problem4-07 - PROBLEM 4.7 KNOWN: Boundary conditions on...

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PROBLEM 4.7 KNOWN: Boundary conditions on four sides of a square plate. FIND: Expressions for shape factors associated with the maximum and average top surface temperatures. Values of these shape factors. The maximum and average temperatures for specified conditions. SCHEMATIC: 0 0 x W y W T 1 T 1 T 1 ′′ s q ASSUMPTIONS: (1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS: We must first find the temperature distribution as in Problem 4.5. Problem 4.5 differs from the problem solved in Section 4.2 only in the boundary condition at the top surface. Defining θ = T – T , the differential equation and boundary conditions are 22 θθ + = 0 xy ∂∂ s yW θ θ (0,y) 0 θ (L,y) 0 θ (x,0) 0 k q y = === = (1a,b,c,d) The solution is identical to that in Section 4.2 through Equation (4.11), n n=1 n π xn π y θ = C sin sinh L L (2) To determine C n , we now apply the top surface boundary condition, Equation (1d).
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problem4-07 - PROBLEM 4.7 KNOWN: Boundary conditions on...

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