Heat guide
The Stefan-Boltzmann Law
The Stefan-Boltzmann law states that ideal blackbody radiant emission is proportional to the fourth power of absolute temperature.
The blackbody equation
The ideal blackbody emissive power is E = σT⁴, where σ is the Stefan-Boltzmann constant and T is absolute temperature in kelvin.
Real surfaces use emissivity
A common grey-surface approximation writes E = εσT⁴. Emissivity ε represents the ratio of real-surface emission to ideal blackbody emission under the model assumptions.
Net exchange requires surroundings
A surface also receives radiation. For a small surface facing large isothermal surroundings, a simplified net relation is Q̇ = εσA(Ts⁴ - Tsur⁴). More complex geometry requires view factors.
Why Celsius cannot be used directly
The fourth-power relation is tied to absolute temperature. Using Celsius would produce physically incorrect results because 0 °C is not zero thermal energy and not the zero point of the thermodynamic temperature scale.
Radiation becomes especially important at high temperature
Because of the fourth-power dependence, radiative heat transfer can dominate in furnaces, combustion systems and other high-temperature applications.
Linearised radiation coefficients are useful over modest temperature ranges
For some engineering calculations, net radiative exchange can be written in a form resembling convection, Q̇ = hrA(Ts − Tsur), where hr is a linearised radiation coefficient derived from the fourth-power relationship around the temperatures of interest. This allows radiation to be combined conveniently with convection in a thermal-resistance model.
The coefficient is not constant over arbitrary temperatures. It depends on emissivity and the surface and surrounding absolute temperatures, so it should be recalculated when operating conditions change substantially.
Radiative emission rises rapidly with absolute temperature
The Stefan-Boltzmann law gives the total emissive power of an ideal blackbody as E = σT⁴. Because temperature appears to the fourth power, a modest percentage increase in absolute temperature can produce a much larger percentage increase in emitted radiative power.
Temperature must be expressed in kelvin. Using Celsius directly inside T⁴ gives physically meaningless results because Celsius is not an absolute temperature scale.
Real surfaces use emissivity
A common engineering approximation multiplies blackbody emission by emissivity, giving E = εσT⁴ for a grey diffuse surface. Emissivity depends on surface condition and can vary with wavelength, direction and temperature.
Polished metals can have low thermal emissivity, while oxidised or coated surfaces may emit much more effectively. Surface finish therefore matters as much as the underlying material name.
Net exchange depends on the surroundings
A surface emits radiation while also absorbing radiation from its environment. For a simple surface facing large surroundings, net exchange can be represented with a difference of fourth powers, εσA(Ts⁴ - Tsur⁴), when the grey-surface assumptions apply.
This is why the same hot object loses less net radiative heat inside a hot enclosure than it would facing much colder surroundings.
Net radiative exchange depends on both temperatures to the fourth power
A surface does not simply emit σT⁴ into nothing. In an environment, surrounding surfaces also radiate toward it. For a small gray surface facing large isothermal surroundings, the ideal net exchange can be written εσA(Ts⁴ − Tsur⁴). This difference is why a warm surface can still receive substantial incoming radiation while losing more than it gains overall.
Using Celsius directly in the fourth-power term produces physically meaningless results because the law is based on absolute thermodynamic temperature. Temperatures must be converted to kelvins before applying the equation.
Radiation becomes increasingly important as temperature rises
Because emitted power scales with the fourth power of absolute temperature, radiative heat transfer can grow rapidly at high temperatures. Furnaces, glowing metals, combustion chambers and spacecraft thermal-control problems therefore often require careful radiation analysis even when convection or conduction are also present.
The fourth-power relationship does not mean temperature alone determines exchange. Emissivity, area, geometry, view factors and surrounding temperatures still control the actual result.