Solar Radiation Modeling — Solar Heat Load and Sol-Air Temperature in CAE Thermal Analysis

It's not just for architecture. Solar radiation modeling is active in roughly four areas.

• Vehicle Solar Load Analysis — The reason a parked car's dashboard exceeds 80°C in midsummer. Calculates solar transmittance through windows and absorption by interior materials.
• Solar Panel Power Generation Prediction — Annual power generation depends on panel temperature. Since efficiency drops by about 0.4% per 1°C temperature rise, thermal design is directly linked.
• Building Envelope HVAC Load Calculation — Solar radiation on exterior walls, roofs, and windows affects heating and cooling energy.
• Spacecraft Thermal Control — In orbit without an atmosphere, solar radiation and infrared radiation are the only heat inputs and outputs.

It can't be explained by convection alone. Window glass transmits most visible light, so direct solar radiation hits the dashboard and heats its surface. This is a problem of radiation transmission and absorption. The common key point is modeling solar radiant energy by separating it into three components: direct solar radiation, diffuse solar radiation, and ground-reflected radiation. And the sun's position can be calculated geometrically from latitude, longitude, and date/time.

The irradiance received by a surface perpendicular to the sun's rays outside the atmosphere is called the Solar Constant. The latest satellite measurement value is:

However, Earth's orbit is elliptical, so the Earth-Sun distance varies seasonally. The extraterrestrial solar irradiance $G_0$ is corrected using the day number $n$ (Jan 1 = 1):

It becomes about 1412 W/m² in early January (perihelion) and about 1321 W/m² in early July (aphelion). The variation range is about ±3.3%.

Yes, that's the case outside the atmosphere when facing the sun directly. By the time it reaches the ground, it's attenuated by atmospheric absorption and scattering, so even on a clear noon, direct solar radiation at the ground surface is reduced to about 900–1000 W/m². The "AM1.5G = 1000 W/m²" used in the Standard Test Conditions (STC) for solar cells is this ground-level standard value.

The sun's position is expressed by two angles: the zenith angle $\theta_z$ (angle from the vertical) and the azimuth angle $\gamma_s$ (angle from south). The zenith angle can be calculated by:

Where:

• $\phi$ — Latitude (positive for north)
• $\delta$ — Declination (seasonal north-south movement of the sun). Approximated by Cooper's formula: $\delta = 23.45° \sin\!\left(\frac{360(284+n)}{365}\right)$
• $\omega$ — Hour angle. 0 at solar noon, 15° per hour. Negative in the morning, positive in the afternoon.

Summer solstice: $\delta = 23.45°$, noon: $\omega = 0$, so:

$\cos\theta_z = \sin(35.7°)\sin(23.45°) + \cos(35.7°)\cos(23.45°)\cos(0°) = 0.232 + 0.744 = 0.976$

$\theta_z \approx 12.3°$ — meaning the sun is almost directly overhead. On the winter solstice, $\delta = -23.45°$, so $\theta_z \approx 59.2°$. The sun's altitude is lower and solar radiation becomes weaker. CAE solvers usually automatically perform this calculation when you input latitude, longitude, and date/time.

The indicator that determines the degree of atmospheric attenuation is Air Mass (AM). It's the relative optical path length of sunlight through the atmosphere, approximated from the zenith angle $\theta_z$:

If the sun is directly overhead ($\theta_z = 0°$), AM = 1.0; if $\theta_z = 60°$, AM = 2.0. The lower the sun, the longer the path, increasing absorption and scattering by the atmosphere. However, for $\theta_z > 80°$, Earth's curvature cannot be ignored, so Kasten's correction formula is used:

A representative model is the Hottel Clear Sky Model. The Direct Normal Irradiance (DNI) is:

Here $a_0$, $a_1$, $k$ are constants determined by altitude and climate type. A simpler approximation is:

Using this formula, for AM=1.5, $G_{bn} \approx 1361 \times 0.7^{1.5^{0.678}} \approx 845\;\text{W/m}^2$, which matches measurements well. In practice, using TMY (Typical Meteorological Year) data is common.

If you only consider direct radiation, you completely ignore solar radiation on cloudy days. Solar energy reaching the ground surface comes via three paths:

• Direct Solar Radiation $G_b$ (Beam/Direct) — The component coming straight from the sun. It creates shadows. Horizontal direct irradiance is $G_b = G_{bn} \cos\theta_z$.
• Diffuse Solar Radiation $G_d$ (Diffuse) — The component scattered by molecules and aerosols in the atmosphere, coming from all sky directions. On cloudy days, most solar radiation is this. Even on clear days, it accounts for 15–25% of total solar radiation.
• Ground-Reflected Radiation $G_r$ (Ground-reflected) — The component reflected by the ground and incident from below onto an inclined surface. It can be significant for snow surfaces (albedo 0.7–0.9).

