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Transient heat conduction in a slab

  1. Feb 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider unsteady state heat conduction through an infinitely wide slab of solid material of thickness 2L. There is no internal heat generation and the thermal properties of the material are independent of temperature and position. Starting from an energy balance, show that the temperature T at a distance z from the central plane at time t is described by an equation of the form

    [itex]\frac{dT}{dt}[/itex] = A [itex]\frac{\partial^{2}T}{\partial\eta^{2}}[/itex]


    where η = z/L and A is a group of parameters. Define A.


    3. The attempt at a solution

    I want to perform a energy balance, which should come out in the form:

    aq = a(q+dq)+[itex]\frac{\partial H}{\partial t}[/itex]

    where a is the area and then I can probably solve it from there
    However, in this case I can't do so as because the slab is infinitely wide I can't get a. Is there a problem with the way I'm visualising it in three dimensions or should I be taking a different approach to the energy balance?
     
  2. jcsd
  3. Mar 13, 2012 #2
    I'm no expert and have just started with thermodynamics but I think you should look at lumped capacity and semi-infinite solids
     
  4. Mar 14, 2012 #3
    Based on the equation you cite, the problem is one dimensional and you are being asked to derive the Fourier equation. Call your control volume area Δa. The areas will all cancel out once you finish the derivation. The derivative with respect to time should be a partial derivative because you have two independent variables, time and space.

    I would not worry about non-dimensionalizing until after you complete the derivation.
     
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