Radiation heat transfer formula
The Stefan-Boltzmann law gives the heat radiated by a surface. It rises with the fourth power of absolute temperature, so radiation is negligible when cool but dominates at high temperatures like furnaces and filaments.
Variables
| Q | Net radiated heat rate | W |
| ε | Emissivity (0 to 1) | dimensionless |
| σ | Stefan-Boltzmann constant | 5.67×10⁻⁸ W/m²K⁴ |
| A | Surface area | m² |
| T, Ts | Surface and surroundings temperature | K |
Rearranged
Black body: ε = 1
Worked example
A surface (ε = 0.9, A = 1 m²) at 400 K radiates to 300 K surroundings.
Q = 0.9 × 5.67×10⁻⁸ × 1 × (400⁴ − 300⁴).
Because temperature is raised to the fourth power, doubling the absolute temperature multiplies the radiated power sixteenfold. Emissivity scales the output from a perfect black body, and the surroundings term subtracts the radiation coming back. Temperatures must always be in Kelvin.
Comparing with the other heat modes?
See the Thermal Conduction Formula and the Convection Heat Transfer Formula.
Stefan-Boltzmann law
The law states that a body radiates power in proportion to its emissivity, its area, and the fourth power of its absolute temperature. A perfect emitter, a black body, has emissivity one; real surfaces are lower. The net exchange with the surroundings subtracts the incoming radiation, so a body in equilibrium with its surroundings radiates no net heat.
Why temperature must be in Kelvin
The fourth-power law only works with absolute temperature, because radiated power depends on the true thermal energy, which is zero at absolute zero. Using Celsius would wrongly imply no radiation at the freezing point. Always convert to Kelvin before raising to the fourth power, or the result is meaningless.
FAQ
What is the Stefan-Boltzmann law?
Radiated power equals emissivity times the Stefan-Boltzmann constant times area times absolute temperature to the fourth power, minus the same for the surroundings.
Why is radiation so strong at high temperature?
Because it depends on temperature to the fourth power. Doubling the absolute temperature increases radiated power sixteen times.
What units does temperature need to be in?
Kelvin. The fourth-power law requires absolute temperature, so Celsius or Fahrenheit values must be converted first.