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Heat conduction problem: sphere

  1. Oct 2, 2006 #1
    A solid sphere of radius a is immersed in a vat of fluid at a temperature T_0. Heat is conducted into the sphere according to

    dT/dt = D(d^2T/dr^2)
    (d-> partial derivative btw)

    If the temperature at the boundary is fixed at T_0 and the initial temperature of the sphere is T_1, find the temperature within the sphere as a function of time.

    My reasoning

    Ok. Here's my reasoning. Use a solution of the form T=X(t)R(r), and plug into the above equation to get


    I get X(t) = C*[exp(-t/(D*k^2))]. (k^2>0 for convergence)

    Then I have R'' +k^2*R = 0

    so R= Acos(kr) + Bsin(kr), since k^2>0.

    The Problem

    Assuming that the above steps are right, we could set the boundary conditions and have Acos(ka)+Bsin(ka)=T_0. That only eliminates one variable, either A or B. I don't know where to go from there however. Should this be like a fourier series or something?
  2. jcsd
  3. Oct 3, 2006 #2

    Meir Achuz

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    Science Advisor
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    The Laplacian in spherical coords is not d^2T/dr^2.
    It is [itex]\frac{1}{r}\left[\partial_r^2(rT)\right][/itex]
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