Worked example — How much a radiator radiates

Problem

A cast-iron radiator has area $A=1{,}5\ \mathrm{m}^2$ and temperature $70\ ^{\circ}$C $=343$ K; emissivity $\varepsilon=0{,}9$. The room is at $20\ ^{\circ}$C $=293$ K. What is the net radiative power? ($\sigma=5{,}67\cdot10^{-8}\ \mathrm{W/(m^2\cdot K^4)}$).

Solution

Setup. The net radiated power (Stefan-Boltzmann law) is $P=\varepsilon\,\sigma\,A\,(T^4-T_{amb}^4)$. Substitution: $P=0{,}9\cdot 5{,}67\cdot10^{-8}\cdot 1{,}5\cdot(343^4-293^4)\approx 0{,}9\cdot 5{,}67\cdot10^{-8}\cdot 1{,}5\cdot 6{,}48\cdot10^{9}$ $\ev{P\approx 495\ \mathrm{W}}$ Almost half a kilowatt from radiation alone; the rest of the emitted heat goes into convection of the air.

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Topics: Termologia Concepts: Trasmissione del calore