The number of stages in an ideal cascade may be evaluated by a procedure similar to that used in deriving Eq. (12.72) for the minimum number of stages at total reflux. The result is

In [yf(l ~xw)l( 1 — yp)xw] In ypQ. — xw)l( 1 —J’pfrK’]

In |5 In a

Thus the number of stages required for a given separation in an ideal cascade is just twice the minimum number needed at total reflux minus 1.

By a similar procedure, the number of stages in the stripping section is found to be

and in the enriching section

A relation between composition and stage number may be derived by a procedure similar to that which led to (12.68):

(12.96)

Because q„ (12.97)

The corresponding equations in the stripping section are

8.2 Reflux Ratio

The reflux ratio required to bring about condition (12.83) defining an ideal cascade may be found as follows. From (12.62),

Nt*i ^ УР ~Уі P Уі-Хі+і

But уі = r,+1 in an ideal cascade, and z,-+i is given in terms of xI+i by (12.19) with a = 0s, so that

Мы yptfxj+i + 1 -*f+i)-flXf+r _ 1 Гyp № — зуЛ

P (fi— l)*r+i(l “■*<♦!) 0-l[xi+l 1 —ЛГ/+1 J

This equation is the same as for minimum reflux (12.79), except that 0 replaces a.

Figure 12.16 is a McCabe-Thiele diagram for an ideal cascade. The equilibrium line, relating

Уі to Xf, is represented by the solid curved line, with the equation

Уі

1 — уі

The operating line, relating у,- to jci+], is represented by the dashed curved line, with the equation

у і _ y/uxj+1 1 — уі 1 — хі+1

The graphic construction shows that with these two lines x,-+1 = уui, as required for an ideal cascade. The straight line connecting the product point (yp, yp) with the point on the operating line (yi, x,+i) has a slope (ур—Уі)І(ур—Хі+1), which equals Ni+l /(jV,+1 +F), the ratio of tails to heads flow at this point in the cascade. Ni+1/P is given by Eqs. (12.101) and (12.102).

In the stripping section, the equation corresponding to (12.102) is

Щ 1 (l-xw folA

IV 0-ll-уі Уі)

An equation for the reflux ratio in the enriching section as a function of stage number may be obtained by substituting x,+1 from (12.99) into (12.102):

^ET =^Tbp(l — U’-") + (1 — ypW1-*- 1)] (12.106)

Similarly, in the stripping section, Eqs. (12.105) and (12.100) lead to

jbi l*wf& -1) + (1 -*wXl — Ґ)]