Two-Component Separations by Diagrams for Determining Operating Conditions A. KLJNKENBERG N. V . De Bataafsche Petroleum Maafschappij, The Hague, Holland
I
In the previous paper an analysis was made of an experiment (number 5 ) by Scheibel (4). In this experiment,
N A recent paper by Klinkenberg, Lauwerier, and Reman ( 3 ) .
the problem of finding the conditions required for a two-component separation using two solvents in countercurrent flow, with given feed and product composition, was solved. An exact and general graphical method was given, and simpler approximations were also indicated. I n the present paper, the solutions are presented in the form of graphs, ready for use. - .
Component Component Solvent A Solvent B =
0.375
XA = XB =
0.12 0.817
XF
The Problem The feed, consisting of two solutes, 1 and 2, is introduced a t an intermediate point in an extraction apparatus in which two immiscible solvents, SA and SB, are flowing countercurrently. Partition coefficients KI and K , (concentration in S Adivided by concentration in S B )are assumed t o be known constants. Their ratio K2 = K1
corresponds to the relative volatility,
The compositions of the feed and of both products are known. The minimum total number of ideal stages required for the separation, the position of the feed point, and the solvent ratio are to be found.
-
ol,
1 and 2, R1= 0.256 and R2 = 8.28. The numbers of stages a t either side of the feed point (both inelusive of the feed stage) are denoted by m and n. The method of counting the stages is shown in Figure 1. Solvent B is intraduced in the 1st stage, the feed is introduced in the mth stage, and solvent A is introduced in the (n m - 1)th stage. so that, from Equations
in distilla-
+
__
S o l v e n t B,
-
Solvent A + ProduetA
- --
.-."=!Q9'i.
-
-
m
4
The given compositions of feed and products are expressed on a solvent-free basis. I t is immaterial whether they are expressed in mole, weight, or volume fractions. Let the fraction of component 1 be XF in the feed, X A in the final solution in solvent A (product A ) , and X B in the final solution in solvent B (product B). Then, for a unit of feed:
_-
+ Solvent B t P r o d u c t B
mtnl
--
*-
-
. .n_ft-a.q.e.s.
.
_ __._ _._
Solvent A
_-
Partition coefficients K1 and K2 are now multiplied by solvent ratio S to obtain extraction factors E1 and Ez. These represent the ratios of the amounts of solute leaving an ideal stage in the two directions. The performance of the extraction system of Figure 1 is then described by the following equations (Bartels and Kleiman, 1, and Klinkenberg, 9):
B - TF Yield of product A = z____ XB
1 = p-chloronitrobenzene 2 = o-chloronitrobenzene = aqueous methanol (15 wt. % water) = Skellysolve (a heptane fraction)
- XA
(3)
- ZA Yield of product B = ____ X B - $4 XF
(4)
E2
If z is expressed as a weight fraction, the yield is also a fraction by weight. The amounts of solute 1 in products A and B may now be divided t o give the ratio R1
=
- XF)XA - ZA)XB
(1)
- X F ) (1 - 5 8 ) - xa) (1 - X B )
(2)
(2-B
(XF
E;
These three equations with the four unknowns, El, El, n, and there are an infinite number of extraction conditions, all producing the same products from the same feed. Thus the extraction factor, E,, might still be m - l), varied, in which case the total number of stages (n when plotted against E,, goes through a minimum ( 3 ) . This minimum condition is used as the fourth relation. This fourth relation may be expressed in algebraic form,.but the set of four equations cannot be solved by algebraic means. m, still leave one degree of freedom-Le.,
+
Similarly, for solute 2
Rg
=
(PB (XR
(5)
= B
653
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
654
The Klinkenberg, Lauwerier, and Reman paper ( 3 ) now gives: 1. A method for drawing the above curve through points rep-
resenting a number of special solutions to Equations 3, 4, and 5, whence the minimum of the curve can be found 2. An exact and general graphical solution for finding the extraction conditions requiring the minimum number of stages 3. Quick approximations for the minimum number, which are valid for not too asymmetrical separations 4. A graph (Figure 4) showing the minimum number of reduced stages for separations requiring up t o 4.4 reduced stages The purpose of the present paper is t o draw attention to this work and t o present the final results for a wide range of variables in the form of graphs, so that no further calculations or constructions are required.
