ENGINEERING AND PROCESS DEVELOPMENT that the interpretation of the laboratory and pilot plant data be made on the same basis as the final design. All of the equations in the present paper have been based on a constant distribution coefficient, and it has been assumed that when the coefficient varies, an average value may be estimated. In selecting average values, care must be taken that the average coefficients selected have the average re!ative distribution, since it is possible in an arbitrary selection of absolute values to obtain ratios outside of the range of their actual relative distributions. The relative distribution controls the number of theoretical stages required for a given separation. In some nonideal systems, the distribution coefficients vary almost logarithmically with concentration, and it is impossible to estimate an average distribution coefficient. In these cases, the trial-and-error procedure of detailed stagewise calculations ( 11 ) must be employed. Probably the best approach would be to evaluate the proper average value of D in nonideal systems to be introduced in the equations given in this paper. This average value of D may possibly be calculated for a linear and a logarithmic variation of the coefficient with concentration. These relationships have not yet been developed and an accurate solution of the nonideal cases requires the tedious stagewise calculation. Nomenclature I) = distribution coefficient
LD E = extraction factor = H Ii‘ = feed quantity per cycle in batch extraction feed rate in
continuous extraction H = heavy solvent quantity per cycle in batch extraction = heavy solvent rate in continuous extraction L = light solvent quantityper cycle in batch extraction = light solvent rate in continuous extraction m = stages below and including the feed stage in fractional liquid extraction = n with center feed n = stages in column in simple extraction = stages above and including feed stage in fractional liquid extraction E p = fraction of component in light solvent = __ E f l 1 p = fraction of component in heavy solvent = E+1
R
=
E’
=
t
= = =
x 21
rejection ratio of component = quantity leaving in light solvent quantity leaving in heavy solvent 1 retention ratio = R number of cycles concentration or quantity in heavy solvent concentration or quantit in light eolvent
-
Jl
relative distribution = D2 6 = fractional deviation from steady state
p
=
Subscripts 1 refers to component more soluble in light phase 2 refers to component more soluble in heavy phase o refers to solute-rich end of extraction system n refers to solute-lean end of extraction system s refers to steady-state conditions t refers to cycle number t Literature Cited
4
(1)Bartels, C. R., and Kleiman, G., Chem. Eng. Progr., 45, 589 (1949). (2) Craig, L. C., J. Bid. Chem., 155, 619 (1944). (3)Craig, L. C., and Craig, D., in “Technique of Organic Chemistry,” Vol. 111, New York, Interscience Publishers, Inc., 1950. (4) Craig, L. C., Hausmann, W., Ahrens, E. H., Jr., and Harfenist, E.J., Anal. Chem., 23, 1236 (1951). (5) Klinkenberg, A., Chem. Ew.Sci., 1, 86 (1951). (6) Klinkenberg, A., IND. ENG.CHEM.,45,653 (1953). (7)Klinkenberg, A., Lauwerier, H. A., and Reman, C. H., Chem. Eng. Sci., 1,93 (1951). (8) Oldshue, J. Y., and Rushton, J. H., Chem. Eng. P r o p . , 48, 297 (1952). (9) Peppard, D. F., and Peppard, M. A., IND.ENG.CHEM.,46, 34 (1954). (10) Perry, J. H.,ed., “Chemical Engineer’sHandbook,” 3rd ed., pp. 713-53, New York, McGraw-Hill Book Co., 1950. (11) Scheibel, E. G., Chem. Eng. Progr., 44, 681-90, 771-82 (1948). (12) Scheibel, E. G.,IND.ENQ.CHEM.,43, 242 (1951). (13)Zbid., 44, 2942 (1952). 46,43 (1954). (14)IW., (15) Scheibel, E. G.,and Karr, A. E., Ibid., 42, 1048 (1950). (16) Underwood, A. J. V., I d . Chemist, 10, 128 (1934). (17)Ibid., 10, 129 (1934). RECEIVED for review March 6, 1953. AOO~PTED September 2,1958.
(BATCHWISE FRACTIONAL LIQUID EXTRACTION)
Double Withdrawal and the EDGAR L. COMPERE‘ AND ADA L. RYLAND2 College of Chemistry and Physics, Louisiana Sfafe University, Baton Rouge, La.
I
NCREASING interest in the use of fractional solvent extraction and countercurrent distribution for the separation of closely related compounds has led to attempts to establish theoretical calculations for use in predicting the results of these experiments. Stene (16) and Craig (4, 17) have made significant contributions to the field by developing theories applicable to 1
Present address, Chemistry Division, Oak Ridge National Laboratory,
Oak Ridge, Tenn. Present address, Polychemimla Department, Du Pont Experimental Station, Wilmington, Del.
24
the basic and single withdrawal operations. Scheibel (1.8) considered the relationship between batchwise and continuous countercurrent extraction processes, and Johnson and Talbot (9, 10) derived equations for predicting the distribution of a fixed quantity of feed in a series of mixers and settlers. The accompanying papers by Auer and Gardner (I), Peppard and Peppard (11), and Scheibel ( 1 4 ) also are concerned with various aspects of the problem. The discrete stage operations considered in this paper represent a n approach to continuous countercurrent extraction processes,
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 46, No. 1
ENGINEERING AND PROCESS DEVELOPMENT and in cases with a moderate number of stages the results should apply quite well. The fact that the equations permit the exact calculation of transient behavior under various conditions is of interest. The work presented in this paper includes a summary of the theories applied to the basic and single withdrawal processes and develops the theory of double withdrawal extraction processes for single and multiple (continued) feed, These three patterns are tjhose principally used in carrying out countercurrent distribution experiments, although others have been suggested ( 4 ) . Discrete Stage Operations Are Approach to Continuous Countercurrent Extraction Processes
Patterns of Operation. The three types of operation are illustrated in Figures l and 2 for experiments using three tubes. The initial stages of all three are identical. A quantity of feed is placed in the first tube with the initial quantities of light and heavy solvents, L1 and HL. The mixture is shaken and allowed to settle. A second tube containing only HZ is added, and Llis transferred into this second tube. A second quantity of light solvent, Le, is added to the first tube, which still contains H I . Aga,in the phases are shaken and allowed to settle. A third tube, containing Ha, is added a t the left and the light phases are decanted to the left. This places L1 in the tube with Ha and Lz in the tube with Hz, and a fresh quantity of light solvent, La, is
HEAVY
LIGHT
SOLVENT
SOLVENT
SOLVENT
S OLVEN T
SOLVENT
Double Withdrawal Extraction Pattern
Single Withdrawal Theory. The theory of the basic operation and the single withdran-a1 procedure has been considered by Stene (16)and Craig (4, 17). The contents of the tubes in the basic operation, can be calculated according t o the terms in the q)', if p is the fraction transferred to the binomial expansion ( p left, p is the fraction transferred to the right, and t is the number of transfers. In the single withdrawal process the fraction of material in a given tube, W,,,, is also expressed in terms of the binomial distribution. If f is the tube number (considering the first tube number zero) and 1 the total number of transfers, then
+
FEED HI,
FEED
Figure 2.
