ENGINEERING AND PROCESS DEVELOPMENT Table IV.
Evaluation of t Calculated by Equations 12, 14, and 17 at E = 1 100' 75 50 23 10 75 50 25 10 75 50 25 10
na 4
5
7
Equation 17 0 0 0 0 0 0 0 0 0
Error in tapprox., % Equation 14 Equation 12
0 0 0
-7 -1
1 1 -11
-3
1
3
- 15 -4 2 4
75 ..
n
- 18
50 25
0 0 0 0 0 0 0 1
-4
10 75 50 25
2 5
- 20 -4 2
5 10 - 23 75 -6 50 0 3 0 25 7 10 0 1 26 75 -7 50 0 0 3 2.5 7 0 10 All calculated errors for n = 2,3 are identically zero.
-
0
0 14 16 17 31 44 49 52 73 96 110 110 140 180 200 210 260 330 360 370 1300 1600 1800 1900 20000 25000 28000 29000
n = number of stage a t which feed is introduced q = number of stage a t which upper phase is introduced Qo = quantity of solute in a single portion of feed t = number of cycles, single-diamond pattern, calculated from appearance of first product phase; in double-diamond pattern, t has same value for any even-numbered product as for immediately preceding odd-numbered product ht = system holdup after equilibration but before moving any phases-Le., product phases still in end stages, a t cycle t H = hm-i.e., steady-state system holdup Q3 = holdup in stage j a t steady state Ft = fraction of feed influent per cycle appearing in upper product phase a t tth cycle F , = value of Ft a t steady state +t = fraction of feed influent per cycle appearing in lower product phase a t tth cycle 6, = value of +t a t steady state pi = numerical multiplier involved in contribution to value of Ft (and 6t) made in i t h cycle T = ratio of p(t + 1) to pt for all values of t sufficiently large for this ratio to be sensibly constant p = smallest value of t a t which ratio of p(t + 1) to pt is considered to be sensibly constant u = ( F , - Fr)/F* = ( 6 8 - + t ) / + a In the systems considered, the upper phase moves to the left, the lower phase to the right. No change in E or U occurs throughout the system or with time.
Nomenclature
literature Cited
IC
= (concentration in upper phase)/(concentration in lower
R
=: (volume of upper phase)/(volume of lower phase), in right
(1) Peppard, D. F., Faris, J. P., Gray, P. R., and hiason, G. W., J. Phys. Chem., 57, 294 (1953). (2) Scheibel, E. G., IND. Esa. CHEX, 43, 242 (1961). (3) Ibid., 44, 2942 (1952). (4) Stene, S., Arkiv. Kemi, Mineral. Geol., 18H,No. 18 (1944).
phase)
section (Figure 1 ) (volume of upper phase)/(volume of lower phase), in left section (Figure 1) E =: Rk, U = Gk; k's are not necessarily equal
G
=
RECEIVED for review June 11, 1953.
ACCEPTEDNovember 4 . 1958
(BATCHWISE FRACTIONALLIQUID EXTRACTION)
Countercurrent Contactors PETER L. AUER A N D CLIFFORD S. GARDNER liverrnore Research laboratory, California Research & Developrnenf Co., livermore, Calif.
C
ALCULATION of batchwise counterqurrent operations has been the subject of considerable study, and references to the earlier literature are given in most standard texts on extraction (7). More recently, Scheibel has discussed, in a series of articles (6),the rate of approach to equilibrium after transient upsets in batchwise contacting units. In most instances the algebraic calculation of approach to equilibrium in countercurrent batchwise contacting units has been based on a type of cascade diagram (6). It is shown in this paper that modification of such a diagram to depict a somewhat more general cascade process leads to a more powerful means of analysis employing the familiar methods of finite difference equations. Cascade Diagram Describes Countercurrent Batchwise Extraction
The following general flow scheme is treated in this paper. A feed stream of unit value is introduced continuously a t some midstage in a batchwise contacting column, while two immiscible solvents serving as extractant and scrub streams are introduced January 1954
countercurrently a t opposite ends of the column. If the column contains N M 1 equilibrium stages, the feed plate is designated 0 and the scrub and extractant are introduced a t - M and N , respectively. While the terminology employed is that used in solvent extraction, the method of analysis described is applicable to any countercurrent separation scheme employing two immiscible phases in a batchwise contacting unit. For purposes of this analysis it is assumed that a given component has a constant extraction factor, El, in stages 0 to N and E2 in stages -1 to - i ! . It is then possible to define four quantities
+
+
+ Ed-1 q = E l ( 1 + Ed-1
s = Ed1
p + q = 1
r + s = l
p = (1
r
=
(1
+
El)-1
+ E*)-1
(I)
such that for unit feed p , q, r, and s describe the amount of the given component in each stream. The cascade diagram appropriate to this flow scheme is given in Figure 1.
