OPTIMIZATION OF OPERATING CONDITIONS IN A PACKED LIQUID-LIQUID EXTRACTION C O L U M N D. H . MOORHEAD AND D. M . H I M M E L B L A U Department of Chemical Engineering, The University of Texas, Austin, Tex.
Several variables affecting the extraction efficiency o f a 3-inch glass column
packed with
'/,-inch
polyethylene
spheres have been studied for the system methyl isobutyl ketone-acetic
acid-water.
The range of mass flow rates
was from about 12 to 240 pounds per hour; flow ratios from
1.5:l t o 2O:l;
and total flows from 100 t o 250 pounds per
hour. To compare the effects o f the variables the lowest
HTU's actually obtainable were estimated from the mathematical models for water phase continuous, extraction from ketone to water; water phose continuous, extraction from water to ketone; ketone phase continuous, extraction from ketone t o water; and ketone phase continuous, extraction from water t o ketone.
The most efficient extraction re-
sulted with the third combination.
Within the experimental
limits the ratio of the mass flow rates o f each phose appeared t o be more important than the total flow rates in determining the efficiency of extraction.
HE DESIGN A N D OPERATION of packed columns have been Tthe subject of considerable investigation. Many of the results have limited utility, since they are applicable only to equipment and operating conditions similar to those used in each study. The objective of this investigation was to carry out a generally applicable statistical analysis to determine if a n optimum combination of certain operating variables could be found and to measure the relative effect of these variables on extraction efficiency. The efficiency of extraction in a packed column is usually measured by the height of a transfer unit (HTU). Many variables affect the efficiency of extraction, and unfortunately some of the basic ones cannot be measured directly. Even precise mathematical analysis of the hydrodynamics of the system presents a n inordinately difficult problem. Therefore, it was necessary to work with variables which can be measured, or at least controlled. These include:
Choice of continuous or dispersed phase (2, 72, 25, 33) Wetting or nonwetting of the packing by either phase (72, 23, 33) Size of packing (27, 22, 24, 33) Shape of packing (70, 77, 33, 34) Height of packing in the column Direction of extraction (79, 33) 68
Holdup of dispersed phase (24, 33, 34) Velocity (mass flow rate) of either phase liquid flows ( 7 , 7 7 , 73, 74, 20, 33, 37, 34) Physical properties of the system Void space in the packing (77, 78) Column diameter (27, 22, 24, 33) Design of liquid distributors ( 7 , 77) Concentration of solute (33)
I&EC FUNDAMENTALS
Factors selected as variables to be studied for their effect on extraction efficiency were choice of continuous or dispersed phase, direction of extraction, and velocity of phase liquid flow. I n considering the packing wetting characteristics, polyethylene was selected as a packing material because it was wet preferentially by the ketone phase. In determining optimum conditions of operation for a piece of equipment, the best combination of quantitative factors is sought, as measured by a maximum or minimum value of the response variable. This problem has been investigated by Box and others (3-7, 9 ) . Of the variables studied, the first two groups are qualitative factors with two possible levels, while the third group, the solute-free velocities (mass flow rates) in the continuous and discontinuous phases, are quantitative factors with an infinite number of levels, since they are continuous variables, within definite operating limits. I n the operation of the experimental apparatus it was not possible to fix selected levels of the quantitative factors in advance without going to a n inordinate amount of trouble. A range of solute-free velocities (the specific number are listed in Table 11) were used, but their levels were not known precisely until calculated from a material balance after each run. T o arrive a t the optimum combination of these variables, the experimental design employed was a 2 X 2 factorial design for the first two variables, giving four possible combinations of these qualitative factors. For each of these four combinations, response surface methods were used to determine the optimum combination of the flow rates only. The combinations in the 2 X 2 factorial design could then be compared to arrive a t the over-all optimum combination of all the factors studied. The most efficient extraction occurred when the ketone phase was continuous, with acid going from ketone to water. I t appeared that the ratio of the flow rate in the discontinuous phase to that in the continuous phase exerted a more important effect on the H T U than the total flow rate, but the experimental range of the latter was relatively less than the former. Although in some cases the experimental error was large enough to prevent meaningful comparisons, the techniques used did demonstrate a convenient and revealing way of comparing results for different combinations of the continuous and fixed level variables.
