problem4-01 - PROBLEM 4.1 KNOWN Method of separation of...

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PROBLEM 4.1 K NOWN: Method of separation of variables for two-dimensional, steady-state conduction. FIND: Show that negative or zero values of λ 2 , the separation constant, result in solutions which annot satisfy the boundary conditions. c SCHEMATIC: A SSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: From Section 4.2, identification of the separation constant λ 2 leads to the two ordinary differential equations, 4.6 and 4.7, having the forms 22 dX dY X 0 Y 0 dx dy += −= λλ (1,2) and the temperature distribution is ( ) ( ) ( ) x,y X x Y y . =⋅ θ (3) Consider now the situation when λ 2 = 0. From Eqs. (1), (2), and (3), find that ( ) ( ) ( ) 12 34 X C C x, Y C C y and x,y C C x C C y . =+ = + + (4) Evaluate the constants - C 1 , C 2 , C 3 and C 4 - by substitution of the boundary conditions: ( ) ( )( ) () ( ) ( ) ( ) ( ) ( ) 1 23 4 3 24 2 4 x 0: 0,y C C 0 C C y 0 C 0 y x,0 0 C x C C 0 0 C 0 x L: L,0 0 C L 0 C y 0 C 0 y W: x,W 0 0 x 0 C W 1 = = +⋅ = = == + + = + + = + + = 0 1 = = The last boundary condition leads to an impossibility (0 1). We therefore conclude that a λ 2 value of zero will not result in a form of the temperature distribution which will satisfy the boundary
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