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Heat Equation

  1. Apr 27, 2013 #1
    1. The problem statement, all variables and given/known data

    A flat plate lies in the region:
    0<x<35, 0<y<inf

    The temperature is steady (not changing with time), and the
    boundary conditions are:
    T = { x if 0<x<35; y=0
    70-x if 35<x<70; y=0
    0 if x=0
    0 if x=70 }

    Enter the temperature at (x = 42, y = 21)

    2. Relevant equations

    heat equation in 2-d : (d^2T/dx^2)+(d^2T/dy^2)=0


    3. The attempt at a solution

    So I non dimensionalized it and solved it down to:
    X=A*cos(k*x)+B*cos(k*x)
    Y=C*e^(k*y)+D*e^(-k*y)
    T=X*Y

    So I solved at the boundary conditions, first one being T(x=0)=0
    From that its true that A must = 0, so X=B*cos(k*x)
    and T = B*cos(k*x)*(C*e^(k*y)+D*e^(-k*y))

    Second boundary condition is T(x=70)=0
     
  2. jcsd
  3. Apr 27, 2013 #2
    didn't mean to post it yet.
    second boundary condition is T(x=70)=0,
    there fore sin(kx) must be an interger multiple of pi so I don't zero old my whole solution.
    Now I have T = (C*e^(k*y)+D*e^(-k*y))+Summation(1->inf)(B*cos(n*pi*x)

    So im having trouble figuring out the boundary conditions for the other two piecewise functions, and dealing with the upper boundary of y being inf. any help would be very appreciated.
     
  4. Apr 29, 2013 #3
    nvm I solved it myself, anyone interested in learning how to solve the heat equation for semi finite plates with steady temp lemme know
     
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