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Ind. Eng. Chem. Fundam. 1982,21, 306-311
Axial Dispersion Coefficients of the Continuous Phase in Liquid-Liquid Spray Towers C. J. Geankoplls," J. B. Sapp,' F. C. Arnold, and 0. Marroquin2 Department of Chemical Engineering, The Ohio State Universify, Columbus, Ohio 43210
Longitudinal dispersion coefficients of the continuous phase were experimentally obtained in spray type liquid-liquid extraction columns. Data for two diameters, different tower lengths, and varled flow rates for the continuous and dispersed phase were obtained. The system used was water as the continuous phase and methyl isobutyl ketone as the dispersed phase: the method used was unsteady-state measurements of a KCI solution as the tracer. Increases in the continuous phase velocity greatly increased the axial mixing coefficient D and increases in the dispersed phase velocity decreased the D,. Small dispersion coefficients were found for small tower lengths and these coefflcients increased as the tower length increased. Also, at long lengths, where the end effects become negligible, D, was independent of the length. A decrease in tower diameter from 35.8 mm to 27 mm caused a decrease in D, of approximately 20% for a range of continuous phase velocities. A comparison of Peclet and Reynolds numbers for spray towers with those for packed beds gave comparable values, indicating that a spray tower could be considered as a diluted packed bed tower with a very high void fraction for the continuous phase.
Introduction Because of their simplicity, low cost, and versatility, unpacked spray columns are still extensively used in extraction processes. They are also a convenient and inexpensive way to experimentally test theoretical models of mass transfer in simple systems. The hydrodynamics of a system represents one of the main difficulties in the scale-up of extractors. For design, difficulties arise mainly because of dispersion in the radial and axial direction; however, in most of the cases, the radial dispersion has less influence. Morello and Poffenberger (1950), Geankoplis and Hixson (1950), and Gier and Hougen (1953) were among the first in noting the influence of axial dispersion in extraction processes. Kreager and Geankoplis (1953) and Vogt and Geankoplis (1954) studied correlations between overall height of transfer unit (HTU) and tower height, observing that the HTU levels off when the tower height is large and end effects become negligible. Sleicher (1959) predicted the effect of backmixing in both phases in an extraction column using an idealized diffusion model. Brutvan (1958) measured the axial dispersion coefficients for the continuous phase (DL) in a spray tower using water as the continuous phase and glass beads as the dispersed phase. He found that an increase in the dispersed phase flow rate corresponded to a slight increase in the dispersion coefficient. However, Brutvan did not measure DL over the entire tower but only in a center section. Hazlebeck and Geankoplis (1963) determined axial dispersion coefficients in a spray liquid-liquid extraction tower with water as the continuous phase and methyl isobutyl ketone (MIK) as the dispersed phase. They found that DL varies directly as the velocity of the continuous phase to the 0.45 power. No correlation was found between the dispersion coefficient and the dispersed phase velocity. Letan and Kehat (1969) studied the residence time distribution (RTD) of the dispersed phase in a spray type extraction column. They found that for low flow rates of the dispersed phase below the onset of coalescence the flow of drops was near plug flow, but for high flow rates above Union Carbide Corp., Charleston, WV. Chemical Engineering Department, National Polytechnic Institute, Mexico.
