I
B. W. MAR' and A. L. BABB Department of Chemical Engineering, University of Washington, Seattle, Wash.
Longitudinal Mixing in a
...
Pulsed Sieve-Plate Extraction Column An empirical correlation has been deve loped for the prediction of continuous phase Peclet numbers for pulsec sieve-plate extraction columns
THE
distribution of flow longitudinally in pipes, reactors, and extraction columns must be determined, to obtain a theoretical understanding of the mass transfer or reaction mechanisms. T h e longitudinal concentration gradients in a continuous countercurrent exraction can be expressed mathematically in terms of four dimensionless groups: Peclet number. number of transfer units, concentration ratio, and length ratio. In this investigation, longitudinal mixing was studied experimentally by both steady-state and delta injection techniques. hlodified Peclet numbers were evaluated and correlated in terms of system properties, flow conditions, column geometry, and pulsing conditions.
Literature Background Eddy diffusivities from 8 t o 160 sq. meters/second in 4- to 20-inch diameter columns ( 1 , 12) Little back-mixing in packed columns, greatest at liquid inlets (11, 19) Axial and radial mixing in packed columns (10, 16) Back-mixing in pulsed sieveplate columns ( 3 , 18, 27) Mathematical models of back( 2 , 22, @) mixing
Experimental This investigation used a steady-state technique primarily (72), although delta injection (ZU) was used to check some data points. The frequency-response method (78, 27, 32) was not used because of mathematical difficulties in interpreting the effect of the pulsing and the sinusoidally varying tracer concentration. Difficulties in obtaining a uniform, instantaneous change in tracer concentration a t the inlet prevented use of the step input method ( 8 ) . Apparatus. The column was 2 inches in diameter and 5 feet long. The calming sections consisted of 3-inch borosilicate glass tees with a 6-inch section of borosilicate glass pipe installed Present address, Boeing Airplane Co., Seattle, Wash.
LO
.5
$
GENERbTDR
The steady-state technique was studied in a borosilicate glass column
between the column and the upper tee for extra capacity. Stainless steel flanges permitted installation of pipe fittings. Conical flange joints joined the end plates and tees. Short lengths of medium-walled glass tubing, 2l ' 4 inches in outside diameter, were used for the column proper. Polyethylene washers (Z3/g inches in outside diameter, 2 inches in inside diameter, 1, inch thick) were inserted between the glass segments as gaskets. Stainless steel plates were sandwiched between polyethylene gaskets, 2l/2-inch squares of 8-inch polyethylene sheet with a 2-inch hole in the center. No. 19 syringe needles were inserted through the gaskets to permit internal sampling. Sample flow through the needles was controlled by three-way syringe stopcocks. A triangular hole spacing with a 2370 free area was selected. Plate thickness and hole size were the only plate geometries investigated. A 1-hp., three-phase 220/440-volt alternating current motor was used to drive a Master Speed Ranger transmission. The output shaft speed could be varied from 20 to 190 cycles per
I
.01
1
1
I
I
/
/
1
1
/
1
1
1
1
1
I
/
1
1
1
Inch08 obove h e d plat0
Figure 1 . Data obtained by the steadystate injection technique show that insertion of three additional plates with smaller hole diameters increases UJE by a factor of 4 and hence reduces the effect of back-mixing
minute. A variable cam which expanded and compressed a Teflon bellows was fastened to the output shaft. .4 driving rod connected the cam to the bellows; its stroke could be varied from 0 to 1 inch, resulting in an amplitude range of 0 to 5 inches in the column. Single-stage stainless steel centrifugal pumps, equipped with mechanical rotary seals, were used in the outlet and tracer feed lines. The lines were polyethylene and flared or Swagelok fittings were used to make pipe or pump connections. Polyethylene is susceptible to attack by some of the solvents studied ( 4 ) ,but the only deterioration observed was a slight swelling. Stainless steel check vahes were installed in each line just before an entrance or exit from the column to prevent back-mixing of column contents into the lines or vice versa. Stainless steel bar stock valves with panel mounts controlled flow. VOL. 51, NO. 9
SEPTEMBER 1959
101 1
~~
Table I. Summary of Analysis of Variance for Eighth Replicates of 2’ Designs Carbon Tetrachloride-Water Hexane-Water BenEene-Rater
Level Variable
-
+
f, m m - 1 30 60 a,in. 0.5 1.0 L‘c, ml./min. 200 400 f, in. 0.032 0.125 I , in. 3 6 d , In. 0.0625 0.125 L-c/t70 0.4 0.8 Gd, ml./min.
