Thermodynamics & Heat Transfer · Radiation exchange
The fourth power, and the zero it will not forgive
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The fourth power, and the zero it will not forgive

Every surface above absolute zero radiates. The rate is P=εσAT4P = \varepsilon \sigma A T^{4}P equals epsilon sigma A T to the fourth. ε\varepsilon (epsilon) is the emissivity, a bare number from 0 to 1: 1 is a perfect black body, oxidised steel and most paints run 0.8–0.95, and polished aluminium foil drops near 0.05, which is exactly why it is wrapped around hot pipes. σ\sigma (sigma) is the Stefan–Boltzmann constant, 5.67×108 W/(m2K4)5.67 \times 10^{-8}\ \mathrm{W/(m^2 \cdot K^4)} — a constant of nature, not something you look up per material. AA is the radiating area and TT is the absolute surface temperature in kelvin.

Absolute, always. A fourth power has no idea where you chose to put your zero: double the kelvin and you get sixteen times the radiation, but double the Celsius reading and you get nonsense. Convert first — T(K)=t(C)+273T(\mathrm{K}) = t(\mathrm{^\circ C}) + 273 — and convert BEFORE the power, never after.

A surface in a room is also being radiated at. The net traffic is Q˙=εσA(T14T24)\dot{Q} = \varepsilon \sigma A (T_1^{4} - T_2^{4}), where subscript 1 is the surface you are standing at and subscript 2 is the surroundings looking back. Note what that is NOT: T14T24T_1^{4} - T_2^{4} is not (T1T2)4(T_1 - T_2)^{4}, and the two are worlds apart.

For modest temperature differences the radiant term can be linearised into a coefficient and simply added to the convective one: ht=hc+εσ(Ts+Tsur)(Ts2+Tsur2)h_t = h_c + \varepsilon \sigma (T_s + T_{sur})(T_s^{2} + T_{sur}^{2}), with hth_t the combined coefficient, hch_c the convective one, TsT_s the surface and TsurT_{sur} the surroundings, both absolute. On a warm lagged pipe in still air the radiant half is often the LARGER of the two — the reason a survey done by convection alone always reads low.