Seoul national university kinetics vi for large x 2

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Unformatted text preview: te Series Solution – Long-time solution 17.1 Introduction - Validity regime of the short time sol’n L 4.6 Dt -L 0 or L2 t 4.6 2 D within 0.1% accuracy L - Complete homogenization is never achieved, because the media is infinite. - What would happen if the media is finite or time is infinitely long? - Expectation Cs t 2 1 1 0 L L L For a short time 4.6 Dt 2 x 1) C Cserfc 2 Dt xL 2) C Cserfc translatio by L n 2 Dt Then reflection with respect to x L L x C C x 1 erf 2 Dt Solid State Ionics Research Lab. As t increases C sin πx t/ e L Seoul National University Kinetics 17.2 Solution -Fick’s 2nd law and IC and BC Co C 2C D 2 t x ( x ) 0 0 C ( 0<x<L, t=0 ) = ф(x) C (x=0, t ) = 0 C ( x=L, t ) = 0 L -Method of separation of variables: C(x,t) = X(x)· T(t) 1 dT d2X X T D dt dX 2 1 dT 1 d 2 X k 2 2 DT dt X dx 1 d ln nT 2 D dt k 2 d X k 2X 0 dx 2 T T0e k 2 C Ansinkn x Bncoskn x e 2 k n Dt n k for any real value Dt X A' sin k x B' cos k x Solid State Ionics Research Lab. Seoul National University Kinetics -Determination of An & Bn ① C(0, t) = 0 ∴ Bn = 0 ② C(L, t) = 0 ∴ sinknL = 0 Boundary (eigen) value problem ∴ k n L nπ ( n = 1, 2, 3, … ) nπ ∴ kn L ( n = 1, 2, 3, … ) 22 n π Dt nππ L2 ③ C Ansin e L nππ C(x,0) A n sin (x) L n From the orthogonality theorem, L o L nππ nππ L (x)sin dx A n sin 2 dx A n 0 L L 2 2 nππ dx ∴ A n 0 ( x) sin L L L Solid State Ionics Research Lab. Orthogonality Theorem y k 2 y 0 (cf : d 2 X / dx 2 k 2 X 0) k 1 , y1 ; k 2 , y 2 2 y k 1 y 1 0 1 2 yy 2 k 1 y 1y 2 0 1 y2 k 2 y 2 0 2 y2y 1 k 2 y 2 y 1 0 2 2 0 yy 2 y2y 1 dx k 1 k 2 1 2 y y 1 2 y 2 y 1 dx y1y 2 y2 y 1 L y1y2 y2 y1 dx 0 0 0 if k1 k 2 , then if k1 k 2 , then L L 0 0 y1 y 2dx 0 y1 y 2dx 0 Seoul National University Kinetics ( x ) Co 17.3 Specific examples Ex. 1: φ(x)=Co 2C0 An L L 0 L 2C0 L nππ nππ sin dx cos L L nπ L 0 0 0 L 2C0 1 cosnπ nπ...
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This document was uploaded on 11/26/2013.

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