Radiation Heat Transfer

The Stefan-Boltzmann law

Radiation is the third mode of heat transfer, alongside conduction and convection. The crucial difference is that it carries energy as electromagnetic waves rather than through a material, so it works perfectly across a vacuum. For an ideal emitter — a blackbody — the total power radiated per unit area over all wavelengths is:

E_b = σ · T⁴       (blackbody emissive power, W/m²)

Q   = ε · σ · A · T⁴   (real surface, total power, W)

where:
  σ = 5.670×10⁻⁸ W/m²K⁴   (Stefan-Boltzmann constant)
  ε = emissivity            (0 ≤ ε ≤ 1, dimensionless)
  A = radiating area        (m²)
  T = absolute temperature  (K, never °C)

The single most important feature is the fourth power. Because Q ∝ T⁴, doubling the absolute temperature multiplies the radiated power by sixteen. A surface at 600 K radiates 16× more than the same surface at 300 K, and a tungsten filament at 3000 K radiates roughly 10,000× more than it would at 300 K. Always work in kelvin — using °C here gives nonsense.

The radiation isn't monochromatic; it spans a spectrum described by Planck's law, and the wavelength of peak emission follows Wien's displacement law, λ_max·T ≈ 2898 µm·K. At human body temperature (310 K) the peak sits near 9.4 µm in the far infrared, invisible to the eye. Heat a steel bar past ~800 K and the spectrum's short-wavelength tail reaches into the visible, and it glows red — the same physics that lets a thermal camera "see" warm objects in the dark.

Emissivity and Kirchhoff's law

Real surfaces are not blackbodies. Emissivity ε scales the blackbody power down to what an actual surface emits. By Kirchhoff's law of thermal radiation, at a given temperature and wavelength a surface's emissivity equals its absorptivity: a good emitter is a good absorber. This single fact governs every radiation design decision — you cannot make a surface shed heat well by radiation without also making it soak up incoming radiation well.

Surface	Emissivity ε (long-wave IR)	Behaviour	Typical use
Polished silver / aluminum	0.02 – 0.05	Reflects, barely emits	Vacuum-flask walls, MLI on spacecraft
Bright stainless steel	0.10 – 0.20	Low emitter	Cryogenic shields, kitchen surfaces
Oxidized / weathered metal	0.6 – 0.85	Emits well once dulled	Engine exhausts, old radiators
Black anodized aluminum	0.80 – 0.88	Strong emitter, light weight	Electronics heat sinks, optics housings
Matte black paint / soot	0.95 – 0.98	Near-blackbody	Radiator panels, solar absorbers, sensor baffles
Human skin	~0.98	Near-blackbody in IR	Why thermal cameras read body temperature accurately

Note that emissivity is wavelength-dependent. A selective surface exploits this: a solar collector coating can have high absorptivity (≈0.95) for the Sun's visible/near-IR spectrum yet low emissivity (≈0.1) in the long-wave IR it would otherwise re-radiate, trapping the energy. Spacecraft white paint does the opposite — low solar absorptivity, high IR emissivity — to stay cool in sunlight.

Net exchange between two surfaces

A surface doesn't just emit — it also receives radiation from everything around it. What matters for heat flow is the net exchange. For a small grey object at T₁ inside large surroundings at T₂ (the surroundings effectively act as a blackbody), the net radiated power is:

Q_net = ε · σ · A · (T₁⁴ − T₂⁴)

When both surfaces are finite and grey, the geometry enters through the view factor F₁₂ and both emissivities. For two opaque grey surfaces forming an enclosure:

           σ · (T₁⁴ − T₂⁴)
Q_net = ───────────────────────────────────────────
        (1−ε₁)/(ε₁A₁) + 1/(A₁F₁₂) + (1−ε₂)/(ε₂A₂)

The three denominator terms are radiation "resistances":
  surface resistance 1  +  space (geometric) resistance  +  surface resistance 2

This radiation-network view is powerful: you treat each emissivity term and each view-factor term as a resistor and add them in series, exactly like an electrical circuit. For two large parallel plates (A₁ = A₂ = A, F₁₂ = 1) it collapses to the form used for vacuum flasks:

         σ · A · (T₁⁴ − T₂⁴)
Q_net = ───────────────────────
          1/ε₁ + 1/ε₂ − 1

View factors

The view factor F₁₂ is the fraction of radiation leaving surface 1 that strikes surface 2 directly. It is pure geometry — it depends only on the sizes, shapes, separation and orientation of the surfaces, never on temperature or emissivity. Two rules make view-factor bookkeeping tractable:

• Reciprocity: A₁·F₁₂ = A₂·F₂₁. A small surface facing a large one has a large view factor toward it, but the large surface sees only a small fraction back.
• Summation: for any surface in an enclosure, the view factors to all surfaces it can see (including itself, if concave) sum to 1: ΣF₁ⱼ = 1.

Concrete cases: two infinite parallel plates have F₁₂ = 1. A small instrument package floating deep in space radiates with F ≈ 1 toward the cold sky (≈ 3 K). A pipe inside a large duct has F = 1 toward the duct, while the duct sees only a sliver of the pipe. Misjudging the view factor is one of the most common errors in radiator sizing — a panel that "sees" a hot neighbouring structure instead of cold space can lose half its rejection capacity.

Worked example: radiative cooling of a satellite panel

A flat radiator panel on a spacecraft is 1.5 m × 1.0 m (A = 1.5 m²), coated matte white with ε = 0.92, held at T₁ = 320 K, and radiates to deep space at T₂ = 4 K with view factor F ≈ 1. How much heat does it reject?