...
*..
1440 487 584 336 697 436
4.6
87
-
1 1 1
5 1
1
... .........
.4 Fisher-Serfass electronic relay was employed as a liquid level controller similar to that described by Rubin and Lehman (26). Orifice meters were specially designed so that the orifice plates could be easily replaced. Each orifice was calibrated and checked daily for accuracy. During a run of 45 minutes, the flow rate fluctuated a maximum of 10%. Analytical Technique. Ferric nirrate was used as the tracer and the amount of ferric ion present in samples was determined colorimetrically, using a 0.75M potassium thiocyanate solution diluted with an equal volume of ferric nitrate. The thiocyanate was made acidic with nitric acid to ensure a stable colored complex. The color intensity of the sample solutions was determined with a Klett-Summerson colorimeter. The sample to be analyzed was diluted with 5%, nitric acid to a final concentration of 1 10 15 p.p.m. of iron. T h e colorimeter was calibrated with standard iron solutions; no deviations from Beer’s law were observed. Procedure. The steady-state technique requires a dual-purpose column which has a section above the feed plate similar to a conventional scrub section and a section below the feed plate similar to a conventional extraction section. By injecting an aqueous feed stream containing the ferric nitrate tracer which is not extracted by the organic phase, rhe eddy diffusivity was determined by observing the longitudinal tracer distribution above the feed plate a t steady state. The order of sampling had little effect on the concentration profile. A steady state was reached in 25 to SO minutes. Some measurements were taken using the delta injection technique. This required a single column section without a feed plate in the central section. The aqueous phase flowed countercurrently to the organic phase, but the tracer was introduced as a single pulse into the aqueous phase at a distance X from the aqueous inlet, Samples of the aqueous phase were obtained with a syringe X)from the aqueous a t a distance ( x inlet at various times after tracer injection. The data were plotted as observed
Level
+
45 0.5 300 0.032 3 0.0625
...
300
60 1.0
600 0.125 6 0.125
...
420
Level
Mean level, %
-
+
30 0.5 350 0.032 3 0.0625
60 1.0 710 0.125 6 0.125
Effect square
7.0 196 6.0 150 -9.9 390 3.0 30 14.0 1274 2.4 20
1 1
1 >5 1 >5
......... 0.1
......
+
concentration at (n X ) us. elapsed time since tracer injection. I t is assumed that the amount of tracer injected is negligible compared to the volume of iquid in the column. A factorial design (23) was used to determine experimental combinations to be investigated. The variables investigated were continuous and dispersed phase flow rates. pulse amplitude and frequency, hole diameter, plate thickness, and plate spacing. A series of liquidscannot befound, the physical properties of which vary in a manner to satisfy a factorial design. To circumvent this, three organic liquids were chosen with desirable physical properties and a fractional factorial design was repeated for each liquid. T h e technical grade organic solvents were obtained from the Phillips Petroleum Co. Calculation of E d d y Diffusivity. Mathematically the steady-state method can be expressed by: J , = - E dC/dx + C’C (1) .issuming that fluid velocity is uniform across the column and eddy diffusivity is constant, integration for steady-state conditions yields: In CIC, = U / E x B (2)
+
Thus, a plot of ln(C/C,) us. distance above the injection point will give a line with a slope equal to U / E (Figure 1). Levenspiel and Smith (20) showed that the continuity equation for the delta injection technique is given by :
... 300 dCjdo
~~
...
Effect
14.0 0.4
...
-12.0
558
1
770 >5
1
...... 14.0 5.4
. I .
500
=
Mean level, %
square
>5 1
1275 100 .
I
.
5
1.0
>5 ,
.
.
>5
v . ( E v C ) = E d2C/dx2 (3’1
which describes the mixing of the injected tracer with the surrounding fluid. Carslaw (5) solves this equation for the injection of a tracer into a stagnant fluid; it can be modified for a flowing fluid to give: (1
-$)2
Levenspiel and Smith showed that a plot of CV/Q us. VOlV yielded a family of curves with E/Ux as the parameter. An increasing deviation from a Gaussian error curve is noted for increasing values of Ux/E, a modified Peclet number. The variance from a Gaussian curve can be obtained from the experimental data
where G is the concentration of the tracer at x = L a t time 0. Where E is independent of position and the vessel is an infinite pipe, Levenspiel and Smiths’ development is applicable. They showed that the variance of the equation, g*, is related to the Peclet number:
1 ( d m- 1)
E = UL 8
(6)
Van der Lann (30) solved the same differential equation for eight different boundary conditions.
