# Transitory heat transfer problem with two materials

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.

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.