Thermal Radiation
Stefan–Boltzmann, emissivity and absorptivity, view factors, radiation between surfaces, shields and solar heating.
Blackbody radiation and Stefan–Boltzmann
Every body above absolute zero emits electromagnetic radiation, with an intensity that grows with the fourth power of its absolute temperature. A perfect emitter and absorber, a blackbody, radiates
q″ = σ T⁴ σ = 5.670 × 10⁻⁸ W/m²·K⁴
Worked example: a hot panel in a cool room
- A blackbody at 600 K (327 °C) emits σ T⁴ = 7.35 kW/m². Surrounded by walls at 20 °C it also absorbs radiation from them, so the net exchange is σ (T⁴ − Tsurr⁴) = 6.93 kW/m².
- A real surface with ε = 0.8 radiates 0.8 of that: 5.54 kW/m².
- Radiation is a strong function of temperature: from a 1 000 K (727 °C) furnace wall at ε = 0.8 to 300 K surroundings, the net flux is 119 kW/m², about 21 times that of the 600 K panel for a temperature 1.7 times higher.
Emissivity, absorptivity and Kirchhoff’s law
A surface can absorb a fraction α of the radiation that reaches it, reflect the rest (and, if it is not opaque, transmit some). Kirchhoff’s law says that at the same temperature and wavelength a surface’s emissivity equals its absorptivity, ε = α: a good absorber is a good emitter. The shiny surface that reflects the sun also radiates little.
| Surface | Emissivity ε (long-wave, near room temperature) |
|---|---|
| Black paint, soot, anodized aluminum | 0.85 to 0.95 |
| Water, human skin, most paints, glass | 0.90 to 0.96 |
| Oxidized steel, rusty iron | 0.6 to 0.85 |
| Concrete, brick, wood | 0.85 to 0.95 |
| Polished aluminum, copper | 0.03 to 0.06 |
| Galvanized steel (new) | 0.2 |
| Aluminum foil, polished silver | 0.02 to 0.05 |
Metals are poor emitters when bright and become much better when oxidized, painted or anodized: painting a bare aluminum heat sink black can raise its radiation heat loss by a factor of ten and cut its temperature by several degrees in natural convection.
Where the energy is: Wien’s law
λmax = 2 898 µm·K ÷ T
The sun (5 772 K) peaks at 0.50 µm in the visible; a furnace at 1 000 °C at 2.3 µm in the near infrared; a room-temperature body (300 K) at 9.7 µm in the long-wave infrared. That is why a surface can be a poor absorber of sunlight and a good emitter at room temperature (white paint) or the reverse (a solar absorber coating), and why glass, which passes sunlight but blocks long-wave infrared, makes a greenhouse.
Radiation between surfaces
The heat exchanged between two surfaces depends on their temperatures, emissivities and how well they “see” each other (the view factor F₁₂, the fraction of the radiation leaving 1 that reaches 2). Two common cases have simple answers:
- A small object in a large enclosure: Q̇ = ε σ A (T₁⁴ − T₂⁴) (view factor 1, the enclosure behaves as a blackbody).
- Two large parallel plates: q″ = σ (T₁⁴ − T₂⁴) ÷ (1 ÷ ε₁ + 1 ÷ ε₂ − 1).
- Concentric cylinders or spheres and general geometries use the view factor and tables of F₁₂ (Hottel crossed-string method, or software).
Worked example: two plates, and a pipe in a room
- Two large parallel plates at 500 K and 300 K, each with ε = 0.8: q″ = σ (500⁴ − 300⁴) ÷ (1 ÷ 0.8 + 1 ÷ 0.8 − 1) = 2 056 W/m².
- The 80 °C bare pipe of Heat, Temperature and the Three Modes (60.3 mm, 1 m, ε = 0.9) in a large room at 20 °C radiates Q̇ = ε σ A (T₁⁴ − T₂⁴) = 79 W per meter.
Radiation shields
A thin sheet of low-emissivity material placed between two hot surfaces adds two more surface resistances to the chain and can cut the radiation heat transfer dramatically: this is how thermos flasks (a vacuum plus silvered walls), foil-faced insulation and the multi-layer blankets on spacecraft work. With 0.05 emissivity foil between the two plates above, the flux falls from 2 056 to 76.2 W/m², a factor of 27. Shields work only against radiation: they need an air gap (or vacuum) because they will conduct if they touch.
Solar heating and cool surfaces
A surface in the sun absorbs αs times the solar flux (about 1 000 W/m² at noon on a clear day, 800 on a typical hot one) and loses heat by convection and radiation. At equilibrium, absorbed sun = convection + net radiation. The result depends mainly on the surface color for sunlight (αs).
Worked example: a flat roof on a hot day
- Sun 800 W/m², air at 30 °C, sky radiative temperature 10 °C, convection h = 10 W/m²·K, long-wave emissivity 0.9, no heat into the building.
- Dark roof (αs = 0.90): equilibrium at 67 °C.
- White roof (αs = 0.25): 36 °C, only 6 K above the air.
- A shiny low-emissivity metal that reflects the sun (αs = 0.10 here, but ε would also be low in practice) would be cooler still in the sun but hold its heat at night. “Cool roofs” and white paint on tanks and on electronics enclosures work by this effect, and cut the heat entering the building.
A radiation coefficient
When the surface and surroundings are not far apart, radiation can be written in the same form as convection, Q̇ = hr A (Ts − Tsurr), with hr = ε σ (Ts + Tsurr)(Ts² + Tsurr²). For the 80 °C pipe in a 20 °C room hr = 6.9 W/m²·K (the value used in the pipe examples), comparable to the natural-convection coefficient, so the two simply add. This linearization is convenient for thermal networks but only works for modest temperature differences.
Related
- Previous: Convection and Heat Transfer Coefficients. Next: Fins and Extended Surfaces.
- Heat Sink and Thermal Resistance Calculator.
- Soldering and Laser Cutting: radiant heat in practice.
- Heat Transfer: the whole guide.
FAQ
Do I always need kelvin for radiation?
Yes. The T⁴ law needs absolute temperature. A temperature difference in the linearized form is the same in K or °C, but the sum Ts + Tsurr inside hr must be in kelvin.
Is radiation important at room temperature?
Yes, about as big as natural convection (both give 4 to 6 W/m²·K) for a surface with high emissivity. It is the reason a shiny heat sink is worse than a black one in still air.
Why is a vacuum flask silvered?
The vacuum removes conduction and convection, and the silver mirror (ε about 0.02) cuts the radiation between the inner and outer walls to a trickle.
Does color matter for infrared?
Not much for long-wave infrared: most non-metallic surfaces, whatever their color, have ε of 0.9 or more. Color matters for sunlight. Metals are the exception: they are poor emitters unless oxidized or coated.
References
- Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley (radiation, view factors, shields).
- Modest, M. F., Radiative Heat Transfer, Academic Press.
- CODATA 2018 (Stefan–Boltzmann constant and Wien displacement constant); ASTM E408, E903 (emissivity and solar absorptance).
- Howell, J. R., A Catalog of Radiation Heat Transfer Configuration Factors (view factors).
Last reviewed: October 2026. Emissivities are typical values: they vary with finish, oxidation and temperature.