# How fast does a surface cool depending on the distance to the source of heat

Say for example I have a sheet of metal which is 1 $m^2$. If I would heat the center of this sheet of metal to a 100 degrees celsius. How warm will a point be which is removed 10 cm from the center? How warm will a point be which 20 cm removed from the surface? and so on ... Is there an equation for this? Also, the rate of cooling does it depend on the distance from the heat source and the temperature at the heat source? (E.g. if the center is a 100 degrees, at 10 cm it is maybe 70 degrees, but if the center is 200 degrees is the temperature at 10 cm 140 degrees? So what's the relationship here?

EDIT:

Ok, Let me give some more info into what I want to learn, never mind the info above. I don't care if the sheet is square, round or whatever. I just want to know given that a surface/sheet is heated how does the heat spread. Let's say, we have a point (in the center of a surface for example) "$c$" which has a certain steady state temperature "$t$", how does the temperature vary with the distance from this point "$c$" ? A point "$p$" which is a certain distance removed from $c$ will be $t-\Delta t$ ... I guess. How would a 2d plot of this temperature distribution look like?

Also, if the temperature at $c$ was higher (let's say 20 degrees) ... would the temperature at $p$ also be 20 degrees higher (assume steady state) ?

I am just trying to get a better intuition into this. Feel free to point me toward relevant material or things to try out ...

• You have neither posted your current attempts at solution nor sufficient information. How is the sheet dumping heat to its surroundings? – Carl Witthoft Sep 23 '15 at 11:32
• There are equations for this, but there is no simple way in which they can be applied. Heat conduction can happen by radiation, diffusion, convection, phonon or electron transport (and then some). In your case you will have several of these mechanism at the same time and it would take a serious amount of effort to apply them to a certain geometry. – CuriousOne Sep 23 '15 at 11:40
• You could model this yourself by assuming pure diffusion of heat and use a spatially dependent heat equation (which would really just be the typical diffusion equation in this case; cf this post for more details)--one diffusion coefficient for each media (the metal & air). You'd likely have to use a computer to solve the problem though. – Kyle Kanos Sep 23 '15 at 11:51
• Also, you don't have sufficient information to really answer this. Has your sheet reached equilibrium? – Sean Sep 23 '15 at 13:53

To get an idea of the complexity of this problem, consider the simplest (due to symmetry) problem of the temperature profile of a radial circular cooling fin with heat source at the centre (and at thermal equilibrium).

For thin fins (much thinner than they are wide), this involves solving the following differential equation:

$\frac{d^2\theta}{dr^2}+\frac{1}{r}\frac{d\theta}{dr}-\frac{h}{kB}=0$

Where $r$ is the distance from centre, $\theta$ the so-called reduced temperature, $h$ the heat transfer coefficient (fin to air), $k$ the thermal conductivity (fin material) and $2B$ the thickness of the fin.

Although solvable, the solution involves those pesky Modified Bessel functions.

So if your sheet was circular that solution could be applied but for square or other shapes it becomes much more complicated.

You can find the full derivation for the radial circular fin problem here.

I hope this helps.