2942
INDUSTRIAL AND ENGINEERING CHEMISTRY = contraction coefficient of stream within port = frictional coefficient
a A v P
= =
v
=
kinematic viscosity of fluid flowing through port density of fluid flowing through port (gms./cm.a) volumetric rate of flow through port, cubic centimeters per second
literature Cited
Vol. 44, No. 12
mechanics," Chap. 111, Kew York, RIcGraw-Hill Book Co., Inc., 1934. (5) Wahlen, Cniv. Illinois Expt. Sta., Bull. 120 (March 1921). ( 6 ) Wilson, C. W., and Hawkins, N. J.,IND.ENG.CHEM.,43, 2129 (1951). ( 7 ) Kohl, K., Kapp, N. AI., and Gaeley, C., "Third Symposium on Combustion, Flame and Explosion Phenomena," pp. 3-21, Baltimore, Md., Williams 8: Xlkins Co., 1949.
(1) Blasius, Physik. Z., 12, 1175 (1911).
Bollinger, L. M., and Williams, D. T., Natl. Advisory Comm. Aeronaut., Tech. Note 1234 (1947). (3) Lewis, B., and von Elbe, G., J . Chem. Phys., 11, 75-97 (1943). (4) Prandtl. L., and Tietjens, 0. G., "Applied Hydro- and Aero(2)
EngFnyring
RECEIVED for review July 12, 1952. ACCEPTED August 20, 1952. Presented before the Diyision of Gas a n d Fuel Chemistry at the Diamond Jubilee ;Meeting of the AMERICANCHEMICAL SOCIETY, New York, September 1931.
Fractional liquid Extraction Approach to Steady State 'Conditions
p*cess development I
EDWARD
G.SCHEIBEL
Hoffmunn-la Roche, Inc., Nutley 10, N. J,
OXSIDERABLE experimental x ork has been desciibed in
C
the literature in which an attempt has been made t o determine the steady state conditions in a continuous countercurrent fractional liquid extraction operation through the use of a batchwise operation on a suitable number of discreet stages. The slow approach t o steadv state conditions, particularly when a large number of stages is involved, has long been recognized, and an empirical equation mas recently presented for estimating the deviation from steady state after a given number of cycles ( 2 ) . The equation showed that the number of cycles for a given approach t o steady state conditions was greatest 17hen the extraction factor approached unity and, under these conditions, varied almost logarithmically with the number of stages in the operation. The reason for this slow approach to steady state conditions is that the feed mixture must be accumulated to a very high concentration in the center stages before the total of the amounts leaving at the ends will approach the feed quantity as required by steady state operation. Compere and Ryland ( 1 ) recognized this effect and in a 33-stage operation added increased amounts of the feed in the initial cycles in an attempt to build up the concentrations as rapidly as possible. A practical design for a fractional liquid extraction process requires operating as closely as possible to the maximum concentration in the center stages in order t o obtain maximum production from a given quantity of solvents. Thus, in building up the concentrations in the center stages, care must be exercised so that this maximum concentration will not be exceeded. Excessive concentrations frequently result in either separation of a third phase if the feed is only partially soluble in both solvents or complete miscibility of the phases if the feed is completely miscible with both solvents. However, even if these extreme effects are not noted, an excessive concentration a t the feed stage would upset the conditions in the multistage operation by decreasing the selectivity of the solvents. Figure 1 shows the conventional triangular pattern for an 11stage fractional liquid extraction with center feed. The quantities noted on the streams represent the amount of one particular component necessary to establish steady state conditions a t the bottom of the triangle based on a total feed of E'" 1. The ideal method for estahlishing steady state conditions would be to in-
+
troduce the appropriate amount of solute in the solvents entering along the sides of the triangle. This is completely impractical because one of the purposes of carrying out the pattern is to determine the proper value of E which is not known accurately in advance. The usual purpose is also to separate two components which cannot be separated by other means; to simultaneously supply the quantities of each component mould require synthetic mixtures of the two components in different ratios than in the feed in all these other solvent streams. Thus, this method of establishing steady conditions is completely impractical arid since the quantities t o be supplied with solvent feeds are hypothetical, the streams have been indicated as broken lines. The figure, however, does suggest a method for varying the quantity of feed on successive cycles to compensate for the fact that the solvents introduced along the sides of the triangle were free of solute. In this case the total amount of feed introduced into the first cycle is the sum of the steady state quantity of fresh feed and the quantity indicated by the broken lines: $'I
= (E'
+ 1) + 2(E6 + E 4 $- E 3 + E' + E )
(1)
and it can be shown that the general expression is
where the total stages in the system are equal to (2% - 1). Increasing the second feed by an amount necessary to compensate for the lack of solute in the solvent stream Ha and LZgives
F*
=
(E6
+ E2 + 1 ) + 2 E5 + E ++ E3 l E4
(3)
and the general equation is (4)
This method will produce steady state conditions in a 5-stage pattern in t-rTocycles-Le., when the second product withdrawal is made. All feeds after the second vould be the steady state quantities.
