Thermodynamics HW Solutions 488

Thermodynamics HW Solutions 488 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction where k = 0.84 W/m. ° C, , T αρ == × kC /. 039 10 6 m/ s 2 i = T o = -3°C h i = 6 W/m 2 . ° C, h o = 20 W/m 2 . ° C, Δ x = 0.002 m, and Δ y = 0.01 m. The upper limit of the time step Δ t is determined from the stability criteria that requires the coefficient of in the T expression (the primary coefficient) be greater than or equal to zero for all nodes. The smallest primary coefficient in the 9 equations above is the coefficient of T in the expression since it is exposed to most convection per unit volume (this can be verified). The equation for node 6 can be rearranged as T m i m i + 1 i 9 T i 6 1 + Δ + Δ + + Δ Δ + Δ + Δ + Δ Δ = + 2 5 2 9 3 0 6 2 2 1 6 2 1 1 2 1 x T y T T T x k h t T x y x k h t T i i i o i o i α Therefore, the stability criteria for this problem can be expressed as Δ + Δ + Δ Δ Δ + Δ + Δ Δ 2 2 2 2 1 1 2 1 t 0 1 1 2 1 x y x k h x y x k h t o o Substituting the given quantities, the maximum allowable value of the time step is determined to be or, s 8 . 4 m) 01 . 0 ( 1 ) m 002 . 0 ( 1 m) 002 . 0 )( C W/m
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This note was uploaded on 01/22/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.

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