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Consider a piece of material of infinite extent (i.e. no boundaries) with a point source of heat at its center which oscillates at frequency $\omega$ about the mean temperature $\langle T\rangle$. If I place a temperature sensor somewhere else in the material at a distance $r$ from the heat source, what will the transfer function between the source and the sensor be?


From experience I expect it to be a single pole low-pass filter of the form $$ \frac{T_{sns}}{T_{src}}=\frac{1}{1+\tau s}, $$ where $s=i\omega$. I also expect that the time constant $\tau$ will be a function of the thermal conductivity $\kappa$, the distance between the source and the sensor $r$, and maybe the heat capacity $Q$. Naive dimensional analysis of these quantities shows that $$ \tau\propto \frac{Q}{\kappa\ r} $$ has the correct units of seconds. Can anyone give a more rigorous answer?

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    $\begingroup$ I'd have to think about this a little more, but my instinct tells me that you'd need some relation to the temperature difference (as opposed to temperature ratio.) It might already exist in a manner I'm not seeing, but the thermal conductivity is linked to a temperature difference, not just an absolute temperature. $\endgroup$ Commented Jun 4, 2015 at 20:31
  • $\begingroup$ I am not quite familiar with transfer functions but would it help to derive the temperature distribution along the material relating $T_{src}$ and $T_{sns}$? I might be able to tackle this one if it's safe to assume a sinusoidal oscillation with an amplitude of $T_{o}$ and then solving the $1D$ heat equation for the material. $\endgroup$
    – Algo
    Commented Jul 11, 2015 at 0:31
  • $\begingroup$ @Algo All of that sounds good. I would be more interested in the solution to the 3D heat equation, but the solution to the 1D equation would still be very illuminating. $\endgroup$ Commented Jul 11, 2015 at 17:23
  • $\begingroup$ @ChrisMueller You must be thinking of a numerical solution then, I didn't look into it but I don't think there exists an exact solution for a 3D heat equation with this kind of boundary condition. $\endgroup$
    – Algo
    Commented Jul 11, 2015 at 17:54

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