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19. 1D_Diffusion_Example

19. 1D_Diffusion_Example - 5.2 1-D Diffusion Equation...

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1 of 6 5.2 1-D Diffusion Equation Example Aluminum rod. 10 cm long, 1 cm diameter Initially at 20ºC Apply BCs for 0 t 1-D Diffusion Equation 2 2 x T t T = α = s cm 2 971 . 0 α I.C. ( ) 20 0 , = x T B.C.s ( ) 100 , 0 = t T ( ) 0 , 10 = t T ( ) 0 t Solve numerically for the rod temperature ( ) t x T , Choose say cm x 2 = Δ Six i x locations Choose say sec 1 = Δ t Time steps 0 0 = t , 1 1 = t , 2 2 = t , … Consider one time step from k t to 1 + k t t t t k k Δ + = + 1 cm x 2 = Δ sec 1 = Δ t Perform a grid- independence check later Notation Arbitrary time k t Arbitrary location i x ( ) k i k i T t x T = , 5 , , 2 , 1 , 0 K = i K , 2 , 1 , 0 = k no limit – predict T over the time range of interest Superscript – time Subscript – position T solution at “new” time 1 + k t (unknown) T solution at “old” time k t (known – starting from the ICs)
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