# Heat Transfer PDE SOV with piecewise BC

1. Mar 11, 2009

### ustink007

1. The problem statement, all variables and given/known data
Heat transfer problem with 3 insulated sides and heat flux in and out on one boundary.
given values: q & k

2. Relevant equations
Governing Equation:
$$\frac{\partial^{2}{T}}{\partial{x}^{2}} + \frac{\partial^{2}{T}}{\partial{y}^{2}} = 0$$

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

3. 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.
$$T(x,y) = \sum_{n=0}^\infty C_n \cos{\frac{n \pi x}{L}} \cosh{\frac{n \pi y}{L}}$$
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.