The fact that the total internal flow rate in a close-separation, ideal cascade is given by Eq. (12.142) may be derived without solving explicitly for the individual internal flow rates by the following development, due originally to P. A. M. Dirac. This procedure is valuable in showing the fundamental character of the separation potential and the separative capacity, and provides a point of departure for the treatment of multicomponent isotope separation.

We consider a close-separation, ideal cascade whose external streams have molar flow rates Xk (positive if a product, negative if a feed), and compositions xk expressed as mole fraction. Let us look for a function of composition ф(хк), to be called the separation potential, with the property that the sum over all external streams, to be called the separative capacity D,

D = 2 Хкф(хк) (12.162)

is proportional to the sum of the flow rate of all internal streams. At this point in the derivation, the nature of ф(хк) is assumed not to be known.

Figure 12.22 represents stages i— 2, і — 1, i, and / + 1 of such a cascade, with the fcth product stream consisting of part of the heads stream of stage і — 1. The total internal flow from stage і is Mt + JV). The separative capacity of the r’th stage, considered as an isolated plant, is

Ді = міФ<Уі) + Nttfxt) — {Mt + NtWzt) = {Mt + ЮІЄіФі) + (1 — віУр(Хі) — фі)]

and

Substitution of these expansions into (12.163) yields

Д/ = [8 i(yі — г,)2 + (1 — віфі — z, f]

where the term in d<j>ldz has dropped out because of the material-balance relations (12.8) and (12.9).

In a close-separation cascade,

Уі ~ z, = (1 — 0/X« — l)z,(l — Zi) (12.167)

as may be seen from (12.18) and (12.21), with (a — 1) and (j3 — 1) considered small relative to unity. Similarly,

Xi — Zi = — в,{a — l)z,-(l — z,)

Figure 12.22 Flow in portion of ideal cascade. Molar flow rates denoted by capital letters, mole fractions by small letters.

(12.173)

where Д,- is defined by (12.163).

When the separation potential satisfies (12.172), the separative capacity of a single stage in a close-separation cascade operated at a cut of ^(Af = iV) from Eq. (12.170) is

Therefore, (12.166) becomes Д/ = (Cc — 1)4(1 — «/)*?( 1 — z, f (z,) In a close-separation, ideal cascade et = so that the total flow leaving the rth stage is 8 Д,

_ M,(a — If

‘І——- A——

We shall now show that the separative capacity of the entire cascade, D, is given by

an

all external

stages streams

D= I A’= £ ХкФ&к) (12.175)

Consider first the sum of the separative capacity of stages і and / — 1.

Ді + ін = + Nttix,) — + NfWz,) +

+ +^нЖгн) 02.176)

The internal streams between this pair of stages, Af,_ t and Nt, may be expressed in terms of

the streams external to this pair of stages Mit Ni+U Xk, Af,_2, and 7V,_ j by means of the

material-balance relations:

Nt = Ni-y. — M,_2 (12.177)

and Nt-Mi-l=Ni+i-M,-Xk (12.178)

*<+i = 2i =Уі~1 (=**) xi = zf-l = Уі-2

Because of the assumption that this is an ideal cascade,

By means of these four equations (12.176) may be expressed as

Д/ + At — і = Міф(уі) — Мі+1ф(хі+1) — М^Фі-т) + + Xкф(хк) (12.181)

This is an example of (12.175) applied to the pair of stages і and і— 1. If Д<+1 is added to this expression, terms in M1 and Nl+, may be eliminated in the same way. By proceeding in this way until the separative capacity of every stage has been included in the sum, terms representing all internal streams cancel out, the only terms that remain on the right are those representing external streams, and Eq. (12.175) results.

Thus, we have shown that the separative capacity of an ideal cascade is the sum of the separative capacities of its component stages. And if the separation potential satisfies the differential equation (12.172), the total internal flow is given by

external

streams

j+K=j^y Y, Xk<Kxk) (12-182)

as was to have been shown.

The general solution of (12.172) is

Ф = (2x — 1) In — p*— + ax + b (12.183)

Here a and b are arbitrary constants, and the general composition variable x has been substituted for Zj. The arbitrary constants a and b do not affect the value of the right side of (12.182) because of the overall material-balance relations

£*fcxfc=0 (12.184)

к

and £ЛГ*=° (12.185)

к

In Eq. (12.157) for the price of uranium, it may be noted that the term in brackets has the general form (12.183) for the separation potential, with

a = 219.5666 (12.186)

and b = —6.4300 (12.187)

The separation potential may be thought of as related to the value of a mixture of isotopes, and has, in fact, been called the “value function” by Cohen [СЗ].