December 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
Similarly, adjusting the third feed for the amount of solute missing in the H 3 and La solvent streams gives
and the general equation for all cycles on this basis is
2943
state conditions instead of the quantity (E" previous equation:
+ 1) used in the
The maximum error in this equation occurs in the second cycle.
It is less with the larger numbers of stages and these required the greatest numbers of cycles to approach steady state conditions. However, this concept does not give the entire mechanism, because the third feed would also have t o be adjusted for the quantities lacking in H z and Lp which move along the sides of the triangle and are also missing from the stages a t which H3 and LOare introduced. Thus, it can be seen that as the number of stages increases, the equation for the feed variation becomes so complex that an arithmetical solution for each individual case would probably be the most convenient approach. A study of several such operations for different values of E and up to an 11-stage operation indicated that the correction term for the feed quantity was approximately a geometric series. For values of E approaching unity and for large numbers of stages, the ratio was approaching as indicated by the slope of a plot of the logarithm of the correction term against the number of cycles. It was then obrepresented the data for all values ( E -Iof E reasonably well. A general equation thus deduced for the served that a ratio of
Thus, the equation is most reliable in those cases where it is most necessary and desirable. I n separations generally under consideration, the equation gives feed stage concentrations which will exceed those a t steady state conditions by less than 10%. I n establishing a practical set of operating conditions, i t would always be desirable t o design for a t least 10% below the maximum allowable concentration a t the feed stage. Thus, the equation can be used t o establish the steady state conditions for a commercial application of liquid extraction in the minimum number of cycles. If, for any reason, i t is desired to operate continually below the steady state feed concentration, the first and second feed quantities can be calculated from the rigorous Equation 6 which, based on a steady state feed of F , instead of the E* 1 quantity in the previous derivat?ion, becomes
+
~
variation in feed quantity for the successive cycles is given as follows, based on a general quantity of feed, F,, for the steady
Figure
Ft
= Fa
(1 +
E" - E t (E" + 1)(E - 1 ) ( E
+ 1)i-I
and the subsequent feeds could be calculated from the geometric series based on the correction term for t = 2 as follows:
STEADY STATE CONDITIONS I. Pattern for 1 I-Stage Fractional Liquid Extraction with Center Feed
INDUSTRIAL AND ENGINEERING CHEMISTRY
2944 Table I.
Feed Cycle
4
1 2 3 4
1
2 3 4
and an average value could be used to estimate the variation of the feed quantity a t successive cycles. If t h ~ Equations 8 and 9 most conservative amounts were desired, Concn. the value of E having the largest numei ideviationa, Ft 4?c tal logarithm, independent of sign, should be used in the equation. This 6.00 0 will ensure maintaining concentrations 3.00 0 2.00 -4 1 below the final steady state concentra1.50 -8 3 tions in the feed stage. The variation of the feed rate in a con2.91 0 1.615 0 tinuous multistage operation may also 1.273 -1 9 1,121 -6 9 be estimated from these equations based on considering a unit cycle as the time 1,666 0 required to change all the solvent holdup 1,133 0 in a single stage at the given solvent 1.043 -0.3 1.014 -0.4 flow rates. Some adjustment is usually necessary because the ratio of light sol1.25 0 vent to heavy solvent in the column 1.025 0 1,0045 -0.02 holdup would only coiucidentally corrp1.0008 -0.03 spond t o the ratio of the flow rates. I t quantity from amount a t is generally not convenient to vary the flow rates continuously according to a given pattern by manual control. Thcrefore, the feed rates may be varied stepwise somenhat similar to the batchwise calculations, as for example, five times the desired feed rate for the first unit of time, three times for the second period, twice for the third period, one and a half for the fourth period, and then finally held at the desired steady state rate. This procedure has been used in obtaining performance data on a large 30-stage continuous estraction column.
