1698
Ind. Eng. Chem. Res. 1993,32, 1698-1705
Dispersed-Phase Residence Times in a Wirz Extraction Column L. M. Rincbn-Rubio,+A. Kumar, and S. Hartland' Department of Chemical Engineering and Industrial Chemistry, Swiss Federal Institute of Technology, CH-8092 Ziirich, Switzerland Residence-time distribution of the dispersed phase has been investigated in a 150-mm-diameter Wirz-I1 agitated extraction column for six different liquid-liquid systems of medium t o low interfacial tension and small density difference, using a two-point transient tracer technique. The flow of the dispersed phase is described by a convective model, and drop velocities and constriction factors are calculated from measured residence-time curves and drop-size distributions. For design purposes, an empirical correlation is presented for the average constriction factor in terms of the agitator speed and density difference between the phases.
Introduction The design of an internally agitated extraction column for a given mass-transfer duty requires, among others, the specification of the appropriate column height. In the past, column heights have been estimated from models assuming ideal plug flow behavior of the liquid phases in the extractor, with the dispersed phase being characterized by the hydrodynamic and mass-transfer behavior of a representative drop of an average diameter, normally the volume surface or Sauter mean diameter. It is, however, known that in practice deviations from plug flow occur that imply efficiency losses and an increased column height to achieve a given separation task (Sleicher, 1959). Particularly, the dispersed phase is known to consist of size-distributed drops, with different velocities with respect to the continuous phase and thus varying mass-transfer rates. In order to develop proper design methods for a given extraction column, knowledge of the dispersed-phase flow behavior is thus of fundamental importance. A rotary-agitated column, the Wirz-I1extractor has been designed to achieve efficient contacting of two countercurrent flowing phases and a high degree of phase interstage separation (Leisibach, 1965;Wirz, 1968). These features have been reported to result in low degree of axial mixing in both liquid phases and, consequently, in small efficiency losses when scaling up from smaller to larger column diameters. Furthermore, the efficient interstage phase separation permits the treatment of liquid-liquid systems with density differences as small as 30 kg/m3 (Leisibach, 1965). There has been one published study regarding the hydrodynamic behavior of a Wirz-I1 extractor whereby a toluene (dispersed)-water (continuous) system of high interfacial tension and moderate density difference was used (Hausler, 1985). No experimental information was provided on the flow behavior of the dispersed phase, and for the purpose of correlating experimental concentration profiles, it was assumed that deviations from plug flow of the organic-phase drops could be characterized by an axial mixing coefficient, determined by optimizing the fit between experimental concentration profiles and profiles predicted by the backflow model (Hartland and Mecklenburgh, 1966). The resulting axial dispersion coefficients for the dispersed phase were found to be about 1order of magnitude larger than measured continuous-phase axial coefficients. Furthermore, optimized overall mass-transfer
* Author to whom correspondence should be addressed. t
Present address: Investigacih y Desarrollo CA, Maracaibo
4002, Venezuela.
coefficients were found to be as much as 50 75 smaller than coefficients calculated from published correlations for single drops or drop swarms (Hliusler, 1985), probably indicating an incomplete description of the dispersedphase motion. In this work, results are reported on determinations of dispersed-phase residence time and drop velocitiescarried out in a 150-mm nominal diameter Wirz-I1 column, as part of a comprehensive study on the hydrodynamics of Wirz-I1 type column extractors (Rincbn-Rubio, 1992). It is endeavored to investigate the effect of liquid physical properties and operating variables on drop translation, and to develop correlations that will be useful in the design of Wirz-I1 columns. Dispersed-Phase Axial Mixing. Deviations from ideal plug flow of the dispersed-phase drops in agitated extraction columns have been mainly attributed to (Aufderheide and Vogelpohl, 1986; Haunold et al., 1990; Houghton et al., 1988; Misek and Haman, 1988; Tsouris et al., 1990)the following factors: backmixing of drops by entrainment in the continuous phase, and different drop velocities arising from the presence of a drop-size distribution, agitation, and column geometry. Backmixing of drops due to entrainment in the continuous phase is expected to grow in importance as the rate of agitation is increased. At high enough agitation intensities it would be expected that droplet axial mixing arising from large-scaleturbulent eddies in the continuous phase would result in similar turbulent dispersion coefficients for the continuous and dispersed phases. Investigators have thus often assumed total entrainment of the dispersed-phase drops by the cohtinuous phase and used, for modeling purposes, a single equation to predict axial mixing in both phases (Haunold et al., 1990; Misek and Haman, 1988). Such an assumption presupposes, among others, the applicability of a dispersion-type model for the treatment of plug flow deviations in the dispersed phase. In many cases, however, measured axial dispersion coefficients for the dispersed phase (based on a onedimensional dispersion model) are found to be up to an order of magnitude larger than those of the continuous phase (Bensalem, 1985; Hlusler, 1985; Kumar, 1985). Furthermore, axial dispersion data for the dispersed phase are found to show rather poor agreements with onedimensional dispersion models (Bauer, 1976; Kumar, 1985). The source of these disagreements is to be in the polydisperse nature of the dispersed phase and the resulting distribution of residence time arising from different drop velocities (Aufderheideand Vogelpohl, 1986, Houghton et al., 1988; Levenspiel and Fitzgerald, 1983;
0888-5885/93/2632-1698$04.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1699 Table I. Physical Properties of Systems Investigated.