BASIC
HEAVY
H I was collected as product. The operation illustrated thus requires only three tubes. The light phases are transferred to the left. Now H , and Lz occupy the tube on the left, Ha and La the center tube, and H z and L4the tube on the right. After shaking, Lp and Hz are withdrawn as products, and the process is continued until any desired number of products has been collected. The double withdrawal pattern can be run using a single feed increment or continued feed increments, as desired. The feed point illustrated is the center stage, but feed may be added at any desired point. The double withdrawal pattern is of particular interest when used with continued feed, since it is the batchwise analog of the continuous countercurrent operation.
LIGHT
I .LI
L5
M SINGLE W l THDR A W A L
The distribution of solute in the products is described by a Pascal distribution. If c designates the product number and 6 is the number of tubes used, then the fraction, W , of the input found in any product is given by the formula
W4.o = p b qc-l
Figure 1. Extraction Patterns
Bb,o
(2)
Bb,o is
then added to the tube containing HI. The machine is shaken and the phases allowed to settle. The basic operation is ended a t this point for the three-tube process, and the tubes are withdram and analyzed for their total contents. In the single withdrawal pattern, after the basic pattern has been run, the light phases are transferred to the left, Llis collected as product, and fresh light solvent, L4, is added to H I in the first tube. The operation can be continued in this fashion until the desired number of products has been collected. No heavy solvent is ever removed as product. In the double withdrawal pattern, after the basic pattern is completed, the light phases are transferred to the left and both LI and H I are collected as products. This leaves tube 2 containing HZand La and tube 3 containing Ha and L2in the machine. These are shaken, and a fourth tube containing H4 is put on at the left. This fourth tube may be tube 1, which was removed when January 1954
a binomial coefficient, which can be calculated for a given number of tubes from the following expression: Bb.c
=
( b + c - 2) ! (a - 1) ! ( c - 1) !
(3)
The calculation of these constants is facilitated by the use of tables of log n!,which are available frommathematical handbooks (I and )specialized sources ( 5 ) . Values of log Bb.. are presented in Table I for 17-tube operation, for various product numbers to 181. Table I also presents values of the percentages of input collected in the products for E = 1 and E = 1/116, where E = p / q . The output curves using these numbers are indicated in Figure 3. Thus the output curves for these two cases would represent the separation obtained a t any period in the process for two solutes having a selectivity factor, p = 1.5, with operation at the indicated extraction factor.
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
25
,
ENGINEERING AND PROCESS DEVELOPMENT
7.0
meets the condition, for solutes I and 11, that VI = WII, and since the Bb,cterms cancel, the following relationships are true at the intersection point:
6.0
OE
(7)
n
W
g
5.0
This equation was tested for the curves shown in Figure 3 and for other cases and was found, as expected, to give the proper values of the intersection points.
W -I
A
0 0
F 4.0
a
Probability Concept of Random Walk Applies to Double Withdrawal Extraction
P
E LL
0
I-
3.0
z W 0
$
e.0
n
1.0
0 PRODUCT NUMBER
Figure 3. Theoretical Single Withdrawal Output Curves with 17 Tubes Two additional formulas are of value in calculating the output curves. If the peak position is visualized as occurring half way between two adjacent products which have equal and maximum values of W , the peak product number may be calculated by the following:
where
A value of W at a point adjacent to one for which the value is known may be calculated
The problem of calculating the results of a double withdrawal extraction process was attacked through the probability concept of the random walk. In the random walk, a series of Bernoulli trials, in which the probabilities of success and failure are fixed, is considered and the results interpreted in terms of the motion of a particle along the z axis. If the particle has an initial position, x = z, and moves a unit step to the left or right at each trial, depending on the success or failure of the trial, the particle is said to execute a random walk. The random walk model has been used as a crude approximation to one-dimensional diffusion processes and Brownian motion. This same attack can be used in dealing with the so-called classical gambler’s ruin problem. If a gambler who has z dollars gambles with an adversary who has a - z dollars, and the probabilities of the gambler’s winning or losing are fixed and equal to q and p , respectively, the gambler’s capital during the course of the game is represented by a particle executing a random walk. I t is possible to calculate the probability of the gambler’s being ruined-that is, reducing his capital to zero-after any number of trials. This process is entirely analogous to the double withdrawal extraction process with fixed distribution ratios. In this process a molecule is introduced at the feed stage, z = z, and has a probability, p , of being transferred into the “given” solvent and a probability, 4, of being transferred into the “other” solvent. It emerges as product in the given solvent if x becomes equal to zero and as a product in the other solvent if z becomes equal to a, where a is the total number of stages plus one. The problem of calculating the probability of the molecule emerging in the given solvent as product after any number of cycles is thus identical with the problem of calculating the probability of the gambler’s ruin at a particular stage of the play. The yield, or fraction of molecules emerging at a particular time, is given by this prob-
(5) Using this formula, the curve may be filled in by stepwise calculation from known points, such as those given in Table I. When two materials are to be separated by single withdrawal, the position of the point of intersection of the two output curves is of interest, If the process is stopped after the number of products corresponding to the intersection point has been collected, the products collected will all have been enriched in the material with the higher E. If the process is continued after this point, the second material begins to appear in the products in a higher ratio than it was introduced in the feed. It would appear that a separation into two fractions of maximum yield might be obtained if a number of products corresponding to the intersection point were collected and the remaining contents of the machine were collected as a second fraction. The material with the higher E would appear in the products in an enriched state, while the material with the lower E would be recovered from the contents of the machine in higher purity than it was fed. It is possible to calculate this intersection point, c, since it 26
Table I. Single Withdrawal Constants and Input Collected in Products Product NO.
