| name | radiation |
| description | Thermal radiation heat transfer — Stefan-Boltzmann law, emissivity, view factors, radiation networks, blackbody/gray body, combined radiation+convection, radiation shields, solar radiation. |
| metadata | {"priority":6,"promptSignals":{"phrases":["radiation","thermal radiation","emissivity","view factor","Stefan-Boltzmann","blackbody","radiation heat transfer"],"minScore":4}} |
Radiation Heat Transfer — Complete Skill
Fundamentals
Thermal radiation: electromagnetic waves, λ = 0.1-100μm (infrared mainly for engineering)
Speed of light: c = 3×10⁸ m/s, c = λν (wavelength × frequency)
Does NOT require medium (works in vacuum)
Stefan-Boltzmann Law (blackbody):
E_b = σT⁴ [W/m²]
σ = 5.67×10⁻⁸ W/(m²·K⁴) [Stefan-Boltzmann constant]
Note: T must be in KELVIN always
Planck's Law (spectral distribution):
E_bλ = C₁/(λ⁵ × (exp(C₂/λT) - 1))
C₁ = 3.742×10⁸ W·μm⁴/m², C₂ = 1.439×10⁴ μm·K
Wien's Displacement Law (peak wavelength):
λ_max × T = 2898 μm·K
Sun (5800K): λ_max = 0.5μm (visible — efficient solar)
Room temp (293K): λ_max = 9.9μm (far infrared)
Real Surfaces (Gray, Diffuse Approximation)
Emissivity ε: fraction of blackbody emission actually emitted
E = ε × σT⁴ (emissive power of real surface)
G = irradiation (incident radiation from all directions)
α = absorptivity, ρ = reflectivity, τ = transmissivity (α+ρ+τ = 1)
Gray surface assumption: ε = α (absorptivity = emissivity, independent of λ)
Diffuse surface: emits/absorbs uniformly in all directions (independent of angle)
Typical emissivities ε:
| Surface | ε |
|---|
| Blackbody | 1.00 |
| Oxidized steel | 0.60-0.80 |
| Polished aluminum | 0.04-0.07 |
| Oxidized aluminum | 0.10-0.30 |
| Painted surfaces | 0.80-0.95 |
| Human skin | 0.95 |
| Brick | 0.85-0.95 |
| Glass (window) | 0.84-0.90 |
| Snow | 0.82-0.92 |
| Polished gold | 0.01-0.03 |
| Polished copper | 0.02-0.05 |
Radiation Between Two Surfaces — View Factors
F₁₂ = fraction of radiation leaving surface 1 that strikes surface 2
Reciprocity: A₁F₁₂ = A₂F₂₁
Summation rule: ΣF₁j = 1 (for enclosure)
Key view factors:
- Infinite parallel plates: F₁₂ = 1
- Small surface in large enclosure: F₁₂ = 1
- Coaxial cylinders (1=inner, 2=outer): F₁₂ = 1, F₂₁ = A₁/A₂
- Infinite concentric cylinders: same as coaxial
- Two small areas, angle θ: F₁₂ = cosθ₁cosθ₂/(πr²)×dA₂
Radiation from disk to disk (coaxial):
F₁₂ = 0.5×(S - √(S²-4(r₂/r₁)²)) where S = 1+(1+R₂²)/R₁², R=r/L
(From radiation handbooks — lookup for complex geometries)
Net Radiation Exchange (Gray, Diffuse Enclosure)
Radiosity J: total radiation leaving surface (emitted + reflected)
J = εσT⁴ + ρG = εσT⁴ + (1-ε)G
Net heat flux from surface:
q = (J-G)×A = (E_b - J)/((1-ε)/(εA)) [surface resistance R_surface = (1-ε)/(εA)]
Space resistance (between surfaces):
R_space,12 = 1/(A₁F₁₂) [geometric resistance]
Net heat transfer:
q₁₂ = (J₁-J₂)/(1/(A₁F₁₂)) = A₁F₁₂(J₁-J₂)
Two Gray Surfaces (Important Cases)
Two infinite parallel plates:
q/A = σ(T₁⁴-T₂⁴) / (1/ε₁ + 1/ε₂ - 1)
Small convex object (1) in large enclosure (2):
q₁ = ε₁A₁σ(T₁⁴-T₂⁴) [F₁₂=1, A₁<<A₂ → surface resistance of enclosure = 0]
Two large concentric cylinders/spheres:
q = A₁σ(T₁⁴-T₂⁴) / (1/ε₁ + (1-ε₂)/ε₂ × A₁/A₂)
General two-surface enclosure:
q₁₂ = σ(T₁⁴-T₂⁴) / ((1-ε₁)/(ε₁A₁) + 1/(A₁F₁₂) + (1-ε₂)/(ε₂A₂))
Radiation Shields
Shield between two parallel plates (same area, shield emissivity ε_s):
With N shields: q = σ(T₁⁴-T₂⁴) / (1/ε₁ + 1/ε₂ - 1 + N×(2/ε_s - 1))
Low ε_s (polished) → dramatic reduction in q
Mylar (aluminized): ε ≈ 0.03 → 1 shield reduces q by ~32× vs. black surfaces
Combined Radiation + Convection
q_total = q_conv + q_rad = h_conv×A×(T_s-T_∞) + ε×σ×A×(T_s⁴-T_surr⁴)
Linearized radiation coefficient:
h_rad = εσ(T_s+T_surr)(T_s²+T_surr²) [W/(m²K)]
Valid for small ΔT only; use exact form (T⁴ difference) for large ΔT
Solar Radiation
G_solar = 1353 W/m² (solar constant at top of atmosphere)
Surface intensity: G_s = G_solar × cos(θ_incidence) × τ_atmosphere
τ_atm ≈ 0.7-0.9 (clear sky), 0.3-0.5 (cloudy)
Solar absorptivity α_solar ≠ thermal emissivity ε_thermal (different wavelength ranges)
White paint: α_solar = 0.12, ε_thermal = 0.90 (reflects sun, emits IR — good for cooling)
Black paint: α_solar = 0.95, ε_thermal = 0.90 (absorbs sun and emits IR)
Polished Al: α_solar = 0.10, ε_thermal = 0.05 (reflects sun, poor IR emitter — hot spacecraft surface)
Output
Provide: q [W], view factors F₁₂, radiosity J, h_rad [W/m²K] if linearized, comparison with convection magnitude.
