Linearity of Heat Conductance - Is Heat Transfer the Same in All Directions?

[Yoni]

Heat conductance - linear?

Hello forum friends,

I have stumbled upon the fallowing heat conduction problem:
Consider a heat source of constant power embedded inside a solid with a constant heat capacity and conductance. Around the source is a box with a constant temperature, which cools the source.
My question is: If the box is a cube, can I conclude that each side contributes equally to the cooling? If I had just one side (out of 6) could I conclude 1/6 cooling?
However if the box is not a cube. Two opposite sides are pulled 2 times further off, can I conclude a cooling of 1/2 about these sides?

Is heat conductance in two or three dimensions a linear problem?

 

[Mapes]

Hi Yoni,

If the box is a cube, can I conclude that each side contributes equally to the cooling? If I had just one side (out of 6) could I conclude 1/6 cooling?

Yes, but it wouldn't be much help in figuring out the answer. You'd have to model conduction in a pyramid, where the base is the original side of the cube and the other sides are adiabatic.

However if the box is not a cube. Two opposite sides are pulled 2 times further off, can I conclude a cooling of 1/2 about these sides?

No, because 3-D conduction is not merely a function of that single dimension.

I don't recall the solution for the geometry you describe, but you can probably find one in one of the handbooks for conduction heat transfer.

 

[Yoni]

Hello,
The fallowing question is troubling me, and I need to fully understand it before I go forth with my experiment:
Is 3-D transfer of heat by conduction linear?

Consider a point in space which is heated. The heated source is r1 distance from a cooling source (which cools by convection), and r2 distance from a second cooling source.
The heat transfer equation: [ dQ/dt = h*dT/dx ] predicts the transfer of heat from one source to the other as a function of the temperature gradient.
So if I calculate the dQ/dt from one cooling source, and the dQ/dt of the other, can I conclude that the total transfer of heat is the sum?
If not, why? Is it because of the transfer of heat between the two cooling sources? Can I neglect this?

I'd appreciate any help,
Yoni

P.S please do not move this to a "homework forum", this is a basic question.

 

[Mapes]

You can assume that the process is linear, as long no coefficients or material properties in your equations are functions of temperature. For example, $h\frac{dT}{dx}$ is linear as long as h isn't a function of temperature. In practice, this means that the temperature difference should be small.

But note that this is a slightly different question from that in your post https://www.physicsforums.com/showthread.php?t=236917" from yesterday. You can't calculate heat transfer results from two differently sized cubes, add them together, and expect to get the correct results for a rectangular box.

 

[Yoni]

Thanks for your help. I have the solution a single tranfer of heat betwin a source and one cooling spot. Since I don't expect the heat coefficient to be dependent on temperature, I understand I can just sum the contributions of all cooling spot to get the over all cooling of the source.
Best to all of you...