## Engineering Reference — EnergyPlus 8.4

### Engineering Reference

Table 25. Variables in Daylighting Calculations
Mathematical Variable Description Units Internal Variable
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
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 Number of exterior windows in a zone
NWD
θskysky 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
ψos 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
Mathematical Variable 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
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 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 y-axis
W21
^W23 Unit vector along window x-axis
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 First-reflected flux lm
ρFW Area-weighted reflectance of floor and upper part of walls
SurfaceWindow%RhoFloorWall
ρCW Area-weighted reflectance of ceiling and upper part of walls
SurfaceWindow%RhoCeilingWall
Er Average internally-reflected illuminance lux EINTSK, EINTSU
A Total inside surface area of a zone m2 ATOT
ρ Area-weighted 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
inc Luminous flux incident on window from sky or ground element lm
Luminous flux from sky or ground element transmitted through window lm
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
φw Window normal altitude angle radians
ΦCW,shFW,sh Upgoing and downgoing portions of transmitted flux through window with shade lm
ΦCW,unshFW,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 x_{R} and y_{R}
DayltgGlarePositionFactor
pH Hopkinson position factor
DayltgGlarePositionFactor
Lb Window background luminance cd/m2 BLUM
Eb Illuminance on window background lm
Er Total internally-reflected 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/reference-point 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.

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 sky-related interior daylight will be proportional to the exterior horizontal illuminance, Eh,sky, due to light from the sky. Similarly, the sun-related interior daylight will be proportional to the exterior horizontal solar illuminance, Eh,sun.

The following daylight factors are calculated:

dsky=IlluminanceatreferencepointduetoskyrelatedlightEh,sky

dsun=IlluminanceatreferencepointduetosunrelatedlightEh,sun

wsky=AveragewindowluminanceduetoskyrelatedlightEh,sky

wsun=AveragewindowluminanceduetosunrelatedlightEh,sun

bsky=WindowbackgroundluminanceduetoskyrelatedlightEh,sky

bsun=WindowbackgroundluminanceduetosunrelatedlightEh,sun

For a daylit zone with N windows these six daylight factors are calculated for each of the following combinations of reference point, window, sky-condition/sun-position and shading device:

[Refpt1Refpt2]⎢ ⎢ ⎢Window1Window2...WindowN⎥ ⎥ ⎥⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢Clearsky,firstsunuphourClear/turbidsky,firstsunuphourIntermediatesky,firstsunuphourOvercastsky,firstsunuphour...Clearsky,lastsunuphourClear/turbidsky,lastsunuphourIntermediatesky,lastsunuphourOvercasesky,lastsunuphour⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥UnshadedwindowShadedwindow(ifshadeassigned)

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+10e3γ+0.45cos2γ)(1e0.32cosecϕsky)0.27385(0.91+10e3(π2ϕsun)+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 48. The angle,* γ*, between sun and sky element is given by

γ=cos1[sinϕskysinϕsun+cosϕskycosϕsuncos(θskyθsun)]

The general characteristics of the clear-sky 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.

The clear turbid sky luminance distribution has the form [Matsuura, 1987]

ψts(θsky,ϕsky)=Lz(0.856+16e3γ+0.3cos2γ)(1e0.32cosecϕsky)0.27385(0.856+10e3(π2ϕsun)+0.3sin2ϕsun)

The intermediate sky luminance distribution has the form [Matsuura, 1987]

ψis(θsky,ϕsky)=LzZ1Z2/(Z3Z4)

where

Z1=[1.35(sin(3.59ϕsky0.009)+2.31)sin(2.6ϕsun+0.316)+ϕsky+4.799]/2.326 Z2=exp[0.563γ{(ϕsun0.008)(ϕsky+1.059)+0.812}]

Z3=0.99224sin(2.6ϕsun+0.316)+2.73852

Z4=exp[0.563(π2ϕsun){2.6298(ϕsun0.008)+0.812}] ClearSkyAngles

Figure 57. Angles appearing in the expression for the clear-sky 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 ).

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 empirically-determined luminious efficacy, is used in the time-step calculation.

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=2π0π/20ψk(θsky,ϕsky)sinϕskycosϕskydθskydϕsky

where* k* = cs, ts, is or os. The integral is evaluated as a double summation:

Eh,k=Nθi=1Nϕj=1ψk(θsky(i),ϕsky(j))sinϕsky(j)cosϕsky(j)ΔθskyΔϕsky

where

θsky(i)=(i1/2)Δθskyϕsky(j)=(j1/2)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 x-y 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 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=(RwinRref)/D

^Wn=windowoutwardnormal=^W21×^W23=W1W2W1W2×W3W2W3W2

Equation becomes exact as dx/Danddy/D0 and is accurate to better than about 1% for dxD/4anddyD/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 Rref, W1, W2, W3 and Rwin are in the building coordinate system.

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 non-reflecting. If Lsky is the sky luminance and tobs is the transmittance of the obstruction (assumed independent of incidence angle), then L = * Lskytobs. Interior obstructions are assumed to be opaque (tobs* = 0).

For the window-plus-shade 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. Closely-woven drapery fabric and translucent roller shades are closer to being perfect diffusers than Venetian blinds or other slatted devices, which usually have non-uniform 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 Eq. (152) with Lw=Lw,sh .

## Internally-Reflected Component of Interior Daylight Illuminance[LINK]

Daylight reaching a reference point after reflection from interior surfaces is calculated using the split-flux method [Hopkinson et al., 1954], [Lynes, 1968]. In this method the daylight transmitted by the window is split into two parts—a downward-going 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 upward-going 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 first-reflected flux, F1, is approximated by

F1=ΦFWρFW+ΦCWρCW

where ρFW is the area-weighted average reflectance of the floor and those parts of the walls below the window midplane, and ρCW is the area-weighted average reflectance of the ceiling and those parts of the walls above the window midplane.

To find the final average internally-reflected 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(1-ρ), where A is the total inside surface area of the floor, walls, ceiling and windows in the room, and ρ is the area-weighted average reflectance of the room surfaces, including windows. From conservation of energy

AEr(1ρ)=F1

or

Er=ΦFWρFW+ΦCWρCWA(1ρ)

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 window-wall is more than three times greater than ceiling height, can lead to substantial inaccuracies in the split-flux 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

The upgoing flux is obtained similarly by integrating over the part of the exterior hemisphere that lies below the window midplane:

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=Awθmaxθminπ/2π/2ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ

The downgoing and upgoing portions of this flux are

ΦFW,sh=Φ(1f)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=0.5ϕw/π

For a vertical window (ϕw=0 ) the up- and down-going 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(Aπ/2)=sinϕtanϕwcosϕ

which gives

A=cos1(tanϕtanϕw)

Thus

θmin=cos1(tanϕtanϕw)θmax=cos1(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(1f)ΦCW,sh=Φf

The luminance of a shaded window is determined at the same time that the transmitted flux is calculated. It is given by

Lsh=1πθmaxθminπ/2π/2ϕwL(θ,ϕ)T(β)cosβcosϕdθdϕ

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=Nyj=1Nxi=1Lw(i,j)NxNy

where Lw(i,j) is the luminance of element* (i,j)* as seen from the reference point.

Similarly,

ω=Nyj=1Nxi=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=Nyj=1Nxi=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 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 pH, 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 xR and yR.

Table 26. Position factor for glare calculation
XR: Horizontal Displacement Factor
0.0 0.5 1.0 1.5 2.0 2.5 3.0 >3.0
yR: 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 ρb is approximated by the average interior surface reflectance of the entire room and

Eb=max(Er,Es)

where Eris the total internally-reflected 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.