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where the constant of integration has been determined by Coppola and LeVan4 to be zero from the relationship in equation 4.23. The axial component of the first order term is determined from R R+(1−R)c∗ g (ζ) = B1 exp 1 − 0 dc∗0 (4.34) [R+(1−R)c∗0]2 c∗0 (1−R)(c∗0 −1) Using the solution found in equation 4.33 and the method of partial fractions, equation 4.34 is solved to give The first and second order terms now have the forms ′ ξ ξ ξ′′f(ξ′′)dξ′′ c∗1=− 0 B1− r found in equations 4.23 and 4.24 the term c∗1 is then written ∂c∗ Pe g (ζ) = B ∂c∗0 1 ∂ζ (4.35) (4.36) 0 dξ′ ∂ζ 20 ξ′ and ′ ∂c∗ Pe ξ ξ ξ′′f(ξ′′)dξ′′ ′ 1 ξ ξ ξ′′f(ξ′′)dξ′′ c∗=B ∂c∗0 2 2∂ζ (4.37) Substituting equations 4.36 and 4.37 into equation 4.28 and using the relationships dξ′−2ξ 0 with the axial Peclet number factored into the term ∂c∗0 as seen in equation 4.30. ∂ζ Subsequently we then determine the solution for this example for two velocity profiles, the first a laminar deviation from plug flow and a plug-flow velocity profile with wall effects. Parabolic Perturbation from Plug Flow A laminar deviation is introduced to the velocity profile by replacing the f (ξ) term found in equation 4.8 with f (ξ) = 1 − 2ξ2 (4.39) c∗=0r 0 1∂ζ20ξ′ 00ξ′ dξ′dξ (4.38) 59PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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