illuminance[LINK]
The direct daylight illuminance at a reference point from a particular window is determined by dividing the window into an xy grid and finding the flux reaching the reference point from each grid element. The geometry involved is shown in Figure 2. The horizontal illuminance at the reference point, →Rref , due to a window element is:
dEh=LwdΩcosγ
where Lw is the luminance of the window element as seen from the reference point.
The subtended solid angle is approximated by:
dΩ=dxdyD2cosB
where
D=∣∣¯Rwin−¯Rref∣∣
CosB is found from:
cosB=^Rray⋅^Wn
where
→Rray=(→Rwin−→Rref)/D
^Wn=window outward normal=^W21×^W23=→W1−→W2∣∣→W1−→W2∣∣×→W3−→W2∣∣→W3−→W2∣∣
Equation [eq:SubtendedSolidAngle] becomes exact as dx/D and dy/D→0 and is accurate to better than about 1% for dx≤D/4 and dy≤D/4 .
The net illuminance from the window is obtained by summing the contributions from all the window elements:
Eh=∑windowelementsLwdΩcosγ
In performing the summation, window elements that lie below the workplane (cosγ<0 ) are omitted since light from these elements cannot reach the workplane directly.
Unshaded Window[LINK]
For the unshaded window case, the luminance of the window element is found by projecting the ray from reference point to window element and determining whether it intersects the sky or an exterior obstruction such as an overhang. If L is the corresponding luminance of the sky or obstruction, the window luminance is:
Lw=LτviscosB
where τvis is the visible transmittance of the glass for incidence angle B.
Exterior obstructions are generally opaque (like fins, overhangs, neighboring buildings, and the building’s own wall and roof surfaces) but can be transmitting (like a tree or translucent awning). Exterior obstructions are assumed to be nonreflecting. If Lsky is the sky luminance and τobs is the transmittance of the obstruction (assumed independent of incidence angle), then L=Lskyτobs. Interior obstructions are assumed to be opaque (τobs = 0).
Shaded Window[LINK]
For the windowplusshade case the shade is assumed to be a perfect diffuser, i.e., the luminance of the shade is independent of angle of emission of light, position on shade, and angle of incidence of solar radiation falling on the shade. Closelywoven drapery fabric and translucent roller shades are closer to being perfect diffusers than Venetian blinds or other slatted devices, which usually have nonuniform luminance characteristics.
The calculation of the window luminance with the shade in place, Lw,sh, is described in [Winkelmann, 1983]. The illuminance contribution at the reference point from a shaded window element is then given by Equation [eq:NetIlluminanceFromWindow] with Lw=Lw,sh .
InternallyReflected Component of Interior Daylight Illuminance[LINK]
Daylight reaching a reference point after reflection from interior surfaces is calculated using the splitflux method [Hopkinson et al., 1954], [Lynes, 1968]. In this method the daylight transmitted by the window is split into two parts—a downwardgoing flux, ΦFW (lumens), which falls on the floor and portions of the walls below the imaginary horizontal plane passing through the center of the window (window midplane), and an upwardgoing flux, ΦCW , that strikes the ceiling and portions of the walls above the window midplane. A fraction of these fluxes is absorbed by the room surfaces. The remainder, the firstreflected flux, F1, is approximated by
F1=ΦFWρFW+ΦCWρCW
where ρFW is the areaweighted average reflectance of the floor and those parts of the walls below the window midplane, and ρCW is the areaweighted average reflectance of the ceiling and those parts of the walls above the window midplane.