the onset of coalescence the RTD increased its variance sharply. However, for very high flow rates, channels of coalesenced dispersed phase were formed decreasing the variance. Vermeulen et al. (1966) analyzed available experimental data and arrived at an empirical expression to estimate the axial dispersion coefficient of the continuous phase as a function of the dispersed flow rate and diameter of tower. Henton and Cavers (1970) found that DL is strongly influenced by the dispersed phase fluid velocity and does not depend upon the continuous phase velocity. They also found a direct correlation between DL and drop size. Henton et al. (1973) extended the mixing-cell packed bed analogy to liquid-liquid spray towers. Erving and Chen (1976) did experimental work in a spray column using a water and MIK system and KC1 as tracer, in one case with saturated water, and in the other case with unsaturated water. They found that less axial mixing occurs with mass transfer than without. They also found that DL is a significant function of the continuous phase velocity and only slightly dependent on the dispersed phase velocity. The data presented in many cases are not directly comparable since the techniques and systems employed and range of parameters used have been different. For example, some authors have used glass beads as the dispersed phase, and others used large distributions of drop sizes, different holdups, exclusion of end effects, etc. The work presented here systematically studied the effect of the following variables in the process: flow rates, holdup, length of contact, diameter of tower, and regions near the nozzles. To study the effect of these parameters glass columns of different diameters and lengths were used with water as the continuous phase and MIK as the dispersed phase. A solution of KC1 was used as a tracer in a descending step function input. Measurements of concentration at various times were taken using a conductivity probe at the end of the test section to obtain the breakthrough curve of the system and the superficial values of the axial dispersion coefficient DL. Theory and Models In order to measure axial mixing, a model has to be used. The most familar model is the diffusion model where the longitudinal mixing is characterized as a random-walk
0196-4313/82/1021-0306$01.25/00 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
diffusion process. Other models are summarized by Wen and Fan (1975), although they do not seem to have more advantages than the first one. For this work the diffusional model was used.
TW
TT
307
TK
This model assumes uniform radial concentration in the continuous phase and that the axial dispersion coefficient DL characterizes the mixing process. The solution is as follows C
- = 1 - (1/2) erfc CO SK
where z = x - Ut. The initial and boundary conditions are c(z,O) = Co (3) (4) lim c(z,t) = co 2--
c(0,t) = 0 (t 1 0) (5) The differentiation of c/co with respect to dimensionless time 8 (V&/hV) when dimensionless time is 1.0 at the end of the contact length, was suggested by Danckwerta (1953) as a simple and convenient way to calculate dispersion coefficients
This method uses the model for an “open-open vessel” system. Other methods to calculate the dispersion coefficient are available. Levenspiel and Smith (1957) as well as Van der Laan (1958), derived expressions to calculate the variance (a2) for the model using different boundary conditions. The variance for the response to a descending step function can be calculated as
t) [ d8 -
u2 = 2 10 - (8
2
de]
(7)
From the variance and depending if the system is “open-open vessel”, “open-closed vessel”, or “closed-closed vessel”, one obtains different values for dispersion coefficients a2
=
a2 =
1 [2N 6, + 81(open-open)
(8)
1 [2N$, + 3](open-closed)
(9)
Nk,2
Nke2 = -[Nke - 1
Nb,2
+
e-”pe]
(closed-closed)
(10)
Expermental Methods Apparatus. The process flow diagram for the spray tower is shown in Figure 1. Columns having different lengths were used. Also two diameters were used of 27 mm and 35.8 mm; the volume between the two nozzles was the contacting region or test section and the length was the distance between the tips of the dispersed phase nozzle and the interface at the water nozzle tips. The dispersed phase nozzle was constructed of Pyrex glass with ten tips each of 2.5 mm inside diameter (i.d.). To ensure the same bubble size regardless of the flow rate, some tips were plugged with Teflon stoppers to give the same velocity in the tips of 0.0822 m/s. A perforated plate inside the nozzle assured a uniform flow through the nozzle tips.
RK
Figure 1. Process flow diagram: CW, CK, CT, storage tanks for water, ketone, and tracer; HW, HK, HT, constant head tanks for water, ketone and tracer; TW, TK, TT, Teflon stopcocks for water, ketone, and tracer; OW, OK, OT, overflow carboys for water, ketone and tracer; RW, RW’, coarse and fine adjustment stopcocks; RK, microregulating valve for ketone flow; FW, FK, FT, rotameters for water, ketone, and tracer flow; SW, SK, Teflon stopcocks to shut off the water and ketone; I, Teflon stopcock to generate the step function; B, microswitch linking stopcock I and the recorder; G, recorder; L, adjustable loop to regulate level of interface; M, measuring device to measure flow r a w P, Florence flasks; D, waste barrel; E, stopcock to empty the column.
