# Linearity of heat conductance

## Main Question or Discussion Point

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?

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Mapes
Homework Helper
Gold Member
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.

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
Homework Helper
Gold Member
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.

Last edited by a moderator:
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...