system 1 2 3 4 5 6 a
dispersed phase n-butyl acetate n-octanol 56 % n-butyl acetate + 44% diethyl carbonate 32% n-butyl acetate + 68% diethyl carbonate n-butanol cyclohexanone
pC,kg/ma
998 998 998 998 984 997
Pd,
k/m3 881 828 923 948 845 950
AP, k / m g 117 170 75 50 139 47
pc, mPa 8
0.98 1.01 0.99 0.99 1.37 1.28
pd,
d as 0.73 9.85 0.78 0.75 3.18 2.48
7 , mN/m
14.1 7.8 14.0 14.0 1.9 4.1
In all cases, the continuous phase was water.
Olney, 1964). This phenomenon is termed the forward mixing effect, and has been shown to cause substantial reduction in the mass-transfer performance of extraction columns due to variations in dispersed-phase solute concentration across column cross sections (Cruz-Pinto and Korchinsky, 1980; Olney, 1964; Rod, 1966). On the other hand, forward mixing does not produce a concentration jump a t the dispersed-phase inlet as observed when axial dispersion arises from fluid backmixing (Rod, 1966). By considering that at high agitation rates the dispersed phase behaves as a continuum, several workers have developed dimensionless correlations for a dispersed-phase axial dispersion coefficient E d , in various types of agitated contactors, of a form similar to expressions developed for correlating axial mixing in the continuous phase (see, e.g., Kumar and Hartland, 1992):
However, the published correlations are often contradictory even when a single type of agitated extractor is considered (Kumar and Hartland, 1992). Levenspiel and Fitzgerald (1983)warnof the different nature of adiffusionlike process, characterized by random velocity fluctuations, and a convective-type process where some fluid elements move faster than others. It was shown that while for small deviations from plug flow a dispersion model predicts that the variance of the residence-time distribution varies proportionally to the length of the vessel, u2 a L,the nature of the convective flow results in g2 a
L~
(2)
and the spread of the fluid increasing linearly with the distance of travel. Analyzing a convective-type process by means of a dispersion model was shown to result in meaningless values of the dispersion coefficient given its dependence on the sampling location. Frequent breakage and coalescence of the dispersedphase drops is expected to reduce the extent of forward mixing in liquid-liquid contactors due to the resulting averaging of solute concentration across the contractor cross section (Houghton et al., 1988; Hamilton and Pratt, 1984; Levenspiel and Fitzgerald, 1983). In the case of a narrow drop-size distribution, very high rates of agitation and high rates of drop coalescence and redispersion, and only then, the one-dimensional dispersion model is thus expected to be appropriate for describing the mixing behavior of the dispersed-phase drops. In all other situations, the convective property of the dispersed-phase drops should be considered and properly quantified.
Experimental Section A pilot plant was constructed comprising a 150-mm nominal diameter Wirz-I1 column, having 22 stages, and accessory equipment including storage tanks, pumps, temperature control systems, and flow measurement devices. Details of the experimental apparatus are given elsewhere (Rinc6n-Rubio,l992). The plant was operated under two-phase countercurrent conditions in the absence
Table 11. Investigated Ranges of Operating Variables
variable NW, rps Qc, ma/(m2h) Qd, m3/(m2h)
range 1.67-6.67 4.1-12.3 0.5-13.8
variable e
R QJQd
range 0.05-0.30 0.30-22.5
Table 111. Geometric Data for Wirz-I1 Column nominal diam 150 mm no. of stages 22 stage height 80 mm active height 1890 mm 1.28 X 10-9 ma stage active vol (av) column active vol 30.5 X 10-9 ma agitator type 6-bladed turbine slanted 45O agitator diam 48 mm blade height 10 mm 8 no. of continuous-phase downcomers downcomer diam 12 mm inner port diam 120 mm
of mass transfer. Six different liquid-liquid systems were used, the organic solvents being dispersed in the aqueous phase. Demineralized water was used as the continuous phase in all experiments, and five organic solvents were used as raw materials for the six different organic mixtures used as dispersed phase. The organic solvents were as follows: n-butyl acetate, purum, 198%;diethyl carbonate, purum, 198%;n-octanol, purum, 198%;cyclohexanone, purum, 198%; n-butanol, technical grade. The physical properties for the various systems used are given in Table I. The dispersed phase corresponding to system 3 was a mixture containing 56% by volume n-butyl acetate and 44% by volume diethyl carbonate, while that corresponding to system 4 contained 32% by volume n-butyl acetate and 68% by volume diethyl carbonate. The properties were determined at a reference temperature of 20 f 1 "C. Interfacial tensions were determined using a drop volume method, densities were determined pycnometrically using a standard density bottle, and viscosities were measured using an Ubbelhode viscometer. All organic solvents were distilled once before their use. For each liquid-liquid system the process variables were systematically varied to determine their influence on the flow behavior of the dispersed phase. The investigated ranges of the operating variables, as well as those of the phase flow ratio, R, and dispersed-phase holdup, e, are given in Table 11. Wirz-I1Column Extractor. The basic specifications and geometric data of the investigated column are given in Table 111. The main column section consisted of two 1000-mm-long QVF precision bore glass tubes with an internal diameter of 152.4 f 0.23 mm enclosing a stack of 22 stages with stainless steel internals. The coalescence section at the top of the column consisted of a 405-mmlong QVF section of maximum nominal diameter 150 mm having three lateral outlets of 50-mm nominal diameter and an upper outlet of 40-mm nominal diameter. At the bottom of the column there was a 180-mm-long glass section of maximum 150-IIUII nominal diameter bored with two lateral openings and a bottom outlet, all of 50-mm
1700 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
Figure 1. Schematic diagram of Wirz-I1column stage showing the flows of light and heavy phases.
