problem4-04 - PROBLEM 4.4 KNOWN: Rectangular plate...

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PROBLEM 4.4 K NOWN: Rectangular plate subjected to prescribed boundary conditions. F IND: Steady-state temperature distribution. SCHEMATIC: A SSUMPTIONS: (1) Steady-state, 2-D conduction, (2) Constant properties. A NALYSIS: The solution follows the method of Section 4.2. The product solution is () ( )( ) ( ) ( ) -y +y 123 4 Tx , y Xx Yy Cc o sx Cs i nx Ce Ce λ λλ =⋅= + + and the boundary conditions are: T(0,y) = 0, T(a,y) = 0, T(x,0) = 0, T(x.b) = Ax. Applying BC#1, T(0,y) = 0, find C 1 = 0. Applying BC#2, T(a,y) = 0, find that λ = n π /a with n = 1,2,…. Applying BC#3, T(x,0) = 0, find that C 3 = -C 4 . Hence, the product solution is ( ) ( ) 24 n T x,y X x Y y C C sin x e e . a π ⎡⎤ ⎢⎥ ⎣⎦ Combining constants and using superposition, find n n1 nx ny T x,y C sin sinh . aa ππ = = To evaluate C n and satisfy BC#4, use orthogonal functions with Equation 4.16 to find 2 n 00 nb C Ax sin dx/sinh sin dx, a πππ =⋅ ∫∫ noting that y = b. The numerator, denominator and C n , respectively, are: [] a 2 22 a n+1 0 0 a a x A a A a A x sin dx
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