Fundamentals

Thermal radiation is electromagnetic energy emitted by matter at finite temperature. All surfaces above absolute zero emit thermal radiation across a continuous spectrum of wavelengths. Understanding radiation fundamentals is essential for HVAC applications including radiant heating, solar load calculations, and building envelope analysis.

Electromagnetic Spectrum

Thermal radiation occupies the portion of the electromagnetic spectrum from approximately 0.1 to 100 μm. The thermal radiation spectrum subdivides into three regions:

For HVAC applications at typical building temperatures (250-350K), thermal radiation occurs almost entirely in the infrared region.

Blackbody Radiation

A blackbody is an ideal surface that:

• Absorbs all incident radiation regardless of wavelength and direction
• Emits the maximum possible radiation at any given temperature
• Has emissivity ε = 1 and absorptivity α = 1

No real surface is a perfect blackbody, but the blackbody represents the upper limit for thermal radiation emission.

Planck’s Law

Planck’s law describes the spectral distribution of radiation emitted by a blackbody:

E_λ,b(λ,T) = (C₁) / [λ⁵(e^(C₂/λT) - 1)]

Where:

• E_λ,b = spectral emissive power of blackbody, W/(m²·μm)
• λ = wavelength, μm
• T = absolute temperature, K
• C₁ = 3.742 × 10⁸ W·μm⁴/m² (first radiation constant)
• C₂ = 1.439 × 10⁴ μm·K (second radiation constant)

Key observations from Planck’s law:

• At any wavelength, emissive power increases with temperature
• As temperature increases, the peak emission shifts to shorter wavelengths
• The area under the curve represents total emissive power

Wien’s Displacement Law

Wien’s displacement law determines the wavelength at which maximum spectral emissive power occurs:

λ_max·T = 2897.8 μm·K

Applications:

This relationship explains why room temperature surfaces emit infrared radiation while the sun emits primarily visible light.

Stefan-Boltzmann Law

The Stefan-Boltzmann law calculates total emissive power from a blackbody by integrating Planck’s law over all wavelengths:

E_b(T) = σT⁴

Where:

• E_b = total hemispherical emissive power, W/m²
• σ = 5.670 × 10⁻⁸ W/(m²·K⁴) (Stefan-Boltzmann constant)
• T = absolute temperature, K

For real surfaces with emissivity ε:

E(T) = εσT⁴

The total radiation emitted is proportional to the fourth power of absolute temperature, making radiation heat transfer highly sensitive to temperature changes.

Radiative Properties

Emissivity (ε)

Emissivity is the ratio of radiation emitted by a real surface to that emitted by a blackbody at the same temperature:

ε = E(T) / E_b(T)

Emissivity ranges from 0 to 1, where:

• ε = 1: Perfect blackbody
• ε = 0: Perfect reflector (no emission)

Absorptivity (α)

Absorptivity is the fraction of incident radiation absorbed by a surface:

α = G_absorbed / G_incident

Where:

• G_absorbed = absorbed irradiation, W/m²
• G_incident = incident irradiation, W/m²

Reflectivity (ρ)

Reflectivity is the fraction of incident radiation reflected by a surface:

ρ = G_reflected / G_incident

Transmissivity (τ)

Transmissivity is the fraction of incident radiation transmitted through a surface:

τ = G_transmitted / G_incident

Energy Balance

For an opaque surface (τ = 0):

α + ρ = 1

For a semitransparent surface:

α + ρ + τ = 1

Kirchhoff’s Law

Kirchhoff’s law states that for a surface in thermal equilibrium:

ε_λ(λ,T,θ,φ) = α_λ(λ,T,θ,φ)

In words: spectral, directional emissivity equals spectral, directional absorptivity at the same wavelength, temperature, and direction.

For diffuse, gray surfaces at thermal equilibrium:

ε = α

This simplification is commonly applied in HVAC calculations when:

• Surface temperature ≈ source temperature (within ~100K)
• Surface properties do not vary strongly with wavelength
• Directional effects are negligible

Emissivity Values for Common Building Materials

Key observations:

• Polished metals: very low emissivity (0.04-0.10)
• Oxidized metals: moderate to high emissivity (0.20-0.90)
• Non-metallic building materials: high emissivity (0.85-0.97)
• Surface finish significantly affects metal emissivity

Spectral vs. Total Properties

Spectral properties vary with wavelength:

• ε_λ(λ): spectral emissivity
• α_λ(λ): spectral absorptivity

Total properties are integrated over all wavelengths:

• ε: total hemispherical emissivity
• α: total hemispherical absorptivity

A gray surface has constant spectral properties (ε_λ = constant). Most non-metallic building materials approximate gray behavior in the thermal infrared spectrum.

Directional vs. Hemispherical Properties

Directional properties depend on emission/incidence angle (θ, φ):

• ε(θ,φ): directional emissivity
• α(θ,φ): directional absorptivity

Hemispherical properties are integrated over all directions:

• ε: hemispherical emissivity
• α: hemispherical absorptivity

A diffuse surface has constant directional properties. HVAC calculations typically assume diffuse surfaces unless dealing with specular reflections from polished metals or glass.

Graybody Approximation

A graybody is a real surface with:

• ε < 1 (emissivity less than blackbody)
• ε_λ = constant (spectral emissivity independent of wavelength)
• ε(θ) = constant (emissivity independent of direction)

For graybody surfaces:

E = εσT⁴

α = ε (when Kirchhoff’s law applies)

Most building materials can be treated as graybodies for HVAC radiation calculations.

Engineering Applications

Radiant heating systems: Stefan-Boltzmann law determines heat output from radiant panels or surfaces.

Solar heat gain: Absorptivity determines how much solar radiation is absorbed by building surfaces and converted to heat.

Low-emissivity coatings: Low-e windows use thin metallic coatings (ε ≈ 0.10-0.20) to reduce radiative heat transfer while maintaining visible light transmission.

Thermal comfort: Mean radiant temperature calculations require emissivity values for surrounding surfaces.

Building envelope: Radiative properties affect heat transfer through walls, roofs, and windows.

Calculation Example

Calculate the radiation emitted by a concrete floor at 75°F (297K) with emissivity ε = 0.95:

E = εσT⁴ E = 0.95 × 5.67×10⁻⁸ × (297)⁴ E = 0.95 × 5.67×10⁻⁸ × 7.78×10⁹ E = 419 W/m² (133 Btu/h·ft²)

For comparison, a blackbody at the same temperature emits:

E_b = σT⁴ = 441 W/m² (140 Btu/h·ft²)

The concrete floor emits 95% of blackbody radiation at this temperature.