358 Fermentation and Biochemical Engineering Handbook Point 0 in Fig. 10 represents a hypothetical quantity obtained by arrangement of the above equation F-E=R-S=Q The material balance for each stage is E=r Thus, a line through Q represents the operating line between stages. The number of stages is obtained by sequentially stepping off first the equilibrium distribution along a tie line, and then to the next stage by a line drawn from point g through the raffinate to locate the next extrac 4.1 Simplified solution If the distribution coefficient is constant, and if there is essentially mutual solubility, the fraction not extracted, Y, can be calculated directly a function of the extraction factor, E, and the number of stages, n X1-Y/ E XE-Ys/ E E≠1 Treybal]discusses the derivation of these equations and presents a graphi solution reproduced here as Fig. ll Even when the two limitations of immiscibility and constant distribu tion coefficient do not quite hold, Fig. 1l 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 ofinsufficient time for diffusion of solute across the phase boundary or insufficient time for complete clarification of each stage
358 Fermentation and Biochemical Engineering Handbook Point Q in Fig. 10 represents a hypothetical quantity obtained by rearrangement of the above equation: F- E= R-S= Q The material balance for each stage is: Thus, a line through Q represents the operating line between stages. The number of stages is obtained by sequentially stepping off first the equilibrium distribution along a tie line, and then to the next stage by a line drawn from point Q through the rafiate to locate the next extract. 4.1 Simplified Solution If the distribution coeficient is constant, and if there is essentially no mutual solubility, the fraction not extracted, Y, can be calculated directly as a function of the extraction factor, E, and the number of stages, n. X,-Y,Im . mS Y= , E=- X, -Y, lm F Tre~bal[~l discusses the derivation ofthese equations and presents agraphical solution reproduced here as Fig. 1 1. 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
Solvent extraction 359 ■o6■ □L■■or TAINUI 0o02 000 00006 renumber of ideal stages igure 11. Countercurrent multistage extraction with immiscible solvents and constar stribution coefficient.(From: Liquid Extraction by R. E. Treybal. Copyrighto 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. Alter- nately, 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. (2J(41
Solvent Extraction 359 Figure 11. Countercurrent multistage extraction with immiscible solvents and constant distribution coeffrcient. (From: Liquid Extraction by R. E. Treybal. Copyright0 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 ba~kmixing.[*1[~]