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Homework Help: Thermal Conductivity

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Rod of cross sectional area A and length l has its left end held at constant temperature t1 and its right end held at t2<t1. If the conductivity varies with distance from the left end, x, according to the relationship k= x/R + k0 (R and k0 are positive), what is the steady state heat flow, H, through the rod.

    2. Relevant equations

    heat flow = kA*((t1-t2)/l)

    3. The attempt at a solution

    I've always had a hard time understanding what to integrate and what to take the derivative of. As far as I know, I need to integrate k from some x to some final x...
     
  2. jcsd
  3. Sep 27, 2010 #2
    Consider a small element dx on the rod, corresponding to a change dT in temperature. Temperature T is a function of position x. We have:

    [tex]H=\frac{dQ}{dt}=kA\frac{dT}{dx}=A(k_o+\frac{x}{R})\frac{dT}{dx}[/tex]

    Therefore: [tex]\int^{T_2}_{T_1}AdT = \int^{L}_{0}H\frac{dx}{k_o +\frac{x}{R}}[/tex]

    Now as H is constant, the above integrals can be solved, right? Then you can deduce H from that.
     
  4. Sep 28, 2010 #3
    Thanks a ton!
     
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