1 5 11 16 21 23 24 31 41 61 61 81 101 121 141 151 161 181
(17-tube operation) Input in Product, % log Bb.a E = l E = 1/1.5 0.00076 0.000017 0.00000 0.231 0.0108 3.68529 3.95 0.552 6.72524 2.43 7.00 8.47790 4.59 5.32 9.86380 4.05 5.03 10.34713 3.43 5.11 10.57648 3.77 0.705 11.99629 0.0290 0.958 13.61960 0.119 0,00058 14.93218 0.0091 16.03464 17.82102 19.23998 ... 20.45736 21.42365 21.87691 22.30237 23.08228
INDUSTRIAL AND ENGINEERING CHEMISTRY
... ... ... ... ... ... ...
... ... ... ... ... ... ...
Vol. 46, No. 1
I
ENGINEERING AND PROCESS DEVELOPMENT ability. In the treatment below the light phase will be the given phase, except as noted. Feller (6) presents a thorough treatment of the problem of the gambler's ruin. There are two formulas available for the calculation. Feller develops the first of these and the second has been derived from equations presented in his treatment. The terminology of the treatment will be that of the extraction problem. usjn
2" (n
+
a P
2
=-
2)
(n
-
2)
a
-1
NO.
Cn,,,.
TY
sin -
x
TYZ
sin - ( 8 ) a
where U,,, represents the fraction of the input material collected in the light phase in a particular product. Either of these equations may be expressed as
... 1 .'i44 1.160 1.164 1.158 1.127 1.081 0.9488 0.7179 0.6296 0.5565 0,4447 0.3648 0.1559
10
11 13 15 20 30 35 40 50 60 120
Table 111.
where C',,,,, represents the factor within brackets in Equation 9, or a suitable similar term in Equation 8. A similar but more useful expression is
2 3 5 7
8
9 10
11
The explicit expremions for the calculation of the constants C,,.,,, nre r
The introduction of the l/,m-l factor helps to reduce the range over which C varies with m, making it easier to interpolate graphically or arithmetically. Under these conditions, the values of C,,,,, rise to a maximum and then fall off, rather than always increase as do the values of Cfn,s,a,Cm.,../2" is then the cycle yield for a single feed increment a t E = 1. Equations 12 and 13 yield identical results for the constants, C, Z , a . It has been found easier to use the trigonometric formula of Equation 13 for the calculations involving large numbers of cycles and the other formula for calculations involving small numbers of cycles. The two are complementary, since the number of terms to be considered for Equation 13 becomes smaller as the number of cycles increases, while the number of terms for Equation 12 becomes larger as the number of cycles increases. Use of the trigonometric formula is further complicated for small numbers of cycles by the fact that a t least seven-place logarithm tables must be used to ensure accuracy. For example, in calculating the constants for the 33-stage process used in this work, sevenplace tables were necessary to around 20 cycles. After this point, values calculated using five- and seven-place tables were in substantial agreement. In calculating the constants with either formula, the use of logarithms seems to be the most rapid method. Logarithms of both the trigonometric functions and the factorials are found in most mathematics handbooks ( 2 ) , as well as in more
12 15 17 20 21 22 30 35 40 45 60 80 100 120 250 500
4.2i3 4.420 4.479 4.434 4.319 3.972 3,149 2.254 1.134 0.8107 0,5842 0.3104 0.1693 0,006247
E=2.25 3.652 7,003 8.950 9,639 9.477 8.825 8.048 6.974 6.030 . 5.154 4.369 3.086 2.150 0.8476 0.1294 0.02023 0,003262
Double Withdrawal Constants and Input Collected in Products for L = 17, a = 34
Product
NO. 1
E=1.5 1,008 2.177 3.009
...
6 7 8 9
v = l
January 1954
E = l 0.1953 0.4395 0.6592 0.9613
Cm,i.a
1 2 3 4 5
cos"-'T_Y X
2
Table II. Double Withdrawal Constants and Input Collected in Products for z = 9,a = 34 Product log10 Input in Product, %
log10
Cm,t,a
Cm.s,a
1 ,000 4.250 10.63 33.60 67.98 88.01 109.1 130.8 152.6 174.2 234.8 270.4 315.5 328.3 340.0 401.0 416.9 421.7 419.3 389.7 335.7 284.4 240.0 79.04 9.318
E = 1 0,00076 0.0032 0.0081 0.02562
Input in Product, % E 1.5 E sa 2.25 0.01693 0.1927 0 . is57 0.4831
0.067lO
... ...
0,09972
1.533
0 . i328
0.1790
0.2405
...
0.3057
0.3ii5 0.2971 0.2559 0.2168 0.1830 0.06026 0.0071
...
2.244 2.382 2.459 2.456 2.442 2.077 1,452 9,5932 0.2259 0.08457 0.03 154 5 . 1 x 10-7
...
1.487 3.414 5.014 5.845 5.968 5.934 5.771 4.813 4,024 2.904 2 . 2i3 0.7447 0.3477 0.1680 0.07056 0.0059
...
... ... ...
...
Table IV. Double Withdrawal Constants and Input Collected in Products for L = 25, a = 34 Product
No. 1 2 3 5
8
10
12 13 15 17 18 19 20 21 23 25 30 35 40 50 60 70 76 78 79 80 85 100 120 200 400 1000
Cm,z,a
log10 Cm,z,a
1.000 6.250 21.88 121.1 601.8 1251 2209 2806 4214 5913 6845 7826 8852 9914 12120 14410 20190 25260 30610 38300 43130 45570 46100 46210 46220 46230 46060 44180 39910 21900 3999 555
INDUSTRIAL AND ENGINEERING CHEMISTRY
Input in Product, % E = 1.5 E 2.25 0.000003 0.00028 0.01017 0.000019 0.000065 0 .i 6 i 5 0.02924 0.6491 1.996 0.2462 0.0037 3.011 ... ... 3.861 4.179 ... 0.6f65 4,557 ... 4.642 ... 4.579 4.461 0.02638 1.iiS 4 I299 4.103 ... 3.643 1.537 0.04295 3.144 0.06017 1.978 1.757 1.792 1.771 0 . 09i22 0,6051 1.473 0,1285 1.103 0.03470 0.1358 0.7746 0.1374 ... 0.1377 ... 0.1377 0.1378 0.52i2 o.ooi5 0.1373 ... 0.1317 0.2207 ... 0.1189 0.0881 0.06527 0.0018 0.01192 ... 0.0016
E
= 1
...