Experimental
The experimental equipment consisted essentially of a glass column packed with polyethylene spheres, four stainless steel tanks, two centrifugal pumps, and two rotameters. The column was a 3-inch outside diameter borosilicate glass tube 4 feet high with a 32.5-inch bed of 3/8-inch polyethylene spheres. The packing was suplported above and below by circular pieces of 4-mesh stainless steel wire cloth attached (by stainless steel wire) above the water and below the ketone distributors, respectively. The polyethylene spheres (Dumond, Inc., San Angelo, Tex.) were inspected before use, and any irregular pieces, or spheres with mold flashing, were removed. The four tanks were used in pairs to hold each of the streams. One tank of each pair contained fresh liquid, the other spent liquid; this order was reversed for succeeding runs. Each pair of tanks was connected via a tee to a stainless steel centrifugal pump, which discharged through a rotameter to the column, the water going to the distributor a t the top of the packing, and the ketone to the distributor a t the bottom. T h e water outlet was a t the bottom of the column from which water was returned to the other tank of the pair (or to a drain), while the ketone outlet was a n overflow via a tee from the top of the column. All piping and connections were of polyethylene, nylon, or stainless steel. Needle valves downstream of each rotameter were used to control the flow rates into the column. The outlet rate was controlled by a needle valve on the water outlet, which therefore maintained the level of the interface, since the ketone left the column by a n overflow. The physical system was methyl isobutyl ketone (4-me thyl-2 -pen tanone)-acetic acid-water (distilled), chosen because equilibrium data were available (75, 76, 27, 28, 29) and because of recent work done with the same system (27, 30). Purity of the methyl isobutyl ketone (Union Carbide Chemicals Co.) was more than 99.8% as measured by a mass spectrometer. The glacial acetic acid (Celanese Chemical Co.) was about 99.9% pure. Cal c ul at ion s a nd ResuIt s
T o facilitate the description of the calculations and results, each of the four combinations of the qualitative variables has been labeled as shown in Table I. T h e object of the experimentation was to determine HTU's corresponding to each ofthe four combinations of the two qualitative factors as functions of the solute-free flow rates, which were constant throughout the column. With the experimentation setup used it was convenient to measure flow rates by timing samples from the column outlets and analyzing these for acetic acid; therefore, the solute-free rates were not known in advance of a run, as previously explained. Thus, instead of using a n orthodox experimental design to determine the optimum flow rates (as measured by a minimum H T U ) for each of the above four combinations, a relatively large number of trials were made for each combination to define the response surface adequately. T h e fact that the levels of the independent variables were scattered a t unequal intervals over the response surface made the calculations more tedious, but the approach was more general as a consequence. This inability to set the flow rates a t known levels in advance also made it impractical to repeat identical runs to obtain a statistical estimate of the error variance. Previous investigations (30-32) have indicated that the controlling resistance to diffusion in this system is in the ketone
Table 1.
Direction of Extraction Ketone to water Water to ketone
Combinations of Factors Continuous Phase Water Ketone 1 3 2 4
phase; so the response variable chosen for this project was the efficiency of extraction as measured by the H T U based on the ketone phase-Le., HTU,,. This meant that increasing efficiency was indicated by decreasing HTU,,, and, further, that the optimum combination of factors would be indicated by a minimum H T U . (This latter value did not necessarily correspond to a n optimum economic HTU.) T h e basic steps in the calculations for each run were to estimate the number of transfer units involved in the extraction, and then to calculate the HTU,, by dividing this number into the height of the packing. T h e original data are given by Moorhead (26) and are available from the AD1 Auxiliary Publications Project. T h e ArtoK were calculated by the following relationship (32):