Data obtained by the delta injection t e c h n i q u e show the time variation of the tracer conc e n t r a t i o n 40 inches below the injection point (run
116)
+
1 01 2
nfieance
Effect rquare yo
18.9 11.0 -12.1 - 9.9 13.2 10.4
Significance
S1p-
S1g-
nficance Mean level,
INDUSTRIAL AND ENGINEERING CHEMISTRY
0
nu
IO
IS 20 since tracor injection
25
. minutes
M
35
E X T R A C T I O N COLUMNS Results Development of Correlating Equation. T h e results of a variance analysis of fractional factorial designs were used to select the variables for a correlation (23). The designs were obtained from Davies ( 9 ) and the variance analyses were made by Yate's method ( 9 ) . T h e main effects of each fractional design are summarized in Table I. Eight effects are confused with each apparent effect. This must be kept in mind when interpreting the data. Pai (25) and Townsend (29) have surveyed the advances in the understanding of turbulence. Hughes (75) confirms the fact that systems such as pulse columns, where bubbles or drops are continually being reformed, are extremely complex mathematically. Klinkenberg and Mooy (77) have discussed the use of dimensional analysis where the governing differential equations can be formulated, but integration is impossible. In this work, dimensional analysis, logarithmic transformation, and multiple regression were used to obtain a preliminary correlation using Ap, f i d , and t as the repeating variables in each dimensionless group. This permitted each remaining variable to appear in only one group. Ap and w d had little effect. Omitting these two and recorrelating, the following correlating equation was obtained: $ i= 0.17
(&)
t
1.46
k
9
0 70
1
( 2 ) . (i)
0t
0.68
(yyyy)'(*) Q 42
fp t 2
Q.36
(;)
0.07
(7)
This equation is limited to systems with water as the continuous phase. A simplified form is:
K E =
1 P . 6 8 u d 0 . 3QfO. 36,Q.Q7dQ. 3 O y O . 41
(8)
u0Q.46t0.06
Figure 2 permits rapid evaluation of E, U,/E. The predicted eddy diffusivities had an average deviation of 17% from experimental values. Comparing the trends of the exponents of the simplified Equation 3 with the level effects in Table I, it is seen that each is of the proper sign and magnitude. P e c k Number. A modified Peclet number for mass transfer can be defined as the product of a Reynolds number and a Schmidt number. Jacques and Vermeulen (76) obtained Peclet numbers in terms of interstitial mean velocity. U/c, and particle diameter, d,-i.e.. UdJtE-which were not constant at 2.0 as predicted by McHenry and Wilhelm (27). Ebach and White (70) reported their Peclet numbers on the same basis but a t much lower Reynolds numbers than Jacques and Vermeulen. T h e longitudinal mixing results are correlated in terms of (Uc/E)-', which can be considered a characteristic distance indicative of the degree of backmixing. In fact, 0.693 (UC/E)-' is the distance above a given point in the 1 )over which the concentration
? g I
of back-mixed tracer decreases by a factor of 2. Modified Peclet numbers based on hole diameter d and the mean velocity through the plate, U,d/tE, varied from 0.02 to 0.4, corresponding to Reynolds numbers, DUpc/ep of 25 to 150. The Peclet number and C,/E values found are lower than those reported for packed column (70, 76), which indicates that the degree of back-mixing is greater in pulsed columns, as expected.
Discussion Observation of the fluid behavior in a pulse column indicates that during a single upstroke, jet formation, drop formation, and wake formation occur. Usually the dispersed phase does not have time to coalesce completely, and the drops are forced through the platesome retain their identity and others break up to form new drops. Some of the continuous phase is also forced upward. If there is sufficient dispersed phase present to occupy the holes on the upstroke continually, the continuous phase cannot be forced backward. Plate Geometry. Experimental results indicate that the Peclet number decreases (increased back-mixing) with increasing hole diameter. On the other hand. plate thickness has a slight tendency to increase the Peclet number: this was interpreted as an added resistance to longitudinal edd>-development.