... ...
... ...
...
...
... ...
... ...
.. .. .. ...
27
ENGMEERING AND PROCESS DEVELOPMENT
IL
0 I-
z
.
W 0
a
W P
0 PRODUCT NUMBER
4.0 0 W
2 -A
3.0
I3 -
a
-z 2.0 lb
0 I-
a
Umm
1.0
W 0.
0 0
90
40 60 80 PRODUCT NUMBER
100
120
Figure 6. Theoretical Double Withdrawal Output Curves forsingle Feed withr = 25,a = 3 4
specialized sources (6, 16). Tables 11, 111, and IV give values of C,.,.,, log C,,,.,, and the percentages of input collected in products for z = 9, 17, and 25, and E = 1, 1.5, and 2.25. An arbitrary ratio of 1.5 between adjacent E values was selected, since this ratio would represent the separation factor for two solutes having a given pair of E's. Many pairs of compounds have separation factors of this approximate value, and the evaluation of the separation for such an EJE,, ratio is of interest. It is usually necessary to consider the yield of each substance in each phase in order to calculate yields and purities. For this purpose, it is sufficient to calculate the values of C,,,,,,, for z = M and z = N , where M = number of stages from feed point to light product and N = stages from feed to heavy product. The
28
BO
100
calculation of heavy phase yields may then be carried out by reversing the above notation as regards phase. Thus, z = N , and Ehesvg product = 1 / E l i g h t product = H / L D . The set of C,,,.,,, values for various cycles depends only on the stage arrangement of the system and is fixed when the number of stages and the location of the feed point is chosen. Once the constants have been determined for each end of the process for a given number of stages and position of the feed stage, it is possible to calculate the fraction of input material emerging in a given product for any value of the p / q ratio. Using the equation
0
3 u
60
Figure 5. Theoretical Double Withdrawal Output Curves for Single Feed with z = 17,a = 3 4
9.0
0
40
PRODUCT NUMBER
Figure 4. Theoretical Double Withdrawal Output Curves for Single Feed with z = 9, a = 3 4
0
20
,
Cm.,,,
p Z(4pqIm-'
(11)
the computations can be made very rapidly using five-place logarithm tables. Figures 4, 5, and 6 indicate typical curves obtained when the per cent of input collected in any product is plotted against the product number for a 33-stage process and z values of 9, 17, and 25, These curves have a characteristic shape similar to that of the Maxwell-Boltzmann distribution curve. Superposition Principle Permits Description of Continued Feed Extraction
If feed is added at every cycle, so that the process becomes a continued feed fractional solvent extraction experiment, identical calculations are valid. In this event, the principle of superposition suggested by Stene (16) permits a description of the result. According to this principle, the addition of a second increment of feed superimposes on a single feed curve B second curve identical with it, except that the curve begins at cycle 1 instead of a t 0. Adding a third increment superimposes a third curve starting from cycle 2, and so on. This indicates that the product for any given cycle, if feed increments are added a t every cycle, will contain the per cent of feed material indicated by the single feed curve for it plus the amounts indicated for each product preceding it. That is, a graphical integration or summation of the single feed curve up to a given point will indicate the results to be obtained in a multiple feed operation. Figures 7,8, and 9 show such
INDUSTRIAL AND ENGIIdEERING CHEMISTRY
Vol. 46, No. 1
ENGINEERING AND PROCESS DEVELOPMENT IO0
100
80
80
0
n
w
e u
W
I-
o
Id1 -J
$
W
j
60
c-
I-
3
3
a
aL
r k
z &
40
G
I-
W (3
W 0
40
z
x
a
E
w
60
0 0
C)
W
eo
I
0 I
IO
1
30 PRODUCT NUMBER
20
I 40
eo
0
so
0
20
40
60
80
100
PRODUCT NUMBER
Figure 7. Theoretical Double Withdrawal Output Curves for Continued Feed with a = 34, z = 9
Figure 8. Theoretical Double Withdrawal Output Curves for Continued Feed with a = 34, I = 17
integral curves for the same E and z as indicated in Figures 4, 5, and 6 . These summation curves show clearly the approach t o the steady state, which is represented by the values approached asymptotically by these curves. The asymptotic values a t the eteady state may be estimated using equations given by Feller for the total probability of the gambler's ruin. These are for P # 4:
where E1 = L D I / H and E I I = L D I I / H . If feed is added in the center, z = a - z, and the equation reduces to
S = (EI/EII)'
(21)
If E I I E I Iis defined asp, then S = p' = pi, and is independent of the values of E I and E I I . Another case of special interest is that in which E I I = l/Ex. For off-center feed and this set of E values, algebraic manipulation shows S = E f . Since E I / E I I = E;, E; = (EI/EII)'. !
