When the distribution coefficients are independent of stage number, an equation can be derived for analytical calculation of the number of stages.

For any extractable component with a constant distribution coefficient, Eqs. (4.27) and (4.28) can be rewritten in terms of the constant extraction factor (3:

Equation (4.45) is a form of the Kremser equation, originally derived for countercurrent gas absorption [S3].

The raffinate concentration х, may be eliminated by an overall material balance,

ЩхР ~ Xi) = E(yN — y0) which combines with (4.45) to yield

■>7v -Уо = 0N ~ 1

DxF — y0 0N+1 — 1

Thus, by specifying /3 for the cascade and the ratio Уо/DxF and recovery p for any one of the extractable components, the required number of equilibrium stages N can be calculated from (4.48).

If we have extractable components A and В in the feed to be separated in a simple extraction cascade, the constant distribution coefficients DA and DB result in extraction factors PA and 0B, and the overall decontamination factor fAB is obtained by applying Eq. (4.48) to each of the components, with

For the special case of y0 = 0, Eqs. (4.48) and (4.49) combine to yield

(4.50)

For a specified number of stages N and specified E/F, the decontamination factor for any two extractable components can be calculated from (4.48) and (4.49) or, for y0 = 0, from (4.50).

To illustrate the use of these equations, consider the extraction of zirconium from an aqueous solution of zirconium and hafnium nitrates, as shown in Fig. 4.12. Although the distribution coefficients do, in fact, depend on the concentration of zirconium and hafnium, as shown later in Sec. 6.6, constant distribution coefficients are assumed here for the purpose of this illustration. The specified feed composition and the specified recovery to be obtained are listed in Table 4.6.

The distribution coefficients assumed above are those observed by Нигё and Saint James

Figure 4.11 Number of equilibrium stages in an extraction cascade. (Adapted from Sherwood et al [S3].)

[H4] for an aqueous solution of the feed of concentration 3.5 ЛЧп NaN03 and 3.0 jV in HNO3, in contact with 60 percent TBP in kerosene. Distribution coefficients will be higher at the bottom of the cascade, where the aqueous zirconium concentration is lower; this will be neglected in the present treatment, but will be taken into account in Sec. 6.6.

By applying Eqs. (4.37) and (4,48) to zirconium,

hr= 1.2

from which N = 12.2.

The zirconium-hafnium decontamination factor is obtained from (4.48) and (4.49) with 0Hf“O.12:

The McCabe-Thiele diagrams for this example are shown in Fig. 4.13.

It is interesting to compare the decontamination obtainable for pZl = 0.98 and E/F = 1.0 with that obtainable with an infinite number of stages, corresponding to operation at the same zirconium recovery but at (£/F)mm — From Eq. (4.35):

= 0.817

and with N = », Eq. (4.50) yields

A more effective way to use an increased number of stages in a simple extracting cascade would be to increase the zirconium recovery. This would occur by allowing the slope of the operating line to approach DZr. In the limit of N-*”0, xZr, i “*■ 0 and p zr_>l — Because Dnt<Dzt, the operating line for hafnium can intersect the hafnium equilibrium line only at x > xfif and not at *нг = 0. In this limit of

Source: Adapted from J. Hure and R. Saint James, “Process for Separation of Zirconium and Hafnium,” Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, vol. 8, United Nations, New York, 1956, p. 551.

we find from (4.50) that

To obtain better decontamination a scrubbing section is added to the cascade, as illustrated in Fig. 4.4 and analyzed in the following section.