To find the final average internallyreflected illuminance, Er, on the room surfaces (which in this method is uniform throughout the room) a flux balance is used. The total reflected flux absorbed by the room surfaces (or lost through the windows) is AEr(1ρ), where A is the total inside surface area of the floor, walls, ceiling and windows in the room, and ρ is the areaweighted average reflectance of the room surfaces, including windows. From conservation of energy:
AEr(1−ρ)=F1
or
Er=ΦFWρFW+ΦCWρCWA(1−ρ)
This procedure assumes that the room behaves like an integrating sphere with perfectly diffusing interior surfaces and with no internal obstructions. It therefore works best for rooms that are close to cubical in shape, have matte surfaces (which is usually the case), and have no internal partitions. Deviations from these conditions, such as would be the case for rooms whose depth measured from the windowwall is more than three times greater than ceiling height, can lead to substantial inaccuracies in the splitflux calculation.
Transmitted Flux from Sky and Ground[LINK]
The luminous flux incident on the center of the window from a luminous element of sky or ground at angular position (θ,ϕ), of luminance L(θ,ϕ), and subtending a solid angle cosϕdθdϕ is:
dΦinc=AwL(θ,ϕ)cosβcosϕdθdϕ
The transmitted flux is:
dΦ=dΦincT(β)
where T(β) is the window transmittance for light at incidence angle β. This transmittance depends on whether or not the window has a shade.
For an unshaded window the total downgoing transmitted flux is obtained by integrating over the part of the exterior hemisphere seen by the window that lies above the window midplane. This gives:
ΦFW,unshaded=Awθmax∫θminπ/2∫0L(θ,ϕ)T(β)cosβcosϕdθdϕ
The upgoing flux is obtained similarly by integrating over the part of the exterior hemisphere that lies below the window midplane:
ΦCW,unshaded=Awθmax∫θmin0∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
where ϕw is the angle the window outward normal makes with the horizontal plane.
For a window with a diffusing shade the total transmitted flux is:
Φsh=Awθmax∫θminπ/2∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
The downgoing and upgoing portions of this flux are:
ΦFW,sh=Φ(1−f)ΦCW,sh=Φf
where f, the fraction of the hemisphere seen by the inside of the window that lies above the window midplane, is given by
f=0.5−ϕw/π
For a vertical window (ϕw=0) the up and downgoing transmitted fluxes are equal:
ΦFW,sh=ΦCW,sh=Φ/2
For a horizontal skylight (ϕw=π/2):
ΦFW,sh=Φ,ΦCW,sh=0
The limits of integration of θ in Equations [eq:PhiFWUnshaded], [eq:PhiCWUnshaded] and [eq:PhiSh] depend on ϕ . From Figure 12 of Winkelmann (1983), we have:
sinα=sin(A−π/2)=sinϕtanϕwcosϕ
which gives:
A=cos−1(tanϕtanϕw)
Thus:
θmin=−∣∣cos−1(−tanϕtanϕw)∣∣θmax=∣∣cos−1(−tanϕtanϕw)∣∣
Transmitted Flux from Direct Sun[LINK]
The flux incident on the window from direct sun is:
Φinc=AwEDNcosβfsunlit
The transmitted flux is:
Φ=T(β)Φinc
where T is the net transmittance of the window glazing (plus shade, if present).
For an unshaded window all of the transmitted flux is downward since the sun always lies above the window midplane. Therefore:
ΦFW,unsh=ΦΦCW,unsh=0
For a window with a diffusing shade:
ΦFW,sh=Φ(1−f)ΦCW,sh=Φf
Luminance of Shaded Window[LINK]
The luminance of a shaded window is determined at the same time that the transmitted flux is calculated. It is given by:
Lsh=1πθmax∫θminπ/2∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
Daylight Discomfort Glare[LINK]
The discomfort glare at a reference point due to luminance contrast between a window and the interior surfaces surrounding the window is given by [Hopkinson, 1970] and [Hopkinson, 1972]:
G=L1.6wΩ0.8Lb+0.07ω0.5Lw
where
G = discomfort glare constant
Lw = average luminance of the window as seen from the reference point
Ω = solid angle subtended by window, modified to take direction of occupant view into account
Lb = luminance of the background area surrounding the window
By dividing the window into Nx by Ny rectangular elements, as is done for calculating the direct component of interior illuminance, we have:
Lw=Ny∑j=1Nx∑i=1Lw(i,j)NxNy
where Lw(i,j) is the luminance of element (i,j) as seen from the reference point.