Assumptions made in deriving this equation were that the part of the equilibrium line relevant to the experimentation was straight and that the solvents were completely immiscible ; the first assumption was approximately true, and to fulfill the latter condition, the water and ketone were saturated with the other before being fed to the column. The H T U was the response variable for each run, but there remained a choice of combinations of the basic independent variables (the two solute-free flow rates) to be used to describe the system. T h e responses could have been mapped simply by plotting them as points on a graph of one flow rate against the other. However, previous work (20, 37) has shown the significance of the ratio of the flow rates, and since the degree of turbulence must have some relationship to extraction efficiency, the total flow rate was expected also to be significant. Therefore, the ratio of the solute-free flow rates of the discontinuous and continuous phases and the sum of these flow rates were selected as the independent variables in plotting the response surface. I n effect then, the equation to be fitted to the response surface was:
Y where y = HTU,,,
=
f(X1,
22)
+ U,
(2)
x1 = UD/Uc,and x z = U D
Polynomials of various degrees in these independent variables were fitted to the experimental points-that is, trial equations were fitted of the form: Y = bozo
'I
+
bixi
Y =
+ + + + bizxixz + bzz.4 + + +bzzd + + b 2 ~ 2
4- bixi b z X z 4- biix? bozo 4- bixl f 62x2 + b i d
= box0
biiix:
622x22
bi2XiXz
(3) (4)
(5)
bizzXiXi
biizX:Xz
Equations 3, 4, and 5 are first-, second-, and third-order models, respectively. Fitting of the second- and third-order models consisted essentially of solving the matrix equation : B = [X+X]-'X+Y.w
(6)
where B is the matrix of the constants in Equation 4 or 5. For the second-order model, B was a 6 X 1 matrix, [X+X]-' was a 6 X 6 matrix, and X + Y M was a 6 X 1 matrix; for the third-order model, these were 10 X 1, 10 X 10, and 10 X 1 matrices, respectively. The elements of these matrices and were obtained by using a n IBM 650 of the matrix [X+X] computer, and were subsequently used to calculate terms in the analysis of variance and also confidence limits. T o locate the point of minimum H T U , as a first step the direction of steepest descent to the minimum was ascertained by fitting to the response surface a first-order equation (Equation 3). T h e effects on the H T U of changing each of the two variables were ascertained from the coefficients in the firstVOL. 1
NO.
1
FEBRUARY 1962
69
Table II.
Equations Fitted to Response Surfaces and Analyses of Variance
COMBINATION 1. WATERCONTINUOUS, EXTRACTION FROM KETONE TO WATER No. of points fitted Fitted equation Second-order Y = 19.43
39 Source
Due to regression + 8.86~1- 0 . 1 4 5 -~ ~2.302~:+ 0.0002934+ 0.04777~1~2Deviations from
Analysis of Variance Degree of Sum o f freedom squares
5
regression Total
33 38
Third-order Y = -24.41 f 87.9421 0.330~2 - 54.26~;- 0.00137xf- 0 . 4 5 6 ~Due ~ ~to~ regression 4- 7.774 f 0.0000019.4 0.187~1~2 0 . 0 0 0 ~ 1 4 Deviations from Canonical form of second-order equation regression Y - 26.63 = 0,00054lXf- 2.302Xi Total Standard deviation of Y Second-order. 0.942 Third-order. 0.580
++
9
+
29
38
COMBINATION 2. WATERCONTINUOUS, EXTRACTION FROM ~ V A T E RTO KETONE No. of points fitted 27 Fitted equation Y = -2.08 f 10.49~1 0.113~2 - 0.988~; - 0.000304~~0.021~1~2 Due to regression 5 Deviations from regression 21 Canonical form Total
+
+
z
Mean square
1868.1 373.6 29.3 1897.4
0.888
1887.7 209.7 9.74 1897.4
634.3
0.336
764.3 152.9 9.23 773.5
F 420.7
347.5
0.440
Y - 91.90 = -0.00019lXf - 0,988Xz Standard deviation of Y 0,663 COMBINATIOK 3. KETONE CONTINUOUS, EXTRACTION FROM KETONE TO ~ Y A T E R No. of points fitted 54 Fitted equation Y = 34.94 - 0 . 4 2 4 ~-1 0.316~2 0.0383~: O.OOlOl$ - 0.00496~1~2Due to regression 5 Deviations from Canonical form regression 48 Y - 0.887 = 0.0384X: - 0.0008473 Total 53 Standard deviation of Y 1 ,442
415.0
COMBINATION 4. KETONE CONTINUOUS, EXTRACTION FROM t'v.4TER TO KETONE No. of points fitted 28 Fitted equation Y = 52,43- 2 , 2 9 9 ~-1 0.270~2 f 0.0146~:-I-0.000476~,? 4- O . O o 6 7 2 Due ~ ~ ~to~ regression 5 Deviations from Canonical form regression 22 Y -18.30 = 0.0154X: - 0.000280Xi Total 27 Standard deviation of Y 1.659
186.6
37.32
60.5 247.1
2.75
+
+
order equation, and in this way the region was indicated where experimentation should be continued. T h e next step was to collect more data in this indicated region, I t was more instructive to use second-order models for subsequent fits. Second-order equations for each combination of qualitative variables were fitted to the new data and usually also to some of the original data, if the latter were near the new region of interest on the response surface. From these second-order equations, the values and positions on the response surface of the minimum HTU's were calculated. The fitted equations adequately described the fitted regions. The minimum H T U calculated in this way for each combination was expected to have been in the range of the data, but in each case the position of the minimum H T U was indicated to be well outside the range of the data, indicating that it was a hypothetical point, so that it was not proper to conjecture about its value or its position. Further experimentation at some distance on the response surface from the region just fitted was assumed necessary to locate the minimum point. More data were collected in the direction of the minimum point for each combination. The indicated areas of experi70
