problem4-46 - PROBLEM 4.46 KNOWN Steady-state temperatures...

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PROBLEM 4.46 KNOWN: Steady-state temperatures at selected nodal points of the symmetrical section of a flow channel with uniform internal volumetric generation of heat. Inner and outer surfaces of channel xperience convection. e FIND: (a) Temperatures at nodes 1, 4, 7, and 9, (b) Heat rate per unit length (W/m) from the outer surface A to the adjacent fluid, (c) Heat rate per unit length (W/m) from the inner fluid to surface B, nd (d) Verify that results are consistent with an overall energy balance. a SCHEMATIC: A SSUMPTIONS: (1) Steady-state, two-dimensional conduction, (2) Constant properties. ANALYSIS: (a) The nodal finite-difference equations are obtained from energy balances on control olumes about the nodes shown in the schematics below. v N ode 1 a b c d g q q q q E + + + + = ± 0 ( ) ( ) ( ) 3 1 2 1 T T T T 0 k y / 2 k x / 2 0 q x y / 4 0 x y + Δ + Δ + + Δ ⋅Δ = Δ Δ ± ( ) 2 1 2 3 T T T / 2 q x / 4 = + + Δ ± k ° ( ) ( ) ( ) 6 3 6 2 1 T 95.47 117.3 C / 2 10 W / m 25 25 10 m / 4 10 W / m K 122.0 C = + ° + × × × = N ode 4 a b c d e f g q q q q q q E
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