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Heat Transfer PDE SOV with piecewise BC

  • Thread starter ustink007
  • Start date
  • #1
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Homework Statement


Heat transfer problem with 3 insulated sides and heat flux in and out on one boundary.
given values: q & k

Homework Equations


Governing Equation:
[tex]
\frac{\partial^{2}{T}}{\partial{x}^{2}} + \frac{\partial^{2}{T}}{\partial{y}^{2}} = 0
[/tex]

Boundary Conditions:
[tex]
@ x = 0 ;
\frac{\partial{T}}{\partial{x}} = 0
[/tex]
[tex]
@ x = L ;
\frac{\partial{T}}{\partial{x}} = 0
[/tex]
[tex]
@ y = 0 ;
\frac{\partial{T}}{\partial{y}} = 0
[/tex]
[tex]
@ y = H ;
\frac{\partial{T}}{\partial{y}} = \frac{q}{k} ; 0 < x < \frac{L}{2}
[/tex]
[tex]
@ y = H ;
\frac{\partial{T}}{\partial{y}} = \frac{-q}{k} ; \frac{L}{2} < x < L
[/tex]

The Attempt at a Solution


I did the separation of variable method and applied the first 3 boundary conditions, @ x=0,L and @ y=0.

I'm stuck at this.
[tex]
T(x,y) = \sum_{n=0}^\infty C_n \cos{\frac{n \pi x}{L}} \cosh{\frac{n \pi y}{L}}
[/tex]
n = 0,1,2,3.....
How do i solve for Cn, and apply the piecewise boundary condition?
I know i have to use the Orthogonality property of Cosine.

Thanks.
 

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