and for p = q
- z/a 1 - (a - z)/a
p. = 1 qs =
(17)
where p , and qe represent the fractions of feed which eventually appear in the light and heavy solvent, respectively. Thus, p , and qs are the steady-state values in the light and heavy solvents. These equations can be used to give a steady-state solution identical with that derived in previous work (3) from material balance considerations. If p / q = E = LD/H and p,/q. = A = the ratio of the amounts of solute appearing in the light and heavy solvents, then
for p # q
for p = q
For the separation of two substances, the over-all separation factor, 8,is AIIAII. For p # q for either solute, the general equation is January 1954
P
Consequently, S = ( E I / E I I ) '= p', which is the same result as that obtained for the center feed operation. It has been noted (3) that it is possible to hasten the approach to the steady state in a continued feed operation by the addition of oversize feed increments for the first few cycles. It is possible to use the output curves to predict the effect of the excess feed increments. A continued feed curve, calculated for unit input, is plotted and the single feed curve is plotted upside down on the same graph in Figure 10. The continued feed curve for unit increments represents the approach to the steady state for ordinary operation. If a double increment is used for the first feed, the situation is represented by a superposition of the single feed curve on the continued feed curve. If a triple feed increment is used, the single feed curve is doubled and superimposed on the continued feed curve, and so on. The requirement for the attainment of the steady state is that the continued feed curve reach the asymptotic value. If a suffi.ciently large feed increment is added a t the first contact, so that the superposition of the single and continued feed curves yielde this asymptotic value, the steady state is attained more rapidly than if unit increments are used throughout the operation. A recent article by Scheibel (IS) indicates that caution should be exercised to avoid exceeding the solubility limit in the feed stage
INDUSTRIAL AND ENGINEERING CHEMISTRY
29
ENGINEERING AND PROCESS DEVELOPMENT IO0
100
80
80
a
0
w
W
I-
I-
W
W .J
o
o A
J
0
g 60
60
o I3
+ 3
f
E
$
40
40
C MULTIPLE FEED, UNIT INCREMENTS
2
z W o a
w
0
20
0 0
2 0
40 60 PRODUCT NUMBER
eo
e0
0
IOO
.
Ea2.25 0.34 z 47
a
W 0.
c
B SINGLE FEED, 7 x UNIT INCREMENT
LL
I-
W 0
UNIT INCREMENT
P
n
0
IO
20
30
40
!56
PRODUCT NUMBER
Figure 9. Theoretical Double Withdrawal Output Curves for Continued Feed with a = 34, z = 25
Figure 10. Use of Oversize Feed Increment to Attain Steady State More Rapidly
when using oversized feed increments to hasten attainment of the steady state. As indicated in Figure 10, a first feed increment of eight times the unit-i.e., seven units in excess-followed by regular unit feed increments, gives a single feed curve which meets the continued feed curve a t 23 products. The space between curves B and C represents the distance from steady state. For an extraction factor of 2.25 and z = 17, the steady state can be obtained after 23 products instead of 46, provided the initial feed increment is made eight times as concentrated as those used subsequently. Similar considerations will apply for any extraction factor and z value, although the curves will, of course, be different. It may be necessary in some cases to use oversize increments for several feeds, rather than for one, to prevent the use of too highly concentrated solutions. The use of large original feed increments, chosen according to these principles, will permit the attainment of the steady state in a substantially reduced number of cycles, providing solubility limits are not exceeded, particularly in the feed stage, The devia-
tion from the steady state will be indicated by the difference between the unit continued feed curve and the enlarged single feed curve, plotted upside down, when this technique is used. In addition to the results for calculating the output curves for double withdrawal processes, the random walk approach provides a method of calculating U ( x ) . The U ( z )indicate the amounts of solute present in the tubes during the course of the extraction and are similar to the results calculated on a continuous basis by Johnson and Talbot (9, IO). The equation of Feller (6) for making this calculation is:
Table
V. Separation of Two Solutes at Comparison Points for a = 34, z = 9 (Separation factora 1.5) Yield-E ualsPurity fSoint
Steady State Leading Cross Point or E Value, Yield Purity, purity Yield, Purity, E1 Cycles % b ’ % E Cycles 7% ’ Cycles % % 1.5 41 90.3 72.5 26 76.3 80 100 57.6 2.25 12 80.0 65,9 10 69.0 30 100 50.0 1.2247 . 150 99.5 86.0
...
0 ~
e
30
..
. ..
. ..
...
.
Separation factor = 9, = EI/EII: !eed of e us1 quantities of two solutes. % Yield = amount of one solute in hght &ase total amount of this solute in input amount of one solute in light phase % Purity = total amount of solute in light phase
~ z , n ( x )=
k
[Yr--2--2ka.
n.
- Ys+x--tka.nl
(22)
where
where z is displacement of position from origin and y is an index number = z =tx - 2ka.
Table VI.
Separation of Two Solutes at Comparison Points for a = 34, z = 17
(Separation factora 1.5) Yield-EqualsPurity Point .. Leading Cross Point Yield Steady State or E Value, Yield, Purity, purity Yield, Purity, zr Cycles % b Cycles % ’ Cycles % % 1.5 78 95.5 83.4 57 86.2 100 99.9 66.6 2.25 22 83.6 72.0 19 75.0 50 100 50.0 1.2247 m 96.9 96.9 m 96.9 m 96.9 96.9 a
e
Separation factor = ,9 = EI/EII ; feed of e ual quantities of two solutes. % Yield amount of one solute in light ,%me loo total amount of this solute in in ut amount of one solute in light &ase x 100 % Purity = total amount of solute in light phase
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 1
ENGINEERING AND PROCESS DEVELOPMENT
Table VII. Separation of Two Solutes at Comparison Points for a = 34, z = 25 (Separation factora 1.5) Yield-EqualsPurity Point Leading E Value, EI 1,5
2.25 1.2247 a
Cross Point Yield, Purity,
Cycles 111 31
.. .
%b
94.0 86.0
. ..
Cycles
89.9 78.9
96 29
...
Steady State Yield, Purity,
or
%c
...
purity,
%
91.0 81.0
. ..