A capillary tube of 0.8 mm i.d. was the tracer injection mechanism and enters in the continuous phase nozzle just before the nozzle tips. This spray nozzle had four tips of 3.96 mm i.d. To ensure that tracer and water were thoroughly mixed and provided even distribution to the nozzle tips, a venturi constriction of 1.98 mm was built in the main tube of the nozzle after the tracer injection and before the four nozzle tips. The electrical conductivity probe had two conducting plates mounted on a phenolic spacer and arranged so that the fluid flowed past the plates. The probe was calibrated using KC1 standard solutions saturated with ketone at different temperatures (Sapp, 1965; Arnold, 1966). The dispersed phase consisted of reagent grade MIK saturated with distilled water and the continuous phase consisted of distilled water saturated with MIK. The tracer was a 0.1 N KC1 solution. A colored solution was used to observe the jetting effect in the continuous phase nozzle and was a 0.05 N KMn04 solution. Run Procedure. Ketone and water were allowed to enter the column and the rotameters were adjusted to give the appropriate flow rates. As the column filled, the interface control loop was used to set the interface of the water and ketone at the end of the water nozzle tips. The tracer flow was started after the column operation was at steady state. When the conductivity probe reading was constant and the interface remained constant, the column was assumed to be at a steady-state operation. At this steady condition, the tracer flow was suddenly stopped and at the same time a signal was sent to the recorder which put a zero blip time on the tracer recording. Flow rates of both phases were taken twice throughout the run and the rates checked within f l %. In order to observe and photograph the jets in the water nozzle, KMnO, dye solution was injected into the nozzle
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Table I . Experimental Data
___
-___ [J,,
mm/s 4.42 7.86 2 63 0.78 7.95 4.48 4.44 7.85 2.64 0.81 7.84 4.42 2.69 3.30 4.49 7.94 10.84 0.87 4.47 4.33 4.12 7.06 2.62 0.74 7.49 4.09 2.57 9.52 12.19 0.19 3.25 2.67 1.68 0.70 0.76 4.04 2.63 4.01 2.63 3.72 2.42 3.66 2.44 2.64 2.59 2.15
Uk, mm/s 88.9 94.8 84.3 80.0 85.4 95.1 88.5 91.3 100.3 89.8 96.5 87.5 88.7 84.6 90.8 82.2 95.6 86.8 73.7 129.6 91.2 92.6 89.9 84.9 94.5 88.6 88.4 96.6 88.6 86.7 95.6 100.9 93.8 102.7 463.1 102.6 102.8 98.4 97.4 120.8 139.5 136.1 149.5 108.6 103.4 129.9
D, x 104, EW
L, m
dt, "
mz/s
NPe
0.9513 0.9535 0.9485 0.9474 0.9485 0.9535 0.9513 0.9535 0.9561 0.9511 0.9546 0.9506 0.9506 0.9475 0.9509 0.9465 0.9519 0.9493 0.9617 0.9909 0.9520 0.9535 0.9523 0.9489 0.9534 0.9513 0.9502 0.9567 0.9524 0.9507 0.9559 0.9569 0.9540 0.9556 0.9982 0.9585 0.9590 0.9585 0.9625 0.9701 0.9722 0.9701 0.9722 0.9722 0.9722 0.9722
0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 0.932 1.747 1.747 1.747 1.747 0.337 0.337 0.337 0.337 0.025 0.025 0.020
35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 27.0 27.0 27.0 27.0 27.0 27.0 27.0 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8
5.95 6.94 5.81 3.34 7.08 6.63 6.53 6.59 6.00 3.46 6.68 6.44 5.93 4.77 5.52 7.26 5.80 3.46 5.73 4.80 5.46 6.14 5.88 3.48 6.32 6.25 5.48 6.32 5.95 0.88 6.39 5.84 5.07 2.95 2.52 8.02 6.42 7.66 6.29 2.61 2.52 3.27 1.98 0.86 1.13 0.52
