Thermodynamics HW Solutions 461

Thermodynamics HW Solutions 461 - heat transfer to be...

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Chapter 5 Numerical Methods in Heat Conduction 5-61 The top and bottom surfaces of an L-shaped long solid bar are maintained at specified temperatures while the left surface is insulated and the remaining 3 surfaces are subjected to convection. The finite difference formulation of the problem is to be obtained, and the unknown nodal temperatures are to be determined. Assumptions 1 Heat transfer through the bar is given to be steady and two-dimensional. 2 There is no heat generation within the bar. 3 Thermal properties and heat transfer coefficients are constant. 4 Radiation heat transfer is negligible. Properties The thermal conductivity is given to be k = 12 W/m °C. Analysis ( a ) The nodal spacing is given to be Δ x = Δ x = l =0.1 m, and all nodes are boundary nodes. Node 1 on the insulated boundary can be treated as an interior node for which . Using the energy balance approach and taking the direction of all
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Unformatted text preview: heat transfer to be towards the node, the finite difference equations for the nodes are obtained to be as follows: 4 node bottom right top left = − + + + T T T T T Node 1: 4 2 120 50 1 2 = − + + T T Node 2: 120 2 50 2 ) ( 2 2 1 2 3 2 2 = − + − + − + − + − ∞ l T kl l T T kl l T T l k l T l k T T hl 120°C 1 2 3 • • • h , T ∞ Insulated 50°C Node 3: 120 2 2 ) ( 3 3 2 3 = − + − + − ∞ l T l k l T T l k T T hl where l = 0.1 m, k = 12 W/m ⋅° C, h =30 W/m 2 ⋅° C, and T ∞ =25 ° C. This system of 3 equations with 3 unknowns constitute the finite difference formulation of the problem. ( b ) The 3 nodal temperatures under steady conditions are determined by solving the 3 equations above simultaneously with an equation solver to be T 1 = 85.7 ° C, T 2 =86.4 ° C, T 3 =87.6 ° C 5-64...
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