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 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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2 of 6 5.2.1 Explicit Scheme 4 4 43 4 4 42 1 4 4 1 difference central order 2nd difference forward order 1st , , 2 2 k t time at i x at k t at i x at x T t T = α 2 1 1 1 2 x T T T t T T k i k i k i k i k i Δ + = Δ + + () k i k i k i k i k i T T T x t T T 1 1 2 1 2 + + + Δ Δ + = Define: Δ Δ = 2 x t F “grid Fourier number” here ( )( ) 243 . 0 2 1 971 . 0 2 2 = = cm s F s cm k i k i k i k i FT FT T F T 1 1 1 2 1 + + + + = ( ¿ ) 514 . 0 2 1 = F Equation ( ¿ ) can be applied step-by-step to solve the problem. There know T values at the “old” time k t give an explicit solution for one unknown value at “new” time 1 + k t . t Δ step Start at 0 = t from given ICs / BCs BCs at ends 100 0 0 = T 0 0 5 = T ICs 20 0 1 = T 20 0 2 = T 20 0 3 = T 20 0 4 = T Now use ( ¿ ) to find all 1 i T values (values at sec 1 1 = t ) 100 1 0 = T (BC) 4 . 39 100 243 . 0 20 243 . 0 20 514 . 0 0 0 0 2 0 1 1 1 = + + = T T T T 0 1 5 = T (BC) = 1 2 T Find all 1 i T . Repeat process for each time step. Advantage – very easy to apply – explicit point-by-point calculation. Disadvantage – the method is stable only if ( ) 0 2 1 > F [Physically, a small increase in k i T should cause a small increase in 1 + k i T – only happens if ( ) 0 2 1 > F ] we require 2 1 2 < Δ Δ = x t F ± A major restriction that forces very small t Δ steps
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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

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