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where γ is the perturbation parameter for which γ ≪ 1, and f (ξ) = O (1) is an axisymmetric term such that 1 ξf (ξ) dξ = 0 (4.9) 0 We solve equation 4.6 using a perturbation method with an assumed solution of the form ∞ c∗ =γkc∗k =c∗0+γc∗1+··· (4.10) k=0 Substituting equation 4.10 into equation 4.6 gives the zero order differential equation −∂n∗0 + ∂c∗0 = 2 ∂2c∗0 (4.11) ∂ζ ∂ζ Pea ∂ζ2 where n∗0 = n∗ (c∗0). We solve this equation following the method outlined by Coppola and LeVan4 resulting in dζ=2 dc∗0 (4.12) P e a ( c ∗0 − n ∗0 ) which they integrated for Langmuir (constant separation factor) and Freundlich isotherms. The first and second order differential equations are obtained by first writing the dimensionless adsorbed-phase concentration n∗ in a Taylor series expansion of the form centered about the plug-flow solution, shows that when taking the partial derivative k ∗ ∗k of the Taylor series expansion with respect to ζ, the terms ∂(d n/dc c∗′)/∂ζ for 0 k > 1 can be neglected for slightly favorable isotherms. The first and second order terms are then described by (4.14) dn∗ (c∗ − c∗′) d2n∗ (c∗ − c∗′)2 ∗∗∗′ 0 0 +··· (4.13) where c∗′ is the point at which the loading is centered. The subsequent example, n =n (c0)+ dc∗c∗′ 1! + dc∗2c∗′ 2! 00 0 dn∗ ∂c∗ ∂c∗ ∂c∗ 21∂∂c∗ 2∂2c∗ 10111 −dc∗c∗ ∂ζ+f(ξ)∂ζ+∂ζ=Perξ∂ξ ξ∂ξ +Pea∂ζ2 0 54PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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