Orthogonality theorem y k 2 y 0 cf d 2 x dx 2 k 2

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Unformatted text preview: 4C0 n 2j 1 (j 0, 1, ...) (2j 1)π 0 n 2j (j 1, 2, ...) (2j 1)2 π 2 Dt 4C0 1 (2j 1)π) ∴ C(x,t) (2j 1) sin L exp π j 0 L2 Solid State Ionics Research Lab. Seoul National University Kinetics Ex. 2: Initial and boundary conditions C1 Steady state distribution (i) C(0<x<L, 0) = 0 (ii) C(0, t) = C1 (iii) C(L, t) = C2 C2 0 0 -Prob. With solution C(x,t) Ansin nππ e L n 2 π 2 Dt L2 ; An L 2L nππ ( x) sin dx L 0 L Fourier integral turns trivial! ∵ condition #1, φ(x)=0. -A Trick transient sol’n C = V(x) + U(x,t) Steady state sol’n V( x) C1 C2 C1 x L -Transform of I.C & B.C. C(0, t) = C1 = C1 + U(0, t) ∴ U(0, t) = 0 C(L, t) = C2 = C2 + U(L, t) ∴ U(L, t) = 0 C(0<x<L, 0) = V(x) + U(x, 0) = 0 ∴ U(x, 0) = -V(x) Solid State Ionics Research Lab. Seoul National University Kinetics -Differential equation C 2C D 2 t x U 2U D 2 t x n x n2 2 Dt / L2 U An sin e L 2L n x An V ( x) sin dx L0 L -If C1=C2=Co ; V(x) = Co 2C L 2C 4Co n x An o cos o cosn 1 L n L 0 n (2 j 1) L ( j 0, 1, 2, ) C V U (2 j 1) 2 2 Dt 1 (2 j 1)x Co (2 j 1) sin L exp j 0 L2 4Co Solid State Ionics Research Lab. Seoul National University Kinetics Ex. 3: In-diffusion from constant surface concentration Co 4C C o 0 ( 2 j 1) 2 2 Dt 1 ( 2 j 1)x (2 j 1) sin L exp L2 reflection (C C) Co Transpose by Co 4C C Co o 0 Solid State Ionics Research Lab. ( 2 j 1) 2 2 Dt 1 ( 2 j 1)x (2 j 1) sin L exp L2 Seoul National University Kinetics 17.4 Some remarks on the sol’n & its variations 1) C( x, t ) s ymmetric about x C 0 x xL / 2 x 0 L/2 L L 2 J L/2 0 L : an impermeable wall 2 2) A new boundary condition c 2C D 2 t x (i) Jx=0-=0 (impermeable wall) (ii) c(L,t) = 0 (iii) c(0<x<L...
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