l&EC FUNDAMENTALS
99.74 514.7
83.0
39.94
2.078
13.57
mentation by this time corresponded to very high ratios of the mass flow rate of water to ketone, and were near the flooding point of the column. Since the column operation \vas restricted to values of U,/LJ, and L', UCwhich could not cause flooding, a limit was imposed on the variable factors beyond which experimentation was impossible. Data were taken as close as practicable to this limit, and second-order equations fitted once more. Again, the new minimum points were indicated to be well outside the range of the data. I n effect. the column operating restrictions meant that the lowest H T U a t the extreme limit of the operating conditions represented the least H T U value it was physically possible to obtain. Thus the physical system had both upper and lower bounds for the range of each of the operating variables and the response variable. This range covered only a small region of the domain of the mathematical model, and this region was far away from the minimum H T U for the mathematical model. The inherent errors in measurement and in determining the flow rates increased as flooding was approached, and thus the error in the calculated HTU's increased.
+
Table 111.
Comparison of Combinations of Factors Using Fitted Equations
Qualitative
HTU,
Trial I
I1
I11
Combination 1 2 3 4 1 2 3 4 1 2 3 4
0.125 0.125
180 180
8.0
180 180 230 230 230 230 220 220 220 220
8.0 0.0833 0.0833 '12.0 '12.0 0.0625 0.0625 l6.0 'L6.0
Inches 4 . 8 0 zt 0 . 4 1 10.16 1 0 . 5 9 2 . 7 7 10 . 8 5 1 1 . 4 3 f 2.11 3 . 4 0 i0 . 6 2 9.08 & 0.77 2 . 5 3 3Z ,174 8 . 5 3 i2 . 8 3 3.11 f0 . 6 4 8.99 3Z0.68 -0.51 3Z 1 . 1 7 6 . 6 5 3Z 1 . 9 8
In view of the above: difficulties, to compare the four combinations of qualitative: factors suitably, it was decided to fit equations to the entire response surface for each combination and use these four more general equations to predict HTU's. T h e predicted HTU's? with corresponding confidence intervals, could then be compared to evaluate the most favorable combination, as indicared by the lowest H T U . T h e final equations employed are shown in Table 11. T h e second-order model provided the best fit for each combination, except for cornbination 1, in which the third-order model was slightly preferable. The highly significant F's in Table I1 showed that the regression of the H T U on the combination of the variables was beyond the range of rseasonable chance variation. Figure 1 shows equal value HTIJ contour lines, drawn for combination 2. The shaded areas represent confidence intervals on the HTU's and are intended to portray a dimension perpendicular to the paper. Finally, to compare leach of the combinations of qualitative factors, HTU's were calculated for various mass flow rates and ratios, using the equatimons shown in Table I1 (the third-order model was employed for combination 1). The mass flow rates and ratios chosen for each combination were near the flooding limits-that is, near the point of lowest attainable H T U . This comparison gave a measure of the most efficient extraction attainable for each of the four combinations of qualitative variables. Sample values of the H T C ' s with 957, confidence intervals, are shown in Table 111. I t is evident from Table I11 that combination 3 of the qualitative factors gave the most efficient extraction in the system. This combination consists of having ketone as the continuous phase and extracting from the ketone to the water. This conclusion confirms previous work ; best extraction was obtained when the continuous phase wet the packing, as the ketone did the polyethylene in this case. Moreover, the higher solubility of acetic acid in the water than in the ketone indicates the probability of better extraction in the direction from ketone to water. Although the best combination of the variables may be chosen with some confidence in this way (since the intervals were determined a t 9 5 ' 3 probability levels), it is apparent from Table I11 that for many combinations no conclusions can be reached in making comparisons between combinations. For example, in trial 1, it may be concluded with reasonable confidence that combination 3 will give the most efficient extraction, and combination 1 the next, but 3 and 4 cannot be ranked apart, because their values including the confidence intervals overlap. For the same reason: in trial 11, 1 and 3
Table IV.