8 ' 0 9
1
i
t Figure 2. Nomograph for calcularing aqueous-phase longitudinal eddy-diffusivities in a 2-inch-diameter pulsed sieve-plate extraction column VOL. 51, NO. 9
SEPTEMBER 1959
1013
Plate Spacing. Decreasing plate spacing appears to decrease back-mixing. Consequently, a configuration of small holes: small plate spacing, and intense power input would cause each plate to serve as a mixer-settler stage with complete mixing on each plate. Pulse Frequency and Amplitude. The pulse frequency determines the velocity of the fluid moving through the holes, and the amplitude determines the distance the fluid travels per stroke. If back-mixing is also caused by tracer carried on a film surrounding the organic bubble, a n increase in drop area caused by increased agitation or pulse input will cause greater axial mixing. An increase in the continuous phase velocity decreased back-mixing. Physical Properties of Liquids. In their review of jet breakup in denser liquids, Christiansen and Hixson (6) reported that viscosity and density have a small effect; interfacial tension contributes the greatest effect, as it controls drop size. The property controlling the bubble size can be very important. Accuracy of Results. Eddy diffusivity values were reproducible within 10%. Because of lack of automatic controls, the largest error was caused by variation of the flow rates; fluctuations as great as 10% were observed. The errors from all other sources were estimated to be under 5%. Seven runs, duplicating the original conditions, were made by the delta injection technique. The error between values is approximately lo%, but definite conclusions cannot be obtained without thorough investigation of both methods. The tracer was not extracted by the organic phase, nor did its concentration affect the results.
2-inch column. In all cases the U,/E values were almost a factor of 2 lower than those obtained under identical conditions using stainless steel plates. This indicates that the plastic plates increase back-mixing for a given column geometry and pulsing conditions. To explore the effects of the continuous phase properties, several experiments were made using an aqueous sucrose solution and hexanr. A 20 weight % aqueous solution of sucrose had a density 10% greater than water, a viscosity 100% greater than water (74):and an interfacial tension identical to Ivater ( 3 7 ) . The Lrc;’Evalues were 25y0lower for the sucrose solutions than for pure water. When the sucrose solution was used, the radial tracer concentration was not uniform. This indicates that the velocity is not uniform across the column, as would be the case for fully developed turbulence. Tichacek, Barkelew, and Baron (28) have shown for liquids flowing in pipes that axial mixing increases rapidly as the velocity profile approaches that of a laminar flow and becomes a function of the velocity profile. Back-mixing can be minimized by inserting above the feed plate a number of plates which are closely spaced and have small holes. Figure 1 sho\vs that insertion of three plates with smaller hole diameters and with a smaller plate spacing increased C‘,.’E by a factor of 4. The effect of such modifications can be ascertained from Figure 2. Acknowledgment
The authors thank P. L. Huffman fot assistance with the experimental work.
Prediction of Longitudinal Concentration Gradients. L’alues of N t have
Nomenclature
been primarily obtained from a log-mean estimation of internal concentration gradients based on terminal conditions (7). The concentration of solute (in this case tracer) is large just inside the column inlet of pure liquid, because of backmixing ( 7 7, 76). Thus, it is not correct to assume that the solute concentration in the entering liquid is zero. when backmixing is appreciable. Recently, mathematical analyses (2, 22, 24) of the problem of simultaneous intra- and interphase mass transfer have been developed to predict internal concentration gradients from a knowledge of the actual number of transfer units, extraction factor. degree of separation, and Peclet numbers for the two phascs. If these relationships can be verified experimentally. it may be possible to obtain a relationship between apparent values based on terminal concentrations and actual A’, values evaluated from internal concentration gradients. Exploratory Investigations. In exploratory runs four polyethylene plates were inserted above the feed plate in the
B = constant of integration
.\;
1 014
u
= pulse amplitude, ft.
