problem4-55

# problem4-55 - PROBLEM 4.55 KNOWN Flue of square cross...

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PROBLEM 4.55 KNOWN: Flue of square cross section with prescribed geometry, thermal conductivity and inner and uter surface convective conditions. o FIND: (a) Heat loss per unit length, q , by convection to the air, (b) Effect of grid spacing and onvection coefficients on temperature field; show isotherms. c SCHEMATIC: Schematic (a) ASSUMPTIONS: (1) Steady-state, two-dimensional conduction, (2) Constant properties. ANALYSIS: (a) Taking advantage of symmetry, the nodal network for a 75 mm grid spacing is shown in schematic (a). To obtain the heat rate, we need first to determine the temperatures T i . Recognize that there are four types of nodes: interior (4-7), plane surface with convection (1, 2, 8-11), internal corner with convection (3), and external corner with convection (12). Using the appropriate relations from Table 4.2, the finite-difference equations are Node Equation 1 ( ) i i 4 2 2 ,i 1 2h x h x 2T T T T 2 2 T k k Δ Δ + + + + = 0 4.42 2 ( ) i i 5 3 1 ,i 2 2h x h x 2T T T T 2 2 T 0 k k Δ Δ + + + + = 4.42 3 ( ) ( ) i i 6 6 2 2 ,i 3 2h x h x 2 T T T T T 2 3 T 0 k k Δ Δ + + + + + = 4.41 4 ( ) 8 5 1 5 4 T T T T 4T 0 + + + = 4.29 5 ( ) 9 6 2 4 5 T T T T 4T 0 + + + = 4.29 6 ( ) 10 7 3

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