problem4-58 - PROBLEM 4.58 KNOWN Bar of rectangular...

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PROBLEM 4.58 K NOWN: Bar of rectangular cross-section subjected to prescribed boundary conditions. FIND: Using a numerical technique with a grid spacing of 0.1m, determine the temperature istribution and the heat transfer rate from the bar to the fluid. d SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction, (3) onstant properties. C ANALYSIS: The nodal network has Δ x = Δ y = 0.1m. Note the adiabat corresponding to system symmetry. The finite-difference equations for each node can be written using either Eq. 4.29, for interior nodes, or Eq. 4.42, for a plane surface with convection. In the case of diabatic surfaces, Eq. 4.42 is used with h = 0. Note that a 2 h x 50W/m K 0.1m 3.333. k 1.5 W/m K Δ × = = Node Finite-Difference Equations 1 -4T 1 + 2T 2 + 2T 4 = 0 2 -4T 2 + T 1 + T 3 + 2T 5 = 0 3 -4T 3 + 200 + 2T 6 + T 2 = 0 4 -4T 4 + T 1 + 2T 5 + T 7 = 0 5 -4T 5 + T 2 + T 6 + T 8 + T 4 = 0 6 -4T 6 + T 5 + T 3 + 200 + T 9 = 0 7 -4T 7 + T 4 + 2T 8 + T 10 = 0 8 -4T 8 + T 7 +T 5 + T 9 + T 11 = 0 9 -4T 9 + T 8 + T 6 + 200 + T 12 = 0 10 -4T 10 + T 7 + 2T 11 + T 13 = 0 11 -4T 11 + T 10 + T 8 + T 12 + T 14 = 0 12 -4T 12 + T 11 + T 9 +200 + T 15 = 0 13 2T
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