A close-separation cascade is one in which a — l < 1. In such a cascade, the condition 0 = /a for an ideal cascade, in which heads and tails fed to a stage have the same composition, may be approximated by

(12.123)

Equation (12.102) for the tails flow rate in an ideal cascade may be approximated by

jxypziL (a-lMl-*)

Р(УР ~ x)
СЗ-1Ж1-Х)

because of (12.123). We shall now show that when the total tails flow rate of a dose-separation cascade is a minimum, the tails flow rate at each stage is given by (12.125).

The difference in composition between stage heads and stage tails, given by (12.76), may be approximated by

Уі+і -*/+1 =(«-1)Уї+і(1 ~Уі+і) (12.126)

in the close-separation case. A relation for the change in heads composition between adjacent stages is obtained by combining this with the material-balance equation (12.62):

Уі+1 ~Уі = (а — 1)У/+і(1 — Л+і)-дА (Ур-Уі) (12.127)

Jvi+i

Because Уі+І, у і, and х,- are nearly equal, this difference equation may be approximated by the differential equation

^ = (a-l)x(l — x)-^(yP-x)

The total tails flow rate in the enriching section is

is a minimum at all x. The optimum value of N that makes this a minimum is that at which the derivative of the denominator vanishes, or at which

	(a-l)x(l-x) 2P, , л : + —: (yp —x) = 0 ДгЗ W >	(12.131)
Thus	r 2P(yp-x) Nopt ~ (a — l)x(l —x)	(12.132)

This is just twice the minimum tails flow rate at which dx/di = 0.

This is identical with (12.125). Thus it has been shown that in the dose-separation case an ideal cascade may be defined in any one of the three following equivalent ways:

РІУР-Х)

" (p-iyxil-x)

N is so chosen that total interstage flow is a minimum

In such a cascade, the heads and tails fed to each stage have the same composition, and the cut в is I. The last may be seen from (12.21), which becomes

13-І =(а-1Х1 — в) (a-l<l) (12.133)

At the optimum flow rate, the change in composition per stage, from (12.128), is

§ = ^*0-*) (12-134)

which is just half the change at total reflux at which P/N = 0. The total number of stages in the enriching section is

Equation (12.95) reduces to this expression, except for terms of the order of unity.
The total tails flow rate in the enriching section at the optimum flow rate is

(12.136)

With-Nop, from (12.132) and dijdx from (12.134), this becomes

The total heads flow rate, from (12.120), reduces to the same expression.

K2**-l)ln —

The total flow rate in both stripping and enriching sections, from (12.122), becomes

(12.139)

The total heads flow rate or total tails flow rate in both sections is one-half this value.

These formulas are extraordinarily useful in roughing out the characteristics of an isotope separation plant without the necessity of designing every one of its stages, which often number in the thousands. As an illustration, the total heads flow rate in the uranium isotope separation example considered in Fig. 12.17 is

-213.57[(2X0.0072) — 1] In ^||j-= (217,343X191.57) = 41,636,000

(12.140)

This is the area within Fig. 12.17. Around 42 million moles of UF6 must be pumped for every mole of 90 percent 235UF6 separated.