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2D Steady State Heat Equation, Mixed Boundary Conditions - Separation of Variables

  1. Sep 22, 2012 #1
    1. The problem statement, all variables and given/known data
    We have a 2D fin that has length L (x-axis), and thickness t, (y-axis). The left side has a fixed temperature, the right side is insulated, and the top and bottom surfaces are subject to convection. Find an analytical solution for the temperature at steady state.

    2. Relevant equations
    Boundary Conditions:
    [itex](0,y) \quad T=T_b[/itex]
    [itex]\displaystyle (L,y) \quad \frac{\partial{T}}{\partial{x}}=0[/itex]
    [itex]\displaystyle (x,t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0[/itex]
    [itex]\displaystyle (x,-t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0[/itex]

    Separation of Variables:
    [itex]T(x,y)=X(x)Y(y)[/itex]

    [itex]X''-\lambda{^2}X=0[/itex]
    [itex]Y''+\lambda{^2}Y=0[/itex]

    [itex]X(x)=c_1 \sinh{(\lambda{x})} + c_2 \cosh{(\lambda{x})}[/itex]
    [itex]Y(y)=c_3 \sin{(\lambda{y})} + c_4 \cos{(\lambda{y})}[/itex]

    3. The attempt at a solution
    Use symmetry to simplify the boundary conditions in the y-axis direction by cutting the fin along the x-axis at [itex]y=0[/itex] where the flux will be zero:
    [itex]\displaystyle (x,0) \quad \frac{\partial{T}}{\partial{y}}=0[/itex]

    I have tried going through it using separation of variables about half a dozen times now but I can never solve for [itex]\lambda[/itex]. Here is my current progress (in Mathematica):

    http://img6.imagebanana.com/img/38bbdmje/Selection_006.png

    Should I be setting my coordinate system up differently? Do I have to treat the convective (mixed) boundary conditions differently?

    Thanks!
     
    Last edited: Sep 22, 2012
  2. jcsd
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