nominal diameter. This section contained the stainless steel dispersed-phase distributor, adapted to the lateral opening by means of a stainless steel fitting and a Teflon ring. The aqueous phase was fed through a 10-mm glass tube located some 150 mm above the top stage, while the outlet for this phase was located at one of the lateral openings of the column bottom section. To minimize the entrainment of organic-phase drops with the exiting aqueous phase, the bottom section was filled with Teflon packing to promote coalescence of fine drops. The organic phase was fed through the dispersed-phase distributor, a funnelshaped piece constructed from stainless steel and closed at its top with a Teflon plate containing 150 stainless steel nozzles of 1.5-mm diameter. The outlet for the organic phase was located at one of the lateral openings of the column top section. Each column section contained 11stages 80 mm high, so there were a total of 22 stages. A schematic diagram of one stage is given in Figure 1. Dispersion of the organic phase in each stage is achieved by means of a 48-mmdiameter turbine agitator comprising six 45' slanted blades. Light phase enters through the lower port F and heavy phase enters through the upper downcomer C as indicated by the dotted and continuous lines. The dispersion produced by the agitator B rotating on the shaft A passes into the chamber D where partial coalescence occurs. Light phase passing through the upper slit leaves the stage via the upper port F, and continuous heavy phase passing through the lower slit leaves the stage via the lower downcomer C. The rate of agitation is varied between 1.33 and 6.67 revolutions per second (rps) by means of a variable-speed drive incorporated into the motor on top of the column.
Measurement of Dispersed-PhaseResidenceTime. The particulate nature of the dispersed phase renders difficult the application of steady-state tracer methods to the determination of axial mixing parameters, owing to the difficulty of initially injecting a tracer into the drops and subsequently sampling the dispersion. On the other hand, several techniques have been reported in the literature for the measurement of dispersed-phase residence-time curves from transient tracer injections. Among others, ultraviolet absorption (Nedungadi, 1991), laser photometric techniques (Tsouris and Tavlarides, 1990), photocolorimetric techniques (Misek and Haman, 1988) and light absorption techniques (Bensalem, 1985;Kumar, 1985)have been used. The use of point-measuring probes located inside the investigated column seems to be appropriate under high agitation conditions, when an even distribution of the dispersed-phase drops is achieved
throughout the column cross section, thus ensuring the samples reaching the detection probes are representative. In cases where there is an uneven distribution of the dispersed-phase drops, some drops may pass by the measuring points undetected by the sampling probes. In such cases, techniques allowing for integral measurements throughout the investigated column cross section are desired. In this work, a nonintrusive light absorption technique was implemented as shown below and was found to give satisfactory results. Measurement Technique. A fixed 10.0-mL volume of organic-soluble dye (Fluka Fettschwarz) with a concentration of lo00 kg/m3 was rapidly injected into the dispersed-phase feed line, and the change in the intensity of dye-colored drops with time was monitored at two positions downstream from the dispersed-phase distributor. At each of the monitoring points, corresponding to stages 3 and 5, a photodiode (BPW 21, Siemens AG) was fixed onto the column wall at a position corresponding to the entrance of the corresponding stage. On the opposite side of the column, illumination was provided by means of a 150-W white lamp. With this setup, it was ensured that the dispersed-phase drops were in the receiving range of the photodiodes at the point where they leave one stage and enter the next. The column section under study was covered with black paper to avoid interference. Before each experimental run, the baselines corresponding to 0% and 100% light absorption were calibrated to equalize the signals emanating from the photodiodes. The output signals were sent to a chart recorder, and the resulting residence-timedistribution curves were digitized and stored for their evaluation. Procedure. For each liquid-liquid system appropriate agitation rates were chosen and phase flow rates were selected from experimentally determined holdup curves (Rincbn-Rubio, 1992)to ensure operation in the 0.05-0.25 holdup range. The rate of agitation and the continuousphase flow rate were then set at the desired values, and the organic phase was gradually admitted into the column up to the required volumetric flow rate. Care was taken to maintain a stable interface at the top of the column throughout each run, and the phases were recirculated for at least 30 min at steady state before any measurement was taken. Photographs were then taken at three different positions along the column for determining drop-size distributions as described elsewhere (Rincbn-Rubio,1992), and the distribution of residence times for the dispersedphase drops between the third and fifth stages was obtained as previously described.
