Treybal[3] discusses the derivation ofthese equations and presents agraphical solution reproduced here as Fig. 11.

Even when the two limitations of immiscibility and constant distribution coefficient do not quite hold, Fig. 11 does allow a quick estimate of the trade-offs between solvent/feed ratio and number of stages required to obtain a desired degree of extraction (raffinate purity).

The above solutions are all based on ideal or theoretical stages. Even in discrete stage systems, like mixer-settlers, equilibrium may not be attained because of insufficient time for diffusion of solute across the phase boundary or insufficient time for complete clarification of each stage.

Figure 11. Countercurrent multistage extraction with immiscible solvents and constant distribution coefficient. (From: Liquid Extraction by R. E. Treybal. Copyright© 1963, McGraw-Hill. Used with the permission.)

In continuous differential extractors (columns) it has been convenient to think in terms of a height equivalent to a theoretical stage (HETS), and to correlate HETS as a function of system and equipment variables. Alternately, correlations may be obtained on the basis of the height of a transfer unit (HTU), which is more amenable to calculations which separately include the effects of backmixing.[21[4]

4.2 Sample Stage Calculation

An aqueous waste stream containing 3.25% by weight phenol is to be extracted with one-third its volume of methylene chloride to produce a raffinate without more than 0.2% phenol. How many stages are required?

Graphical Solution. Figure 12 is constructed using the equilibrium data for the distribution of phenol between methylene chloride and water from Fig. 6.

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