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Substituting this into equation 4.38, the concentration profile is solved to give γPe dc∗ 1 ξ4 c∗=c∗0− r 0 −ξ2− (4.40) We evaluate the first order correction term for the fluid-phase concentration for an isotherm with R = 0.95, axial and radial Peclet numbers of 100, and a velocity profile perturbation parameter γ of 0.1. Here we assume that the velocity is low enough that the effects of molecular diffusion dominate those of dispersion, i.e., the axial and radial Peclet numbers are equal. Figure 4.2 shows the first order correction term plotted against the dimensionless axial distance ζ. As ζ approaches −∞ and +∞, the first order correction term becomes negligible and the concentration wave approaches asymptotic values determined by the feed and presaturated concentra- tions. However, near the stoichiometric center of the adsorbed-phase transition, the first order correction term has a sizable impact on the fluid-phase concentration. A comparison of concentration profiles evaluated from the plug-flow model and the plug-flow model with a laminar deviation is shown in Figure 4.3. The dimension- less concentrations c∗0 and c∗ have been plotted versus the dimensionless axial distance ζ, where they have been centered stoichiometrically using the adsorbed-phase transi- tion. As expected, when there is a slight deviation from plug flow, a radial gradient of concentration forms based on the shape of the velocity profile. For the case shown in Figure 4.3, the concentration at the centerline moves slightly ahead of the plug-flow solution while the concentration at the wall is slightly delayed. In Figure 4.4 the lines of constant concentration are similar in shape to that of the quadratic plug-flow perturbation used. Because of this, breakthrough will occur at different times over the radial position in the bed, occurring earlier at the centerline and later at the wall. It is interesting to note that taking the cross-sectional average of equation 4.40 collapses the solution for the fluid-phase concentration to that of the zeroth 4 2 dζ 3 2 60PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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