I'm having a tough time trying to figure this question out mainly because I haven't any formal training in "scaling" of a problem.
An infinite cylindrical rod (radius $a$) is initially at temperature $T = T_0$ and its surface temperature is reduced to $T = 0$. Determine the time required for the cylinder to cool.
Write down the defining equations and scale the problem to reduce it to the form:
$$\frac{1}{r}\left ( rT_{r} \right )_{r}=T_{t}.$$
In cylindrical coordinates we have:
$$T_{t} = \kappa \nabla ^{2}T(r,t)$$ which gives $$T_{t} = \kappa \frac{1}{r}\frac{d}{dr}\left ( r\frac{dT}{dr}\right )$$ assuming heat diffusion is independent of $\theta$ and $t$.
You have $$\frac{dT}{dt} = \kappa \frac{1}{r}\frac{d}{dr}\left ( r\frac{dT}{dr}\right )$$ now nondimensionalise by choosing characteristic scales. For $T$, which will vary between $0$ and $T_0$, let $T=T_0\hat T$, where $\hat T$ is the nondimensional variable that varies between $0$ and $1$. For $r$, choose $r=a\hat r$ as the radius of the rod seems like a good length scale for this problem. For time, there is no obvious scale to use, so let $t=t_c\hat t$ where $t_c$ is some unknown for the being. Putting this into the equation and simplifying gives $$\frac{dT}{dt} = \frac{\kappa t_c}{a^2} \frac{1}{r}\frac{d}{dr}\left ( r\frac{dT}{dr}\right )$$ where everything is nondimensional except for $\kappa$, $t_c$ and $a$, so I didn't put all the hats on everything. We have free choice over what we make $t_0$, so let's choose $t_c=a^2/\kappa$ which reduces the equation to $$\frac{dT}{dt} = \frac{1}{r}\frac{d}{dr}\left ( r\frac{dT}{dr}\right ).$$
Having $t_c=a^2/\kappa$ makes sense, because we'd expect that if the rod got bigger it would take longer for the temperature to diffuse, (and ought to be proportional to the square) and increasing the thermal conductivity should increase the rate of diffusion and decrease the time scale for the temperature change.