INDUSTRIAL AND ENGINEERING CHEMISTRY
39
ENGINEERING AND PROCESS DEVELOPMENT Each circle in the cascade diagram represents an equilibrium stage. The alternate dotted circles are imaginary in that no streams are fed to these stages physically and they are introduced purely for the sake of mathematical symmetry. Index k designates the location of a given stage and is in this sense a space
and the series is assumed to converge for small enough t. Furthermore, it is convenient to transform Fk(t) to a new variable, Gk(t),where
F d t ) = (p/q)"/"Gk(t)
k 2 0
= (r/S)k/2Gk(t)
Fk(t)
k
O
Gk+i)
Gk = vt(Gii--l Gay = utG.v-1
k < -1
Gk+i)
k
Go G-1
=
s
k = -M
G-,M = vtG-(.w--l) 1 + utG1 + __ 1 - t = vtG-2 + WtGo
VtG-1
k = O k = -1
where
u=dG
v=dG
1
Figure 1.
3
The above set of equations may be solved by wellknown methods used to solve finite difference equations (4). Solving specifically for the end stages of the column
Cascade Diagram
GAT
,w - 2(~t)2R.w-1
1
2 v ' - [ K
coordinate index. Index n designates the number of cycles performed and is in this sense a time coordinate index. In view of the alternate imaginary stages which have been introduced, the number of physical cycles performed is equal to ( ! / ~ ) T Zand not n. (In this terminology a cycle is performed every time material is transferred from one stage to another; a physical cycle occurs every time fresh feed is introduced to the system. Scheibel (6) uses cycles to denote the number of times feed material leaves the unit.) The p , p, Y, and s indicate the manner in which the contents of a given stage partition to the other stages in the cascade. Difference Equations. It is now possible to write a set of difference equations for the above cascade process. ( I t has been brought to the attention of the authors that sets of difference equations similar to the one described here occurs in the treatc ment of random walk problems in the theory of probability. The accompanying article by Compere and Ryland (3) contains an interesting application of random walk theory to solvent extraction problems.) Letting An+1.k denote contents of stage k in cycle n 1
+
An+l,k
=
PAn,k--l
+ qAn.k+l
k
>0 < -1
An+l,k = rAn,k--l SAa,k+l k The following boundary conditions are obtained: A n + i ,= ~~A~.N-I
k = N
A ~ + I , - - .= M~ A n , - c ~ - 1 )
k = -M
An+l,o = 1
+ ? A ~ , - +I q A n , ~
A n + ~ , -= l qA,,,
+ rAn,-2
(2)
= --I
A t start-up, Ao,o
1
Ao,k = 0
40
A = ~ / ~ P N + Z R .W VWt2P.v+IR,tf +I
a = d l
- 4U2t2
p = d l -
(7)
4v2t2
When the expressions for the G's obtained above are expanded as power series in t, association of the proper expansion coefficients with the An.kJsprovides the required solution to the problem. The application of Equations 6 and 7 to specific flow schemes is demonstrated in the following discussion. Modifled Equations May Be Applied to End Fed Column
In an end fed column only two streams are fed to the unit. The feed plate is a t stage 0 and the extractant is fed to stage N . The product stream emerges at the feed plate and only those solutions are required which give the composition of the product stream. The total number of stages in the column is now *V 1 , since stages -iM to - 1 do not appear in a simple end fed column. The equations must be modified slightly to conform with the new boundary conditions. After start-up the appropriate difference equations for the end fed column are
An+l.k = PAn,k-l
k#O
An+l.k
+ AI + t2Az,k f . . . . k
where
k=O (3)
Equation 2, along with the boundary conditions of Equation 3, constitutes a set of difference equations in two indices. They may be reduced to a set in a single index by eliminating the time index, 72, through the use of a generating function, F k ( t ) , where t is a dummy variable.
Fk(t) = Ao,k
1
A
+
k=O
IC
(6)
w - q m s
=0
AnTl o = 1
+ qAn
qAn.k+l
k>Q
k
