problem4-70 - PROBLEM 4.70 KNOWN Straight fin of uniform...

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PROBLEM 4.70 K NOWN: Straight fin of uniform cross section with insulated end. FIND: (a) Temperature distribution using finite-difference method and validity of assuming one- dimensional heat transfer, (b) Fin heat transfer rate and comparison with analytical solution, Eq. 3.76, (c) ffect of convection coefficient on fin temperature distribution and heat rate. E SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in fin, (3) Constant roperties, (4) Uniform film coefficient. p ANALYSIS: (a) From the analysis of Problem 4.43, the finite-difference equations for the nodal arrangement can be directly written. For the nodal spacing Δ x = 4 mm, there will be 12 nodes. With A >> w representing the distance normal to the page, ( ) 2 2 2 2 3 3 c hP h 2 h 2 500W m K 2 x x x 4 10 mm 0.0533 kA k w kw 50W m K 6 10 m × ⋅Δ Δ = Δ = × = × × A A Node 1: or -2.053T ( ) 2 100 T 0.0533 30 2 0.0533 T 0 + + × + = 1 1 + T 2 = -101.6 Node n: or n 1 n 1 n T T 1.60 2.0533T 0 + + + = n 1 n n 1 T 2.053T T 1.60 + = − Node 12: ( ) ( ) 11 12 T 0.0533 2 30 0.0533 2 1 T 0 + + = or 11
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