Analyses of Variance for Each Variable in Fitted Equations Degrees of Sum of Mean F Source Freedom Squares Square COMBINATION 1 (3RD-ORDER MODEL) 5 . 84a 1.963 1.963 XI 1 0.41 0.139 x2 1 0.139 19.11a 6.419 6.419 x? 1 0.32 0.108 0.108 X2 1 1.73 1 0.580 0.580 46. 34a 1 15.566 15.566 0.25 1 0,084 0.084 9.414 1 3,160 3.160 0.72 x1xf 1 0,243 0.243 Deviations 29 9.74 0.336 ( F for 1/29 d.f. = 7 . 6 0 for 99% probability level) = 4 . 1 8 for 95% probability level
COMBINATION 2 1 4.144 4.144 1 0.596 0.596 1 398.88 398.88 1 0 680 0 680 1 0.561 0.561 Deviations 21 9.23 0.4395 (Ffor 1/21 d.f. = 8 . 0 2 for 99% probability level)
9.43" 1.36 907.6" 1.55 1.28
COMBINATION 3 1 3.613 3.613 1 17,801 17,801 1 40.97 40.97 1 16.63 16.63 1 7.832 7.832 Deviations 48 99.74 2.078 ( F f o r 1/48 d.f. = 7.22 for 99% probability level)
1.74 8.57O 19.72a 8 . OOa 3.77
COMBINATION 4 1 3.697 3,697 1 2.280 2.280 1 1.560 1.560 1 1.128 1.128 1 1,313 1.313 Deviations 22 60.50 2.750 ( F for 1/22 d.f. = 2 . 9 5 for 90 yo probability level) 5 Statistically significant at least at 95% level.
1.34 0.83 0.57 0.41 0.48
1.2
1.0
0.8 0
3 \
n 0.6 3
0.4
14
0.2
0.0
I
4
I 1 140
160
180
U, Figure 1,
+ U,
200
I
I
220
240
LB./HR.
HTU contour lines for combination 2 VOL. 1
NO. 1
FEBRUARY 1962
71
cannot be differentiated, nor can 2 and 4, although either of the former gives more efficient extraction than either of the latter. Similar reasoning can be applied to the other trials, and the general conclusion can be drawn that combination 3 gave the most efficient extraction, combination 1, in most cases, the next best, followed by combinations 2 and 4 together, Since the equations in Table I1 were accepted as the most suitable, an attempt was made to separate the effects of each variable on the calculated response. T h e method followed in doing this is available ( 8 ) . The detailed analyses of variance are given in Table IV. From the results shown in Table I V for combination 1 the xI--i.e., UD/Lr,-xf, x;, and ~ 2 x 2variables are statistically significant; for combination 2, XI and x:; for combination 3, x z (U, Uc), xf and x i , but for combination 4 the results are inconclusive. T h e F tests demonstrate the effect of these variables on the H T U , but no valid conclusions can be made as to whether the other variables do or do not affect the H T U . I t is difficult to draw general conclusions from the results i n Table I V ; it would appear that the ratio of mass flow rates has a pronounced effect on the extraction efficiency. Also, only for combination 3 was the sum of flow rates demonstrated to be significant. I t is important, however, that the range of variation of the total flow rates was relatively small (less than twofold). T h e lower limit was imposed by the size of the apparatus and the magnitude of the acceptable experimental error, and the upper limit by the flooding conditions, as discussed earlier. I n contrast, the relative variation in the ratio of flow rates was greater than 20-fold.