Total horizontal irradiance (GHI: Global Horizontal Irradiance) is $G = G_b + G_d$. Meteorological data often measures GHI and Diffuse Horizontal Irradiance (DHI).

Exactly! The albedo (reflectance) of snow is 0.7–0.9, over 5 times that of asphalt (0.1–0.15). In solar power plants in Hokkaido or Tohoku, the ground-reflected component in winter can boost annual power generation by 3–5% in some cases. To evaluate this correctly in CAE, a model that separates the three components is essential.

This is the core part of solar radiation modeling. The total irradiance on an inclined surface $G_T$, in the simplest form using Liu's isotropic diffuse model, is:

Where:

• $\beta$ — Tilt angle (angle from horizontal)
• $R_b$ — Geometric correction factor for direct radiation. $R_b = \cos\theta_i / \cos\theta_z$ ($\theta_i$ is the angle of incidence on the inclined surface)
• $\rho_g$ — Ground albedo (reflectance). Grass ≈0.2, Concrete ≈0.3, Fresh Snow ≈0.8
• $(1 + \cos\beta)/2$ — View factor of the sky from the inclined surface
• $(1 - \cos\beta)/2$ — View factor of the ground from the inclined surface

Letting the surface azimuth angle be $\gamma$ (south-facing=0, west=positive):

It's long, but essentially it's the dot product of the sun's direction vector and the surface's normal vector. CAE solvers automatically recognize the surface normal direction, so users don't input this formula directly. However, the isotropic diffuse model has slightly lower accuracy for cloudy conditions. More precise models like the Perez model or Klucher model consider sky anisotropy (increased brightness around the sun, increased brightness near the horizon).

Sol-Air temperature is a boundary condition that consolidates the effects of solar radiation and longwave radiation into an "equivalent outdoor air temperature." It's very convenient for heat transfer calculations of walls and roofs:

The meaning of each term:

• $T_{\text{air}}$ — Outdoor air temperature [°C]
• $\alpha_s$ — Surface solar absorptance. Black wall ≈0.9, white wall ≈0.3, aluminum foil ≈0.1
• $G_T$ — Total irradiance on the inclined surface [W/m²]
• $h_o$ — Overall exterior surface heat transfer coefficient (convection + radiation). Typically 15–25 W/(m²·K)
• $\varepsilon$ — Surface longwave emissivity
• $\Delta R$ — Correction term due to longwave radiative exchange between the surface and the sky. About 63 W/m² for horizontal surfaces, about 0 for vertical surfaces.

Good question. Let's calculate for Tokyo, August noon, outdoor air temperature 35°C, total horizontal irradiance 800 W/m²:

Black roof ($\alpha_s = 0.9$, $h_o = 22$, $\varepsilon = 0.9$, $\Delta R = 63$)

$T_{\text{sol-air}} = 35 + \frac{0.9 \times 800}{22} - \frac{0.9 \times 63}{22} = 35 + 32.7 - 2.6 = 65.1°\text{C}$

This means the exterior surface of a black roof receives the same thermal load as if it were in an environment with an air temperature of 65°C. Simply changing to white paint ($\alpha_s = 0.3$):

$T_{\text{sol-air}} = 35 + \frac{0.3 \times 800}{22} - 2.6 = 43.3°\text{C}$

It drops by about 22°C. This is the principle of cool roofs (high-reflectance paint), and evaluating Sol-Air temperature with CAE directly connects to energy-saving HVAC design.

The most common is the Ray Tracing Method (Solar Ray Tracing). It shoots parallel rays from the sun's direction and geometrically calculates incidence and shading on each surface mesh:

1. Sun Direction Vector Calculation — Calculate $(\theta_z, \gamma_s)$ from date/time and latitude, generate a 3D sun direction vector $\hat{s}$.
2. Shadow Determination — Backtrace from the center of each surface mesh. Evaluate shading of direct solar radiation by intersection tests with other surfaces.
3. Incidence Angle Calculation — Dot product of surface normal vector $\hat{n}$ and sun direction $\hat{s}$: $\cos\theta_i = \hat{n} \cdot \hat{s}$.
4. Absorbed Heat Flux Calculation — $q_{\text{solar}} = \alpha_s \cdot G_{bn} \cdot \max(0, \cos\theta_i)$.

Ansys Fluent's "Solar Load Model" and STAR-CCM+'s "Solar Radiation" adopt this method.

Diffuse radiation doesn't have a fixed direction (comes from the entire sky), so you can't do ray tracing like for direct radiation. There are two implementation methods:

• Isotropic Diffuse Model — Assumes the sky is uniform over a hemisphere and multiplies by the sky view factor from the surface $(1 + \cos\beta)/2$. Fast calculation.
• Multi-Directional Ray Tracing — Divides the sky into many patches and shoots rays from each patch to evaluate shading. Can accurately calculate the influence of diffuse radiation on complex shapes like cities or factories, but computational cost is high.