I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where ##\kappa_1=a_1T+b_1## for material 1 and ##\kappa_2=a_2T+b_2## for material 2. ##a_1, a_2, b_1, and \;b_2## are constants and T is temperature. Essentially ##\kappa## depends on both temperature and space since we have two materials.(adsbygoogle = window.adsbygoogle || []).push({});

I will explain my approach to solving this. I am hoping to see what you think about its correctness and if you identify it as an approach with a specific name that I am unaware of. Thank you. I should specify that using the described method I have obtained an analytical solution in Mathematica.

First, solve ##\frac{\partial}{\partial x}(\kappa\frac{\partial T}{\partial x})=0##. This will give you a temperature profile as a function of x with two unknown coefficients that are to be determined by the boundary conditions.

Then substitute ##\kappa## with ##\kappa_1\; and\; \kappa_2## to obtain two separate temperature profiles for each section. Now you will have to determine four coefficients. To do this use the following boundary conditions:

##1)\; T_i=T_h\;##

##2)\; T_f=T_c## where ##T_h## and ##T_c## are known.

##3)\;\kappa_1\frac{\partial T_1}{\partial x}=\kappa_2\frac{\partial T_2}{\partial x}## (constant heat flux)

4) the boundary temperatures equate.

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Thank you

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# Non-linear non-constant coefficient second order ODE

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