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problem4-42 - PROBLEM 4.42 KNOWN One-dimensional fin of...

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PROBLEM 4.42 KNOWN: One-dimensional fin of uniform cross section insulated at one end with prescribed base emperature, convection process on surface, and thermal conductivity. t FIND: Finite-difference equation for these nodes: (a) Interior node, m and (b) Node at end of fin, n, here x = L. w SCHEMATIC: A SSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction. ANALYSIS: (a) The control volume about node m is shown in the schematic; the node spacing and control volume length in the x direction are both Δ x. The uniform cross-sectional area and fin perimeter are A c and P, respectively. The heat transfer process on the control surfaces, q 1 and q 2 , represent conduction while q c is the convection heat transfer rate between the fin and ambient fluid. erforming an energy balance, find P ( ) in out 1 2 c m-1 m m+1 m c c E E 0 q q q 0 T T T T kA kA hP x T T 0. x x = + + = + + Δ Δ Δ ± ± m = M ultiply the expression by Δ x/kA c and regroup to obtain 2 2 m-1 m+1 m c c hP hP T T x T 2 x T 0 1<m<n kA kA + + ⋅Δ
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