Seoul national university kinetics ex 3 in diffusion

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Unformatted text preview: , 0) = C0 0 Solid State Ionics Research Lab. L Solution, left as an exercise. Seoul National University Kinetics ( x ) Co 17.5 More insight into the series solution ( 2 j 1)2 2 Dt 4Co 1 ( 2 j 1)x Cx, t sin exp j 0 ( 2 j 1) L L2 1) Diffusion profile ----- a linear combination of Fourier sine waves 0 L L Odd number of half waves j=0 C C0 0 Boundary value problem j=1 at t =0 ; Cx,0 4Co 1 ( 2 j 1)x sin CO j 0 ( 2 j 1) L 2) Each wave component decays exponentially with time: Higher terms decay faster! Solid State Ionics Research Lab. Seoul National University Kinetics 3) How faster? - Ratio test ( j 1) th term R= jthterm ( 2 j 3) 2 2 Dt 1 ( 2 j 3)x sin exp 2j 3 L L2 ( 2 j 1) 2 2 Dt 1 ( 2 j 1)x sin exp ( 2 j 1) L L2 8( j 1) 2 Dt exp L2 After a short period of time elapsed, the infinite series can be satisfactorily represented by only first few terms Solid State Ionics Research Lab. Seoul National University Kinetics 4) First term approximation 8 2 Dt 2nd 1 R st exp 0.001 ( 0.1% accuracy) 1 3 L2 Dt 0.27 L L 3.7 Dt 4 Dt or L2 t 0.074 D Then, 2 Dt 4C0 x C( x, t ) sin exp 2 L L (within 0.1% accuracy) Long-time solution Solid State Ionics Research Lab. Seoul National University Kinetics 5) Comparison of the validity regimes Short-time sol’n Long -time sol’n (for 0.1% accuracy) L 4.6 Dt 2 L 3.7 Dt L2 t 14D L2 t 84D L2 84 D 0 t Shorttime sol’n (erf soln) Solid State Ionics Research Lab. L2 14 D Long-time sol’n (1st term approximation) Seoul National University Kinetics 6) Average concentration and its 1st term approximation ( 2 j 1)2 2 Dt 4Co 1 ( 2 j 1)x C( x, t ) sin exp 2j 1 L L2 ( 2 j 1)2 2 Dt 8Co 1 1 C(t ) C( x, t )dx 2 exp 2 L0 ( 2 j 1) L2 _ L -1st term approximation 2 Dt 8Co 8Co t C(t ) 2 exp 2 2 exp L _ L2 where 2D Solid State Ionics Research Lab. : the relaxation time of the system Seoul National University Kinetics 7) Determination of the relaxation time and hence, D 0 0.8 -2 ln( 1.0 0.6 0.4 0.2 -4 -1/τ -6 -8 Conductivity rlxn 0.0 -200 -10 0 200 400 600 800 1000 1200 1400 1600 t/s -200 0 200 400 600 800 1000 1200 1400 t/s c (t ) `c(t 0) ( t ) ( t 0) t exp c(t ) c(t 0) (t ) (t 0) Solid State Ionics Research Lab. Seoul National University...
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This document was uploaded on 11/26/2013.

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