Solar radiation calculation using the atmospheric extinction coefficient

Direct radiation

Direct radiation refers to solar radiation that reaches the Earth's surface without being scattered or reflected by the atmosphere. It is selectively attenuated by atmospheric components and is a key input in solar heating simulations.

The thermal solver models direct radiation using the method which defines the direct normal irradiance I_DN [3] as:

where:

• A is the apparent extraterrestrial irradiation at air mass m = 0.
• B is the atmospheric extinction coefficient, which depends on the date and atmospheric conditions, and accounts for seasonal Earth-Sun distance variations and atmospheric water vapor.
• β is the solar altitude angle, the angle of the sun above the horizon.
• m=1/sinβ is the air mass at sea level, representing the relative path length that solar radiation travels through the atmosphere (compared to when the sun is directly overhead).

The computed irradiance is further adjusted by a clearness number to account for local atmospheric clarity, including water vapor content and elevation effects.

Diffuse radiation

The thermal solver models diffuse sky radiation using a simplified approach based on [3]. The model assumes that:

• The sky has a uniform radiation distribution.
• The diffuse radiation intensity is proportional to direct radiation via a constant called the diffuse sky factor.
• The shadowing effects are not considered.

Diffuse radiation I_diff incident on a tilted or vertical surface is calculated as:

where:

• C is the diffuse sky factor, a constant that scales the direct radiation to estimate diffuse intensity.
• I_DN is the direct normal irradiance.
• F_ss is the sky view factor, which accounts for the surface tilt angle Σ and is given by:

The tilt angle is the angle between the surface and the horizontal plane. A tilt angle of 0° indicates a horizontal surface, 90° indicates vertical.

Ground surface reflection

Ground surface reflection accounts for the portion of solar radiation that is reflected from the ground onto a surface. This is particularly important for inclined or vertical surfaces exposed to reflected sunlight.

The thermal solver models the reflected radiation I_r from the foreground using [3]:

where:

• I_tH is the total horizontal irradiation, composed of diffuse and direct components.
• ρ_g is the reflectance of the ground surface.
• F_sg is the view factor from the surface to the ground, given by:

The total horizontal irradiation is given by:

This formulation assumes the maximum available ground reflection and does not account for shadowing effects. It is suitable for estimating reflected solar gains in clear-sky conditions and open terrain.

You can also model the sky dome and ground explicitly to calculate view factors and shadowing effects. For more information see, Explicit sky and ground modeling.

Application

Use this approach when:

• Atmospheric clarity and solar position vary significantly over time or location.
• Accurate differentiation between direct, diffuse, and reflected components is needed.
• Shadowing effects are either negligible or can be approximated separately.