Daylight Factor Calculation[LINK]

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
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
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
${\overrightarrow{R}}_{ref}$	Vector from zone origin to reference point	m	RREF
${\overrightarrow{R}}_{win}$	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 rat from reference point to window element	radians
${\hat{R}}_{ref}$	Unit vector from reference point to window element		RAY
${\hat{W}}_n$	Unit vector normal to window element, pointing away from zone		WNORM
${\hat{W}}_{21}$	Unit vector along window y-axis		W21
${\hat{W}}_{23}$	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
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
{min, θ{max	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 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

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[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/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.

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

Daylight Factors[LINK]

The following daylight factors are calculated:

$$d_{sky}=\frac{Illuminanceatreferencepointduetosky-relatedlight}{E_{h,sky}}$$

$$d_{sun}=\frac{Illuminanceatreferencepointduetosun-relatedlight}{E_{h,sun}}$$

$$w_{sky}=\frac{Averagewindowluminanceduetosky-relatedlight}{E_{h,sky}}$$

$$w_{sun}=\frac{Averagewindowluminanceduetosun-relatedlight}{E_{h,sun}}$$

$$b_{sky}=\frac{Windowbackgroundluminanceduetosky-relatedlight}{E_{h,sky}}$$

$$b_{sun}=\frac{Windowbackgroundluminanceduetosun-relatedlight}{E_{h,sun}}$$

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:

$$\left[\begin{array}{c}Refpt1\\ Refpt2\end{array}\right]\left[\begin{array}{c}Window1\\ Window2\\ ...\\ WindowN\end{array}\right]\left[\begin{array}{c}Clearsky,firstsun-uphour\\ Clear/turbidsky,firstsun-uphour\\ Intermediatesky,firstsun-uphour\\ Overcastsky,firstsun-uphour\\ ...Clearsky,lastsun-uphour\\ Clear/turbidsky,lastsun-uphour\\ Intermediatesky,lastsun-uphour\\ Overcasesky,lastsun-uphour\end{array}\right]\left[\begin{array}{c}Unshadedwindow\\ Shadedwindow\\ \left(ifshadeassigned\right)\end{array}\right]$$

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)

$${\psi }_{cs}({\theta }_{sky},{\varphi }_{sky})=L_z\frac{(0.91+10e^{-3\gamma }+0.45{\cos }^2\gamma )(1-e^{-0.32cosec{\varphi }_{sky}})}{0.27385(0.91+10e^{-3(\frac{\pi }{2}-{\varphi }_{sun})}+0.45{\sin }^2{\varphi }_{sun})}$$

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

$$\gamma ={\cos }^{-1}\left[\sin {\varphi }_{sky}\sin {\varphi }_{sun}+\cos {\varphi }_{sky}\cos {\varphi }_{sun}\cos ({\theta }_{sky}-{\theta }_{sun})\right]$$

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.

Clear Turbid Sky[LINK]

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

$${\psi }_{ts}({\theta }_{sky},{\varphi }_{sky})=L_z\frac{(0.856+16e^{-3\gamma }+0.3{\cos }^2\gamma )(1-e^{-0.32cosec{\varphi }_{sky}})}{0.27385(0.856+10e^{-3(\frac{\pi }{2}-{\varphi }_{sun})}+0.3{\sin }^2{\varphi }_{sun})}$$

Intermediate Sky[LINK]

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

$${\psi }_{is}\left({\theta }_{sky},{\varphi }_{sky}\right)=L_z{Z}_1{Z}_2/\left(Z_3{Z}_4\right)$$

where

$Z_1=\left[1.35\left(\sin \right(3.59{\varphi }_{sky}-0.009)+2.31)\sin (2.6{\varphi }_{sun}+0.316)+{\varphi }_{sky}+4.799\right]/2.326$ $Z_2=\exp \left[-0.563\gamma \left\{\right({\varphi }_{sun}-0.008\left)\right({\varphi }_{sky}+1.059)+0.812\}\right]$

$$Z_3=0.99224\sin (2.6{\varphi }_{sun}+0.316)+2.73852$$

$$Z_4=\exp \left[-0.563(\frac{\pi }{2}-{\varphi }_{sun})\left\{2.6298\right({\varphi }_{sun}-0.008)+0.812\}\right]$$

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

Overcast Sky[LINK]

The overcast sky luminance distribution has the form [Moon & Spencer, 1942]

$${\psi }_{os}\left({\varphi }_{sky}\right)=L_z\frac{1+2\sin {\varphi }_{sky}}{3}$$

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 (${\varphi }_{sky}=\pi /2$ ) is three times brighter than the horizon (${\varphi }_{sky}=0$ ).

