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Heat diffusion with source

  1. Oct 28, 2009 #1
    1. The problem statement, all variables and given/known data

    a material occupies -L < x < L and has uniform ambient temperature T_a. A chemical reaction begins within the body leading to the 1-d heat equation:

    [tex] pc \frac {\partial{T}}{\partial{t}} = k \frac {\partial^2{T}}{\partial{x^2}} + pQAe^\frac{-E}{RT} [/tex]

    with BC and IC

    [tex] T(+/- L, t) = T_a [/tex] and [tex] T(x,0) = T_a [/tex]

    2. Relevant equations



    3. The attempt at a solution

    The book gives the following substitutions without any justification

    [tex] \theta = \frac{E}{RT_a^2}(T-T_a) [/tex] and [tex] x = L*b [/tex]

    and arrives at the following linear equation

    [tex] L^2 \frac {pc}{k} { \frac {\partial{\theta}}{\partial{t}} = \frac {\partial^2{\theta}}{\partial{b^2}} + z*e^\frac{\theta}{1+y\theta} [/tex]




    then it asks to give the equations for z and y.

    However, before I begin to solve this problem.. I would love to understand where the substitutions

    [tex] \theta = \frac{E}{RT_a^2}(T-T_a) [/tex] and [tex] x = L*b [/tex]

    came from. I presume that x = L*b is used to make x dimensionless.. but what about theta?

    Im pretty sure I know how to solve the problem (seperation of variables on the last eq.) , but I really don't understand the substitutions. It seems that whoever wrote the book solved the problem first and then realized the substitutions were a good fit. Any information (about the substitutions) is appreciated.
     
    Last edited: Oct 28, 2009
  2. jcsd
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