Analyzing 2D Fin Temperature with Separation of Variables

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Homework Statement

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.

Homework Equations

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

Separation of Variables:
$T(x,y)=X(x)Y(y)$

$X''-\lambda{^2}X=0$
$Y''+\lambda{^2}Y=0$

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

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 $y=0$ where the flux will be zero:
$\displaystyle (x,0) \quad \frac{\partial{T}}{\partial{y}}=0$

I have tried going through it using separation of variables about half a dozen times now but I can never solve for $\lambda$. 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!

 

[KataKoniK]

Thank you for your post. I am a scientist and I would be happy to help you with your problem.

First, let's review the problem statement and boundary conditions. We have a 2D fin with length L along the x-axis and thickness t along the y-axis. The left side of the fin has a fixed temperature, while the right side is insulated. The top and bottom surfaces are subject to convection, with a heat transfer coefficient h and an ambient temperature T∞.

To find the analytical solution for the temperature at steady state, we will use separation of variables. This means that we will assume that the solution can be written as the product of two functions, one depending only on x and the other depending only on y. This is represented by the equation T(x,y)=X(x)Y(y).

Now, let's look at the boundary conditions. At the left side of the fin, we have a fixed temperature T=Tb. This means that at x=0, the temperature is Tb, or in other words, X(0)=Tb. At the right side of the fin, we have insulation, which means that there is no heat flux through this boundary. This can be represented by the condition X'(L)=0.

For the top and bottom surfaces, we have the convective boundary conditions. These can be written as k∂T/∂y+h(T-T∞)=0, where k is the thermal conductivity. We can simplify these conditions by using symmetry, as you have correctly mentioned. By cutting the fin along the x-axis at y=0, we can see that the heat flux must be zero at this point, which means that ∂T/∂y=0 at y=0. This can be rewritten as Y'(0)=0. For the bottom surface, we can use the same reasoning and conclude that Y'(t/2)=0.

Now, let's move on to solving the problem using separation of variables. We can start by substituting T(x,y)=X(x)Y(y) into the heat equation and the boundary conditions. This will give us two ordinary differential equations, one for X(x) and one for Y(y). For X(x), we have X''-λ^2X=0, where λ is a constant that we need to solve for. This is a second-order linear differential equation, and its general solution is X(x)=