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Non-linear non-constant coefficient second order ODE

  1. Nov 23, 2015 #1
    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.

    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.

    ----

    Thank you
     
  2. jcsd
  3. Nov 28, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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