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

$$E_{h,k}=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi /2}{\int }}{\psi }_k({\theta }_{sky},{\varphi }_{sky})\sin {\varphi }_{sky}\cos {\varphi }_{sky}d{\theta }_{sky}d{\varphi }_{sky}$$

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

$$E_{h,k}=\sum _{i=1}^{N_{\theta }}\sum _{j=1}^{N_{\varphi }}{\psi }_k\left({\theta }_{sky}\left(i\right),{\varphi }_{sky}\left(j\right)\right)\sin {\varphi }_{sky}\left(j\right)\cos {\varphi }_{sky}\left(j\right)\Delta {\theta }_{sky}\Delta {\varphi }_{sky}$$

where

$$\begin{array}{c}{\theta }_{sky}\left(i\right)=(i-1/2)\Delta {\theta }_{sky}\\ {\varphi }_{sky}\left(j\right)=(j-1/2)\Delta {\varphi }_{sky}\\ \Delta {\theta }_{sky}=2\pi /N_{\theta }\\ \Delta {\varphi }_{sky}=\pi /2N_{\varphi }\end{array}$$

$N_{\theta }=18$ and $N_{\varphi }=8$ were found to give a $\pm 1\%$ accuracy in the calculation of ${E_h}_{,k}$ .

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, ${\overrightarrow{R}}_{ref}$ , due to a window element is

$$d{E}_h=L_{w}d\Omega \cos \gamma$$

where L_w is the luminance of the window element as seen from the reference point.

The subtended solid angle is approximated by

$$d\Omega =\frac{dxdy}{D^2}\cos B$$

where

$$D=\left|{\overline{R}}_{win}-{\overline{R}}_{ref}\right|$$

CosB is found from

$$\cos B={\hat{R}}_{ray}\cdot {\hat{W}}_n$$

where

$${\overrightarrow{R}}_{ray}=({\overrightarrow{R}}_{win}-{\overrightarrow{R}}_{ref})/D$$

$${\hat{W}}_n=windowoutwardnormal={\hat{W}}_{21}\times {\hat{W}}_{23}=\frac{{\overrightarrow{W}}_1-{\overrightarrow{W}}_2}{\left|{\overrightarrow{W}}_1-{\overrightarrow{W}}_2\right|}\times \frac{{\overrightarrow{W}}_3-{\overrightarrow{W}}_2}{\left|{\overrightarrow{W}}_3-{\overrightarrow{W}}_2\right|}$$

Equation becomes exact as $dx/Danddy/D\to 0$ and is accurate to better than about 1% for $dx\le D/4anddy\le D/4$ .

The net illuminance from the window is obtained by summing the contributions from all the window elements:

$$E_h=\sum _{\frac{window}{elements}}{L}_{w}d\Omega \cos \gamma$$

In performing the summation, window elements that lie below the workplane ($\cos \gamma <0$ ) 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, W₁, W₂, W₃ 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

$$L_w=L{\tau }_{vis}\left(\cos B\right)$$

where ${\tau }_{vis}$ 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 L_sky is the sky luminance and t_obs is the transmittance of the obstruction (assumed independent of incidence angle), then L = * L_skyt_obs. Interior obstructions are assumed to be opaque (t_obs* = 0).

Shaded Window[LINK]

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, 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 $L_w=L_{w,sh}$ .

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, ${\Phi }_{FW}$ (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, ${\Phi }_{CW}$ , 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

$$F_1={\Phi }_{FW}{\rho }_{FW}+{\Phi }_{CW}{\rho }_{CW}$$

where ρ_FW is the area-weighted average reflectance of the floor and those parts of the walls below the window midplane, and ρ_CW 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, 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(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

$$A{E}_r(1-\rho )=F_1$$

or

$$E_r=\frac{{\Phi }_{FW}{\rho }_{FW}+{\Phi }_{CW}{\rho }_{CW}}{A(1-\rho )}$$

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 $(\theta ,\varphi )$ , of luminance $L(\theta ,\varphi )$ , and subtending a solid angle $\cos \varphi d\theta d\varphi$ is

$$d{\Phi }_{inc}=A_{w}L(\theta ,\varphi )\cos \beta \cos \varphi d\theta d\varphi$$

The transmitted flux is

$$d\Phi =d{\Phi }_{inc}T\left(\beta \right)$$

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

$${\Phi }_{FW,unshaded}=A_w\underset{{\theta }_{min}}{\overset{{\theta }_{max}}{\int }}\underset{0}{\overset{\pi /2}{\int }}L(\theta ,\varphi )T\left(\beta \right)\cos \beta \cos \varphi d\theta d\varphi$$

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

$${\Phi }_{CW,unshaded}=A_w\underset{{\theta }_{min}}{\overset{{\theta }_{max}}{\int }}\underset{\pi /2-{\varphi }_w}{\overset{0}{\int }}L(\theta ,\varphi )T\left(\beta \right)\cos \beta \cos \varphi d\theta d\varphi$$

where ${\varphi }_w$ is the angle the window outward normal makes with the horizontal plane.