Comparison of Equations with Calculated Feed Quantities and Feed Staqe Concentrations for an 1 1 -Stage - Operation . Equation 6 Concn. deviationa,
Equation 7 Concn. deviationa, Ft %
Ft
70
6.00 3.00 2.25 1.875
6.00 3.00 1.75 1.25
0
-
6.00 3.50 2.25 1.625 E - 2
0 +8.3 +4.1 -1.0
2.91 1.615 1,327 1.199
2.91 1.615 1.191 1.055
0
-
0 +7.8
1.666 1.133 1,052 1.025
1.666 1.133 1.027 1.005
-
2.91 1.847 1.376 1.167 E = 4 1.666 1.211 1.067 1.021
Theory
E - 1 1 2 3
Vol. 44, No. 12
0 8.3 -14.5
0 4.6 -10.4
-
0 0 1.3 2.3
+4.8
-0.6 0 +4.? +2.0
+1.1
E = 9 0 1.25 1.25 1 1.25 0 1.045 2 1.025 1.025 1.0081 0.2 3 1 0048 1.0025 1.0015 1.0003 - 0.1 4 1.0012 a Concentration deviation expressed as percentage deviation of steady state conditions.
-
E"
0 $1.6 +0.6 $0.3
feed stage
- E2
This equation does not give excessive concentrations in the feed stage in the range of conditions considered in this work which covered values of E from 0.1 to 10 and up to eleven stages. Table I summarizes the comparison of the equations x i t h the theoretical amounts of feed required to maintain the final concentration a t the feed stage for an 11-stage operation a t different values of E. The table also gives the deviation of the feed stage concentration from the steady state conditions. It is apparent that the maximum deviation occurs in the second cycle when E = 1. I n general the concentration in the feed stages decreases in all cases to a minimum in the region of t = n. With Equation 7 the minimum is less than 10% below the steady state concentration. Sfter 11 cycles with this technique for the case of E = 1, the average approach of the stages to steady state conditions was better than 9070, with the end stages at 88.4% and the feed stage a t 92.5%. d b o u t 70 cycles would be required for a similar approach using the usual constant feed method. I n systems involving more stages the advantage of this method is even more impressive. On the other hand, calculations based on Equations 8 and 9 never exceed the steady state concentration a t the feed stage. This may be considered a more conservative technique, but will require somewhat more cycles for a given approach to steady state conditions. Under conditions of a large number of stages and a large value of E, these latter equations may exceed the desired concentration. For practical purposes, a large number of stages would never be employed with a large value of E because the separation would be too complete for accurate evaluation. A 5-stage operation Kith a value of E = 10 would give 99.9% of the solute in the light solvent. The primary objective of the previous equations was to develop a n improved technique for studying fractional liquid extraction in the batchwise operations of discreet stages. Examination of 011 the equations will indicate that the correction factor will be the same for values of E and 1/E. These are the optimum conditions for separating two components when the feed, introduced into the center stage, is free of both solvents. Thus, the equations are ideally suited for studying fractional liquid extraction a t the optimum solvent ratios. When it is necessary to introduce some solvent with the feed, it is no longer possible to maintain the value of E for one component equal to the reciprocal of the value of E for the other component,
Summary Equations have been developed for different techniques of varying the feed quantities in a batchwise fractional liquid extraction operation which allow the attainment of steady state conditions in a fraction of the time required by the conventional method of uniform feed quantities. The quantity of feed is adjusted to maintain the steady state concentration in the feed stage. Thus, a 33-stage operation on tn-o components with E' values of 0.88 and 1.14 would require over 500 product cj~Aesto approach 90% of steady state conditione, u-hereas by the present method 33 to 50 product cycles would probably give a better approach to steady conditions. I n a previous work, Compere and Ryland ( 1 ) were able to establish steady state conditions in such a unit in 70 cycles by adding ten times the desired amount of feed in the first nine cycles. The present equations which provide a method for continuallj- varying the feed quantity with each cycle should produce steady conditions in a n even shorter number of cycles. Thus, this technique of operation can decrease the amount of work required in studying fractional liquid extraction to less than one tenth of that required by the usual method.
Nomenclature
D = distribution coefficient
=
concentration in light s o w concentration in heavy solvent
LD E = extraction factor = H Ft = feed quantity a t any cycle, t F. = feed quantity a t steady state conditions H = quantity of heavy solvent to each stage L = quantity of light solvent to each stage
n = feed stage location such t h a t (212 - 1) is the total number t
of stages = number of cycles considering first feed introduction as the first feed cycle and first product drawoff as first product cycle
Literature Cited (1) Compere, E. L., and Ryland, A , , unpublished manuscript. (2) Scheibel, E. G., IND. ENG.CHEM.,43, 242 (1951). RECEIVED for review February 9, 1952.
ACCEPTEDSeptember 5, 1952.
