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Engineering and Comp Sci Homework Help
Analyzing 2D Fin Temperature with Separation of Variables
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[QUOTE="blink-, post: 4084308, member: 438786"] [h2]Homework Statement [/h2] 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. [h2]Homework Equations[/h2] Boundary Conditions: [itex](0,y) \quad T=T_b[/itex] [itex]\displaystyle (L,y) \quad \frac{\partial{T}}{\partial{x}}=0[/itex] [itex]\displaystyle (x,t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0[/itex] [itex]\displaystyle (x,-t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0[/itex] Separation of Variables: [itex]T(x,y)=X(x)Y(y)[/itex] [itex]X''-\lambda{^2}X=0[/itex] [itex]Y''+\lambda{^2}Y=0[/itex] [itex]X(x)=c_1 \sinh{(\lambda{x})} + c_2 \cosh{(\lambda{x})}[/itex] [itex]Y(y)=c_3 \sin{(\lambda{y})} + c_4 \cos{(\lambda{y})}[/itex] [h2]The Attempt at a Solution[/h2] Use symmetry to simplify the boundary conditions in the y-axis direction by cutting the fin along the x-axis at [itex]y=0[/itex] where the flux will be zero: [itex]\displaystyle (x,0) \quad \frac{\partial{T}}{\partial{y}}=0[/itex] I have tried going through it using separation of variables about half a dozen times now but I can never solve for [itex]\lambda[/itex]. Here is my current progress (in Mathematica): [PLAIN]http://img6.imagebanana.com/img/38bbdmje/Selection_006.png[/PLAIN] Should I be setting my coordinate system up differently? Do I have to treat the convective (mixed) boundary conditions differently? Thanks! [/QUOTE]
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Analyzing 2D Fin Temperature with Separation of Variables
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