Cycles 150 60
250
%
%
97.4 99.9 83.9
78.6 50.6 99.4
Separation factor = P = E I I E I I ;feed of equal quantities of two solutes. % Yield = amount of one solute in light phase x 100 total amount of this solute in input Purity = amount of one solute in light phase loo total amount of solute in light phase ~
Random Walk Calculations Evaluate Yields and Purities at Selected Cycles
In using the results of the random walk calculations to determine the separation to be obtained when two solutes are fractiona ted according to the double withdrawal pattern, several comparisons may be used. It is possible to evaluate the separation a t any selected cycle by comparison of the single feed curves, but three points seem of special interest: The point a t which the two curves intersect, or the cross point, for single feed operation The point a t which the yield is equal to the purity for each solute, for single feed operation The steady state, for multiple feed operation
and VI1 for three pairs of E ratios, 2.25/1.5, 1.5/1, and 1.2247/ 0.8165. The latter is the symmetrical case, EI = l/E11. The factor, p = 1.5, is the same in all cases. The E's are adjusted in practice by changing the solvent volume ratios or by causing a change in distribution coefficient by adjustment of pH, etc. I t can be seen from a comparison of these results that a number of factors must be considered in selecting optimum conditions for a separation. Particular conditions may be used because they require fewer cycles, or result in higher purities or in higher yields. Stopping the operation a t the yield-equals-purity point requires that fewer cycles be run and produces a sample of higher purity than that obtained at the cross point, although the yields are somewhat lower. In all cases indicated, except for the symmetrical case (leading E = 1.2247), the purities obtained under steady-state operation are considerably inferior to those obtained under other conditions. The steady-state operation with E I = ~ / E I Ihowever, , yields the highest purity obtainable for each value of z for the cases considered. Under ordinary operating conditions for continued feed, many cycles are required to reach the steady state. I t is possible to hasten this approach by the addition of excess feed material a t the start of the operation. In the special case of a center feed operation, with EI = l/En, as indicated by Scheibel ( l a ) , the approach of the two components t o the steady state is identical; consequently, every product contains the same purity material, although the yield increases as the system approaches the steady state. For fractionating a small quantity of material in the shortest possible time, it is probably preferable to use a single feed operation and cut a t the yield-equals-purity point. For larger quantities, the steady-state operation is preferable with z and E values adjusted to give high purities. In this case, it may be also advis-
In order to compare the degree of separation in terms of yields and purities for these different cases, an arbitrary p value of 1.5 was selected, and the results were calculated for a 33-stage operation using z values of 9, 17, and 25. Unit feed to all cycles, including the first, is assumed. The results are shown in Table Ti, VI,
V
I
I
I
7.0
6.0
.
BUFFER COMPOSITION: 0.436 MOLES NoOH 0.365 MOLES NoH,P04*YO NoH,P04*Y0 PER LITER
0 W
I-
;5.0 3 3
s
g
4.0
a.
f LL
,-z 3.0 0
-
Y 0
a
w 2.0 a.
I .o
0
I
I
I
I
I
I
_
0.5 1.0 1.5 2.0 2.5 3.0 CONC. IN BUFFER, MOLES PER LITERx103
Figure 1 1. Distribution of p-Cresol between Benzene and Phosphate Buffer for Single Feed Experiment January 1954
I
I
I
I
I
5
IO
15
20
25
P R O D U C T NUMBER
Figure 12. Double Withdrawal Output Curves for Partitioning p-Cresol between Benzene and Phosphate Buffer a = 3 4 , r = 17
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
31
ENGINEERING AND PROCESS DEVELOPMENT
1.6
r’
0 X a W
1.4
-
1.2
-
1.0
-
J K W
0
I
1
I
I
I
BUFFER COMPOSITION: 0.414 MOLES NaOH 0.365 M OLE S N a Y P O,.H,O PER LITER
0
.
P 0.e
THEORETICAL FOR E a 2 . 6 4 CURVE
.
80
0
EXPERIMENTAL POINTS
0
+ 0
W J
$
60
I3
-
ti
f
W
z W
/ -I
L -
0 W
v)
W J
IO0
,I
LL
0.6
-
I-
40
t
W
W 0
m
.
a Y
a
0
1.0 2.o 3.0 4.0 SO CONC. IN BUFFER, MOLES PER L l T E R x 1 0 4
0
Figure 13. Distribution of p-Cresol between Benzene and Phosphate Buffer for Continued Feed Experiment
-
30
I
0
0
5
IO
Partitioning p-Cresol between Benzene and Phosphate Buffer Is Experimental Study
For the experimental demonstration of the applicability of the random walk calculations, p-cresol was selected as the solute and benzene and an aqueous phosphate buffer as solvents. Work had previously been done by Golumbic (7, 8) and by the authors (3) on the system m- and p-cresol, using these solvents, and it was believed that the data obtained in these further studies would be of intrinsic importance in establishing a separation procedure for the cresol isomers. The p-cresol used in this work was a practical grade chemical (Matheson Co.) and was distilled before use. Its purity was estimated as 96% by freezing point measurements, and the sample was used without further purification. Two different phosphate buffers were used in the work. These buffers were made by mixing appropriate molar quantities of sodium hydroxide and monobasic sodium phosphate monohydrate, and had pH’s of approximately 11. Because of the high pH and the high content of sodium ion, attempts to determine the pH’s of these solutions precisely did not yield satisfactory results. The buffers have been characterized by their molar concentrations and by the distribution ratios obtained for p-cresol. The p-cresol was determined in the benzene phase by measurement of the light absorption a t 286 mb, using benzene as a blank. All measurements were made on a Beckman Model DU quartz spectrophotometer. A11 the extraction experiments were performed using the machine built by the authors in previous work on this problem (5). This machine consists of a series of glass tubes, individually mounted, which can be plased on a plywood bed for simultaneous shaking and manipulation. The two solvents are placed in the lower section of the tube and agitated. After the phasea have settled, the light phase is transferred into a reservoir by tilting 32
I 20
25
PRODUCT NUMBER
Figure 14. Double Withdrawal Output Curves for Partitioning p-Cresol between Benzene and Phosphate Buffer a
able to attempt a more rapid approach to the steady’state by the use of oversize feed increments a t the beginning of the operation.
I IS
= 34,z = 17
the plywood bed. As the device returns to its normal position, the light phase flows into the next mixer to the left. Transfer of the heavy phase to the right is effected by moving the tube along the bed. The present machine has 17 tubes which provide 33 stages or contacts in the extraction. In the initial experiment, a single feed increment of p-creeol was partitioned in the 17-tube, 33-stage apparatus according to the double withdrawal pattern. The products were collected and the benzene phases were analyzed for p-cresol so that a comparison of the theoretically predicted and experimental yields could be made. The buffer used for this experiment was 0.436M in sodium hydroxide and 0.365M in monobasic sodium phosphate monohydrate. The distribution curve given in Figure 11 was obtained for p-cresol, using this buffer. Table VI11 indicates the conditions used for the experiment. The solvent volumes were adjusted to give an extraction factor of 2.86, so that essentially all the cresol would be waehed out within 30 cycles.