For a window with a diffusing shade the total transmitted flux is

$${\Phi }_{sh}=A_w\underset{{\theta }_{min}}{\overset{{\theta }_{max}}{\int }}\underset{\pi /2-{\varphi }_w}{\overset{\pi /2}{\int }}L(\theta ,\varphi )T\left(\beta \right)\cos \beta \cos \varphi d\theta d\varphi$$

The downgoing and upgoing portions of this flux are

$$\begin{array}{c}{\Phi }_{FW,sh}=\Phi (1-f)\\ {\Phi }_{CW,sh}=\Phi f\end{array}$$

where f, the fraction of the hemisphere seen by the inside of the window that lies above the window midplane, is given by

$$f=0.5-{\varphi }_w/\pi$$

For a vertical window (${\varphi }_w=0$ ) the up- and down-going transmitted fluxes are equal:

${\Phi }_{FW,sh}={\Phi }_{CW,sh}=\Phi /2$ .

For a horizontal skylight (${\varphi }_w=\pi /2$ ):

$${\Phi }_{FW,sh}=\Phi ,{\Phi }_{CW,sh}=0$$

The limits of integration of θ in Equations (153), (154) and (155) depend on $\varphi$ . From [Figure # 12 - Winkelmann, 1983] we have

$$\sin \alpha =\sin (A-\pi /2)=\frac{\sin \varphi \tan {\varphi }_w}{\cos \varphi }$$

which gives

$$A={\cos }^{-1}\left(\tan \varphi \tan {\varphi }_w\right)$$

Thus

$$\begin{array}{cc}{\theta }_{min}& =-\left|{\cos }^{-1}\right(-\tan \varphi \tan {\varphi }_w\left)\right|\\ {\theta }_{max}& =\left|{\cos }^{-1}\right(-\tan \varphi \tan {\varphi }_w\left)\right|\end{array}$$

Transmitted Flux from Direct Sun[LINK]

The flux incident on the window from direct sun is

$${\Phi }_{inc}=A_w{E}_{DN}\cos \beta f_{sunlit}$$

The transmitted flux is

$$\Phi =T\left(\beta \right){\Phi }_{inc}$$

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

$$\begin{array}{c}{\Phi }_{FW,unsh}=\Phi \\ {\Phi }_{CW,unsh}=0\end{array}$$

For a window with a diffusing shade

$$\begin{array}{c}{\Phi }_{FW,sh}=\Phi (1-f)\\ {\Phi }_{CW,sh}=\Phi f\end{array}$$

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

$$L_{sh}=\frac{1}{\pi }\underset{{\theta }_{min}}{\overset{{\theta }_{max}}{\int }}\underset{\pi /2-{\varphi }_w}{\overset{\pi /2}{\int }}L(\theta ,\varphi )T\left(\beta \right)\cos \beta \cos \varphi d\theta d\varphi$$

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]:

$$G=\frac{L_w^{1.6}{\Omega }^{0.8}}{L_b+0.07{\omega }^{0.5}{L}_w}$$

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

$$L_w=\frac{\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}{L}_w(i,j)}{N_x{N}_y}$$

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

Similarly,

$$\omega =\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}d\omega (i,j)$$

where dω(i,j) is the solid angle subtended by element* (i,j)*with respect to the reference point.

The modified solid angle is

$\Omega =\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}d\omega (i,j)p(x_R,y_R)$

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

$$\begin{array}{cc}{x}_R(i,j)& =\frac{\sqrt{A^2-(YD{)}^2}}{RR}\\ y_R(i,j)& =\left|YD/RR\right|\end{array}$$

where

$$\begin{array}{cc}RR& =D({\hat{R}}_{ray}\cdot {\hat{v}}_{view})\\ A^2& =D^2-(RR{)}^2\\ YD=R_{win}\left(3\right)-R_{ref}\left(3\right)& \end{array}$$

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 $p=p_H^{1.25}$ . 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

		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

$$L_b=E_b{\rho }_b$$

where ρ_b is approximated by the average interior surface reflectance of the entire room and

$$E_b=max(E_r,E_s)$$

where E_ris the total internally-reflected 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

$$G_I=10{\log }_{10}\sum _{i=1}^{\frac{numberof}{windows}}{G}_i$$

where G_i is the glare constant at the reference point due to the i^th window