Heat Transfer PDE SOV with piecewise BC

[ustink007]

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.

 

[mighty2000]

Thank you for posting your heat transfer problem. It seems like you have made good progress so far by using the separation of variables method and applying the first three boundary conditions. To solve for Cn and apply the piecewise boundary conditions, you can use the orthogonality property of cosine as you mentioned. This property states that:

\int_{a}^{b} \cos{\frac{m \pi x}{L}} \cos{\frac{n \pi x}{L}} dx = \begin{cases} 0, & m \neq n \\ \frac{L}{2}, & m = n \end{cases}

Using this property, you can solve for Cn by multiplying both sides of the governing equation by \cos{\frac{m \pi x}{L}} and integrating over the domain from 0 to L. This will give you an equation for Cn in terms of q, k, H, and L. Then, you can use the piecewise boundary conditions to solve for Cn for both regions of x, and finally combine the two solutions to get the overall solution for T(x,y).

I hope this helps. Good luck with your problem!