Radiation Heat Transfer

All bodies above absolute zero emit thermal radiation. It needs no medium, and it grows with the fourth power of temperature, so it dominates at high temperatures.

Black body laws

• Stefan–Boltzmann law: a black body emits E_b = σT⁴, with σ = 5.67 × 10⁻⁸ W/m²·K⁴ (T in kelvin).
• Wien's displacement law: the wavelength of peak emission λ_max T = 2898 µm·K. The sun (≈ 5800 K) peaks at about 0.5 µm (visible light).
• Planck's law gives the full spectrum.

Real surfaces

• Emissivity ε = E/E_b (0 to 1). A grey body has the same ε at all wavelengths.
• Incident radiation is absorbed, reflected or transmitted: α + ρ + τ = 1 (τ = 0 for opaque bodies).
• Kirchhoff's law: at thermal equilibrium, ε = α. Good absorbers are good emitters.

View factors

F₁₂ is the fraction of radiation leaving surface 1 that reaches surface 2.

• Reciprocity: A₁F₁₂ = A₂F₂₁.
• Summation: Σ F₁ⱼ = 1 for an enclosure.
• A flat or convex surface cannot see itself: F₁₁ = 0.
• Small body inside a large enclosure: F₁₂ = 1.

Radiation exchange

Between two grey surfaces, use the network of resistances: surface resistance (1 − ε)/(εA) and space resistance 1/(A₁F₁₂).

• Two large parallel plates: q = σ(T₁⁴ − T₂⁴) / (1/ε₁ + 1/ε₂ − 1).
• Small body in a large enclosure: Q = ε₁ A₁ σ (T₁⁴ − T₂⁴).
• Radiation shields: with n shields of the same emissivity as the plates, the heat flow is divided by (n + 1).

Worked example

Two large parallel plates at 800 K and 400 K have ε = 0.8 each.

• σ(T₁⁴ − T₂⁴) = 5.67 × 10⁻⁸ × (4.096 × 10¹¹ − 2.56 × 10¹⁰) = 21,773 W/m².
• Factor 1/(1/0.8 + 1/0.8 − 1) = 1/1.5 = 0.667.
• q = 14,515 W/m². With one shield of ε = 0.8 in between: 7,258 W/m².

Practice