results were plotted in Figure 2 in the form of curves for conM ) and for constant ( N - M ) . The correspondstant ( N ing x values were plotted in Figure 3. The minimum values of ( N M ) can be obtained with great accuracy (to within about 0.01). The values of ( N - M ) and x are far less exact, since they depend on the abscissa of the miniM ) values mum, which cannot be read accurately. The (N may be in error by about 0.1. This fact is of no practical importance because in the vicinity of the minimum the effect of changing the feed point may be compensated by a change in solvent ratio. Example. For Scheibel's experiment 5 (R1 = 0.256; X2 = 8.28), the minimum total number of stages corresponds to:
+
+
-
.Ir
In the earlier paper, a t the suggestion of Lauwerier, new variables \yere introduced. These variables, in slightly modified form, are: =
JI =
11
log
??llog
-+-
*1f = 3.00 = 0.2 = 0.555
-Y - M x
Special Solutions
.IT
Vol. 45, No. 3
p
(6)
p
(7)
For the separation studied by Scheibel, log p Total number of stages, n
=
0.2095 ( 3 ) .so
-.
that:
3 00 + ~n- 1 = 0.2095 - 1 = 13.4
0.2 Asymmetry of feed point, n - ?n = -_ = 1.0 0.2095 Extraction fartors, log E2 = 0 555 X 0.2095; E2 = 1.307 log E1 = -0.145 X 0.2095; El = 0.826
Approximations to Exact Solutions using common or Briggsian logarithms. These new variahles will be called the reduced numbers of stages and the reduced extraction factors. Two equations with three unkno\ms are now obtained:
10-(1
- 2)M(10-(1 10-
(1
- z).V - 1) = - z).M - 1
Ri
(9)
I n drawing Figures 2 and 3, use was also made of certain approximations which the exact curves must approach. The (A? M ) curves as drawn are independent of the approximations. For the (iV - M ) and x curves the aqymptotic approximations were used, since i t became apparent t h a t the deviations from those curves were less than the accuracy of the calculated points. Higher Estimate for ( N M ) . A highrr estimate for (3' M) is obtained by using the (Y 3f) value found for special solutions 3 and 4 (Table 1):
+
+
N
+ .If
+
+
RP R1
= 2 log -
(11)
There are a number of simple additional conditions which toEstimates for (N - M) and x may be obtained by taking the gether with Equations 9 and 10 allow a solution in terms of x, S, arithmetic means of the (S- Jf) and x values for those soluand M ; the sum 111 N , in general, however, i s not as low as tions : possible. These additional conditions specify either the ratio of n n 1 - = (viz., - = 2, 1, or ) or the value of the extraction factors ?n 1M m 2 ( E p = 1, El = 1. or ElE2 = 1, i.e., x = 0, 1 x = 1, or x = 2 respectively). The methods of solving Equations 3, 4, Table I. Six Special Solutions to Equations 9 and 10 and 5 with each one of these additional Case s *If X conditions viere given in the earlier paper. 1 s = n?.ii Root of 0 lo-" ( 1 o - R 2 V - 1) -Similar treatment of Equations 9 and 10 10-u - 1 = Ri gives the results in terms of the reduced quantities shown in Table I.
+
i"
-.
2
A- = 2 J1
Graphs for Computing N, Ikl, and x For a given separation-Le., for given R, and R2, in general, a set of six corresponding S,M , and x values is thus obtained. Plots mere now made of S, M , and (S -11) against x, the position of the minimum of ( N M ) determined graphically and the .V, M,and z values read. By plotr M ) and (Ll'- 31) against R1 ting (S for constant Rp, combinations of RI and R2 were found for which (S M) and (S - M ) assume round value-. The
+
Rz . X R i
1 2 log Re ___.___
R log -z RI
-
log Rz
+
+
+
6
S = RIM
Root of 10" ( 1 0 R I M 1o.v 1
-
-
1 1) = R 2
log K I
March 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
655
+
Lower Estimate for ( N M ) . A lower estimate for (-V M ) is obtained by assuming: In the logarithmic plots of Figures 2 and 3, consequently, lines for constant ( N M ) are straight lines under 45' and lines for constant x are straight lines through the origin (upper left-hand corner, Figure 3). This approximation is exact on the diagonal RtRz = 1 when the
+
extraction system is symmetrical
(N
M; x
N
+M
= (m
+ n ) log p
= 2 log ( l
- zB)xA ZA)ZB
(1 -
&>>I;
ET>>1
ET E,"
= -
+
Er:> =