Similarly,
ω=Ny∑j=1Nx∑i=1dω(i,j)
where dω(i,j) is the solid angle subtended by element (i,j) with respect to the reference point.
The modified solid angle is:
Ω=Ny∑j=1Nx∑i=1dω(i,j)p(xR,yR)
where p is a “position factor” [Petherbridge & Longmore, 1954] that accounts for the decrease in visual excitation as the luminous element moves away from the line of sight. This factor depends on the horizontal and vertical displacement ratios, xR and yR (Figure 3), given by:
xR(i,j)=√A2−(YD)2RRyR(i,j)=YD/RR
where
RR=D(^Rray⋅^vview)A2=D2−(RR)2YD=Rwin(3)−Rref(3)
The factor p can be obtained from graphs given in [Petherbridge & Longmore, 1954] or it can be calculated from tabulated values of pH, the Hopkinson position factor [Hopkinson, 1966], since p=p1.25H . The values resulting from the latter approach are given in Table 2. Interpolation of this table is used in EnergyPlus to evaluate p at intermediate values of xR and yR.
Position factor for glare calculation [table:positionfactorforglarecalculation]


xR: Horizontal Displacement Factor 









0 
0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
>3.0 
yR: 
0 
1.00 
0.492 
0.226 
0.128 
0.081 
0.061 
0.057 
0 
Vertical 
0.5 
0.123 
0.119 
0.065 
0.043 
0.029 
0.026 
0.023 
0 
Displacement 
1.0 
0.019 
0.026 
0.019 
0.016 
0.014 
0.011 
0.011 
0 
Factor 
1.5 
0.008 
0.008 
0.008 
0.008 
0.008 
0.006 
0.006 
0 

2.0 
0 
0 
0.003 
0.003 
0.003 
0.003 
0.003 
0 

>2.0 
0 
0 
0 
0 
0 
0 
0 
0 
The background luminance is:
Lb=Ebρb
where ρb is approximated by the average interior surface reflectance of the entire room and
Eb=max(Er,Es)
where Er is the total internallyreflected component of daylight illuminance produced by all the windows in the room and Es is the illuminance setpoint at the reference point at which glare is being calculated. A precise calculation of Eb is not required since the glare index (see next section) is logarithmic. A factor of two variation in Eb generally produces a change of only 0.5 to 1.0 in the glare index.
The net daylight glare at a reference point due to all of the windows in a room is expressed in terms of a glare index given by
GI=10log10number ofwindows∑i=1Gi
where Gi is the glare constant at the reference point due to the ith window.
Daylight Factor Calculation[LINK]
The daylighting factor calculation using the SplitFlux daylighting method is described below.
Overview[LINK]
There are three types of daylight factors: interior illuminance factors, window luminance factors, and window background luminance factors. To calculate these factors the following steps are carried out for each hourly sun position on the sun paths for the design days and for representative days^{1} during the simulation run period:
1. Calculate exterior horizontal daylight illuminance from sky and sun for standard (CIE) clear and overcast skies.
2. Calculate interior illuminance, window luminance and window background luminance for each window/referencepoint combination, for bare and for shaded window conditions (if a shading device has been specified), for overcast sky and for standard clear sky.
3. Divide by exterior horizontal illuminance to obtain daylight factors.
Interior Illuminance Components[LINK]
To calculate daylight factors, daylight incident on a window is separated into two components: (1) light that originates from the sky and reaches the window directly or by reflection from exterior surfaces; and (2) light that originates from the sun and reaches the window directly or by reflection from exterior surfaces. Light from the window reaches the workplane directly or via reflection from the interior surfaces of the room.
For fixed sun position, sky condition (clear or overcast) and room geometry, the skyrelated interior daylight will be proportional to the exterior horizontal illuminance, Eh,sky, due to light from the sky. Similarly, the sunrelated interior daylight will be proportional to the exterior horizontal solar illuminance, Eh,sun.
