I need to solve the heat equation in a complicated situation but before I want to refresh (and test if it is still there) my knowledge with a simpler problem:
- Infinitely long wall composed by two materials of different properties (thermal diffusivity).
Schematic:
Exterior (fixed 0 temp) [ Material 1 | Material 2 ] Exterior (fixed 0 temp) ------------------------x=0------------x=x0-----------x=L------------------>
I hope you understand. Besides the existence of the two materials, it is the same as the classic 1D Heat Equation problem.
Regarding boundary condition, I have to add this equations:
- T1(x0, t) = T2(x0, t)
- $ k1 \cdot \frac{dT1}{dx} = k2 \cdot \frac{dT2}{dx} $ evaluated at x=x0 for all t
I solved the equation for each material as if they were independent and later tried to find conditions for the eigenvalues and integration constants using the conditions stated before, but I could not.
So, I'm starting to wonder whether this is the right approach or if I lack arithmetic skills... and that is why I'm here posting this and begging for your help.
Thanks in advance.
PD: I found no solution or mention of this problem in google. Maybe I'm not searching with the right keywords (I'm not a natural english speaker). So, if what I'm asking can be found in some other place, I will appreciate if you can point me to it.