Thermodynamics HW Solutions 478

Thermodynamics HW Solutions 478 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction Time Time Nodal temperatures, ° C step, i T 0 T 1 T 2 T 3 T 4 T 5 T 6 0 h (7am) 0 20.0 16.7 13.3 10.0 6.66 3.33 0.0 6 h (1 pm) 24 17.5 16.1 15.9 18.1 24.8 38.8 61.5 12 h (7 pm) 48 21.4 22.9 25.8 30.2 34.6 37.2 35.8 18 h (1 am) 72 22.9 24.6 26.0 26.6 26.0 23.5 19.1 24 h (7 am) 96 21.6 22.5 22.7 22.1 20.4 17.7 13.9 30 h (1 pm) 120 21.0 21.8 23.4 26.8 34.1 47.6 68.9 36 h (7 pm) 144 24.1 27.0 31.3 36.4 41.1 43.2 40.9 42 h (1 am) 168 24.7 27.6 29.9 31.1 30.5 27.8 22.6 48 h (7 am) 192 23.0 24.6 25.5 25.2 23.7 20.7 16.3 The rate of heat transfer from the Trombe wall to the interior of the house during each time step is determined from Newton’s law of cooling using the average temperature at the inner surface of the wall (node 0) as QQ t h A T T t h A T T T ii i i i Trumbe wall Trumbe wall in 0 in in in == = + & () [ ( ) / ] ΔΔ 00 1 2 t Δ T t Therefore, the amount of heat transfer during the first time step ( i = 1) or during the first 15 min period is Btu 8 . 96 h) 25 . 0 ]( F 70 2 / ) 70 3 . 68 )[( m 7 C)(2.8 . W/m 1 . 9 ( ] 2 / ) [( 2 2 in 0 0 1 0 in 1 wall Trumbe = ° + × ° = Δ + = t T T T A h Q The negative sign indicates that heat is transferred to the Trombe wall from the air in the house which represents a heat loss. Then the total heat transfer during a specified time period is determined by adding
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