Diffusion of energy by heat flow

In summary, the heat flow into the element from each neighbour is: $$\frac{D_H}{dx}\left( T(x-dx, t) \right)$$
  • #1
Nacho Verdugo
15
0

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


[/B]
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?
 
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  • #2
Nacho Verdugo said:
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.
 

1. What is diffusion of energy by heat flow?

Diffusion of energy by heat flow is the process in which heat energy is transferred from a region of higher temperature to a region of lower temperature, resulting in a more uniform distribution of thermal energy.

2. What is the driving force behind diffusion of energy by heat flow?

The driving force behind diffusion of energy by heat flow is the difference in temperature between two regions. Heat energy naturally flows from areas of higher temperature to areas of lower temperature in an attempt to reach thermal equilibrium.

3. How does heat energy transfer through a material?

Heat energy is transferred through a material through a process called conduction, in which heat is transferred from molecule to molecule without the actual movement of the molecules themselves.

4. What factors affect the rate of diffusion of energy by heat flow?

The rate of diffusion of energy by heat flow is affected by the temperature difference between the two regions, the type of material through which the heat is transferring, and the distance over which the heat is transferring.

5. How is diffusion of energy by heat flow different from other forms of energy transfer?

Diffusion of energy by heat flow is different from other forms of energy transfer because it does not involve the physical movement of matter, but rather the transfer of thermal energy from one region to another. This is in contrast to other forms of energy transfer such as convection and radiation, which involve the movement of matter or electromagnetic waves, respectively.

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