+
Nomenclature
b
= matrix of constants (bo, b,, , . . b,) in Equations 4 and 5 = estimated regression coefficient
F
=
B
distribution of ratio of two estimates of variance from two independent samples from the same normal universe H T U = height of transfer unit, inches M.S. = mean square (or variance) Nt = number of transfer units = ratio of molecular weight of nonsolute to molecular C weight of solute U = solute-free mass flow rate, pounds per hour = weight ratio of nonsolute to solute W = matrix of experimental values of independent variX ables = transpose of X matrix A+ X I , X I ?= variables in canonical equation-Le., principal axes = dummy variables = 1.0 xo = ratio of solute-free mass flow rates, UD/Uc x1 x2 = sum of solute-free mass flow rates, ( U D Uc), pounds per hour = calculated value of response variable Y YM = matrix of experimentally determined response (dependent) variable values
+
SUPERSCRIPT
*
= equilibrium concentration
SUBSCRIPTS 1 = end of column with concentrated solutions 2 = end of column with dilute solutions
72
l&EC FUNDAMENTALS
C = continuous phase = discontinuous phase K = ketone phase 0 = over-all basis
D
literature Cited
(1) (2) (3) (4) (5)
Appel, F. J., Elgin, J. C., IND.ENG.CREW29, 451 (1937). Ballard, J. H., Piret, E. L., Ibid., 42, 1088 (1950). Box, G. E. P., Biometrics 10, 16 (1954). Box, G. E. P., Biometrika 39, 49 (1952). Box, G. E. P., Hunter, J. S., Ibid., 41, 190 (1954). (6) Box, G. E. P., Wilson, K. B., J . Roy. Stat. Soc. B13, 1 (1951). (7) Burkhart, L. E., Fahien, R. W., ‘U. S. At. Energy Comm., AEC Doc. ISC-860 (June 1956). (8) Chew, V., ed., “Experimental Designs in Industry,” Wiley, New York, 1958. (9) Davies, 0. L., ed., “Design and Analysis of Industrial Experiments,” Oliver and Boyd, London, 1954. (10) Fenske, M. R., Tonberg, C. O., Quiggle, D., IND. ENG. CHEM.26, 1169 (1934). (11) Gaylor, R., Pratt, H. R. C., Tfans. Inst. Chem. Enzrs. 31, 78 (1953). (12) Jacques, G. L., Cotter, J. E., Vermeulen, T., U. S. At. Energy Comm. UCRL-8658 (1959). (13) Jacques, G. L., Vermeulen, T., Ibid., UCRL-8029 (1957). (14) Kafarov, V. V., Planovskaya, M. A,, Zhur. Priklad. Khim. 24, 624 (1951). (15) Karr, A. E., Schiebel, E. G., IND.ENG.CHEM.42, 1048 (1950). (16) Karr, A. E., Schiebel, E. G., Chem. Eng. Progr. Symp. Series No. 10, 50, 73 (1953). (17) Leva, M., “Tower Packings and Packed Tower Design,” United States Stoneware Co., Akron, Ohio, 1953. (18) Leva, M., Wen, C . Y , Chem. Eng. 64, 267 (.4pril 1957). (19) Lewis, J. B., Chem. Eng. Sci. 3, 248 (1954). (20) Lewis, J. B., Pratt, H. R. C., Nature 171, 1155 (1953). (21) Liebson, I., Beckmann, R. B., Chem. Eng. Progr. 49, 405 (19 53). (22) Ibid., p. 1885. (23) Livingston, N. J. (to Standard Oil Co. of Indiana), U. S. Patent 2,215,359 (Sept. 17, 1950). (24) Markas, S. E., Beckmann, R. B., A . I Ch.E. Journal 3, 223 (19 57). (25) Meissner, H. P., Stokes, C. A., Hunter, C. M., Morrow, G. M., IND.ENG.CHEM.36, 917 (1944). (26) Moorhead, D. H., M.S. thesis, University of Texas, Austin, Tex., 1961. (27) Osman, F. O., M.S. thesis, University of Texas, Austin, Tex., 1960. (28) Othmer, D. F., White, R. E., Treuger, E., IND.ENG.CHEW 33, 1240 (1941). (29) Sherwood, T. K., Evans, J. E., Longcor, J. V. A., Ibid., 31, 1144 .~ (1939). (30) Sobotik, R., M.S. thesis, University of Texas, Austin, Tex., 1959. (31) Treybal, R. E., IND.ENG.CHEM.47, 538 (1955). (32) Treybal, R. E., “Liquid Extraction,” McGraw-Hill, New York, 1951. (33) Treybal, R. E., “Mass Transfer Operations,” McGraw-Hill, New York, 1955. (34) Wicks, C. E., Beckmann, R. B., A.I.Ch.E. Journal 1, 425 (1955). RECEIVED for review March 27, 1961 ACCEPTED September 11, 1961 Division of Industrial and Engineering Chemistry, 140th Meeting, ACS, Chicago, Ill., September 1961, Material supplementary to this article has been deposited as Document 6964 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $1.25 for photoprints or $1.25 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress. I~
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,
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