I'm trying to figure out the time required for a blackbody to cool assuming it only looses heat via radiation.
I can estimate the mass, specific heat, surface area, emissivity, initial temperature, final temperature, etc.
I know the amount of heat which has to be lost is:
$ Q = mc\Delta T$,
where $m$ is mass, $c$ is specific heat, $\Delta T$ is a temperature_change
I know the rate of heat loss is:
$ P = \epsilon kST^4$,
where $k$ is Stefan Boltzmann, $T$ is the body temperature, $S$ is the surface area, $\epsilon$ is the emissivity.
The body temperature changes as the blackbody cools so I cannot simply use the initial or final temperature or I get very different results. I think I have to somehow integrate $P$ from $t = 0$ (when body is at initial temperature) to when the body is at the final temperature.
Any help would be greatly appreciated.
Best Answer
Let us assume temperature of black body is $T_1$ and of surrounding is $T_2$, also $T_1>T_2$ , at time $t =0$. Heat lost by body at any instant is
$F= k*S_{area}*(T^4 -T_2^4)$.
Where T is temp at that instant and $k$ is the Boltzmann constant and $S_{area}$ is the surface of the body. Now this lost heat can be written as
$F= - ms \frac{dT}{dt}$
$-ms \frac{dT}{dt} = \sigma(T^4 - T_2^4 )S_{area}$
Integrate this expression and put the limits and solve.