Theory of Countercurrent Equilibrium Extraction

The calculation of the concentration of extractable components in a countercurrent cascade of equilibrium solvent extraction stages is first developed for the simple countercurrent extraction section of Fig. 4.3. The theory is then extended to the extracting-scrubbing system of Fig. 4.4 for fractional extraction and is illustrated by a numerical calculation for the separation of zirconium from hafnium, using TBP in kerosene as solvent. 

Concentrations in the organic and aqueous phases leaving a stage are related by the equilibrium relation 

Assume that the cascade is to reduce the concentration of the extractable component from x to X] by extraction with organic of relative volume /F. The point Xi, yo is thereby specified. Beginning at Xi, jvo and projecting upward in vertical and horizontal steps, the compositions for all of the other equilibrium stages are determined. The number of vertical projections between the operating line and equilibrium line necessary to step from Xi to x gives the required number of equilibrium stages. 

Given this number of stages, construction of a similar McCabe-Thiele diagram for other components in the feed, such as impurities, allows the calculation of the extent to which these impurities extract into the organic phase. If two or more extractable components are each in sufficient concentration to affect the distribution coefficient of the other species, e.g., TBP extraction of U02(N03)j and HNOa, the equilibrium lines for the two components cannot be specified in advance but must be calculated by an iterative procedure, similar to that to be illustrated in Sec. 6.6 for the zirconium-hafnium separation. 

From Eq. (4.27) or from the construction of Fig, 4.9 it is apparent that the ratio of organic flow rate to aqueous flow rate is given by 

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