Solar radiation calculation using altitude and turbidity

The thermal solver computes solar flux by modeling direct, diffuse components as a function of altitude and turbidity.

This solar heating model enhances accuracy by incorporating parameters such as altitude, turbidity, and overcast conditions into the calculation of direct and diffuse radiation. It also includes precise view factor and shadowing calculations for diffuse and ground surface reflected radiation.

Direct radiation

The method described in [40] models direct solar radiation by incorporating the effects of altitude and atmospheric turbidity. It refines the estimation of direct radiation transmittance through the atmosphere using empirical corrections and air mass formulations.

The direct solar irradiance at the surface is:

where G₀ is the normal extraterrestrial irradiance, which is given by:

where:

G_sc ≅ 1367 W/m² is the solar constant.
n is the day of the year, for example, Jan 1 = 1, Dec 31 = 365 or 366.

The atmospheric transmittance for direct radiation, τ_D, is calculated as:

where:

T_LK is the corrected dimensionless Linke turbidity factor, [22].
m is the relative optical air mass.
δ_R(m) is the Rayleigh optical thickness at air mass m.

The relative optical air mass m is calculated using the following formula [23]:

where:

h₀^ref is the solar altitude h_o in degrees corrected for atmospheric refraction component Δh₀^ref.
p/p₀= exp(-z/8434.5) is the pressure correction for elevation z in meters.

The Rayleigh optical thickness δ_R(m) is computed as follows [22]:

For m ≤ 20:

For m > 20:

Values for the Linke turbidity factor T_LK can be obtained from online databases or literature sources [24, 35].

Diffuse radiation

This model [40] computes diffuse solar radiation based on the Linke turbidity factor. As atmospheric turbidity increases under cloudless skies, diffuse irradiance increases while direct irradiance decreases.

The diffuse irradiance on a horizontal surface, I_diff [W/m²], is calculated as:

where:

T_n(T_LK) is the diffuse transmission function dependent on the Linke turbidity factor.
F_d(h₀) is the diffuse solar altitude function dependent on the solar altitude angle h₀ [35].

The transmission function is given by a second-order polynomial:

The solar altitude function is expressed as:

The coefficients A₁, A₂, and A₃ are functions of the Linke turbidity T′_LK defined in the following expressions:

Note:

You can select another method that defines overcast conditions by using the clearness index. For more information, see Computing radiation under overcast conditions.

Ground surface reflection

The thermal solver uses a similar approach for computing ground surface reflection described in Solar radiation calculation using the atmospheric extinction coefficient.

The model also includes optional explicit calculations for sky dome and ground geometry to enhance view factor and shadowing accuracy. For more information see, Explicit sky and ground modeling.

Application

Use this approach when:

Environmental accuracy is important such as in high-altitude locations, areas with variable atmospheric clarity (turbidity), or cloudy conditions.
Performing thermal modeling of buildings, solar panels, or outdoor equipment where shadowing, surface orientation, and ground reflection significantly affect solar heat gain.
Shadowing effects are significant.