Solving 2D Heat Conductivity Problem - Where to Start?

[dinaharchery]

Please Help,

I am not an mechanical engineer but am trying to learn Finite Element Methods to solve a 2D Steady State Heat Conductivity problem.

Can anyone point me to an exact solution to the 2D Heat conductivity problem? Please be gentle, I have not had a physics class since 8th grade although I have taken calculus. My idea is to solve a specific 2D model analytically and then compare this exact solution to a solution that I will generate using FEM approximation equation.

Thanks.

 

[Saladsamurai]

Given your limited background, I would highly suggest going through a 1D problem. If that it something you are interested in, I would be happy to help. Let me know.

 

[Mapes]

There is no single solution to the steady-state heat equation (which, for uniform thermal conductivity and no heat generation, is just Laplace's equation $\nabla^2T=0$). Various boundary conditions will result in different solutions.

As Saladsamurai suggested, 1D is a good way to get your feet wet. The equation is identical, but the solution is always a line (or, if heat generation exists, a parabola).

 

[dinaharchery]

Thanks for the reply's.

I took your suggestions and did the 1D Heat Conduction problem. I finished it and it looks correct. Of course there is a lot of information on the web on the 1D problem, but the 2D is another issue.

 

[Mapes]

I was thinking about this a little bit more, and I think the first nontrivial 2D configuration to try would be a square; hold two adjacent sides at a temperature gradient from 0 to 0.5 (i.e., each side goes linearly from 0 at the common corner to 0.5 at the ends) and the other two sides held from 0.5 to 1 (so that the corner opposite from 0 is held at 1). Then you can verify that one diagonal is a uniform 0.5 and the other is a linear gradient from 0 to 1. The temperature profile is simply T(x,y) = x + y, which can be verified to solve Laplace's equation. Know what I mean?

 

[dinaharchery]

Thank you.

I think I understand what you are saying - the temperature profile T(x, y) = x + y is "plugged-into" the Laplace equation and solved? To solve the Laplace equation I will need to take the second order partial derivatives of T(x, y)?

Is this correct?

 

[Mapes]

Thank you.

I think I understand what you are saying - the temperature profile T(x, y) = x + y is "plugged-into" the Laplace equation and solved? To solve the Laplace equation I will need to take the second order partial derivatives of T(x, y)?

Is this correct?

Yes, exactly. And the solution also needs to solve all the boundary conditions: $T(x,0)$, $T(x,L)$, $T(0,y)$, and $T(L,y)$ if the temperatures are specified at the edges, $\partial T(x,0)/\partial y$ and so on if the heat fluxes are specified at the edges, a combination of these, or conditions set within the domain. That's why no single, simple solution exists.

 

[dinaharchery]

Cool, I am glad I understood something.

The 2D model that I am trying to solve consists of the a square structure with the temperature T(0) = 1000 and T(L) = 100 where L = 100. I think this should be relatively simple but I am a bit rusty on differential equations.

Thank you again.

 

[Mapes]

If the sides aren't constrained and there's negligible convection in the third dimension and no heat generation, this is a 1D problem: $T(x,y)=100x/L$.

EDIT: Nope, please see below.

 

[dinaharchery]

Sorry but I should have specified the boundaries better.

Temperature at Node 1 (x=0, y=0) is 1000, Temperature at Node 4 (x=0, y=100) is 1000, Temperature at Node 2 (x=100, y=0) is 100, Temperature at Node 3 (x=100, y=100) is 100.

Does this make any difference?

Also can you please show me how you arrived at the equation T(x, y) = 100x/L?

Thank you for all your help.

 

[Mapes]

Whoops, I must have looked at your earlier post too fast; make that $T(x,y)=9(100-x)+100$. Does that make more sense?

 

[dinaharchery]

Whoops, I must have looked at your earlier post too fast; make that T(x,y)=9(100-x)+100. Does that make more sense?

The results make more sense, thank you. I just seems strange that the y coordinates have no influence on the final temperature at a given node.

Can you provide any links to explain this kind of equation? I am really trying to learn this stuff.

 

[Mapes]

Have you googled "Laplace's equation"? That's where I'd start.

 

[dinaharchery]

Thanks Mapes.

You have been very helpful, I really appreciate it.

 

[Mapes]

Any time!

 

[dinaharchery]

Mapes,

Once again, thank you. I derived the linear equation from the original T(x, y) = x + y and got T(x, y) = -9(100 - x) + 1000. This equation gives the same results as the equation you derived but I must have arrived at it in a different fashion.

Please excuse my ignorance with Laplace equations, but as I said I am trying to learn. Can you please give me the details at how you arrived at your equation?

Thank you.

 

[Mapes]

Actually, I just did it by inspection. I knew y wouldn't appear, as there was nothing in the boundary conditions to cause heat flux in the y direction. (Any temperature change in a certain direction implies heat transfer in that direction; that's the physical interpretation of the conduction equation.) Then I just played around until I had the equation for a line with $T(x=0)=1000$ and $T(x=100)=100$.

But to do it systematically, you'd integrate Laplace's equation in 1D, $\nabla^2 T=d^2T/dx^2=0$, to get $T(x)=C_1x+C_2$, then find $C_1$ and $C_2$ by plugging in the boundary conditions. If this is what you did, you're in great shape.

 

[dinaharchery]

Mapes,

Yes, that is exactly what I did. I just wanted to be sure I was correct.

Again, a million thanks for all your help.

 

[Mapes]

You're welcome!