Daylight Factors[LINK]
The following daylight factors are calculated:
dsky=Illuminance at reference point due to sky−related lightEh,sky
dsun=Illuminance at reference point due to sun−related lightEh,sun
wsky=Average window luminance due to sky−related lightEh,sky
wsun=Average window luminance due to sun−related lightEh,sun
bsky=Window background luminance due to sky−related lightEh,sky
bsun=Window background luminance due to sun−related lightEh,sun
For a daylit zone with N windows these six daylight factors are calculated for each of the following combinations of reference point, window, skycondition/sunposition and shading device:
[Ref pt 1Ref pt 2]⎡⎢ ⎢ ⎢⎣Window 1Window 2...Window N⎤⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Clear sky, first sun−up hourClear/turbid sky, first sun−up hourIntermediate sky, first sun−up hourOvercast sky, first sun−up hour...Clear sky, last sun−up hourClear/turbid sky, last sun−up hourIntermediate sky, last sun−up hourOvercase sky, last sun−up hour⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢⎣Unshaded windowShaded window(if shade assigned)⎤⎥⎦
Sky Luminance Distributions[LINK]
The luminance distribution of the sky is represented as a superposition of four standard CIE skies using the approach described in (Perez et al. 1990). The standard skies are as follows.
Clear Sky[LINK]
The clear sky luminance distribution has the form (Kittler, 1965; CIE, 1973)
ψcs(θsky,ϕsky)=Lz(0.91+10e−3γ+0.45cos2γ)(1−e−0.32cosecϕsky)0.27385(0.91+10e−3(π2−ϕsun)+0.45sin2ϕsun)
Here, Lz is the zenith luminance (i.e., the luminance of the sky at a point directly overhead). In the calculation of daylight factors, which are ratios of interior and exterior illumination quantities that are both proportional to Lz, the zenith luminance cancels out. For this reason we will use Lz = 1.0 for all sky luminance distributions.
The various angles, which are defined in the building coordinate system, are shown in Figure 1. The angle, γ, between sun and sky element is given by:
γ=cos−1[sinϕskysinϕsun+cosϕskycosϕsuncos(θsky−θsun)]
The general characteristics of the clearsky luminance distribution are a large peak near the sun; a minimum at a point on the other side of the zenith from the sun, in the vertical plane containing the sun; and an increase in luminance as the horizon is approached.
Clear Turbid Sky[LINK]
The clear turbid sky luminance distribution has the form [Matsuura, 1987]:
ψts(θsky,ϕsky)=Lz(0.856+16e−3γ+0.3cos2γ)(1−e−0.32cosecϕsky)0.27385(0.856+10e−3(π2−ϕsun)+0.3sin2ϕsun)
Intermediate Sky[LINK]
The intermediate sky luminance distribution has the form [Matsuura, 1987]:
ψis(θsky,ϕsky)=LzZ1Z2/(Z3Z4)
where
Z1=[1.35(sin(3.59ϕsky−0.009)+2.31)sin(2.6ϕsun+0.316)+ϕsky+4.799]/2.326
Z2=exp[−0.563γ{(ϕsun−0.008)(ϕsky+1.059)+0.812}]
Z3=0.99224sin(2.6ϕsun+0.316)+2.73852
Z4=exp[−0.563(π2−ϕsun){2.6298(ϕsun−0.008)+0.812}]
Angles appearing in the expression for the clearsky luminance distribution. [fig:anglesappearingintheexpressionfor]
Overcast Sky[LINK]
The overcast sky luminance distribution has the form [Moon & Spencer, 1942]:
ψos(ϕsky)=Lz1+2sinϕsky3
Unlike the clear sky case, the overcast sky distribution does not depend on the solar azimuth or the sky azimuth. Note that at fixed solar altitude the zenith (ϕsky=π/2 ) is three times brighter than the horizon (ϕsky=0 ).
