Calculate heat loss from a heated volume inside a larger unheated volume 2

[GeorgeEllwood]

Hello,
I was wondering if someone could help me witha problem. I've attached an image which shows a volume which is blue, a smaller part of this volume which is shown as a red sphere is generating heat whilst the rest of the volume isn't. Can someone tell me how I can calculate the heat loss from the red sphere into the rest of the volume? Or point me in the right direction of where to start?
Cheers
George

 

[MintJulep]

All of it. Where else is it going to go?

 

[GeorgeEllwood]

I should have been more specific with my question. I'm working on a problem to do with superconducting magnets, where below a critical temperature they have zero resistance and therefore zero joule heating, and above a the critical temperature they have resistance and joule heating.
If a small volume within a larger volume goes above the critical temperature it will start generating heat, however the heated volume will be cooled by the surrounding superconducting volume that isn't generating heat. If the heated volume is small, the heat generated will be removed by conduction to the surrounding area, this will then reduce the temperature below the critical temperature and the heated volume will disappear. If the heated volume is very large, then the heat won't be able to be removed by conduction and the heated volume will grow. There will also be a certain size of heated volume when it is in equilibrium where the heat generated is balanced by the heat being removed.
I know how to calcuate the heat generation per unit volume, but I'm struggling to calculate the cooling. I think I need to find the cooled perimeter which I can calculate and the cooling per unit area, which I don't know how to find. Most cooling problems I've come across have been a heated solid being convection cooled by a solid. But I think this is different because I've got a heated volume within an unheated volume but they're the same solid material, not a fluid flowing past a solid.

 

[racookpe1978]

The resistance to heat f;low will be the material coefficient, right? It "might" be proper to assume this heat transfer coef. is constant with temperature, but I can't be sure for exotic material at that low a temperature.

So the amount of heat energy transferred per unit time will be a function of the area (surrounding the hot spot) AND the delta-T between hot and cold.

But - big but here! - the area will change as the radius^2 gets larger (but, the mass at that temperature gets larger by the radius ^3).

Your best bet? Treat your hot spot as a series of layered spheres: each layer will have the same thickness, but each layer will have a different temperature at a different time, a different mass, a different area-inside, and a different area-outside. Iterate.

to "learn" the method, assume you don't have a bunch of nested spheres, but a long insulated bar of uniform area. (The insulation assumption is "force" your logic to solve the heat in = heat out part of the problem with worrying about losses to the atmosphere around the bar. When you can solve that problem, then address the problem of different areas for each nested layer.

 

[IRstuff]

Biot number needs to be checked.

TTFN
faq731-376

 

[ko99]

Hi George, I assume you want to know the temperature profile of the cube. If so, I suggest you define the thermal path from the red sphere to the external environment.

The transfer from the red sphere through the blue cube appears to be simple thermal conduction:
dT at a given point = (power x distance from center)/(thermal conductivity x cross sectional area)

That gives you the deltas but not the absolute temperatures. The next step is to determine the surface temperature of the blue cube. The answer to this depends on the surrounding fluid and how the cube physically supported.

Hope this helps

ko (

http://www.MSthermal.com)

 

[chicopee]

Looks like a problem when I took heat transfer.

 

[GeorgeEllwood]

Thanks for the help, especially racookpe1978 and ko99. I think I understand what I doing now, I've found one of my errors that was leading to erroneous results. I was depositing heat into the sphere and then at the end of the time step I was seeing how large the resistive volume had become and then calculating and adding the new heat generation for the next time step. One problem was that my time steps were too large, so the resistive volume had increased a significant amount within 1/10 of a time step, but didn't start generating heat until the start of the next time step. I can fix this by either making the time steps smaller or calculating the heating as function of temperature.
Thanks George