C = concentration = tracer concentration a t x = L D = diameter of column? ft. d = diameter of hole in plate, ft. E = longitudinal eddy diffusivitv, sq. ft./hr. f = pulse frequency, hr.-I J = quantity of tracer transferred per unit area per unit time K = constant L = length of column or test section, ft. I = plate spacing, ft. Art = number of transfer units Q = volume of tracer injected, cu. ft. t = plate thickness, ft. C = superficial linear velocity, fr./hr. 1’ = volume of test section or column! cu. ft. u = volume velocity, cu. ft.,’hr. X = distance from aqueous inlet, ft. .Y = distance from injection point, ft. y = interfacial surface tension, Ib.,’sq. hr. e = packing void fraction or plate free space B = time, hr. p = viscosity. lb. (ft.)(hr.) p = density, lb. cu. ft. u = variance c
INDUSTRIAL AND ENGINEERING CHEMISTRY
SUBSCRIPTS = initial conditions = continuous phase d = dispersed phase Y = position x
o c
Literature Cited 1 I ) .\skins, J. W., Chem. Eng. Progr. 47, 401 (1951). 821 Babb, A . L., Hanford Works Notebook, HWN-1301 (1955). , > I Beyer, G. H., Edwards, R. B., U. S. Atomic Energy Comm., Rept. ISC-553 (1954). (,41 Bockhoff, F. J., Roth, R. F., Chem. Eng. Progr. 51, 251 (1955). ( 5 ) Carslaw, H. S., “Introduction to .Mathematical Theory of Conduction of Heat in Solids,” 2nd ed., p. 153, Dover Publications, New York, 1945. (6) Christiansen, R. M., Hixson, A. N., IND. ENG.CHEM.49, 1017 (1957). 17) Colburn, A. P., Zbid., 33, 459 (1941). ( 8 ) Danckwerts, P. V., Chem. Eng. Sci. 2, 1 (1953). (,9) Davies, 0. L., “Design and Analysis of Industrial Experiments,” Hafner Publication Co., New York, 1954. 110) Ebach, E. A., White, R. R., ,4~Z,Ch.E‘. Journal 4. 161 11958). (11) Gier, ’T. k.,Hougen, J. o., IhD. ENG.CHEM.45, 1362 (1953). (12) GiIliland, E. R., Mason, E. A , Ibid., 41, 1191 (1949). (13) Gnffith, W. L., Joeny, G. R., Tupper, H. T., Mass. Inst. Technology, Enginet-riner Practice School. Oak Ridge. Tenn.. RGpt. KT-114 (1952). 114) Hodgman, C. D.: “Handbook of Chemistry and Physics,” Chemical Rubber Pub. Co., Cleveland, Ohio, 1949. (15) Hughes, R. R., IND. ENG. CHEM. 49, 947 (1957). (16) Jacques, G. L., Vermeulen, T., U. S. Atomic Energy Comm., Rept. UCRL8029 (1957). (17) Klinkenberg, ,4., Mooy, H. H., Chem. Eng. Progr. 44, 17 (1948). (18) Kramcrs, H., Alberda, G., Chrrn. Ene. Sci.2. 173 11953). (19) lapidus, L.‘, Amundson; N. K., J . Phys. Chem. 56, 984 (19521. (20) Levenspiel, D., Smith, W. K., Chem. Eng. Sci.6,227 (1957). (21) McHenry, K. W., Wilhelm, R . H., A.I.Ch.E. Journal 3, 83 (1957). 122) McMullen. A,. Mivauchi. l.,Vermeulen. T.. U’. S. Atomic Enerw C’omm., Rept. UCR’L-3911 (Suppl.) (1958). (23) Mar, B. W., Ph.D. thesis, University of Washington, 1958. (24) Miyauchi, T., U. S. Atomic Energy Comm. Rept. UCRL-3911 (1953). (25) Pai, S. I., “Viscous Flow Theory,” 11, “Turbulent Flow,” Van Nostrand, Princeton, N. J., 1957. (26) Rubin, B., Lehman, H. R., U. S. Atomic Energy Comm., Rept. UCRL718 (1950). (27) Swift, W. H., Burger, L. L., Ibid.: HW-29010 (1954). (28) Tichacek, L. J., Barkelew, C. H., Baron, T., A.I.CI2.E. Journaf‘3,439 (1957). (29) Townsend, A. A., Structure of Turbulent Shear Flow,” Cambridge University Press, Cambridge, 1956. (30) Van der Laan, E., Chem. Eng. Sei. 7 , 187, (1958). (31 I Vavruch, I., Listj, Cukrocar 65, 81 (1948). (321 Winsche, W.E., Rosen, J. B., J . C h m . Phys. 18, 1587 (1950). RECEIVED for review September 15, 1958 ACCEPTED April 17, 1959 Based on work performed for the U. S. Atomic Energy Commission, Contract KO. AT(45-1)-1053, at Cniversity of Washinqton.
