problem4-44 - PROBLEM 4.44 KNOWN Nodal temperatures from a...

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PROBLEM 4.44 KNOWN: Nodal temperatures from a steady-state, finite-difference analysis for a one-eighth symmetrical section of a square channel. FIND: (a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4 and 7, and determine T 2 , T 4 and T 7 , and (b) Heat transfer loss per unit length from the channel, . q SCHEMATIC: Node T( ° C) 1 430 3 394 6 492 8,9 600 ASSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction, (3) No internal volumetric generation, (4) Constant properties. ANALYSIS: (a) Define control volumes about the nodes 2, 4, and 7, taking advantage of symmetry where appropriate and performing energy balances, in out E E 0 = ± ± , with Δ x = Δ y, Node 2 : a b c d q q q q + + + = 0 ( ) ( ) ( ) 3 2 6 2 1 2 2 T T T T T T h x T T k y 2 k x k y 2 0 x y x Δ + Δ + Δ + Δ = Δ Δ Δ ( ) ( ) 2 1 3 6 T 0.5T 0.5T T h x k T 2 h x k = + + + Δ + Δ ( ) [ ] 2 2 T 0.5 430 0.5 394 492 50W m K 0.01m 1W m K 300 K 2 0.50 = × + × + + × + < 2 T 422K = Node 4 : a b c q
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