We wish to find a value function V, a generalization of the separation potential ф for a two-component mixture, now a function of xs and x6, which can be used to evaluate the separative capacity, and from it, the total flow rate. The difference equation (12.313) for V is obtained by writing a V balance for the stage, in which the difference between the separation potential carried by the stage effluents and the stage feed is equated to the separative capacity of the stage, given by Eq. (12.174) а&Мф2І4:

MV(ys, y6) + MV(xs, x6) — 2AfV(zs, z6) = (12.313)

For a close-separation cascade with a cut of |,

(12.314)

When Eq. (12.313) is expanded in a Taylor series about xs and x6, the following differential equation is obtained:

^ -*s)2 0 + 20,5 “*5)(Уб ~*б) гВк + 0,6 ~X6? U = ф2 (12-315)

Terms in V, dV/bXs, and bV/bx6 have dropped out because of material-balance relations. Substitution of ys — xs from (12.311) and y6 —x6 from (12.312) into (12.315) leads to

We wish to find a solution of Eq. (12.316) that can be used to evaluate total flow rates, as was done for two components in Sec. 11. To do this, it is necessary to arrange that there be no loss of V when two streams are mixed. In a two-component system this was done by requiring the two streams to have the same composition. In a three-component system this is not generally possible. The mole fractions of only one component in the two streams may be made equal, or one function of the mole fractions in the two streams may be made equal. For the present derivation, we shall require that the abundance ratio R of the two principal components, 235 U and 238 U, be equal whenever two streams are mixed.

== *5

1 — XS — x6

de la Garza et al. [Dl, D2] have shown that this leads to a cascade with nearly the minimum total internal flow as long as the fraction of other components is small, and have called such a cascade a matched R cascade. We then need to find the most general solution of Eq. (12.316) that has the property that when two streams are mixed, V is conserved.

If the streams being mixed have flow rates M1 and M" and compositions (R, x’6) and (R, х’б), the condition that V be conserved is

{M’ + M") V(R, x6)=M’V(R, x’f) + M"V(R, xl) (12.318)

with the 236 U fraction in the mixed stream x6 given by material balance

M’x6 + M"x’s *6 M’ + M"

To satisfy (12.318) and (12.319), V(R, x6) must be a linear function of x6:

V(R, xt) = a(R) + b(R)x6 (12.320)

The most general solution of (12.316) of the form (12.320) is

V(R, x6) = k0 + ksxs + k6x6 + + (2×5 + 4×6 — 1) In R (12.321)

К

к0, к5, к6, and к are arbitrary constants. The fact that (12.321) satisfies the differential equation (12.316) may be verified by direct substitution.

When interstage flows are adjusted so that the abundance ratios R of 235 U to 238 U of each pair of streams being mixed are equal, the separative capacity D of an entire cascade whose feed, product, and tails are

is

D = PV(RP, y6J>) + WV(Rw, x6<w) — FV(Rf, zttF) (12.322)

This may be shown by a development similar to that of Sec. 11.

When Eqs. (12.321) for feed, product, and tails are substituted into (12.322), the coefficients of к0, k5, and k6 vanish because of material-balance relations. The coefficient of the remaining arbitrary constant к in Eq. (12.321) for the separative capacity may be made to vanish by requiring that

Equation (12.323) and the material-balance equation for 236 U make possible evaluation of the distribution of 236 U between product and tails in terms of the specified fraction of 236 U in feed z6F and the specified abundance ratios RP, Rw, and Rp of “U to 238U in product, tails, and feed, respectively. It should be noted that it is not possible to specify in advance the distribution of the third component, 236 U in this case. The distribution of only two components, called key components, 235 U and 238 U in this case, are the only ones that can be specified in advance.

With the distribution of 236 U between product and tails thus determined, the separative capacity of the entire cascade, from Eqs. (12.321) and (12.322), becomes

D = P{2y%j> + 4y6J> — 1) In RP + W(2×5>w + 4x6jW — 1) In Rw — F(2zStp + 4z6fF — 1) In Rp

Thus, it has been shown that the separation potential, or value function, for an ideal cascade treating a mixture of 235 U, 236 U, and 238 U in which the ratios of 235 U to 238 U in each pair of streams being mixed are made equal, is