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order solution. However, the mixing-cup concentration given by ⟨u⟩A f (ξ) = Solving for the concentration profile with this velocity profile gives (4.42) (4.43) (4.44) c∗ = A (4.41) uc∗ dA differs from the plug-flow solution. As the fluid flow increasingly diverges from plug flow, the mixing-cup concentration continues to broaden, diverging from the plug-flow model prediction. Wall Effect To model a plug-flow velocity profile with wall effects, we exchange the quadratic function found in 4.39 for that of a piecewise-continuous function written as (a2−1) a2 , 0 ≤ ξ < a 1, a≤ξ≤1 c∗−γdc∗0Per h(a)−a2−1ξ2, ∗0dζ2 a24 with h (a) = 18 + 12 ln a − 18 a2 As before we evaluate the first order correction term for the fluid-phase con- centration for an isotherm with R = 0.95, axial and radial Peclet numbers of 100, and a velocity profile perturbation parameter γ of 0.1. Again, we assume that molecular diffusive forces dominate and can equate the axial and radial Peclet numbers. Fig- ure 4.5 depicts the first order correction term plotted against the dimensionless axial distance ζ. Similarly to the previous velocity profile the first order correction term becomes negligible and the concentration wave approaches asymptotic values deter- mined by the feed and presaturated conditions for values of ζ approaching −∞ and +∞, but has a significant impact on the fluid-phase concentration near the stoichio- metric center of the adsorbed-phase transition wave. Though when Figures 4.2 and 4.5 are compared with each other it can be seen that the magnitude and the sign in c = c∗−γdc∗0 Per h(a)−ξ2 +1lnξ−1lna+1, 0dζ2 4224 0≤ξPDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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