0.0253 0.0385 0.0154 0.0080 0.0382 0.0230 0.0231 0.0405 0.0150 0.0080 0.0399 0.0233 0.0154 0.0235 0.0277 0.0372 0.0636 0.0086 0.0265 0.0307 0.0256 0.0391 0.0155 0.0072 0.0403 0.0222 0.159 0.0512 0.0696 0.0074 0.0173 0.0155 0.0113 0.0081 0.0103 0.0171 0.0139 0.0178 0.0142 0.0484 0.0326 0.0380 0.0419 0.1040 0.0779 0.1400
using the KC1 tracer line. Pictures and visual observations were taken at the same time and the apparent length was measured. In order to measure the axial mixing within the jetting section the shortest column (0.495m) was used in the same manner as previously described. In this case the test section (the region between the water nozzle and the ketone nozzle) was made as long as the jetting section. The probe was set at the ketone nozzle. This length was determined from photographic and visual observations. The temperature of the runs was between 23 and 28 "C.
Calculations and Results The void fractions of the dispersed phase were obtained by using quick-opening valves in shutting off the inlet ketone and water flows simultaneously. The difference in level at the interface of the ketone section accounted for the MIK volume with an error of &8%. The results essentially agree with those of Johnson and Bliss (1946). Two general methods were used to calculate DL: the method of slopes, eq 6, and the method of moments with three different boundary conditions, eq 7,8,9, and 10. For the method of slopes the breakthrough curve was drawn and the slope at one residence time was determined; DL
-
NRe
16.83 29.87 10.13 2.94 30.28 17.06 16.87 29.57 10.03 3.08 30.54 17.03 10.25 11.93 16.08 28.50 39.27 3.14 16.27 15.76 16.96 28.53 10.70 3.05 30.66 16.60 11.47 36.97 47.16 0.73 13.00 10.88 6.69 2.75 2.92 15.98 11.38 14.99 10.61 13.56 9.89 13.50 10.03 9.90 9.83 8.65
was calculated using eq 6. Using this value of DL, the model was checked by comparing the predicted breakthrough curve with the experimental curve in Figure 2. The second method to calculate the dispersion coefficients was the method of moments. Using the computer, numerical integration was performed to obtain the variance (a2) with eq 7. Then the dispersion coefficients were calculated using eq 8, 9, and 10. The moments method was used only in 22 runs since data having long tails at large times were not available for the other runs. Interstitial velocities relative to the wall, Peclet numbers, and Reynolds numbers were calculated. The densities of saturated flows were determined experimentally and are within &0.2% of those of Fish et al. (1974). The Hayworth and Treybal (1950) correlation was used to predict the bubble size diameter as 3.4 mm, which is very close to the experimental value of Henton and Cavers for similar conditions. Works of Keith and Hixson (1955) indicated that the drop size was uniform within the velocity range of the work. Photographic and visual evidence confirmed this. Table I presents the results of the 46 runs with the continuous phase interstitial velocity ranging from 0.20 mm/s to 12 mm/s. The dispersed phase velocity varied
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
309
Ql--1.0
D L = ~ 2 5 x 1.2 m2 s
0
0 0
04 ~ -
0
\, 1
0’’
2
e
3
4
t
5
Figure 2. Comparison of models with experimental curve.
from 73 mm/s to 460 mm/s. Different lengths and two diameters of 35.8 mm and 27.0 mm were used. The DL results in Table I used in the correlations were the values of dispersion coefficients obtained using the method of slopes. For approximately constant run conditions, seven replication runs gave an average deviation *5% from the mean with a maximum of *lo%.