Results Operating conditions were selected to cover a wide range of dispersed-phase holdups, rates of agitation, and continuous-phase flow rates, as these variables were thought to influence the flow of the dispersed-phase drops. In all cases, the Wirz-I1column was operated far from conditions of maximum capacities (Rincbn-Rubio, 1992) and at holdups E < 0.25. Well-defined tracer concentration curves were in general obtained, characterized at times by spikes as more intensively colored drops passed within the range of the detecting photodiodes. In some cases, particularly at high agitation rates and with systems of low density difference, the recorded residence-time curves had to be discarded due to large colored drops remaining stationary for long periods of time below the plate beneath the impeller, thus impeding a proper return to the baseline. To avoid this inconvenience, further design work is needed to modify
Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1701 Table IV. Dispersed-Phase Residence-Time-Distribution Measurements for n-Butyl Acetate (Dispersed)-Water (Continuous) System
N,, l/e 3.33 3.33 3.33 5.00 6.00 5.00 5.00 5.00 5.83 5.83 5.83 6.67 6.67 6.67 6.67 6.67
Qc,
m3/(m2h) 4.1 7.4 12.3 4.1 7.4 7.4 7.4 12.3 7.4 12.3 12.3 7.4 7.4 7.4 12.3 12.3
Qd,
ms/(mzh) 4.7 6.2 6.2 4.7 3.5 5.5 8.2 8.2 4.7 4.7 6.2 4.7 6.2 12.6 4.8 6.2
tl, 8
U12,SZ
16.9 21.1 18.3 21.9 21.3 21.4 17.6 22.4 31.5 27.9 27.4 29.7 30.7 31.2 33.2 33.6
64.5 96.2 68.5 101.9 159.3 129.3 86.9 158.5 211.8 228.2 223.1 217.4 297.9 250.0 349.3 309.2
tz, 8 23.5 27.5 23.9 31.8 32.5 29.7 27.7 33.2 44.1 42.6 42.1 45.1 44.2 46.2 55.0 56.9
U22,82
tz1, 8
96.8 115.8 84.4 156.9 203.1 142.6 112.3 184.8 302.1 322.2 302.3 324.2 397.0 331.9 454.3 485.6
6.6 6.5 5.6 9.9 11.2 8.4 10.1 10.8 12.6 14.7 14.7 15.4 13.5 14.9 21.9 23.3
Table V. Dispersed-Phase Residence-Time-DistributionMeasurement6 for 32% n-Butyl Acetate (Dispersed)-Water (Continuour) System
N,, l/e
Qc, m3/(m2h)
3.33 3.33 3.33 4.17 4.17 4.17 5.00 5.00 5.00 5.00 5.00 5.83 5.83 5.83 5.83
4.1 4.1 12.3 4.1 12.3 12.3 4.1 4.1 4.1 7.4 12.3 4.1 7.4 12.3 12.3
Qd,
m3/(mzh) 1.2 4.7 3.5 4.7 1.2 3.5 1.2 3.5 4.7 3.5 1.2 3.5 3.5 1.2 2.4
tl, 8
u12, 82
18.7 14.8 19.5 13.2 56.1 35.0 31.7 29.3 28.6 41.9 70.5 37.8 51.6 70.3 56.2
69.4 49.8 86.8 35.2 757.1 342.2 267.0 235.7 223.6 573.7 1288.8 495.6 658.6 1226.9 603.2
the measurement setup, bringing the detecting photodiodes inside the Wirz-I1 column, right at the edge of the inner port F, near the point where dispersed-phase drops enter each stage. The digitized concentration versus time curves were fed to a FORTRAN computer program, and the mean value and variance of each curve were calculated in the usual way according to (Levenspiel, 1962): (3)
(4) where Cj stands for the tracer concentration at time ti and Ati is the time interval between two adjacent digitized points. By polynomial interpolation of the digitized points, normalizing and subsequent discretization of the curves into equal-size time intervals, eqs 3 and 4 were simplified
to
The resulting values of mean time and variance for the concentration c u g e a t the first location, &,q2, and a t the second location, tz,u22, together with the mean residence time and variance between both measuring locations,
-
-
-
t,, = t 2- t ,
(7)
tz, 8 35.6 32.7 46.9 34.7 133.3 84.3 97.2 65.3 54.5 102.2 170.2 83.3 99.7 220.4 142.0
QZ1,S
0212,
32.3 19.6 15.9 55.0 43.9 13.3 25.4 26.2 90.3 93.9 79.2 106.9 99.0 81.9 104.9 276.5
5.7 4.4 4.0 7.4 6.6 3.6 5.0 5.1 9.5 9.7 8.9 10.3 9.9 9.1 10.2 13.3
+ 68% Diethyl Carbonate
U22,82
tZl,8
u212,
s2
Ql, 5
230.5 131.9 268.6 195.7 2145.7 994.4 1093.2 579.6 476.3 2078.2 3600.3 1406.9 1586.1 5576.5 2996.5
17.0 18.0 27.4 21.5 77.2 49.3 65.5 35.9 25.9 60.3 99.6 45.4 48.1 150.1 85.8
