# Solute Extraction Cascade with Solvent Recycle

This Demonstration analyzes a solute extraction process. A raffinate stream with volumetric flow containing an unwanted solute enters a five-stage extraction cascade that is connected to a tank holding the extraction solvent. The solvent container is modeled as a perfectly mixed tank such that the solute composition leaving the tank at any time is the same as the composition in the tank at that time. This problem was solved analytically by Ugo Lelli [1]. Here a numerical solution using Mathematica is given that lets you examine how two key parameters for the extraction process influence the system performance. You can see the effect of varying different values of the solvent container residence time (expressed as a dimensionless parameter ) and the dimensionless parameter (related to the volumetric flow rates and in the raffinate and extract phases).
The Demonstration shows how the equilibrium composition (in the raffinate phase) of each stage in the cascade varies with extraction time, expressed in terms of the dimensionless variable . At the start , there is no solute in the solvent stream and thus . Then at , the raffinate stream has a step change in solute concentration . For very large values of the residence time or dimensionless parameter , we get the usual equilibrium cascade behavior without recycle and for all and . In contrast, for small values of residence time (or ), solvent saturation is observed at large values of the dimensionless time .

### DETAILS

A material balance over the solute phase for an arbitrary stage is:
, ,
where and are the stage holdup volumes expressed in for the raffinate and extract phases; and are the solute mass fractions in the raffinate and extract phases leaving the stage, respectively; and are the volumetric flow rates for the extract and raffinate phases expressed in ; and is the volumetric rate of solute that is transferred between phases at the stage. If equilibrium is established at each stage so that , the solute material balances can be combined to give a single equation for the change in mass fraction of solute in the raffinate at the stage:
.
For computational purposes it is convenient to work with the dimensionless form of this equation, which for the stage is given by:
,
where is the dimensionless solute composition given by . For the dilute system approximation to hold, we require that .
The specified dimensionless inlet composition of the raffinate stream for is . The dimensionless time is defined as , while the flow rate ratio parameter is given by . Finally, the dynamic behavior of the solvent tank is governed by:
, , or equivalently, , ,
where the dimensionless parameter and is the residence time of the solvent tank with volume .
References
[1] U. Lelli, Annali di Chimica, 56, 1966 pp. 113–125.
[2] J. Ingham, I. J. Dunn, E. Heinzle, and J. E. Prenosil, Chemical Engineering Dynamics, 2nd ed., Weinheim, Germany: Wiley-VCH, 2000.

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