Discussion and Conclusions Comparison of Calculation Methods for Obtaining Dispersion Coefficients. The dispersion model used with the method of slopes has been suggested and/or used successfully by several investigators (Danckwerts 1953; Brutvan, 1958, Hazlebeck, 1963). It has an advantage since it does not require a measurement of tracer concentration in the tails at long times, which is very difficult to measure accurately at low concentrations. The disadvantage of the method of slopes is in the difficulty of calculating the slope accurately. In twenty-two runs the method of moments in addition to the method of slopes were compared. The average deviation of the axial coefficient using eq 8 (open-open vessel) with respect to the slope method using eq 6 (open-open vessel) is f10.4%, with a maximum difference of -28%. Since eq 6 and eq 8 both use an open-open vessel, the data should check if the data are consistent and represent an open-open system. The average deviation using q 9 (open-closed vessel) is f19.7% with a maximum difference of +54%. Comparison with eq 10 (closed-closed vessel) gave the largest deviations with *134% average and +500% maximum. The results of the open-open vessel model using the variance or method of moments check reasonably well with the results using the method of slopes. This can be explained as follows. The experimental system at the continuous phase settling section end resembles an open vessel since the sampling is continuous or “through the wall” as described by Levenspiel (19721, and the continuous phase flows to the exit from the settling section. The boundary condition at the top is not clear and could be considered as in between open and closed. Figure 2 shows the experimental results of a typical run when compared to the predicted of the model using a DL obtained from the slope method and a DL obtained with the method of moments for an open-open vessel (eq 8). For purposes of comparison some points of the model with DL obtained with eq 9 and eq 10 are also shown. The
1.51” 0
’
2
’
’
4
u,.
’
’
6
’
0
10
mm/s
Figure 3. Effect of the continuous phase velocity on DL.(All of the runs are V, = 100 mm/s except for L = 0.33 m, where V, = 136 mm/s).
model with DL obtained with the method of slopes for the open system model approximates closely the moments method of eq 8 for the open system model. Effect of the Continuous Phase Velocity. The influence of the continuous phase velocity on the axial dispersion coefficient when the dispersed phase velocity is constant is shown in Figure 3 for columns having a 35.8 mm diameter and three different lengths. The data of Hazlebeck and Geankoplis (1963) for L = 0.93 m are also shown. For the case of the column with 0.93 m length, it was found that an increase in the continuous phase velocity corresponds to an increase in the dispersion coefficient, but after a certain point, DL appeared to remain constant. This effect was not observed for lengths of 0.33 m and 1.74 m since the data did not cover velocities larger than 4.5 mm/s. Erving and Chen (1976) also found the same tendency to level out as the continuous phase velocity increased. As the velocity is increased, more eddies and back trapping occur, which increases the axial mixing. Using the data of this work and Hazlebeck and Geankoplis, the following relationship is obtained using the method of least squares
DL = 3.43 X 10-4Uw0.42(U, 5 4 . 5 mm/s)
(11)
The correlation coefficient is 0.94 with an average deviation of 433%. Hazlebeck and Geankoplis did not cover velocities larger than 4.5 mm/s to observe that the curve levels off. Effect of the Dispersed Phase Velocity. Figure 4 shows the effect of the dispersed phase velocity on DL. There is a trend on the DL flattening out at relatively high velocities of U,. However, at relatively low velocities, the values of DLappear to decrease when U, is increased. This effect can be explained, since at higher U,, the velocity profile is made flatter, and hence, DL decreases. Data of Henton and Cavers, Erving and Chen, and Kim and Baird (1976) when plotted versus interstitial velocity showed similar trends. Effect of Tower Length and Comparison with Packed Beds. In Figure 5 is shown the large effect of
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7
N
E 6
-
P 0
i l d ( = 2 7 mm u e 4 . 4 mm s L ~ 0 . 9 3m 0 dt
U,,=O0.7 m m s ~ ~ 0 . 9m 3
l t
200
300
400
500
s
S P a c k e d Tower
0.4
08 L .
12
16
,
4
U,
-6 mm
L a
U s
-.