161.1 82.1 181.8 160.6 1388.6 652.2 826.2 343.9 252.9 1504.5 2311.4 911.3 927.5 4349.6 2393.3
12.7 9.1 13.5 12.7 37.2 25.5 28.7 18.5 15.9 38.8 48.1 30.2 30.5 65.9 48.9
are given elsewhere for all the conducted runs (Rinc6nRubio, 1992),and the results corresponding to systems 1, 4, and 6 are included here in Tables IV-VI. The mean time, 121, might be interpreted as the average residence time of the dispersed-phase drops in the investigated column section, while the variance, u212, quantifies the broadness of the residence-time distribution. It is seen that, for a given rate of agitation and dispersed-phase flow rate, increasing the continuous-phase flow rate increases the mean residence time of the drops in the investigated section. Considering the system cyclohexanone dispersed in water (Table VI), increasing the continuous-phase flow rate from 4.1 to 12.3 m3/(m2h) at a dispersed-phase flow rate of 1.1-1.6 m3/(m2 h) and an agitation rate of 3 rps results in a 3-fold increase in the dispersed-phase mean residence time. The behavior was similar for all the liquidliquid systems investigated, but the effect of Qcwas much stronger in systems with low density difference, as one would have expected from the small terminal velocities of drops of density close to that of the continuous phase. For all systems investigated, at given flow rates of the continuous and dispersed phases, increasing the rate of agitation increased the residence time of the dispersedphase drops, with the effect being much more apparent in systems of low density difference and low interfacial tension and at high continuous-phase flow rates. As an illustration, and considering the system of cyclohexanone dispersed in water, increasing the rate of agitation from 3 to 3.33 rps, a t a continuous-phase flow rate of 4.1 m3/(m2 h) and a dispersed-phase flow rate of 1.6 m3/(m2 h), increases the mean residence time by a factor of 1.37(Table VI). When the rate of agitation is increased from 2.5 to 3 rps, this time a t a higher continuous-phase flow rate of 12.3 m3/(m2 h) and a similar dispersed-flow rate of 1.1-1.6 m3/ (m2 h), the dispersed-phase mean residence time
1702 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 Table VI. Dispersed-Phase Residence-Time-Distribution Measurements for Cyclohexanone (Dispersed)-Water (Continuous) System
NO^, l/s 2.00 2.00 2.50 2.50 2.50 3.00 3.00 3.33 3.33 3.33 3.67 3.67 3.67
Qc,
m3/(m2h) 4.1 12.3 7.4 12.3 12.3 4.1 12.3 4.1 7.4 12.3 7.4 7.4 12.3
Qd, ms/(m2h)
tl, 8
u12,s2
tz, 8
UZZ,S2
t21, 8
0212,
s2
U218
6.0 4.4 3.8 1.6 2.7 1.6 1.1 1.6 2.7 0.6 1.1 2.2 0.6
27.4 29.1 21.8 41.4 37.4 33.7 52.3 43.2 45.0 87.0 44.7 47.9 92.6
237.2 245.0 113.1 312.2 310.7 250.1 648.7 430.9 565.7 2186.8 504.3 702.7 2407.8
43.5 55.4 43.4 74.7 72.6 64.7 145.4 85.7 91.6 195.8 108.4 104.2 217.8
307.3 417.0 187.8 644.5 547.9 388.6 1595.9 730.6 1182.9 2425.8 960.5 1534.0 3995.3
16.1 26.3 21.5 33.6 35.1 31.0 93.1 42.5 46.6 108.8 63.7 56.3 125.2
70.1 172.0 74.7 332.2 237.2 137.9 947.1 299.7 617.1 239.0 456.2 831.3 1587.5
8.4 13.1 8.6 18.2 15.4 11.7 30.8 17.3 24.8 15.5 21.4 28.8 39.8
increases by a factor of 2.77. As seen from Tables IV-VI, the dispersed-phase flow rates were found, on the other hand, not to influence in a significant way the measured drop residence times. As seen from Table IV, the standard deviations of the residence-time distributions between the two measuring probes tend to be comparable to the respective mean residence times for systems of high interfacial tension and high density difference, indicating a broad distribution of drop residence times for these systems. For systems of low density difference (Tables V and VI) the standard deviations seem to be somewhat lower than the corresponding mean residence times, possibly indicating more uniform drop velocities.