Table VIII. Operating Conditions for Single Feed Partition of p-Cresol Solvent volume, ml. Benzene t o first stage 40.0 28.0 Aqueous phosphate buffer to last stage 40.0 Feed solution in benzene 0.00588 p-Cresol concn. in feed solution, molo/l. Feed increment added instead of pure benzene at first contaot
Thirty products were collected and the benzene phases, without dilution, were analyzed on the spectrophotometer. The comparison of the theoretical and experimental results is shown in Figure 12. The per cent of input material found in each product is plotted against the product number. The curve indicates the
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 1
-
ENGINEERING AND PROCESS DEVELOPMENT theoretical result for an extraction factor of 2.64, and the experimental results are indicated by the points. The extraction factor of 2.64 wm selected to fit the curve peak position because it was realized that the intended value of 2.86 could be affected by a number of factors. The second experiment was designed to demonstrate the washout of continued feed increments of p-cresol in a double withdrawal process. This experiment, consequently, served to verify the principle of superposition which was assumed in predicting the approach to the steady state. For this work the buffer mas 0.414M in sodium hydroxide and 0.3652Min phosphate. The distribution curve is shown in Figure 13 and the operating conditions are indicated in Table IX. The feed volume was kept small to prevent aberrations in the results. The theoretical calculations are completely valid only when no solvent is introduced with the feed, but it was believed that if the ratio of solvent in the feed to total amount of light solvent was kept c>mall,the deviations would not be appreciable.
Table IX. Operating Conditions for Multiple Feed Partition of p-Cresol Solvent volume ml. Benzene t o fiist stage Aqueous phosphate buffer to last stage Feed solution in benzene p-Cresol concn. in feed solution, mole/l.
25.0 28.0 1.0 0.0317
4 total of 25 cycles were run and the benzene phases analyzed on the spectrophotometer. Products 1 through 7 were analyzed without dilution, 8 through 14 were diluted 10 to 25 ml., and 15 through 25 were diluted 5 to 25 ml. To determine the results of the experiment in terms of the fraction of input collected, it was necessary to determine the volumes of benzene collected as products. The average volume was 22 ml. for each product. Approximately 2 ml. of benzene appeared with each heavy product, and 2 ml. were lost, primarily through evaporation and poor drainage. The results of the experiment are shown in Figure 14 in terms of the per cent of input material collected as products. The agreement of the experimental points with the theoretical curve for E = 2.64 is good, although the experimental points seem to define a curve which is slightly skewed with respect to the theoretical. Agreement of Theory and Observations Indicates Validity of Calculations
The close agreement between the theoretical and observed results for the experiments performed in this work indicated the validity of the random walk calculations. This agreement is noteworthy because of the following conditions assumed by the theory: 1. The solutes have constant distribution ratios. 2. Equilibrium is established between the two liquid phases a t every contacting. 3. There is complete separation of the phases a t each stage in the operation. 4. The liquid phases are immiscible, or have a miscibility which is constant and not affected bv the Dresence of the solutes. 5. If two or more solutes are to he u&d, they do not interact because their distribution ratios must be independent. 6. There is no loss of solvent in the system through evaporation or a similar process. 7. There is no solvent input except at the ends of the systemthat is, no solvent may be introduced a t the feed stage.
In designing the experiments to verify the theory, attempts were made to comply as closely as possible with these conditions. It was observed previously ( S ) , however, that the extraction machine used in this work was inherently incapable of operation in exact accord with conditions 3 and 6. There was a slight January 1954
amount of solvent carry-over and incomplete separation of the phases and there was a slight loss of light solvent in the transfer process through poor drainage and evaporation, I n previous work it was observed that these conditions did not affect the results of continued feed extraction experiments appreciably. The results shown in Figures 12 and 14 indicate the substantial agreement of the experimental results with theory for the single and continued feed partitions of the p-cresol. It has been observed by Craig (4)that the major effect of solvent carry-over is to broaden the single feed output curves without shifting the peak. This appears to be true in the present case. The value anticipated for the extraction factor in the single feed experiment was 2.86. The experimental results, however, match closely a theoretical curve for E = 2.64. The decrease in the distribution ratio of p-cresol with decrease in concentration probably accounts for this discrepancy. The situation is similar in the continued feed experiment. The experimental points give a reasonably good fit to the curve for an extraction factor of 2.64, with slight deviations requiring a factor of 2.86 in the early stages and a factor of 2.4 to 2.5 in the final stages. This apparent shift in the extraction factor is probably accounted for by the drifting distribution ratio, by carryover, and by the fact that a small amount of light solvent was introduced with the feed. I t is possible that the introduction of solvent with the feed has the effect of skewing the curve slightly. This effect has not been studied experimentally and an exact theoretical treatment appears to be very difficult. These effects are minor, however, and do not alter the substantial agreement with the theoretical mode. This theoretical treatment would be of interest to those who are interested in countercurrent extraction as a separation technique. Prior to this, no satisfactory general method of calculation had been devised for double withdrawal processes except for the case of continued feed and steady state operation. The theory presented here has the advantage of describing the situation for single as well as continued feed operations, and of being applicable to any position of the feed stage. The theory is of greater value because of the analogy between the continued feed double withdrawal process and the continuous countercurrent extraction process. By using solvent flow rates instead of solvent volumes, it should be possible to adapt the equations presented here to, describe the results in a continuous countercurrent column. Nomenclature
A