Q_net = ε · σ · A · (T₁⁴ − T₂⁴)

  ε = 0.92
  σ = 5.670×10⁻⁸ W/m²K⁴
  A = 1.5 m²
  T₁⁴ = 320⁴   = 1.049×10¹⁰ K⁴
  T₂⁴ = 4⁴     = 256 K⁴  (negligible)

Q = 0.92 × 5.670×10⁻⁸ × 1.5 × 1.049×10¹⁰
  = 0.92 × 5.670×10⁻⁸ × 1.574×10¹⁰
  = 0.92 × 892
  ≈ 821 W

The panel sheds about 821 W with no fan, no coolant, no air — pure radiation. Drop the panel temperature to 280 K and the T⁴ term falls to 6.15×10⁹, cutting rejection to ≈ 481 W: a 12.5% drop in temperature costs you 41% of your cooling. This steep penalty is exactly why spacecraft thermal engineers fight to keep radiators as hot as the payload will tolerate.

Worked example: a silvered vacuum flask

Compare the radiant heat leak through a vacuum gap between two walls, both at A = 0.05 m², inner wall T₁ = 363 K (hot coffee, 90 °C), outer wall T₂ = 293 K (20 °C). First with matte-black walls (ε = 0.95), then with silvered walls (ε = 0.04):

Q = σ·A·(T₁⁴ − T₂⁴) / (1/ε₁ + 1/ε₂ − 1)

  σ·A·(T₁⁴ − T₂⁴) = 5.670×10⁻⁸ × 0.05 × (363⁴ − 293⁴)
                   = 5.670×10⁻⁸ × 0.05 × (1.736×10¹⁰ − 7.37×10⁹)
                   = 5.670×10⁻⁸ × 0.05 × 9.99×10⁹
                   ≈ 28.3 W   (numerator)

Black walls:   1/0.95 + 1/0.95 − 1 = 1.105   →  Q ≈ 25.6 W
Silvered:      1/0.04 + 1/0.04 − 1 = 49.0    →  Q ≈ 0.58 W

Silvering the walls cuts the radiant leak from ≈25.6 W to ≈0.58 W — a 44× reduction — which is why a real thermos keeps liquid hot for hours. The vacuum already kills conduction and convection; lowering emissivity is the only lever left for the radiation path.

Radiation versus conduction and convection

	Conduction	Convection	Radiation
Carrier	Lattice / electrons	Moving fluid	Electromagnetic waves
Needs a medium?	Yes (solid/fluid)	Yes (fluid)	No — works in vacuum
Governing law	Fourier: q = −k∇T	Newton: q = h·ΔT	Stefan-Boltzmann: q = εσT⁴
Temperature scaling	∝ ΔT	∝ ΔT (h may vary)	∝ (T₁⁴ − T₂⁴)
Dominant when	Solids, small gaps	Fluids in motion	High T, vacuum, large ΔT
Key material property	Thermal conductivity k	Heat-transfer coeff. h	Emissivity ε
Speed of action	Diffusive (slow)	Bulk flow	Speed of light

At everyday temperatures around 300 K, natural-convection and radiative heat-transfer coefficients are comparable — both roughly 5–10 W/m²K — so neither dominates. The decisive crossover is temperature: because radiation scales as T⁴ while convection scales roughly as ΔT, radiation overtakes everything once surfaces glow. By 1000 K a furnace wall sheds most of its heat radiantly; in a vacuum, radiation isn't merely dominant, it's the only path.

Where radiation rules the design

• Reentry heat shields. A capsule at Mach 25 sits in a shock layer near 10,000 K; the ablative or ceramic surface re-radiates a huge fraction of the incoming flux back out — radiation is part of the survival strategy, not just a loss.
• Rocket nozzles and afterburners. Walls run at 1000–1800 K, where radiation is the leading loss; nozzle extensions are often "radiatively cooled" — designed to glow and dump heat to space rather than carry coolant.
• Spacecraft thermal control. With no atmosphere, every watt must leave by radiation. Engineers tune emissivity with coatings and stack reflective multi-layer insulation (MLI) to throttle it.
• Furnaces and kilns. Above ~700 °C the load is heated mostly by radiation from the walls and flame, and view factors decide which parts of the charge heat fastest.
• Electronics and LEDs. Black-anodized heat sinks add a radiative path on top of convection; in still air, radiation can carry 20–40% of the total from a finned sink.
• Building energy. Low-emissivity (low-e) window coatings reflect long-wave IR back into a room in winter and out in summer, cutting radiative loss through glass.

Failure modes and design traps

• Designing in °C instead of K. The T⁴ law only works in absolute temperature. Plugging in Celsius is the classic beginner error and can be wrong by orders of magnitude.
• Ignoring the surroundings' temperature. A surface emits Q = εσAT₁⁴ but also absorbs from its surroundings; only the net (T₁⁴ − T₂⁴) matters. Forgetting the second term overestimates cooling when surroundings are warm.
• Emissivity drift in service. A bright surface that oxidizes, gets contaminated, or accumulates micrometeoroid pitting can see its emissivity climb from 0.1 to 0.6 over a mission, wrecking the original thermal balance — radiator and shield coatings are chosen for end-of-life ε, not pristine ε.
• View-factor blunders. Pointing a radiator where it "sees" a hot adjacent structure, the Sun, or a planet instead of cold space slashes net rejection; this is a frequent cause of spacecraft overheating.
• Forgetting the selective-surface trade. Solar absorptivity and IR emissivity are different numbers. A coating chosen to absorb sunlight will also be a strong IR emitter unless engineered to be spectrally selective.
• Assuming radiation is negligible at "low" temperature. In a vacuum or a still, evacuated gap, radiation may be small in absolute terms but it is the only leak — and over weeks it dominates a cryostat's heat budget.