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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.

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1 Answer

I would start by thinking about the energy flows to define the problem. What is the heat transfer between your materials and the exterior? Is it insulated, so there is no heat flow? Or is there a contact resistance, so $\frac{dE_1}{dt}=h_1T_1$ and similarly on the other interface? Usually you would model a contact resistance at the interface of your materials, which means there can and will be a temperature difference across the interface. For the simplest model, you could consider the conductivity of the materials to be very high, so each material is isothermal. Then at the interface you have $\frac{dE_1}{dt}=h_i(T_1-T_2)$ If the conductivity is not so high, you are right about the temperature slopes being inversely proportional to the conductivities, but you also need to impose the slope at the interface to the outside world.

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I was thinking in a perfect contact between materials with no insulation (that would split the problem in two independent parts). It is like when with the wave equation you solve the behavior of a plane wave when passes from one material to another. –  falbani Feb 7 '12 at 3:03
    
The eigenvectors in the single material case are sine waves with a decay time IIRC proportional to the square of the thickness and inverse to the conductivity and heat capacity. The wavelength must be such that an integral number of half-waves fit in the material. In this case they will look much the same but with a bend at the interface due to the different material properties. –  Ross Millikan Feb 7 '12 at 21:23
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