3
Modeling of Dispersed-Phase Axial Mixing It has been pointed out above that care must be taken at the time of determining axial mixing parameters from dispersed-phase residence time measurements. Preliminary attempts to correlate the dispersed-phase residence time curves by means of a dispersion model were unsuccessful. This finding is not surprising since under the agitation levels used in this work the movement of the dispersed-phase drops as they rise through the Wirz-I1 column is hardly random in nature. Furthermore, no backflow of dispersed-phase drops was observed, except for systems of low density difference and a t continuousphase flow rates higher than critical velocities for drop entrainment (Rinc6n-Rubio, 1992). In order to test the applicability of a convective model to the flow of the dispersed-phase drops in the Wirz-I1 column, the functional dependence of the spread of the residence-time distribution between the two measuring probes on the distance between the measuring probes or else on the’ mean residence time was investigated (Levenspiel and Fitzgerald, 1983). In the present work the distance between probes was kept constant for all runs, while experiments were conducted a t different agitation rates and phase flow rates, thus resulting in different dispersed-phase residence times. For a convective-like process, elements of fluid move past each other at constant velocities and, for a Gaussian residence-time distribution, the dimensional standard deviation of the residence-time distribution curve is given by
4
-
u = uet
(9) where ue is the dimensionless (and constant) flow parameter of the convective-Gaussian process (Levenspiel and Fitzgerald, 1983). It is thus seen from eq 9 that (10) The calculated variances of the residence-time distributions, u212, were thus coppared with the corresponding mean residence times, t 2 1 , as shown in Figure 2 for q2 a
j2
t L
0.8
1
1.2
1.4
log Residence Time (s)
1
1.2
1.4
1.6
1.8
2
2.2
log Residence Time (s)
Figure 2. Variance of the dispersed-phase residence-time distribution versus residence time. (top) n-Butanol (dispersed)-water (continuous)system: +, N , = 1.67 rpe; 0, N , = 2.17 rps; X, N , = 2.50 rps; *, N , = 3.00 rps. (bottom) Cyclohexanone (dispersed)water (continuous)system: +, N , = 2.00 rps; 0 ,N , = 2.50 rps; X, N , = 3.00 rps; *, N , = 3.33 rps; 0 , N , = 3.67 rps. Table VII. Least-Squares Linear Fit of log ua+ versus log b1 system 1 3 2 4 5 6 slope 1.70 2.49 1.62 1.76 1.81 1.60
n-butanol (dispersed)-water (continuous) and cyclohexanone (dispersed)-water (continuous) systems. The slopes resulting from a least-squares fit of the data by means of straight lines are given in Table VII, where values close to 2.0 are observed. This indicates that, under the operating conditions used in the present work, the flow of the dispersed-phase drops seems to approximate a convective process. No further attempts were therefore made to interpret the experimental data in terms of a onedimensional dispersion model, and a convective approach was adopted to estimate actual velocities of the dispersedphase drops in the Wirz-I1 column. Under two-phase operation, the actual velocity of a drop of size di with respect to the column, u(di), may be expressed as (Olney, 1964)
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1703 u(di) = CR(di)u,(di)(l-
e)
- [VJ(l-
(11) where Ut(di) stands for the terminal velocity of a single drop of size di in a stagnant medium and CR(di) is a constriction factor accounting for the slowing effect due to agitation and to the presence of column internals. The form of the equation above would be useful for determining actual drop velocities in the Wirz-I1column from physical data of the liquid-liquid system [~t(di)]and operating variables [di, CR(di), V,, and €1. No information is available, however, regarding the values of constriction factors in Wirz-I1 columns and their dependence on operating variables such as the rate of agitation. A calculation procedure to determine constriction factors from the data of this work, based on a convectiveapproach, may then be visualized by determining actual drop velocities from experimentally measured residence-time curves, and the computation of drop-size dependent CR(di) as E)]
for the different liquid-liquid systems investigated and for different operating conditions. The real behavior of a liquid-liquid dispersion in the Wirz-I1 extraction column is very complex, and in order to estimate actual drop velocities from experimental dispersed-phase residence-time-distributioncurves, one is bound to make some simplifying assumptions. In the following, an approach based on a model proposed by Aufderheide and Vogelpohl(1986) will be used to analyze the experimental dispersed-phase residence-time curves to obtain actual drop velocities. The following assumptions, similar to those considered by Aufderheide and Vogelpohl (1986), will be made: 1. The flow of the dispersed-phase drops is purely convective. 2. Drop velocities are proportional to the drop size. 3. There is no drop coalescence or breakage in the investigated section. 4. There is a constant drop-size distribution within the investigated section. 5. There is no backflow of drops. From the experimental number drop-size distributions corresponding to the third column stage, volumetric dropsize distributions were calculated by fitting the frequency histograms to a r distribution function (thus obtaining the parameters a and p of the continuous number dropsize distributions), discretizing the number distribution into 250 intervals, and computing the discretized volumetric distributions according to Ad = d,,/250
(13)
di = (i - 1)Ad
(14)
vol(di) = pn(di) d:
(16)
vol(di) PVOl(di) =
(17) zvol(di)Ad I
where pn(di) and pvol(di) represent the number and volume density of drops size di and Ad is the class interval. The subsequent calculation procedure for each set of operating conditions involved the discretization of the normalized residence-time-distribution curves and the volumetric drop-size distribution into 15intervals of equal
area. (For normalized curves the latter means dividing each curve in intervals of area 1/15.) The drop diameters and residence times delimiting each area interval were calculated by numerically integrating the corresponding curves according to Simpson's rule and marching from di 0 and t l = tz = 0 to dm, and t2,m,. The drop size characterizing each area interval was calculated as the algebraic mean of the drop sizes delimiting the given area interval. Followingthe assumption that drops reach the detection points according to their size, and considering that their rise velocity is directly proportional to the drop diameter, each drop class is assigned to a corresponding interval in the first and second normalized tracer concentration versus time curves. The time at whicih half of the drops of a given size interval pass the first and second detection point is then calculated for each size interval as
In order to compute mean residence times according to eqs 18 and 19, each ith time interval was subdivided into 40 subintervals of size Atl,k
(20)
= (tl,i+l- t1,i)/40
= (t'&i+l- t2,i)j40 (21) From eqs 18and 19 the residence time between detection points 1and 2 of drops belonging to the ith size interval may be calculated from
With the known length of the test section between points 1 and 2,1512, the actual velocity of drops in the ith size interval with respect to the Wirz-I1 column follows from u(dJ = L12/?12,i
(23) Upon estimation of single drop terminal velocities from the correlations given by Klee and Treybal(1956) and use of the experimental values of dispersed-phase holdups, constriction factors corresponding to each drop-size interval were calculated for each mean drop size according to eq 12. An average constriction factor, CR,was computed for each run as pvol(di)Ad + VJ(1C,=C(1u(dJ- t)Cut(di) pvol(di)Ad
E)
(24)
A listing of the FORTRAN computer program used for the calculation of drop velocities is given elsewhere (Rinc6n-Rubio, 1992). Actual drop velocities in the WirzI1 column calculated from eq 23 and drop velocities estimated for no effect of either agitation or column internals from eq 11 with CR set equal to unity, u*(di) = ut(di)(l- t) - [ V J ( l - e)] (25) are compared in Figure 3 for liquid systems 3 and 4 (Table I). It is observed that the combined effect of agitation and hindrance to drop flow due to the presence of column internals results in significant reductions in drop rise velocities, as compared to values estimated for unconstricted flow. For all the studied liquid-liquid systems, increasing the rate of agitation is seen to decrease drop rise velocities, an effect more noticeable in systems of low density difference. In some cases, particularly with
1704 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
6or--l +
Table IX. Average Values of Constriction Factor for 32% Butyl Acetate 68% Diethyl Carbonate (Dispersed)-Water (Continuous) System.
