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Math Help - Flux of a uniform rod

  1. #1
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    Flux of a uniform rod

    The end x = 0 of a uniform rod is maintained at a constant temperature and the end x = L is maintained at temperature zero. After the steady state has been reached, it is found that the flux out of the end of the rod at x = L has the value F. What is the temperature at x = 0?

    \displaystyle\text{B.C.}=\begin{cases} u(0,y)=C\\ u(L,y)=0\end{cases}

    How do I obtain the flux at x = L?
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    The end x = 0 of a uniform rod is maintained at a constant temperature and the end x = L is maintained at temperature zero. After the steady state has been reached, it is found that the flux out of the end of the rod at x = L has the value F. What is the temperature at x = 0?

    \displaystyle\text{B.C.}=\begin{cases} u(0)=C\\ u(L)=0\end{cases}

    How do I obtain the flux at x = L?
    \displaystyle\text{D.E.}: \ \frac{d^2u}{dx^2}=0\Rightarrow u(x)=xk+g

    u(L)=Lk+g=0\Rightarrow g=-Lk

    u(0)=g=C

    Also, "...empirical physical law which states that the heat flux at any point is proportional to the temperature gradient at that point."

    \displaystyle F(x,t)=-\kappa\frac{\partial u}{\partial x}(x,t), \ \ \kappa>0

    The solution to my question is \displaystyle\frac{FL}{\kappa}

    How can I obtain the solution from all this?
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  3. #3
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    From BC we get

    g=C

    k_1=-C/L

    and

    <br />
u(x)=-(C/L)x+C.<br />

    Flux is

    <br />
F=-\kappa \frac{du}{dx}|_{x=L}=\kappa \; C/L<br />

    <br />
C=FL/ \kappa .<br />
    Last edited by zzzoak; February 7th 2011 at 03:19 PM.
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