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lecture7-10

# lecture7-10 - At short times many terms are required for...

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Unformatted text preview: At short times, many terms are required for obtaining the accurate concentration profile. However, at long times, a single term is sufficient as each successive term is smaller than the preceding one. Consider the ratio of the maximum values of the first and second terms, ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 8 2 1 2 2 1 1 2 3 1 2 1 1 2 h t D h t D h t D e e e R π π π − + × − + × − = + × + × = For Dt h 75 . 4 ~ = R is about 100, so that for Dt h ~ < The error in using the first term to represent c ( x,t ) is less than 1% Experimentally, the average concentration is much easier to measure than the actual concentration profile. Using the first term ( ) ( ) ∫ = = − h h Dt o e c dx t x c h t c 2 2 2 8 , 1 π π Single Term Solution in Different Geometries For plate geometry, 2 2 2 8 h Dt s o s e c c c c π π − = − − For cylindrical geometry (R – Radius), 2 2 405 . 2 2 405 . 2 4 R Dt s o s e c c c c − = − − For spherical geometry (R – Radius), 2 2 2 6 R Dt s o s e c c c c π π − = − − Note: 2.405 is the first root of the Bessel function of zero order Method of Laplace Tranform While the separation of variables produces two ordinary differential equations, the Method of Laplace transform eliminates the time derivative and thus transforms the diffusion equation to an ordinary differential equation in space....
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lecture7-10 - At short times many terms are required for...

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