Chapter 2.3, Problem 20P

Heat transfer from a body to its surrounding by radiation, based on the Stefan-Boltzmann law, is discribed by the differntial equation

$\frac{du}{dt}=-\alpha \left(u^4-T^4\right)$ (i)

where $u(t)$ is the absolute temperature of the body at time $t$, $T$ is the absolute temperature of the surroundings, and $\alpha$ is a constant depending on the physical parameter of the body. However, if $u$ is much larger than $T$, then solutions of Eq.(i) are well approximated by solutions of the simpler equation

$\frac{du}{dt}=-\alpha u^4$. (ii)

Suppose that a body with initial temperature $2000\text{K}$ is surrounded by a medium with temperature $300\text{K}$ and that $\alpha =2.0\times {10}^{-12}{\text{K}}^{-3}/s$.

Determine the temperature of the body at any time by solving Eq.(ii)

Plot the graph of $u$ versus $t$.

Find the time $\tau$ at which $u(\tau )=600$, that is, twice the ambient temperature. Up to this time, the error in using Eq.(ii) to approximate the solutions of Eq.(i) is no more than $1\%$.