## Engineering Reference — EnergyPlus 8.4

### Engineering Reference

In EnergyPlus the calculation of diffuse solar radiation from the sky incident on an exterior surface takes into account the anisotropic radiance distribution of the sky. For this distribution, the diffuse sky irradiance on a surface is given by

Where

Diffuse Solar Irradiance is the diffuse solar irradiance from the sky on the ground.

surface is the surface being analyzed.

AnisoSkyMultiplier is determined by surface orientation and sky radiance distribution, and accounts for the effects of shading of sky diffuse radiation by shadowing surfaces such as overhangs. It does not account for reflection of sky diffuse radiation from shadowing surfaces.

The sky radiance distribution is based on an empirical model based on radiance measurements of real skies, as described in Perez et al., 1990. In this model the radiance of the sky is determined by three distributions that are superimposed (see Figure 37)

1. An isotropic distribution that covers the entire sky dome;

2. A circumsolar brightening centered at the position of the sun;

3. A horizon brightening.

Figure 37. Schematic view of sky showing solar radiance distribution as a superposition of three components: dome with isotropic radiance, circumsolar brightening represented as a point source at the sun, and horizon brightening represented as a line source at the horizon.

The proportions of these distributions depend on the sky condition, which is characterized by two quantities, clearness factor and brightness factor, defined below, which are determined from sun position and solar quantities from the weather file.

The circumsolar brightening is assumed to be concentrated at a point source at the center of the sun although this region actually begins at the periphery of the solar disk and falls off in intensity with increasing angular distance from the periphery.

The horizon brightening is assumed to be a linear source at the horizon and to be independent of azimuth. In actuality, for clear skies, the horizon brightening is highest at the horizon and decreases in intensity away from the horizon. For overcast skies the horizon brightening has a negative value since for such skies the sky radiance increases rather than decreases away from the horizon.

Table 21. Variables in Anisotropic Sky Model and Shadowing of Sky Diffuse Radiation
Mathematical variable Description Units FORTRAN variable
Isky Solar irradiance on surface from sky W/m2
Ihorizon Solar irradiance on surface from sky horizon W/m2
Idome Solar irradiance on surface from sky dome W/m2
Icircumsolar Solar irradiance on surface from circumsolar region W/m2
a, b intermediate variables
F1, F2 Circumsolar and horizon brightening coefficients
F1, F2
α Incidence angle of sun on surface radians IncAng
Z Solar zenith angle radians ZenithAng
Δ Sky brightness factor
Delta
ε Sky clearness factor
Epsilon
m relative optical air mass
AirMass
I Direct normal solar irradiance W/m2 Material%Thickness
Fij Brightening coefficient factors
F11R, F12R, etc.
SunLitFrac
DifShdgRatioIsoSky
DifShdgRatioHoriz
θ Azimuth angle of point in sky radians Theta
φ Altitude angle of point in sky radians Phi
Ii Irradiance on surface from a horizon element W/m2
Iij Irradiance on surface from a sky dome element W/m2
SF Sunlit fraction
FracIlluminated

The following calculations are done in subroutine AnisoSkyViewFactors in the SolarShading module.

In the absence of shadowing, the sky formulation described above gives the following expression for sky diffuse irradiance, Isky, on a tilted surface:

Isky=Ihorizon+Idome+Icircumsolar

where

In the above equations:

Ih = horizontal solar irradiance (W/m2)

a = max(0,cosα)

b = max(0.087, cosZ)

F1 = circumsolar brightening coefficient

F2 = horizon brightening coefficient

where

α = incidence angle of sun on the surface (radians)

Z = solar zenith angle (radians).

The brightening coefficients are a function of sky conditions; they are given by

F1=F11(ε)+F12(ε)Δ+F13(ε)ZF2=F21(ε)+F22(ε)Δ+F23(ε)Z

Here the sky brightness factor is

Δ=Ihm/Io

where

m = relative optical air mass

Io = extraterrestrial irradiance (taken to have an average annual value of 1353 W/m2);

and the sky clearness factor is

ε=(Ih+I)/Ih+κZ31+κZ3

where

I = direct normal solar irradiance

κ = 1.041 for Z in radians

The factors Fij are shown in the following table. The Fij values in this table were provided by R. Perez, private communication, 5/21/99. These values have higher precision than those listed in Table # 6 of Perez et al., 1990.

Table 22. Fij Factors as a Function of Sky Clearness Range.
ε Range 1.000-1.065 1.065-1.230 1.230-1.500 1.500-1.950 1.950-2.800 2.800-4.500 4.500-6.200 > 6.200
F11 -0.0083117 0.1299457 0.3296958 0.5682053 0.8730280 1.1326077 1.0601591 0.6777470
F12 0.5877285 0.6825954 0.4868735 0.1874525 -0.3920403 -1.2367284 -1.5999137 -0.3272588
F13 -0.0620636 -0.1513752 -0.2210958 -0.2951290 -0.3616149 -0.4118494 -0.3589221 -0.2504286
F21 -0.0596012 -0.0189325 0.0554140 0.1088631 0.2255647 0.2877813 0.2642124 0.1561313
F22 0.0721249 0.0659650 -0.0639588 -0.1519229 -0.4620442 -0.8230357 -1.1272340 -1.3765031
F23 -0.0220216 -0.0288748 -0.0260542 -0.0139754 0.0012448 0.0558651 0.1310694 0.2506212

Sky diffuse solar shadowing on an exterior surface is calculated as follows in subroutine SkyDifSolarShading in the SolarShading module. The sky is assumed to be a superposition of the three Perez sky comp1onents described above.

For the horizon source the following ratio is calculated by dividing the horizon line into 24 intervals of equal length:

where* Iiis the unobstructed irradiance on the surface from the ith interval,SFi* is the sunlit fraction from radiation coming from the ith interval, and the sums are over intervals whose center lies in front of the surface. SFi is calculated using the beam solar shadowing method as though the sun were located at the ith horizon point. Here

Ii=E(θi)dθcosαi

where

E (θi) = radiance of horizon band (independent of θ)

= 2π/24 = azimuthal extent of horizon interval (radians)

θi = 0O, 15O, … , 345O

αi = incidence angle on surface of radiation from θi

The corresponding ratio for the isotropic sky dome is given by

where (i,j) is a grid of 144 points (6 in altitude by 24 in azimuth) covering the sky dome, Iij is the unobstructed irradiance on the surface from the sky element at the ijth point, SFij is the sunlit fraction for radiation coming from the ijth element, and the sum is over points lying in front of the surface. Here

Iij=E(θi,ϕj)cosϕjdθdϕcosαij

where

E (θi,φj) = sky radiance (independent of θ and φ for isotropic dome)

= 2π/24 = azimuthal extent of sky element (radians)

= (π/2)/6 = altitude extent of sky element (radians)

θi = 0O, 15O, … , 345O

φj = 7.5O, 22.5O, … , 82.5O

αij = incidence angle on surface of radiation from (θi,φj)

Because the circumsolar region is assumed to be concentrated at the solar disk, the circumsolar ratio is

where SFsun is the beam sunlit fraction. The total sky diffuse irradiance on the surface with shadowing is then

Isky=RhorizonIhorizon+RdomeIdome+RcircumsolarIcircumsolar

Rhorizon and Rdome are calculated once for each surface since they are independent of sun position.