Combined Convection and Radiation Coefficient

Radiation is a fourth-power law, but T₁⁴ − T₂⁴ factors exactly into (T₁ − T₂)(T₁ + T₂)(T₁² + T₂²), so over a modest temperature range you can hide everything but the linear ΔT inside an equivalent coefficient h_r = εσ(T_s + T_sur)(T_s² + T_sur²). Add it to the convective film and a single h_t drives the whole surface with plain old Q̇ = h_t A ΔT — which is exactly what makes building-envelope and insulation software tractable.

The size of the radiation term surprises people. A painted surface (ε = 0.9) at 350 K facing 293 K surroundings has h_r = 6.8 W/(m²·K), larger than the 5 W/(m²·K) of still-air natural convection beside it, giving h_t = 11.8. That is why a bare hot pipe in a still basement loses more than half its heat by radiation, and why a low-emissivity foil wrap — ε ≈ 0.05 — kills that channel almost completely while doing nothing about convection. The two traps: h_r depends on both temperatures, so it is not a constant and must be recomputed if the surface moves far; and the surroundings temperature is the temperature of the walls seeing the surface, not the air temperature, which on a clear night can be 20 K colder than the air and is why cars frost over at 4 °C.