For more efficient fractional extraction of two or more extractable components, the extracting — scrubbing cascade of Fig. 4.4 is employed. Nomenclature for flow rates, concentration, and stage number is shown in Fig. 4.14. With the same assumptions and approach as in Sec. 6.1, a material balance for any one of the components in the portion of the cascade below stage n in the extracting section is

Eyf + (S + F)xf = Eyf_, + {S + F)xf

Л- — Уо = (X%-xf) (4.51)

The extracting-section operating line shown in the McCabe-Thiele diagram of Fig. 4.15 passes through the point (yf, xf) and has the slope (5 + F)jE.

A material balance around the portion of the plant above stage m of the scrubbing section is

5*o + Eysm = Sxsm _ j + £>?

or /„ — yf = ;£(*m_i -*o) (4.52)

E

In Fig. 4.15 this is represented by the operating line for the scrubbing section, which passes through (*o, Уі) and has the slope S/E.

As illustrated in Fig. 4.15, different equilibrium lines can exist for the extracting and scrubbing sections, as might occur if the scrub solution contains a different salting agent

concentration than the feed solution, or from the effect of one extractable component on the distribution coefficients of other extractable components (cf. Sec. 6.6).

In Fig. 4.15 a graphic solution is illustrated for a specified number of stages N and M in the extracting and scrubbing sections, respectively. Proceeding upward by vertical and hori­zontal steps from the point xf, yf for N vertical steps between the extracting-section operating and equilibrium lines, the concentration in the solvent leaving the extracting section is identical with that entering the scrubbing section; i. e.,

У%=А+і (4.53)

where Ух is found at (x%, yfj) on the equilibrium line for the extracting section. The horizontal projection of yf/ onto the scrubbing section operating line yields (xff, yff+1), and this point is then projected downward to the equilibrium line for the scrubbing section.

The number M of equilibrium stages in the scrubbing section specifies the number of vertical steps between the operating line and the equilibrium line, starting at y^+ and ending at У.

By thus determining the extract concentration for one of the extractable components in the feed, a similar graphic or numerical calculation is made for each of the other extractable components so that the composition of the organic product and aqueous raffinate can be determined. When two or more extractable components are each present in sufficient concentration to affect distribution coefficients of the other species, equilibrium lines must be calculated by an iterative procedure similar to that illustrated in Sec. 6.6 for TBP extraction in the Zr-Hf-HN03 system.

The operating lines for a given component in the extracting and scrubbing sections intersect at the feed composition x?. This can be demonstrated by defining xmn, ym„ as the intersection point, such that at the intersection

xmn ~ *n ~ xm -1 (4.54)

and ymn =yf_, =ysm (4.55)

Substituting Eqs. (4.54) and (4.55) into (4.51) and (4.52) and solving, we find that

Fxmn + EyE0 + Sxs0 =(S + F)xf + EyS (4.56)

However, an overall material balance for the extractable component written for the entire separation cascade of Fig. 4.14 is

Fx* + Eyf + Sxf = (S + F)xf + Eyf (4.57)

Comparison of Eqs. (4.56) and (4.57) shows that the operating lines intersect at

xmn = ^ (4-58)

In an extracting-scrubbing cascade with a finite number of stages, none of the aqueous streams entering or leaving a stage is at a concentration as high as xF. The intersection of the two operating lines represents only an extrapolated point that is useful in graphic construction of the operating lines.