0 l x seoul national university kinetics 166

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Unformatted text preview: Cs 0 C’ From C(x,t) C’/2 C' x 1 erfc 2 2 Dt C 1 x 1 erfc C' 2 2 Dt 0 Solid State Ionics Research Lab. Seoul National University Kinetics 2) Cs By reflection w.r.t. x=0; x C Cserf 2 Dt 0 3) Cs Cs C 0 C - C0 2 x 1 erf 2 Dt or Co C - C0 1 x 1 erf Cs C 0 2 2 Dt 0 Solid State Ionics Research Lab. Seoul National University Kinetics 4) Cs By reflection C C0 1 x erfc Cs C0 2 2 Dt Co 5) Co Surface conc. fixed x C C0erfc 2 Dt 0 6) Co Surface conc. fixed x C C0 1 erf 2 Dt 0 Solid State Ionics Research Lab. Seoul National University Kinetics 16.7 Generalization of the error function solutions x C A Berf 2 Dt ex) C(x 0, t 0) Cs C(x 0, t 0) C0 A B Cs A B C0 cs co 0 1 C 1 (Cs C0 ) 1 (Cs C0 )erf x A (Cs C0 ) 2 2 2 Dt 2 1 C C0 1 x B (Cs C0 ) or 1 erf 2 Cs C 0 2 2 Dt Solid State Ionics Research Lab. Seoul National University Kinetics 16.8 Math. interlude: some properties of erf(z) 2 erf ( z ) z 0 e 2 d i) erf(z) z for z 0.6 From the identity 2 ii) erf(z) π 2 π ( 1)n n!(2n 1) z 2n1 1 15 z z3 z .. .... for s mall z 3 10 ( x) n ex n 0 n! d 2 - z 2 dz iii) erf(z) e dx dx π iv) i n erfc(z) i n 1erfc(z) d z Solid State Ionics Research Lab. Seoul National University Kinetics ierfc(z) 1 v) e rfc (η )dη ierfc(z) z erf(z) when z 0, i.e., ierfc(0) ? 0 1 erf(z)dz z erfc(z) 0 0 0 f z 2 z2 z e dz π g' 2 π 2 π 1 π 0 ze z dz 2 1 z2 2 e 0 For more, see Carslaw & Jeager Appendix Solid State Ionics Research Lab. Seoul National University Kinetics vi) For large x 2 erfc(x) π x e ξ 2 dξ By integrating by parts 1 π x 2ξ e f' 1 π 1 π -ξ 2 1 dξ ξ g 1 ξ 2 1 ξ 2 -e e 2 dξ ξ x ξ x 2 e x 1 1 1 3 1 3 5 x2 1 ξ 2 1 3 2 5 3 7 ... e ξe dξ 2x 2x xx ξ3 π x 2x 1 ξ 2 1 1 ξ 2 -e e 3 4 dξ x 2 ξ3 x ξ Solid State Ionics Research Lab. Seoul National University Kinetics 17. Infini...
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This document was uploaded on 11/26/2013.

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