Thermodynamics HW Solutions 457

Thermodynamics HW Solutions 457 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction 5-59E The top and bottom surfaces of a V-grooved long solid bar are maintained at specified temperatures while the left and right surfaces are insulated. The temperature at the middle of the insulated surface is 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 are constant. Analysis The nodal spacing is given to be Δ x = Δ y = l =1 ft, and the general finite difference form of an interior node for steady two-dimensional heat conduction with no heat generation is expressed as 0 4 0 4 node bottom right top left 2 node node bottom right top left = + + + = + + + + T T T T T k l g T T T T T & There is symmetry about the vertical plane passing through the center. Therefore, T 1 = T 9 , T 2 = T 10, T 3 = T 11, T 4 = T 7 , and T 5 = T 8 . Therefore, there are only 6 unknown nodal temperatures, and thus we need only 6 equations to determine them uniquely. Also, we can replace the symmetry lines by insulation and utilize the mirror-image concept when writing the finite difference equations for the interior nodes.
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