18.08.2015   NUCLEAR CHEMICAL ENGINEERING

Minimum Number of Stages: Constant Separation Factor

The number of stages required to separate feed into product and tails of specified composition is a minimum at total reflux, when N(+1/P-*». Under this condition we have seen that

*i+i =Уі (12.63)

Abundance ratios in these two streams are also equal:

?,+,=% (12.64)

Because of the definition of separation factor (12.15), abundance ratios on adjacent stages at total reflux are related by

t? i’+i = ar>i (12.65)

When applied to stage 1, this equation is

i?2=ar? i (12.66)

When applied to stage 2 it is, for constant a,

%=at? j=a2r?1 (12.67)

By proceeding in this way through the entire cascade, we find

TJ„ =o"-1r}1

(12.68)

But

УР

(12.69)

and

„ °«W Vi — aSt ~ , v 1 — xw

(12.70)

so that

УР _ cfxw 1 — yp 1 — xw

(12.71)

or

In [УрО — xw)l( 1 ~yp)xw]

n — —————————-

In a

(12.72)

This is the familiar Underwood [Ul]-Fenske [FI] equation for total reflux. The ratio of abundance ratios appearing in (12.72) is the overall separation (П) of the recycle cascade:

■УрО — xw)

(1 — yp)x w

Equation (12.72) gives the minimum number of stages for a particular overall separation. The minimum number of stages requires that the ratio of interstage flow rate to product be infinite.

The minimum number of stages increases as the overall separation increases and as the separation factor approaches unity. Because both these conditions hold in a typical isotope separation plant, the minimum number of stages is often very large. For example, in a 235 U gaseous diffusion plant (a = 1.00429) making product containing 90 percent 233 U and tails 0.3 percent,