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The Taylor series expansion of the Langmuir isotherm about the zeroth order solution, c∗0, is given by n∗ = c∗0 + R γc∗ +γ2c∗+ R(1−R) γc∗ +γ2c∗2+··· R+(1−R)c∗0 [R+(1−R)c∗0]2 1 2 [R+(1−R)c∗0]3 1 2 Taking the axial derivative of equation 4.28 gives ∂n∗ R ∂c∗0 R ∂c∗1 ∂ζ = [R+(1−R)c∗0]2 ∂ζ +γ[R+(1−R)c∗0]2 ∂ζ −γ 2R(1−R) dc∗0c∗ +Oγ2 [R+(1−R)c∗0]3 dζ 1 (4.28) (4.29) After substituting equations 4.10 and 4.29 into equation 4.6 and rearranging, the zeroth, first, and second order differential equations are ∂c∗0 = Pea c∗0 (1−R)(c∗0 −1) ∂ζ 2 R+(1−R)c∗0 (4.30) (4.31) (4.32) Perξ∂ξ ∂ξ Pea ∂ζ2 respectively. The third term of equation 4.31 and the second, third, fourth, and fifth terms of equation 4.32 are O (1 − R)2 or greater and are therefore negligible since as R approaches unity they approach zero faster than the other terms. This assumption is only valid for the conditions in which the quantity (1 − R)2 ≤ 0.0025, i.e., 0.95 ≤ R ≤ 1. The zeroth order term is solved by decomposing equation 4.30 by the method of partial fractions. These are then integrated to give 2 1 1−c∗0 ζ=Pe 1−R ln c∗R (4.33) a0 58 R ∂c∗1 ∂c∗0 2R(1−R) dc∗0∗ 1−[R+(1−R)c∗]2 ∂ζ +f(ξ) ∂ζ +[R+(1−R)c∗]3 dζc1 0∗2∗0 =21∂ξ∂c1 +2∂c1 Perξ∂ξ ∂ξ Pea ∂ζ2 R ∂c∗2 ∂c∗1 2R(1−R) dc∗0∗ and 1−[R+(1−R)c∗0]2 ∂ζ +f(ξ) ∂ζ +[R+(1−R)c∗0]3 dζc2− 2Rc∗1 (1−R) dc∗1c∗ + 3R(1−R)2 dc∗0c∗2 [R+(1−R)c∗]3 dζ 2 [R+(1−R)c∗]4 dζ 1 0 ∗ 2∗0 =21∂ξ∂c2 +2∂c2PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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