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Thermal Diffusion with Heat Source

  1. Nov 2, 2008 #1
    1. The problem statement, all variables and given/known data
    A cylindrical wire of thermal conductivity k, radius a and resistivity p uniformly
    carries a current I. The temperature of its surface is fixed at T0 using water cooling.
    Show that the temperature T(r) inside the wire at radius r is given by
    T(r) = T0 + p(I^2)(a^2 - r^2)/4pi^2a^4k

    2. Relevant equations
    del squared(T) = (c/k)dT/dt - H/k

    Where H is heat generated per unit volume


    3. The attempt at a solution
    I took the system to be in a steady state as the temperature is fixed. So:

    d2T/dr2 = -H/k

    I took H = I^2 (pl/A)(1/lA)
    = I^2p/pi^2a^4

    Subbing this into my differential equation and integrating twice wrt r, I get:

    T = -(I^2.p.r^2)/(2pi^2.a^4.k) + ba + c

    Where b and c are integration constants. I think b must be zero but am not sure why. Then putting in T=T0 at r=a I would get a close answer but with a 2 on the bottom of the second term instead of a 4.

    Where did I go wrong?!?

    Thanks
     
    Last edited: Nov 2, 2008
  2. jcsd
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