Daylight Factor Calculation[LINK]
Table 25. Variables in Daylighting Calculations
Mathematical Variable

Description

Units

Internal Variable

E_{h,sky}

Exterior horizontal illuminance due to light from the sky

lux

GILSK

E_{h,sun}

Exterior horizontal illuminance due to light from the sun

lux

GILSU

d_{sky},d_{sun}

Interior illuminance factor due


DFACSK,DFACSU

w_{sky},w_{sun}

Window luminance factor due to sky, sun related light

cd/lm

SFACSK, SFACSU

b_{sky},b_{sun}

Window background luminance factor due to sky, sun related light

cd/lm

BFACSK, BFACSU

N Number of exterior windows

Number of exterior windows in a zone


NWD

θ_{sky},φ_{sky}

Azimuth and altitude angles of a point in the sky

radians

THSKY, PHSKY

ψ_{cs}

Clear sky luminance distribution

cd/m2


ψ_{ts}

Clear turbid sky luminance distribution

cd/m2


ψ_{os}

Intermediate sky luminance distribution

cd/m2


ψ_{os}

Overcast sky luminance distribution

cd/m2


φ_{sun}

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

L_{z}

Sky zenith luminance

cd/m2

ZENL

Mathematical Variable

Optical air mass of the atmosphere

m

AM

h

Building altitude

m

ELEVATION

E_{h,k}

Exterior horizontal illuminance for sky type k

lux


N_{θ},N_{φ}

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

L_{w}

Luminance of a window element as seen from reference point

cd/m2

WLUMSK, WLUMSU

L_{w,shade}

Luminance of window element with shade in place

cd/m2

WLUMSK, WLUMSU

dE_{h}

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 rat from reference point to window element

radians


^Rref

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

τ_{vis}

Glass visible transmittance


TVISB

L

Luminance of sky or obstruction

cd/m2

ELUM, 

Φ_{FW}

Downgoing luminous flux from a window

lm

FLFW –

Φ_{CW}

Upgoing luminous flux from a window

lm

FLCW –

F_{1}

Firstreflected flux

lm


ρ_{FW}

Areaweighted reflectance of floor and upper part of walls


SurfaceWindow%RhoFloorWall

ρ_{CW}

Areaweighted reflectance of ceiling and upper part of walls


SurfaceWindow%RhoCeilingWall

E_{r}

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–

A_{w}

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

dΦ_{inc}

Luminous flux incident on window from sky or ground element

lm


dΦ

Luminous flux from sky or ground element transmitted through window

lm


dΦ_{FW, dΦCW
}

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

lm


{min, θ{max

Azimuth angle integration limits

radians

THMIN, THMAX

φ_{w}

Window normal altitude angle

radians


Φ_{sh},Φ_{unsh}

Transmitted flux through window and shade, without shade

lm


Φ_{CW,sh},Φ_{FW,sh}

Upgoing and downgoing portions of transmitted flux through window with shade

lm


Φ_{CW,unsh},Φ_{FW,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

Φ_{inc}

Flux incident on glazing from direct sun

lm


f_{sunlit}

Fraction of glazing that is sunlit


SunLitFrac

Φ

Transmitted flux from direct sun



L_{sh}

Luminance of window with shade

cd/m2


L_{b}

Window background luminance

cd/m2

BLUM

G

Discomfort glare constant


GTOT

G_{i}

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

N_{x},N_{y}

Number of elements in x and y direction that window is divided into for glare calculation


NWX, NWY

p(x_{R},y_{R})

