How to Calculate Thermal Radiation

Finds the heat a body radiates as emissivity × σ × surface area × absolute temperature⁴. Because it goes with the fourth power, a small rise in temperature raises the radiation sharply. What the surroundings send back is subtracted to give the net loss.

Every object gives off electromagnetic waves according to its temperature, and loses heat by doing so. This is thermal radiation.

P=εσAT4P = \varepsilon\sigma A T^4

ε\varepsilon is the emissivity, σ\sigma the Stefan-Boltzmann constant, 5.67×1085.67 \times 10^{-8}, AA the surface area and TT the absolute temperature. The object also receives radiation from its surroundings, so the net loss is the difference εσA(T4Ts4)\varepsilon\sigma A(T^4 - T_s^4).

The fourth power bites

Going with the fourth power of temperature, radiation climbs sharply for a modest rise in heat. Doubling the absolute temperature multiplies the radiation by sixteen. A heater glowing red warms a room out of all proportion for the same reason.

Example

A person of 1.8 m² surface area with a skin temperature of 33°C stands in a room at 20°C. Taking the emissivity as 0.98, they radiate 879 W and receive 739 W back, losing 140 W net. Since a person at rest produces roughly 100 W of heat, radiation clearly carries a large share of the body's heat balance.

Notes

The temperature must be absolute. Raising a Celsius figure to the fourth power gives a completely different answer.

Emissivity is a property of the surface. A black rough surface is close to 1, while polished metal may be as low as 0.05. A vacuum flask is mirrored inside precisely to lower the emissivity and keep the heat in.