We have this experiment where a metal bar is heated and then we have to make a model for the cooling that occurs. We get numbers for how long it takes the metal bar to cool from 200 to 100 degrees Celsius, and we have to calculate how long it takes for the object to cool to 50 degrees. Now, I want to calculate what the cooling would be like if it was only radiative. I know that $T(t)=T_0+(T_s−T_0)e^{−kt}$ is the formula for cooling by convection, so I'm wondering what the equivalent of that is for radiative cooling. In other words, what is the formula for radiative cooling?
[Physics] Radiative Cooling of metal object
thermal-radiationthermodynamics
Best Answer
If you assume that the object radiates as a blackbody, then the Stefan-Boltzmann Law tells you that the power radiated by the object will be
$P = A \sigma T^4$
where $A$ is the surface area, $\sigma$ is the Stefan-Boltzmann constant, and $T$ is the temperature. If your object is not a blackbody, you can parameterize the above equation with an additional constant, $\epsilon$, called the emissivity.
From this you should be able to figure out the temperature as a function of time by knowing that the thermal energy is proportional to temperature and that the power is equal to the time derivative of energy.