Position factor for horizontal and vertical displacement ratios x_{R} and y_{R}


DayltgGlarePositionFactor

p_{H}

Hopkinson position factor


DayltgGlarePositionFactor

L_{b}

Window background luminance

cd/m2

BLUM

E_{b}

Illuminance on window background

lm


E_{r}

Total internallyreflected component of daylight illuminance

lm


E_{s}

Illuminance setpoint

lm

IllumSetPoint

G_{i}

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[8] 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, E_{h,sky}, due to light from the sky. Similarly, the sunrelated interior daylight will be proportional to the exterior horizontal solar illuminance, E_{h,sun}.
Daylight Factors[LINK]
The following daylight factors are calculated:
dsky=Illuminanceatreferencepointduetosky−relatedlightEh,sky
dsun=Illuminanceatreferencepointduetosun−relatedlightEh,sun
wsky=Averagewindowluminanceduetosky−relatedlightEh,sky
wsun=Averagewindowluminanceduetosun−relatedlightEh,sun
bsky=Windowbackgroundluminanceduetosky−relatedlightEh,sky
bsun=Windowbackgroundluminanceduetosun−relatedlightEh,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:
[Refpt1Refpt2]⎡⎢
⎢
⎢⎣Window1Window2...WindowN⎤⎥
⎥
⎥⎦⎡⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣Clearsky,firstsun−uphourClear/turbidsky,firstsun−uphourIntermediatesky,firstsun−uphourOvercastsky,firstsun−uphour...Clearsky,lastsun−uphourClear/turbidsky,lastsun−uphourIntermediatesky,lastsun−uphourOvercasesky,lastsun−uphour⎤⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦⎡⎢⎣UnshadedwindowShadedwindow(ifshadeassigned)⎤⎥⎦
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, L_{z} 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 L_{z}, the zenith luminance cancels out. For this reason we will use L_{z} = 1.0 for all sky luminance distributions.
The various angles, which are defined in the building coordinate system, are shown in Figure 48. 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}]
Figure 57. Angles appearing in the expression for the clearsky luminance distribution.
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/m^{2}. 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 58. The horizontal illuminance at the reference point, →Rref , due to a window element is
dEh=LwdΩcosγ
where L_{w} 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=windowoutwardnormal=^W21×^W23=→W1−→W2∣∣→W1−→W2∣∣×→W3−→W2∣∣→W3−→W2∣∣
Equation becomes exact as dx/Danddy/D→0 and is accurate to better than about 1% for dx≤D/4anddy≤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.
Figure 58. Geometry for calculation of direct component of daylight illuminance at a reference point. Vectors R_{ref}, W_{1}, W_{2}, W_{3} and R_{win} are in the building coordinate system.
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=Lτvis(cosB)
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 L_{sky} is the sky luminance and t_{obs} is the transmittance of the obstruction (assumed independent of incidence angle), then L = * L_{sky}t_{obs}. Interior obstructions are assumed to be opaque (t_{obs}* = 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, L_{w,sh}, is described in [Winkelmann, 1983]. The illuminance contribution at the reference point from a shaded window element is then given by Eq. (152) 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, F_{1}, is approximated by
F1=ΦFWρFW+ΦCWρCW
where ρ_{FW} is the areaweighted average reflectance of the floor and those parts of the walls below the window midplane, and ρ_{CW} 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, E_{r}, 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 AE_{r}(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 (153), (154) and (155) depend on ϕ . From [Figure # 12  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
L_{w} = 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
L_{b} = luminance of the background area surrounding the window
By dividing the window into N_{x} by N_{y} 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 L_{w}(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, x_{R} and y_{R} (Figure 59), 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)
Figure 59. Geometry for calculation of displacement ratios used in the glare formula.
The factor p can be obtained from graphs given in [Petherbridge & Longmore, 1954] or it can be calculated from tabulated values of p_{H}, the Hopkinson position factor [Hopkinson, 1966], since p=p1.25H . The values resulting from the latter approach are given in Table 26. Interpolation of this table is used in EnergyPlus to evaluate p at intermediate values of x_{R} and y_{R}.
Table 26. Position factor for glare calculation