Direct Normal Solar Illuminance[LINK]
For purposes of calculating daylight factors associated with beam solar illuminance, the direct normal solar illuminance is taken to be 1.0 W/m2. The actual direct normal solar illuminance, determined from direct normal solar irradiance from the weather file and empiricallydetermined luminious efficacy, is used in the timestep calculation.
Exterior Horizontal Illuminance[LINK]
The illuminance on an unobstructed horizontal plane due to diffuse radiation from the sky is calculated for each of the four sky types by integrating over the appropriate sky luminance distribution:
Eh,k=2π∫0π/2∫0ψk(θsky,ϕsky)sinϕskycosϕskydθskydϕsky
where k = cs, ts, is or os. The integral is evaluated as a double summation:
Eh,k=Nθ∑i=1Nϕ∑j=1ψk(θsky(i),ϕsky(j))sinϕsky(j)cosϕsky(j)ΔθskyΔϕsky
where
θsky(i)=(i−1/2)Δθskyϕsky(j)=(j−1/2)ΔϕskyΔθsky=2π/NθΔϕsky=π/2Nϕ
Nθ=18 and Nϕ=8 were found to give a ±1% accuracy in the calculation of Eh,k.
Direct Component of Interior Daylight Illuminance[LINK]
The direct daylight illuminance at a reference point from a particular window is determined by dividing the window into an xy grid and finding the flux reaching the reference point from each grid element. The geometry involved is shown in Figure 2. The horizontal illuminance at the reference point, →Rref , due to a window element is:
dEh=LwdΩcosγ
where Lw is the luminance of the window element as seen from the reference point.
The subtended solid angle is approximated by:
dΩ=dxdyD2cosB
where
D=∣∣¯Rwin−¯Rref∣∣
CosB is found from:
cosB=^Rray⋅^Wn
where
→Rray=(→Rwin−→Rref)/D
^Wn=window outward normal=^W21×^W23=→W1−→W2∣∣→W1−→W2∣∣×→W3−→W2∣∣→W3−→W2∣∣
Equation [eq:SubtendedSolidAngle] becomes exact as dx/D and dy/D→0 and is accurate to better than about 1% for dx≤D/4 and dy≤D/4 .
The net illuminance from the window is obtained by summing the contributions from all the window elements:
Eh=∑windowelementsLwdΩcosγ
In performing the summation, window elements that lie below the workplane (cosγ<0 ) are omitted since light from these elements cannot reach the workplane directly.
Geometry for calculation of direct component of daylight illuminance at a reference point. Vectors Rref, W1, W2, W3 and Rwin are in the building coordinate system. [fig:geometryforcalculationofdirectcomponent]
Unshaded Window[LINK]
For the unshaded window case, the luminance of the window element is found by projecting the ray from reference point to window element and determining whether it intersects the sky or an exterior obstruction such as an overhang. If L is the corresponding luminance of the sky or obstruction, the window luminance is:
Lw=LτviscosB
where τvis is the visible transmittance of the glass for incidence angle B.
Exterior obstructions are generally opaque (like fins, overhangs, neighboring buildings, and the building’s own wall and roof surfaces) but can be transmitting (like a tree or translucent awning). Exterior obstructions are assumed to be nonreflecting. If Lsky is the sky luminance and τobs is the transmittance of the obstruction (assumed independent of incidence angle), then L=Lskyτobs. Interior obstructions are assumed to be opaque (τobs = 0).
Shaded Window[LINK]
For the windowplusshade case the shade is assumed to be a perfect diffuser, i.e., the luminance of the shade is independent of angle of emission of light, position on shade, and angle of incidence of solar radiation falling on the shade. Closelywoven drapery fabric and translucent roller shades are closer to being perfect diffusers than Venetian blinds or other slatted devices, which usually have nonuniform luminance characteristics.
