problem4-02

# problem4-02 - PROBLEM 4.2 KNOWN Two-dimensional rectangular...

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Unformatted text preview: PROBLEM 4.2 KNOWN: Two-dimensional rectangular plate subjected to prescribed uniform temperature boundary conditions. FIND: Temperature at the mid-point using the exact solution considering the first five non-zero terms; assess error resulting from using only first three terms. Plot the temperature distributions T(x,0.5) and (1,y). T SCHEMATIC: A SSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: From Section 4.2, the temperature distribution is ( ) ( ) ( ) ( ) n 1 1 2 1 n 1 1 1 s i n h n y L T T 2 n x x, y sin T T n L s i n h n W L θ π π θ π π + = − + − ⎛ ⎞ ≡ = ⋅ ⎜ ⎟ − ⎝ ⎠ ∑ . (1,4.19) Considering now the point (x,y) = (1.0,0.5) and recognizing x/L = 1/2, y/L = 1/4 and W/L = 1/2, ( ) ( ) ( ) ( ) n 1 1 2 1 n 1 1 1 s i n h n T T 2 n 1,0.5 sin T T n 2 s i n h n 2 θ π π θ π π 4 + = − + − ⎛ ⎞ ≡ = ⋅ ⎜ ⎟ − ⎝ ⎠ ∑ ....
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