I
ALFRED J. SUROWIEC Maplewood,
N. J.
McCabe-Thiele Diagram for the Ideal Cascade Induced reflux in the enriching section of the distillation column may improve isotopic separations I
ideal cascade employed in the theory of isotopic separations is defined as one whose individual separation stages have feed streams of the same composition so that a mixing effect is avoided. This report shows the construction of a McCabe-Thiele diagram (4)for the ideal cascade. The treatment given here is general and exact. Only the downflow from the first stage is specified and not the reflux to the top stage. The method of generation of reflux is immaterial. I t can be continuously generated with zero top reflux. I n the case of a distillation tower, this would mean zero heat removal from the top of the tower. Only the enriching section has received consideration here. A similar procedure can be used for the waste end. T
H
E
Theoretical The defining equations for an ideal cascade are readily given as [x/(l
- x)In-:
=
- Y)ln
=
b/(l - Y ) l n + l
(1)
and cy41
a[./(1
-
X)l.
(2)
The combination of Equations 1 and 2 yields [~/’(1 - Y ) ] + I
= d ” [ ~ / (l x)ln
(3’
and
SI
0
I
X
This is the diagram developed for an ideal cascade tion at y = yl. At y = yz, begin a second total reflux construction. The diagram is now composed of two interlacing total reflux steps whose intersections are given by Equation 4. The slopes of the tie lines drawn from y l on the 4 5 O diagonal to the intersections give the net downflows from the individual separation stages. Although each stage is fitted with some type of reflux interchanger or generator, it is the net downflow, L,, and net upflow, Vn, from each stage that is obtained from the diagram. A material balance around the first n stages of the cascade gives Vn+:yn+:
Lnxn
+ DYI
(6)
and T h e minimum number of stages for any separation is given by the Fenske total reflux equation ( 3 ):
4. = Y I - yni.1 D
yn+:
Diagram Construction The McCabe-Thiele diagram is readily constructed for the ideal cascade from two total reflux diagrams. As shown (see figure), begin a total reflux construc-
(7)
The minimum downflow is given by
- Y: - Yn+: - y n + 1 - xn+:
(%)min
Comparison of Equations 4 and 5 shows that the ideal cascade requires twice the minimum number of stages.
- Xn
n
= yn+l --x,+l Xn
(gmin.)
------)
a-1
242 =2
(11)
Thus, the ideal cascade in isotope separations employs twice the minimum number of stages, and twice the minimum downflow at all interstage points (7, 2 ) . As a result, plant size is kept to a minimum.
Nomenclature D = net overhead product, moles L, = net downflow from stage n, moles n = stagenumber N = minimum number of stages V , = net upflow from stage n, moles xn = mole fraction of the light component in the downflow from stage n yn = mole fraction of the light component in the upflow from stage n
(8)
while the ratio of the two
(gmin.)
I n the limit as a! approaches unity L’Hospital’s rule gives
(9)
Substituting Equations 2 and 3 in Equation 3 :
cy
= relative volatility or distribution
ratio
literature Cited (1) Benedict, B., Boas, A., Chem. Eng. Progr. 47,111 (1951). (2) Cohen, K., “Theory of Isotopic Separations,” McGraw-Hill, New York, 1951.
( 3 ) Fenske, M. R., IND. ENC.CHEM.24, 482 (1932). (4) McCabe, W.L., Thiele, E. W., Zbid., 17, 605 (1925). RECEIVED for review December 3, 1956 ACCEPTED May 12, 1960 VOL. 52, NO. 9
SEPTEMBER 1960
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