ratio of total amounts of solute in light and heavy product phases, respectively B = binominal coefficient C’, C = constants characterizing the o u t m t for a niven number of stages and prodYucts D = distribution ratio of solute between light and heavy phases E = extraction factor = L D / H = p/q H = quantity of heavy phase per cycle I, = quantity of light phase per cycle M = stages from feed point to light product A T = stages from feed point to heavy product over-all separation factor = AI/AII sL7 = = chance of gambler’s ruin on a given trial; or fraction of single cycle feed leaving system in a given product W = fraction of input in a given tube or product (single withdrawal) a = total of gambler’s and opponent’s capital; or total extraction stages plus 1; or stage number of heavy product b = total number of tubes employed c = product number (single withdrawal) f = tube number I C = integral running summation index m = product number (double withdrawal) for given phase n = 2 + 2 ( m - I’, P = probability of ’gambler’s loss for single trial; or fraction of solute transferred to given solvent (light) q = probability of gambler’s winning for single trial; or fraction of solute transferred to other solvent (heavy) =
INDUSTRIAL AND ENGINEERING CHEMISTRY
A
I
33
ENGINEERING AND PROCESS DEVELOPMENT n&z
term in binomial coefficient = - ka 2 t = number of transfers r = stage number, counted from light product as zero y = index number in evaluating U ( z ) z = gambler’s original capital; or stage number of feed stage, counted from light product as zero 0 = single stage separation factor = DI/Drr = Er/EII Y = integral running summation index; also used by Feller as a term in evaluation of U ( z ) n! = binomial coefficient = T ! (n - T ) ! =
(4)
~
r;)
Subscripts 1, 2 indicate transfer or cycle number I, I1 indicate solute number literature Cited
(5) (6) (7)
(8) (9) (10)
(11)
Craig, L. C., and Craig, D., in Weissberger, ed. “Technique of Organic Chemistry,” Vol. 111, Chap. IV, New York, Interscience Publishers, 1960. Duarte, F. J., “Nouvelles Tables de Log n,” Geneva, Imprimerie Albert Kundig, 1927. Feller, William, “Introduction to Probability Theory and its Applications,” Vol. I, New York, John Wiley & Sons, 1950. Golumbic, C., J. Am. Chem. Soc., 71, 2627 (1949). Golumbic, C., Orchin, M., and Weller, S., Ibid., 71, 2624 (1949). Johnson, J. D. A., J . Chem. SOC.,1950, 1743. Johnson, J. D. A,, and Talbot, A., I b i d . , 1944, 1068. Peppard, D. F., and Peppard, M . A., IND.ENQ.CHEM.,46, 34
(1954). (12) Scheibel, E. G . , I b i d . , 43, 242 (1951). (13) Ibid., 44, 2942 (1952). (14) I b i d . , 46, 43 (1954).
(15) Stene, S., Arkiv Kemi, Mineral. Geol., 18A, No. 18 (1944). (1 6 ) Vegs, G., “Logarithmisch-TrigonometrischesHandbuch,” Berlin, Weidmannsche Buchhandlung, 1857. (17) Williamson, B., and Craig, L. C., J.B i d Chem., 168,687 (1947).
Ib
(1) Auer, P. L., and Gardner, C. S., IND.ENQ. CWEM.,46, 39 (1954).
Burington, R. S., “Handbook of Mathematical Tables and Formulas,” 2nd ed., Sandusky, Handbook Publishers, 1947. (3) Compere, E. L., and Ryland, .4.,IND.ENQ.CHEM.,45, 1682 (2)
(1953).
RECIZIVED for review July 1 , 1983. ACCEPTED October 21, 1953. Presented before the Division of Physical and Inorganic Chemistry at the 121st Meeting of the AMERICAN CHEMICAL SOCIETY, Buffalo, N. Y. This work was taken from a dissertation presented by Ada L. Ryland in partial fulfillment of the requirements for the degree of doctor of philosophy, Louisiana State University, Baton Rouge, La.
(BATCHWISE FRACTIONAL LIQUID EXTRACTION)
Holdup and Approach to Steady State DONALD F. PEPPARD Argonne National Laboratory, Lemonf, 111.
MARVILA A. PEPPARD 715 Thomas, Oak Park, 111.
I
N THE evaluation of a countercurrent liquid-liquid extraction
process the problems of holdup and of departures from steady-state operation arise concomitantly. For example, there exist systems which may be operated satisfactorily for a given number of cycles but in which, a t the following cycle, a third phase appears, owing to the build-up of one or more components to a critical value a t some point. And from an elementary consideration of the mechanism of approach to steady state it may be demonstrated that the purity of the product obtained during nonsteady-state operation may be much greater than that of the product obtained during steady- state operation. Specifically, determination of the number of cycles required for a given approach to steady-state operation of a system with a large number of stages is an important problem in the investigation of liquid-liquid fractionation of rare earths (i), since sharp fractionation in the solvent systems now under study requires the use of 20 to 30 stages. This particular fractionation of adjacent rare earths necessarily involves a t least one E value which is very close to unity, a critical value. Since the systems employed closely approximate the ideal double-diamond systems (Figure 4), utilizing an odd number of contactors with feed entry a t the center contactor and with a single value of E throughout the battery of stages, exact calculations were made, by means of Equation 8, for such ideal systems. This is equivalent to the rigorous summation method introduced by Stene (4). Although exact values may be obtained for the ideal system by means of Equation 8, this equation is not subject to ready usage, owing both to its form and to the necessity for preparing a table of pi values. Therefore, expressions by means of- which the approach to steady-state operation may be calculated in an ap-
34
proximate but rapid manner have been derived, in the one case involving use of the interrelationship of holdup and the approach to steady state. Expression for System Holdup at Steady State Is Developed
In Figure 1, a derivation of expressions for stage holdup and product composition for a general system, operated a t steady state, is given. The method of operation is the double diamond, 90 that all stages are active a t each cycle. In the derivation, the product compositions with respect to solute are defined as &4 and &,UB, respectively. From consideration of material balance and the fact that the quantity of solute in the upper phase is E . times that in the corresponding equilibrated lower phase, expressions for &I and &, may be derived. In like manner, expressions for Q2 and &(* - 1) are obtained, etc. These expressions are then generalized to the forms given in Figure 1. In order to evaluate A and CB (which are identically & and F,, respectively), the sum of the entries to stage n is set equal to Qn (as given by the second expression for Qj), and the identity A U B 3 1 is introduced. The system holdup, H, is the sum of the & j values for j = 1, 2,
+
....q.
The remaining discussion is restricted to systems in which - 1 and E = U. For such systems, F , (i.e., U B ) and (Le., A ) are, respectively, E”/( 1 E”) and 1/(1 C ) ,as pointed out by Scheibel (I,9), and the Q , / Q o expressions are, therefore, q = 2n
INDUSTRIAL A N D ENGINEERING CHEMISTRY
+
+
j
=
1,2,
. . ., (n
-
11, n
Vol. 46, No. 1
*
