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Problem solving Heat Diffusion Equation

  1. Dec 22, 2011 #1
    1. The problem statement, all variables and given/known data

    One face of a thick uniform layer is subject to a sinusoidal temperature variation of angular frequency ω. SHow that the damped sinusoidal temperature oscillation propagate into eh layer and give an expression for the decay length of the oscillation amplitude.

    A cellar is built underground covered by a ceiling which 3m thick made of limestone. The Outside temperature is subject to daily fluctuations of amplitude 10 C and annual fluctuations of 20 C. Estimate the magnitude of the daily and annual temperature variation within the cellar.

    2. Relevant equations



    3. The attempt at a solution

    I am unable to solve the first part of this question.

    Take the diffusion equation

    [tex] \frac{\partial T}{\partial t} = -D\frac{\partial^2T}{\partial x^{2}} [/tex]

    Using separation of variable method:

    Let [tex] T=X(x)F(t) [/tex]

    [tex]X \frac{dF}{dt} = -DF\frac{d^2X}{DX^2} [/tex]

    [tex]-D\frac{dF}{dt}=\frac{1}{X}\frac{d^2X}{DX^2}= k [/tex]

    where k is the separation constant.

    These separate into two equation which I solve to give

    [tex] X=Ae^{\sqrt{k}x}+Be^{-\sqrt{k}x} [/tex]

    [tex]F=Ce^{-Dkt} [/tex]

    By superpositon principle

    [tex] T=\sum (A_ke^{\sqrt{k}x}+B_ke^{-\sqrt{k}x})e^{-Dkt} + A_0 + B_0x [/tex]

    where C has been absorbed into A and B.

    Then taking the boundary conditions:

    At x →∞ T→0, which shows A_k → 0

    [tex] T=\sum B_ke^{-\sqrt{k}x}e^{-Dkt} + B_0x [/tex]
    Then apply conditon that at x=0 T [itex]\propto[/itex] sinωt.
    [tex] e^{-Dkt} = sinwt [/tex]

    However here is where I am stuck, I do not see how proceed further. Is my solution so far correct, it does not seem so as I seem to have the wrong form. Or have I chosen the wrong form for my separation coefficient?

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Dec 22, 2011 #2
    The transient heat equation is generally written without the negative sign as you have written it. When you use separation of variables, use -k as your separation constant. That causes the X(x) portion to be sines and cosines.
     
  4. Dec 23, 2011 #3
    Ahh that makes better sense. Thanks for the help, I was able to solve to give:

    [tex]Tcos(wt-x\sqrt{\frac{w}{2D}}) e^{-x\sqrt{\frac{w}{2D}}}[/tex]
     
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