The calculation of the window luminance with the shade in place, Lw,sh, is described in [Winkelmann, 1983]. The illuminance contribution at the reference point from a shaded window element is then given by Equation [eq:NetIlluminanceFromWindow] with Lw=Lw,sh .
InternallyReflected Component of Interior Daylight Illuminance[LINK]
Daylight reaching a reference point after reflection from interior surfaces is calculated using the splitflux method [Hopkinson et al., 1954], [Lynes, 1968]. In this method the daylight transmitted by the window is split into two parts—a downwardgoing flux, ΦFW (lumens), which falls on the floor and portions of the walls below the imaginary horizontal plane passing through the center of the window (window midplane), and an upwardgoing flux, ΦCW , that strikes the ceiling and portions of the walls above the window midplane. A fraction of these fluxes is absorbed by the room surfaces. The remainder, the firstreflected flux, F1, is approximated by
F1=ΦFWρFW+ΦCWρCW
where ρFW is the areaweighted average reflectance of the floor and those parts of the walls below the window midplane, and ρCW is the areaweighted average reflectance of the ceiling and those parts of the walls above the window midplane.
To find the final average internallyreflected illuminance, Er, on the room surfaces (which in this method is uniform throughout the room) a flux balance is used. The total reflected flux absorbed by the room surfaces (or lost through the windows) is AEr(1ρ), where A is the total inside surface area of the floor, walls, ceiling and windows in the room, and ρ is the areaweighted average reflectance of the room surfaces, including windows. From conservation of energy:
AEr(1−ρ)=F1
or
Er=ΦFWρFW+ΦCWρCWA(1−ρ)
This procedure assumes that the room behaves like an integrating sphere with perfectly diffusing interior surfaces and with no internal obstructions. It therefore works best for rooms that are close to cubical in shape, have matte surfaces (which is usually the case), and have no internal partitions. Deviations from these conditions, such as would be the case for rooms whose depth measured from the windowwall is more than three times greater than ceiling height, can lead to substantial inaccuracies in the splitflux calculation.
Transmitted Flux from Sky and Ground[LINK]
The luminous flux incident on the center of the window from a luminous element of sky or ground at angular position (θ,ϕ), of luminance L(θ,ϕ), and subtending a solid angle cosϕdθdϕ is:
dΦinc=AwL(θ,ϕ)cosβcosϕdθdϕ
The transmitted flux is:
dΦ=dΦincT(β)
where T(β) is the window transmittance for light at incidence angle β. This transmittance depends on whether or not the window has a shade.
For an unshaded window the total downgoing transmitted flux is obtained by integrating over the part of the exterior hemisphere seen by the window that lies above the window midplane. This gives:
ΦFW,unshaded=Awθmax∫θminπ/2∫0L(θ,ϕ)T(β)cosβcosϕdθdϕ
The upgoing flux is obtained similarly by integrating over the part of the exterior hemisphere that lies below the window midplane:
ΦCW,unshaded=Awθmax∫θmin0∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
where ϕw is the angle the window outward normal makes with the horizontal plane.
For a window with a diffusing shade the total transmitted flux is:
Φsh=Awθmax∫θminπ/2∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
The downgoing and upgoing portions of this flux are:
ΦFW,sh=Φ(1−f)ΦCW,sh=Φf
where f, the fraction of the hemisphere seen by the inside of the window that lies above the window midplane, is given by
f=0.5−ϕw/π
For a vertical window (ϕw=0) the up and downgoing transmitted fluxes are equal:
ΦFW,sh=ΦCW,sh=Φ/2
For a horizontal skylight (ϕw=π/2):
ΦFW,sh=Φ,ΦCW,sh=0
The limits of integration of θ in Equations [eq:PhiFWUnshaded], [eq:PhiCWUnshaded] and [eq:PhiSh] depend on ϕ . From Figure 12 of Winkelmann (1983), we have:
sinα=sin(A−π/2)=sinϕtanϕwcosϕ
which gives:
A=cos−1(tanϕtanϕw)
Thus:
θmin=−∣∣cos−1(−tanϕtanϕw)∣∣θmax=∣∣cos−1(−tanϕtanϕw)∣∣
Transmitted Flux from Direct Sun[LINK]
The flux incident on the window from direct sun is:
Φinc=AwEDNcosβfsunlit
The transmitted flux is:
Φ=T(β)Φinc
where T is the net transmittance of the window glazing (plus shade, if present).
