Axial dispersion in time‐variable laminar flow in a tubular reactor is analyzed using an exact procedure for the case of a homogenous first‐order reaction. For the first time since the Taylor Dispersion model was originally introduced for the modeling of reactors, its validity is examined over a wide range of the reaction rate parameter by comparison against an exact analysis. It is shown that a constant coefficient dispersion model can be obtained from first principles for large values of time only for initial distribution problems; however, this simple approximate model also is reasonably good for describing concentration distributions for the present inlet distribution problem for slow reactions and for axial locations sufficiently far away from the inlet. For rapid reactions, while the dispersion model is inaccurate in describing axial concentration distributions, it is surprisingly good for predicting the reactor length required for complete conversion. In contrast to the conclusion of a recent article, it will be shown that the dispersion coefficient is independent of the reaction rate constant.
The swelling and shrinking of ion exchange resins due to ion exchange and/or electrolyte sorption is a commonly observed phenomenon (Helfferich, 1962). However, there have been no attempts to include the effects of this phenomenon in design models for column sorption and ion exchange processes even though 20% changes in resin volume are not unusual (Dow Chemical Company, 1971).In this paper a design model for the special case of ideally sharp fronts is developed. Additionally the results are used to describe, at least in a qualitative manner, constant pattern front situations.As a resin particle changes volume it also changes mass. The change in mass of a bed of resin causes a difference between the mass flow rate at the inlet and outlet of the bed. The amount of solute in the bed depends on the inlet and outlet mass flow rates and hence is coupled to the swelling and shrinking phenomenon. If the mass of the resin bed decreases during a saturation operation, then the mass flow rate at the inlet of the bed will be less than the mass flow rate at the outlet of the bed. If the operation is at constant effluent flow rate and one uses a constant density, constant mass flow rate approach based on the effluent flow rate to describe the operation, then the predicted amount of solute in the bed at any time will be greater than the actual amount. Furthermore, the predicted breakthrough time will be less than the actual breakthrough time.In order to quantitatively determine the effect of the swelling and shrinking phenomenon one must simultaneously solve a solute mass balance and an overall mass balance on the resin bed. This is most easily done by assuming that equilibrium exists and that the front is ideally sharp. Under these assumptions a homogeneous region between the inlet of the bed and the concentration discon-where VN is the volume of the region described above and Vo is the volume that region would occupy if the resin were in its initial state (see Figure 1). VN and VO are related by the equation VN = VO(DN/DO)~ where DN is the diameter of a resin particle in equilibrium with the feed solution and Do is the initial particle diameter. The length of the homogeneous region described above is AZ so that VN = SAZ and VO = (Do/DN)~SAZ. Using these relations, the overall mass balance becomesIf one further assumes that the feed solution concentration is constant and that the bed is initially uniform with respect to concentrations, then a solute balance yields rate of dt -A0 $' rate of dt 0 mass out Once the inlet or outlet flow rate is specified the unknown flow rate is eliminated by combining Equations (2) and From this, the expression for the breakthrough time for the constant effluent flow rate case is and for the constant influent flow rate case isThe case of the saturation of an initially solute free bed at constant effluent flow rate was discussed briefly above. It was qualitatively argued that the shrinking and swelling phenomenon would have some effect on this process. This can be further illustrated using the ab...
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