-1
1 0 11
Figure 6. Effect of continuous phase velocity and diameter of columns.
Figure 4. Effect of dispersed phase velocity on DL. S p r a y Tower C U w = 4 0 m m
mm
0 di:27
2
U,, mm/s
0
L2 0
100
0
= 3 5.8 n i m
2.0
m
Figure 5. Effect of length for different flow rates and comparison with packed beds.
contacting tower length on DL with a column of 35.8 mm i.d. for two velocities of water flow. There is a 6 to 8-fold increase in the value of DL in increasing the column length from 0.02 m to a column length of 1.74 m. The DL values for very short columns (length equal to the jet length of continuous phase nozzle) were very small. Also the DL reaches a constant value as length increases. The small amount of axial mixing in the region of the nozzle jet can be due to the nature of the flow from the nozzle tips. A jetting effect appears in photographs and visual observations at the continuous phase nozzle. These jets caused the liquid to move in part of the column without experiencing much axial mixing but to move in plug flow. The jetting lengths observed were in the range of 0-30 mm from the nozzle, but this effect could have been present even farther than could be observed on the photographs. The jetting length appeared to increase with increasing water velocity. This is a probable reason for the effect of column length on DL as observed in this work. That is, when near plug flow existed in a large fraction of the length of the column (short column) there was little mixing or a small DL; when near plug flow existed in a relative small fraction of the column length (long column) its effect was not significant and, hence, large DL values were found. This is in agreement with the findings of Kreager and Geankoplis and Vogt and Geankoplis, who found that the height of a transfer unit increases with length and levels off when the
end effects become negligible. Henton and Cavers also found no effect of length on DL when there were no end effects. In packed beds in Figure 5 for the data of Liles and Geankoplis (1960) also tends to reach a constant value as the length of the bed increases; however, for this case there is an inverse relationship of DL vs. L. The opposite effect between spray towers and packed bed towers can be explained by the fact that the packed towers did not have jets at the entrance as in the spray towers. Furthermore, in a spray tower at the entrance the void fraction of the continuous phase experiences only a small change from t, = 1.0 to t, = 0.95; in packed beds the change is considerable and is from t = 1 to t = 0.35. Hence, in packed beds there is a larger velocity change at the entrance that could account for the large DL; on the other hand, in the spray columns, the nozzle jet gives near plug flow or small dispersion coefficients at the entrance. Effect of Tower Diameter. Data plotted in Figure 6 show that a column with a diameter of 27 mm has values of DL approximately 20% less than for a column with a diameter of 35.8 mm when the water velocity is higher than 2 mm/s. The walls can dampen the recirculation of the continuous phase and could account for the lesser amount of mixing in the smaller diameter column. Henton and Cavers also found a reduction in the axial mixing in a smaller diameter column. They found that for a column with a diameter of 76.2 mm, DL ranged from 6.3 to 17.3 times the corresponding values of a column with a diameter of 38.1 mm. They observed that the drop motion in the smaller diameter column, although very erratic, was always in the upward direction. In the larger diameter column even larger drops showed a swirling motion, often with a reversal of direction. Henton and coworkers (1970, 1973) point out that in both towers small-scale or diffusional mixing described by the axial diffusivity was present. In addition, in the larger diameter tower large-scale axial mixing was present in the continuous phase whereby the drops reversed direction for a short distance. There is need for experimental and theoretical work to separate these small-scale and large-scale effects. Reynolds and Peclet Numbers in Spray Towers. In the lower part of Figure 7 data in this study are shown for a NRerange of 0.7 to 50. The NPebased upon the equivalent spherical diameter of the ketone bubbles range from 0.007 to 0.07. The continuous phase water velocity U,' relative to the wall was used for the NPeand NRe. The
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Peclet number, U,l d,'/DL or (U,l+ U
