This type of cascade may have practical application in a Becker nozzle plant or centrifuge plant for producing low-enriched uranium, with individual stages operated at a cut of around Figure 12.26 is a schematic diagram of stage connections showing the nomenclature to be used in solving the enrichment equations for such a cascade. Olander [01] has solved the enrichment equations for such a cascade.

The cascade receives feed of fraction zF at flow rate F and produces an upper product of fraction yP at flow rate P, a lower product of fraction yg at flow rate Q, and tails of fraction Хцг at flow rate W. For this two-up, one-down cascade, p = 2, <7=1, the heads separation factor (3 is

(3 = ctpl(p+q) = a2/3 (12.263)

and the tails separation factor 7 is

7 = aql(p+q) = a113 (12.264)

By counting the number of stages ns in the stripping section of Fig. 12.26, it is seen that

(1+є)5 («"у) -«(«-у) +*[(2 + 5)е(б-^) — о * *>» (‘ — т)]| ■ ТТТГГ, [-‘ + f ■ ~ 8,‘ + 0U61)

ZF _ У ПК— 1 — yHg +1 xw

— zF —y„s—x 1 — xw

Similarly, the total number of stages n satisfies

Ур _ n+l xw — Ур 1 — xw

For this low-enrichment case in which 1 — x <* Г,

_ j. , _ HzfIxw ) ”s + 1——— і——- In 7 _ + о _ fa(yp/xw) In 7 „+ 1 = lnQg/**) In 7

The separative capacity D of the two-up, one-down cascade is D = Щур — 1) In -2*— + Q(2ye — 1) In 1 —yp v 1-у + W(2xw — 1) In

Enriching section. Material balances above line BB in Fig. 12.26 are М,+Мі-і = P + Q + JV/+! and МіУі + Л/,_! у і — , = РуР + QyQ +Ni+l хі+ j but Уі = X*+l =уЙ+2

{[—(7 "*■!)] n~i — 7n—*)

f = H7 + 1)]

7(7 + Q/P)

(7 — 1)(27 + 1)

This may be confirmed by substitution of (12.287) into (12.286). For the three top stages, Eq. (12.187) gives

w—4 II ^ck	(12.288)
m„- 1 _ Q p p	(12.289)
— = 1 + 7 + 73	(12.290)

which are also obtainable by inspection of Fig. 12.26.

External flow rates. One relation among the external flow rates W, Q, and P is obtained by equating the heads flow rate from the top stage of the stripping section Mj evaluated from Eq. (12.278) with / = ns to the heads flow rate into the bottom stage of the enriching section Mj evaluated from Eq. (12.287) with і = ns. From Eqs. (12.278) and (12.280),

where r, s, and t are functions of n, ns, and 7. A second relation between W, P, and Q in terms of these variables may be obtained from the material-balance relations

F = W + P + Q (12.297)

and Fzf = Wxw + Pyp + Q}>q (12.298)

Inspection of Fig. 12.26 shows that

zF=XjyynS+1 (12.299)

yP=xwyn+2 (12.300)

and y<2=*w 7n+1 (12.301)

The result of eliminating F, zF, Xw, yP, and yg from Eqs. (12.297) through (12.301) is

(7"s+1 ~ 1)Н’ = (7И+2 — тЛ5+1)7>+ (7n + 1 “7ns+1)Q (12.302)

Equations (12.296) and (12.302) make it possible to determine the flow ratios Q/P and W/P as functions of n, ns, and y:

Q _ Kt”*2 — і)

P tiys+i — l)-r(y»+1 — yns+i)

W _ f(r"+2 -T,,S+»)-s(y> + 1 — ys+1) P t(ynS+1 — 1) — r(yn + i — ynS+1)

Design example. The foregoing equations will be applied to the two-up, one-down ideal cascade considered by Olander [01] having three stripping stages (ns = 3), seven total stages (n = 7), and a tails separation factor (7) of 1.3027. Values of r, s, and t then are

	Equation	Value
r	(12.292)	0.592674
s	(12.294)	15.99783
t	(12.295)	-2.99973

Table 12.12 gives compositions and flow rates relative to top product calculated from the preceding equations for feed containing 0.71 percent 235 U.