X_{R}: Horizontal Displacement Factor

0.0

0.5

1.0

1.5

2.0

2.5

3.0

>3.0

y_{R}: Vertical Displacement Factor

0

1.00

0.492

0.226

0.128

0.081

0.061

0.057

0

0.5

0.123

0.119

0.065

0.043

0.029

0.026

0.023

0

1.0

0.019

0.026

0.019

0.016

0.014

0.011

0.011

0

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

1.5

0.008

0.008

0.008

0.008

0.008

0.006

0.006

0

The background luminance is
Lb=Ebρb
where ρ_{b} is approximated by the average interior surface reflectance of the entire room and
Eb=max(Er,Es)
where E_{r}is the total internallyreflected component of daylight illuminance produced by all the windows in the room and E_{s} is the illuminance setpoint at the reference point at which glare is being calculated. A precise calculation of E_{b} is not required since the glare index (see next section) is logarithmic. A factor of two variation in E_{b} 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=10log10numberofwindows∑i=1Gi
where G_{i} is the glare constant at the reference point due to the i^{th} window
Daylight Factor Calculation[LINK]
Table 25. Variables in Daylighting CalculationsOverview[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[8] 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, E_{h,sky}, due to light from the sky. Similarly, the sunrelated interior daylight will be proportional to the exterior horizontal solar illuminance, E_{h,sun}.
Daylight Factors[LINK]
The following daylight factors are calculated:
dsky=Illuminanceatreferencepointduetosky−relatedlightEh,sky
dsun=Illuminanceatreferencepointduetosun−relatedlightEh,sun
wsky=Averagewindowluminanceduetosky−relatedlightEh,sky
wsun=Averagewindowluminanceduetosun−relatedlightEh,sun
bsky=Windowbackgroundluminanceduetosky−relatedlightEh,sky
bsun=Windowbackgroundluminanceduetosun−relatedlightEh,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:
[Refpt1Refpt2]⎡⎢ ⎢ ⎢⎣Window1Window2...WindowN⎤⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣Clearsky,firstsun−uphourClear/turbidsky,firstsun−uphourIntermediatesky,firstsun−uphourOvercastsky,firstsun−uphour...Clearsky,lastsun−uphourClear/turbidsky,lastsun−uphourIntermediatesky,lastsun−uphourOvercasesky,lastsun−uphour⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢⎣UnshadedwindowShadedwindow(ifshadeassigned)⎤⎥⎦
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, L_{z} 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 L_{z}, the zenith luminance cancels out. For this reason we will use L_{z} = 1.0 for all sky luminance distributions.
The various angles, which are defined in the building coordinate system, are shown in Figure 48. 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}]
ClearSkyAngles
Figure 57. Angles appearing in the expression for the clearsky luminance distribution.
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/m^{2}. 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 58. The horizontal illuminance at the reference point, →Rref , due to a window element is
dEh=LwdΩcosγ
where L_{w} 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=windowoutwardnormal=^W21×^W23=→W1−→W2∣∣→W1−→W2∣∣×→W3−→W2∣∣→W3−→W2∣∣
Equation becomes exact as dx/Danddy/D→0 and is accurate to better than about 1% for dx≤D/4anddy≤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.
GeomDirComp
Figure 58. Geometry for calculation of direct component of daylight illuminance at a reference point. Vectors R_{ref}, W_{1}, W_{2}, W_{3} and R_{win} are in the building coordinate system.
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=Lτvis(cosB)
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 L_{sky} is the sky luminance and t_{obs} is the transmittance of the obstruction (assumed independent of incidence angle), then L = * L_{sky}t_{obs}. Interior obstructions are assumed to be opaque (t_{obs}* = 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, L_{w,sh}, is described in [Winkelmann, 1983]. The illuminance contribution at the reference point from a shaded window element is then given by Eq. (152) 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, F_{1}, is approximated by
F1=ΦFWρFW+ΦCWρCW
where ρ_{FW} is the areaweighted average reflectance of the floor and those parts of the walls below the window midplane, and ρ_{CW} 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, E_{r}, 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 AE_{r}(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 (153), (154) and (155) depend on ϕ . From [Figure # 12  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
L_{w} = 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
L_{b} = luminance of the background area surrounding the window
By dividing the window into N_{x} by N_{y} 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 L_{w}(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, x_{R} and y_{R} (Figure 59), 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)
GeomGlare
Figure 59. Geometry for calculation of displacement ratios used in the glare formula.
The factor p can be obtained from graphs given in [Petherbridge & Longmore, 1954] or it can be calculated from tabulated values of p_{H}, the Hopkinson position factor [Hopkinson, 1966], since p=p1.25H . The values resulting from the latter approach are given in Table 26. Interpolation of this table is used in EnergyPlus to evaluate p at intermediate values of x_{R} and y_{R}.
Table 26. Position factor for glare calculationThe background luminance is
Lb=Ebρb
where ρ_{b} is approximated by the average interior surface reflectance of the entire room and
Eb=max(Er,Es)
where E_{r}is the total internallyreflected component of daylight illuminance produced by all the windows in the room and E_{s} is the illuminance setpoint at the reference point at which glare is being calculated. A precise calculation of E_{b} is not required since the glare index (see next section) is logarithmic. A factor of two variation in E_{b} 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=10log10numberofwindows∑i=1Gi
where G_{i} is the glare constant at the reference point due to the i^{th} window
Documentation content copyright © 19962015 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.