For an unshaded window all of the transmitted flux is downward since the sun always lies above the window midplane. Therefore:
ΦFW,unsh=ΦΦCW,unsh=0
For a window with a diffusing shade:
ΦFW,sh=Φ(1−f)ΦCW,sh=Φf
Luminance of Shaded Window[LINK]
The luminance of a shaded window is determined at the same time that the transmitted flux is calculated. It is given by:
Lsh=1πθmax∫θminπ/2∫π/2−ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ
Daylight Discomfort Glare[LINK]
The discomfort glare at a reference point due to luminance contrast between a window and the interior surfaces surrounding the window is given by [Hopkinson, 1970] and [Hopkinson, 1972]:
G=L1.6wΩ0.8Lb+0.07ω0.5Lw
where
G = discomfort glare constant
Lw = average luminance of the window as seen from the reference point
Ω = solid angle subtended by window, modified to take direction of occupant view into account
Lb = luminance of the background area surrounding the window
By dividing the window into Nx by Ny rectangular elements, as is done for calculating the direct component of interior illuminance, we have:
Lw=Ny∑j=1Nx∑i=1Lw(i,j)NxNy
where Lw(i,j) is the luminance of element (i,j) as seen from the reference point.
Similarly,
ω=Ny∑j=1Nx∑i=1dω(i,j)
where dω(i,j) is the solid angle subtended by element (i,j) with respect to the reference point.
The modified solid angle is:
Ω=Ny∑j=1Nx∑i=1dω(i,j)p(xR,yR)
where p is a “position factor” [Petherbridge & Longmore, 1954] that accounts for the decrease in visual excitation as the luminous element moves away from the line of sight. This factor depends on the horizontal and vertical displacement ratios, xR and yR (Figure 3), given by:
xR(i,j)=√A2−(YD)2RRyR(i,j)=YD/RR
where
RR=D(^Rray⋅^vview)A2=D2−(RR)2YD=Rwin(3)−Rref(3)
Geometry for calculation of displacement ratios used in the glare formula. [fig:geometryforcalculationofdisplacement]
The factor p can be obtained from graphs given in [Petherbridge & Longmore, 1954] or it can be calculated from tabulated values of pH, the Hopkinson position factor [Hopkinson, 1966], since p=p1.25H . The values resulting from the latter approach are given in Table 2. Interpolation of this table is used in EnergyPlus to evaluate p at intermediate values of xR and yR.
The background luminance is:
Lb=Ebρb
where ρb is approximated by the average interior surface reflectance of the entire room and
Eb=max(Er,Es)
where Er is the total internallyreflected component of daylight illuminance produced by all the windows in the room and Es is the illuminance setpoint at the reference point at which glare is being calculated. A precise calculation of Eb is not required since the glare index (see next section) is logarithmic. A factor of two variation in Eb generally produces a change of only 0.5 to 1.0 in the glare index.
Glare Index[LINK]
The net daylight glare at a reference point due to all of the windows in a room is expressed in terms of a glare index given by
GI=10log10number ofwindows∑i=1Gi
where Gi is the glare constant at the reference point due to the ith window.
The sun positions for which the daylight factors are calculated are the same as those for which the solar shadowing calculations are done.↩
Documentation content copyright © 19962021 The Board of Trustees of the University of Illinois and the Regents of the University of California through the Ernest Orlando Lawrence Berkeley National Laboratory. All rights reserved. EnergyPlus is a trademark of the US Department of Energy.
This documentation is made available under the EnergyPlus Open Source License v1.0.