CI
2
6o
3
40
'G
3 a P 0
20 0
0
5
0
bm)
Drop Diometer
1
2
3
Drop Diometer (mm)
1.2 4.1 4.7 4.1 12.3 3.5 4.7 4.1 1.2 12.3 3.5 12.3 1.2 4.1 3.5 4.1 4.7 4.1 3.5 7.4 1.2 12.3 4.1 3.5 3.5 7.4 12.3 1.2 12.3 2.4 CY and @ are parameters of
3.33 3.33 3.33 4.17 4.17 4.17 5.00 5.00 5.00 5.00 5.00 5.83 5.83 5.83 5.83
::r
5.0 2.62 0.7941 1.1678 0.25 15.5 2.91 2.4199 1.7285 0.25 17.5 3.36 2.1754 1.7235 0.22 15.0 2.08 4.7785 4.0838 0.29 8.5 1.79 2.6350 2.7347 0.15 19.0 2.44 5.1551 3.5578 0.23 6.0 1.79 4.1161 3.7127 0.10 12.5 2.19 2.7170 2.7080 0.16 16.5 2.16 3.7710 3.1720 0.23 17.5 2.11 6.0001 4.3242 0.17 10.0 2.09 3.6895 3.9165 0.18 16.5 1.64 5.7357 5.4701 0.18 23.5 1.67 2.4270 2.7551 0.24 16.0 1.74 4.2062 3.8394 0.17 30.0 2.56 6.3458 3.6106 0.22 r number drop-size distribution.
Table X. Average Values of Constriction Factor for Cyclohexanone (Dispersed)-Water (Continuous) System. ..
-
0
0
2
4
6
0
Drop Diometer b m )
2
4
Drop Diameter (mm)
Figure 3. Comparison of drop velocities in Wirz-I1 column (0) and in unagitated, unconstricted flow (+I. (top) 56% n-butyl acetate + 44% diethyl carbonate (dupewed)-water (continuous) system: left, N, = 4.17 rps, Qc = 4.1 ms/(mz h); right, N, = 5.83 rps, Qc= 12.3 m3/(m2 h). (bottom) 32% butyl acetate 68% diethyl carbonate (dispersed)-water (continuous) system: left, N, = 3.33 rps, Qc = 4.1 m3/(m2 h); right, N, = 5.83 rps, Qc= 4.1 m3/(m2h).
+
Table VIII. Average Values of Constriction Factor for m-Butyl Acetate (Dispersed)-Water (Continuous) System. ..
.
.
.
4.7 4.1 6.2 7.4 6.2 12.3 4.7 4.1 3.5 7.4 5.5 7.4 8.2 7.4 8.2 12.3 7.4 4.7 12.3 4.7 6.2 12.3 4.7 7.4 6.2 7.4 12.6 7.4 4.7 12.3 12.3 6.2 0 a and fl are parameters of
3.33 3.33 3.33 5.00 5.00 5.00 5.00 5.00 5.83 5.83 5.83 6.67 6.67 6.67 6.67 6.67
8.5 2.31 1.3147 1.3613 0.47 11.0 2.82 0.8798 0.9183 0.38 7.9 2.37 3.2541 2.3406 0.43 8.9 1.78 5.5841 4.5341 0.36 5.0 1.84 4.5992 4.3988 0.32 12.0 1.75 4.4810 4.2890 0.37 24.0 2.48 3.5463 2.9712 0.33 17.3 1.89 5.8825 4.6558 0.34 10.4 1.67 7.7155 6.5288 0.32 11.1 1.80 7.7150 6.6290 0.31 14.0 1.98 7.8019 6.7890 0.32 10.1 1.44 6.5897 7.0570 0.33 13.5 1.49 6.6604 6.7489 0.35 24.0 1.91 4.5720 4.0454 0.31 10.4 1.51 7.8876 7.8003 0.28 15.0 1.52 6.6604 6.7484 0.27 r number drop-size distribution.
systems of low density difference, actual drop velocities in the Wirz-I1 column are estimated to be up to nearly 1 order of magnitude smaller than those calculated from eq 25. Average constriction factors computed by using eq 24 have been given in Tables VIII-X for three of the investigated liquid-liquid systems. Values of CR were observed to be in the range 0.10-0.66, with the smaller values corresponding to systems of low density difference under conditions of high agitation rates. Under these conditions, the upward movement of dispersed-phase drops may be strongly hindered by the continuous-phase turbulent flow patterns, increasing the residence time of the drops in the investigated section and thus resulting in lower actual rise velocities.
