Diffusion of energy by heat flow

[Nacho Verdugo]

Homework Statement

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 $\rho$.

Homework Equations

The Attempt at a Solution

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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?

 

[haruspex]

rewriting this and replacing it in the second equation:

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.