Sky Radiance Model[LINK]

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

\[AnisoSkyMultiplie{r_{surface}}\cdot DiffuseSolarIrradiance\]

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
Ih	Horizontal solar irradiance	W/m2
S	Surface tilt	radians	Surface(SurfNum)%Tilt*DegToRadians
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
\[\Delta\]	Sky brightness factor		Delta
ε	Sky clearness factor		Epsilon
m	relative optical air mass		AirMass
IO	Extraterrestrial solar irradiance	W/m2
I	Direct normal solar irradiance	W/m2	Material%Thickness
κ	constant = 1.041 for Z in radians	radians-3
Fij	Brightening coefficient factors		F11R, F12R, etc.
Rcircumsolar	Shadowing factor for circumsolar radiation		SunLitFrac
Rdome	Shadowing factor for sky dome radiation		DifShdgRatioIsoSky
Rhorizon	Shadowing factor for horizon radiation		DifShdgRatioHoriz
E	Sky radiance	W/m2
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
I’	Sky solar irradiance on surface with shadowing	W/m2

Sky Diffuse Solar Radiation on a Tilted Surface[LINK]

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, I_sky, on a tilted surface:

\[{I_{sky}} = {I_{horizon}} + {I_{dome}} + {I_{circumsolar}}\]

where

\[\begin{array}{rcl} I_{horizon} & = \rm{irradiance on surface from sky horizon} & = I_h F_2\sin S \\ I_{dome} & = \rm{irradiance on surface from sky dome} & = I_h (1 - F_1)(1 + \cos S)/2 \\ I_{circumsolar} & = \rm{irradiance on surface from circumsolar region} & = I_h F_1 a/b \end{array}\]

AnisoSkyMult is then I_sky /DifSolarRad.

In the above equations:

I_h = horizontal solar irradiance (W/m²)

S = surface tilt (radians)

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

\[\begin{array}{rl} F_1 & = F_{11}(\varepsilon ) + F_{12}(\varepsilon )\Delta + F_{13}(\varepsilon )Z \\ F_2 & = F_{21}(\varepsilon ) + F_{22}(\varepsilon )\Delta + F_{23}(\varepsilon )Z \end{array}\]

Here the sky brightness factor is

\[\Delta = {I_h}m/{I_o}\]

where

m = relative optical air mass

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

and the sky clearness factor is

\[\varepsilon = \frac{{({I_h} + I)/{I_h} + \kappa {Z^3}}}{{1 + \kappa {Z^3}}}\]

where

I = direct normal solar irradiance

κ = 1.041 for Z in radians

The factors F_ij  are shown in the following table. The F_ij 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.  F_ij 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

Shadowing of Sky Diffuse Solar Radiation[LINK]

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:

\[{R_{horiz}} = \frac{{{\rm{Irradiance from horizon with obstructions}}}}{{{\rm{Irradiance from horizon without obstructions}}}} = \frac{{\sum\limits_{i = 1}^{24} {{I_i}S{F_i}} }}{{\sum\limits_{i = 1}^{24} {{I_i}} }}\]

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

\[{I_i} = E({\theta_i})d\theta \cos {\alpha_i}\]

where

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

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

θ_i = 0^O, 15^O, … , 345^O

α_i = incidence angle on surface of radiation from θ_i

The corresponding ratio for the isotropic sky dome is given by

\[{R_{dome}} = \frac{{{\rm{Irradiance from dome with obstructions}}}}{{{\rm{Irradiance from dome without obstructions}}}} = \frac{{\sum\limits_{i = 1}^{24} {\sum\limits_{j = 1}^6 {{I_{ij}}S{F_{ij}}} } }}{{\sum\limits_{i = 1}^{24} {\sum\limits_{j = 1}^6 {{I_{ij}}} } }}\]

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

\[{I_{ij}} = E({\theta_i},{\phi_j})\cos {\phi_j}d\theta d\phi \cos {\alpha_{ij}}\]

where

E (θ_i,φ_j) = sky radiance (independent of θ and φ for isotropic dome)

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

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

θ_i = 0^O, 15^O, … , 345^O

φ_j = 7.5^O, 22.5^O, … , 82.5^O

α_ij = incidence angle on surface of radiation from (θ_i,φ_j)

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

\[{R_{circumsolar}} = \frac{{{\rm{Irradiance from circumsolar region with obstructions}}}}{{{\rm{Irradiance from circumsolar without obstructions}}}} = S{F_{sun}}\]

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

\[{I'_{sky}} = {R_{horizon}}{I_{horizon}} + {R_{dome}}{I_{dome}} + {R_{circumsolar}}{I_{circumsolar}}\]

R_horizon and R_dome are calculated once for each surface since they are independent of sun position.

With shadowing we then have:

AnisoSkyMult = I’_sky /DifSolarRad.

Shadowing of Sky Long-Wave Radiation[LINK]

EnergyPlus calculates the sky long-wave radiation incident on exterior surfaces assuming that the sky long-wave radiance distribution is isotropic. If obstructions such as overhangs are present the sky long-wave incident on a surface is multiplied by the isotropic shading factor, R_dome, described above. The long-wave radiation from these obstructions is added to the long-wave radiation from the ground; in this calculation both obstructions and ground are assumed to be at the outside air temperature and to have an emissivity of 0.9.