# Under which circumstances is the thermalization time proportional to the square of the length?

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I had heard that the thermalization time (i.e. the time required for an object to reach its final temperature, or say 99% of it) was proportional to its length squared. I hadn't given it much thoughts until I found out this statement in Onsager's famous 1931 free access paper, on page 419:

We may also recall that the time needed for equalization of temperature in a body is proportional to the square of its linear dimensions

But now that I think about it, I do not think this holds under any circumstances, and I would like to know when this holds, and when this doesn't.

Here are my thoughts:

• By length, I assume they mean the length in the direction of heat propagation.
• Intuitively, I think the statement holds for objects to which Dirichlet boundary conditions are applied. For example, let's take a cylinder of height L (let's say it's a metallic rod), whose ends are kept at different constant temperatures, and initially it has a uniform temperature. In that case, if I solve the PDE (which is simplified to a 1 dimensional PDE since there's no radial nor angular dependence of the temperature) $\kappa \frac{\partial T}{\partial z} = C_p\frac{\partial T}{\partial t}$ and if I cheat by considering a temperature as being equal to absolute $0$ (else I do not get a nice quantization of eigenvalues), I am almost able to reach that the solution is an infinite series of terms containing $\exp (\alpha_n t/ L^2)$. Note that the radius and cross section area of the cylinder does not enter into play. Thus the statement holds, in agreement with intuition.
• But also intuitively, if instead of keeping the cylinder ends at fixed temperature, if one applies a constant heat flux, which would correspond to Neumann boundary conditions, then the statement should not hold anymore. Why? Because heating an end with, say a $10 W$ power source, will certainly not have the same effect on a small radius cylinder compared to a bigger radius cylinder. Clearly, a huge metal cylinder is going to heat up more slowly if its radius is much bigger than a cylinder with a tiny radius, assuming they have equal length. In fact, since the volume of the cylinder goes as the radius squared, I expect a cylinder with a radius twice as big as the one of a smaller cylinder to take 4 times more time to thermalize. However, I see no way whatsoever for the cross sectional area, or the cylinder of the radius, to appear in the heat equation (and its boundary conditions), even with Neumann b.c., which would be something of the sort: $\frac{\partial T}{\partial x}_{\text{one end}}=\text{constant}$. I am therefore unable to formally work out what my intuition claims. Can someone clear this stuff up? Am I right in claiming that for constant power source applied on the end(s) of an object, Onsager statement does not hold? If so, how can I see it mathematically? How come I am unable to make the radius of the cylinder appear for the solution to the heat equation under these boundary conditions? How does the radius appear?
• Clearly, radiation effects are neglected. It is clear to me that the statement would hold when radiation effects are not taken into account. These are nonlinear effects that are proportional to the surface area.
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