Figure 12.12 illustrates flow through a simple cascade, fed at rate F with material containing zp fraction of desired component, to produce product at rate P containing yp fraction of desired component. Feed for one stage consists of heads from the next lower stage, so that

In a cascade in which 04 and & are independent of stage number, (12.43) and (12.44) reduce to

n

and (12.47) to

co = /3"

The relation between recovery, overall enrichment, and number of stages then is

/a — uvnn Г = — )

Figure 12.14 illustrates the variation of r with co for a simple cascade of electrolytic cells with a = 7 and n = 1, 2, or 3. The recovery is greater the greater the number of stages. In the limit, as the number of stages increases indefinitely, the recovery from Eq. (12.50) approaches

lim r =

n-> «О

The line for л-к» is also shown in Fig. 12.14. This is the highest recovery that can be obtained in a simple cascade, with a = 7.

Such a simple cascade, with an infinite number of stages each performing an infinitesimal amount of separation, is equivalent to type A differential stage separation. Equation (12.51) is equivalent to the form of the Rayleigh equation (12.35), when one recognizes that ш in the simple cascade is equivalent to the heads separation factor 0 in differential stage enrichment, and a in the simple cascade is equivalent to the local separation factor a.