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Diffusion of energy by heat flow

  1. Mar 27, 2017 #1
    1. The problem statement, all variables and given/known data
    This problem belongs to the Intermediate Physics for Medicine and Biology, Hobbie Chapter 4.

    The heat flow equation in one dimension

    $$ j_H=-\kappa \partial_x T $$

    where ## \kappa ## is the termal conductivity in ## Wm^{-1}K^{-1}##. One often finds an equation for the diffusion of energy by heat flow:

    $$ \partial_t T=D_H \partial^2_x T $$

    The units of ## j_H## are ## Jm^{-2}s^{-1}##. The internal energy per unit volumen is given by ##u=\rho CT##, where C is the heat capacity per unit mass and ##\rho## is the density of the material. Derive the second equation from the first and show ## D_H ## depends on ## \kappa, C## and ##\rho##.

    2. Relevant equations


    3. The attempt at a solution

    I tried this:

    As ## u=\rho CT ##, I can write the temperature as ##T=\frac{u}{C\rho}##, so in the first equation:

    $$ j_H=-\kappa \partial_x \left( \frac{u}{C\rho} \right) $$

    and rewriting this and replacing it in the second equation:

    $$\partial_t T=D_H\partial_x(j_H/\kappa) $$

    which is similar to

    $$\partial_t T=D_H \partial_x \partial_x ({u}{C\rho}) $$

    but I got stucked here because I can't derivate this. Any ideas on how to move on?
     
    Last edited by a moderator: Mar 27, 2017
  2. jcsd
  3. Mar 27, 2017 #2

    haruspex

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    The second equation is the thing to be proved, no? So you cannot use it in the proof.
    Consider a small element at position x length dx, temperature T(x,t). Neighbouring elements are at temperatures T(x-dx, t) and T(x+dx, t).
    What is the heat flow into the element from each neighbour? Approximate T(x-dx, t) etc. using the usual f(x+dx)=f(x)+f'(x)dx+ ... rule, but taking into account the second order terms.
     
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