2.00 2.00 2.50 2.50 2.50 3.00 3.00 3.33 3.33 3.33 3.67 3.67 3.67
4.1 12.3 7.4 12.3 12.3 4.1 12.3 4.1 1.4 12.3 7.4 7.4 12.3
.
.
.
6.0 4.4 3.8 1.6 2.7 1.6 1.1 1.6 2.7 0.6 1.1 2.2 0.6 a a and fl are parameters of
15.5 2.02 1.4029 3.6354 0.54 15.5 2.02 0.9825 2.0065 0.35 16.0 1.30 1.8232 4.4754 0.42 9.5 1.36 1.2301 4.2145 0.36 15.0 1.21 3.0738 4.4526 0.39 7.5 0.87 2.1342 5.8262 0.29 9.0 1.10 1.0499 3.1096 0.24 8.5 1.02 2.9377 5.4087 0.23 16.0 0.94 1.9827 4.4916 0.32 7.5 1.20 2.1722 3.5724 0.21 8.5 0.96 4.8219 7.5889 0.19 16.0 1.04 3.0538 5.1827 0.27 10.0 0.98 2.4044 4.8215 0.25 F number drop-size distribution.
Since only one Wirz-I1 column was used in this study, it was not possible to investigate the relationship between the average constriction factors and the geometry of the column internals. On the other hand, given the observed dependence of the constriction factors on rate of agitation and system physical properties, it is considered convenient for design purposes to quantify such dependence in terms of power functions of the relevant variables. I t is assumed that CRis a function of the agitator speed, Nw, and density difference, Ap. The parameters of the power-law function were estimated by regression on the data for all liquidliquid systems combined (77 observations) to give with an average absolute value of the relative deviation of 15.67%. The drop diameter is also a function of Nw and Ap, so it does not appear directly in this equation. The equation could be reexpressed in terms of the agitator Reynolds number as (27)
However,sincethe agitator diameter, d, was held constant and only water was used as the continuous phase, the physical properties p e and p , did not change, so there is no direct proof that they affect the values of &d in the way indicated.
Conclusions The flow behavior of the dispersed phase in a Wirz-I1 column extractor having a nominal diameter of 150 mm
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1706 has been studied for six different liquid-liquid systems differing in interfacial tension and density difference, under a limited range of phase flow rates and intensities of agitation. The average residence time of the dispersed-phase drops in the column was found to be strongly dependent on the system density difference, the continuous-phase flow rate, and the intensity of agitation. The flow of the dispersedphase drops in the Wirz-I1 column was found to closely approach a convective process, and by making a series of simplifying assumptions, individual drop velocities in the column were estimated as a function of drop size. Individual and average constriction factors for the flow of the dispersed-phase drops were computed, and the latter were correlated in terms of the intensity of agitation and the density difference between the continuous and dispersed phases. Drop axial velocities in the Wirz-I1 column were found to be as much as 10 times smaller than estimated velocities under unconstricted flow, indicating a major effect of column internals and agitation on drop translation.
Acknowledgment The financial support by Hoffman La Roche AG, Sulzer AG, and the Kommission zur Forderung der Wissenschaftlichen Forschung, in Switzerland, is gratefully acknowledged. L.M.R. would like to thank Investigacih y Desarrollo C.A. for sponsoring his doctoral work a t the ETH in Ziirich.
Nomenclature CR = average constriction factor, dimensionless CR(di) = constriction factor for drops of the ith size fraction, dimensionless d, = agitator diameter, m dk = column diameter, m d32 = Sauter mean diameter, m Ed = dispersed-phase axial mixing coefficient, m2/s k , = parameters of empirical equations, dimensionless L = column length, m LIZ = distance between points 1 and 2, m Nw = agitator speed, revolutions/s pn(di) = drop number distribution, Us pvol(di) = drop volume distribution, Us Q = throughput per unit cross-sectionalarea of column, m3/ (m2h) R = flow ratio = QJQd, dimensionless tl = first signal measured time, s t 2 = second signal measured time, s &,z = residence time, s u(dJ = velocity of drop size di in Wirz-I1 column, m/s u*(di) = velocity of drop size di in empty column, m/s V = superficial velocity, m/s ut = drop terminal velocity, m/s vol(di) = volume of drop size 4,m3 vol(d32) = volume of drop size d32, m3 Greek Symbols
parameter of I' distribution, dimensionless j3 = parameter of r distribution, dimensionless y = interfacial tension, N/m c = dispersed-phase fractional holdup, dimensionless p = viscosity, Pa s p = density, kg/m3 u = standard deviation of drop-size distribution, m u2 = variance of drop-size distribution, m2 a=
Subscripts c = continuous phase
d = dispersed phase
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Received for review November 23, 1992 Revised manuscript received May 11, 1993 Accepted May 20, 1993
