I have been exploring finite differences and heat transfer using the 2D heat equation to further expand my knowledge. So far I think it is going well.
I am running into some confusion around grid spacing for the finite difference method.
Basically if I have a metal plate measuring 100mm x 100mm it would seem natural to establish a grid spacing of 1mm which would yield 100 x 100 nodes or so (I haven't fully explored the different grid spaces yet to see what works best). If I run the simulation I get results I would expect after a certain time given the initial conditions. At time, t, the system has evolved to a certain state.
Now, I want to try a larger metal plate made of the exact same material with the exact same properties. The only difference is that it is larger, 1m x 1m. For this plate I would establish the grid spacing at 1cm yielding the same number of notes, 100 x 100. I might be naive here but I can't see how the scale would affect the finite differences. I would expect the plate to take longer to reach the same state (or pretty close, provided the initial conditions and boundary conditions were the same) as the smaller plate. But from what I see in my calculations is that it doesn't. I might be missing something and would appreciate any pointers to the relevant literature.
For the purposes of this question there are constant heat sources defined at the boundaries, nothing fancy. It really is a very basic problem definition. I had set the $\alpha$ value to 1 do that I could investigate the algorithm. I didn't pay attention to the units that it was defined with.
I think my problem is in regards to the unit that are used to define the spacing. In the initial problem I had the units in my mind as millimeter. I really should have paid closer attention to the units used in defining the heat capacity, material density and thermal conductivity. They were defined with meters. To me this would imply that I need to define the grid spacing in terms of meters. This would give me the difference that I am looking for. For example the first plate of 100mm x 100mm would have a spacing of 1mm = 0.001m. The second plate of 1m x 1m would have a spacing of 1 cm = 0.01m.
Is this logic correct?