Representative drop sizes and drop size distributions in A/Odispersions in continuous flow stirred tank

K.K. Singh a, S.M. Mahajani a,⁎, K.T. Shenoy b, S.K. Ghosh b

a Department of Chemical Engineering, Indian Institute of Technology, Powai, Mumbai, 400076, Indiab Chemical Engineering Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India

Abstract

This work presents experimental studies of drop size distributions in aqueous in organic (A/O) dispersions produced in acontinuous flow stirred tank agitated by a four-bladed top shrouded turbine with trapezoidal blades. The organic phase is a mixtureof n-paraffin, tributyl phosphate (TBP) and di-2-ethyl hexyl phosphoric acid (D2EHPA), the aqueous phase is dilute phosphoricacid. Drop size measurements have been performed for different values of impeller speed, feed phase ratio and mean residence timeat two locations in the tank, near the wall. Surfactant stabilization of the dispersion has been used as the drop size measuringtechnique. Log-normal distributions are found to fit the experimental drop size distributions. Experimental results have been usedto obtain the empirical correlations for representative drop sizes — Sauter mean diameter and maximum stable diameter.

Keywords: Liquid–liquid dispersion; Phosphoric acid; D2EHPA; TBP; Surfactant stabilization; Drop size

1. Introduction

Liquid–liquid dispersions in continuous flow stirredtanks play an important role in hydrometallurgicalplants using mixer–settlers, wherein the objective is topreferentially extract a valuable component from oneliquid phase into another immiscible liquid phase. Theoverall extraction affected by the mixer or the stageefficiency depends, among other things, on specificinterfacial area available for mass transfer that in turndepends on sizes of the drops of the dispersed phase.The sizes of the drops depend on several factors such asimpeller geometry, impeller speed, impeller location in

the tank, feed phase ratio and physical properties of thephases. For an optimum design, a quantitative descrip-tion of the effect of all these factors on the drop sizes isrequired. In a stirred tank due to inhomogeneousdissipation of power (Cutter, 1966), drop sizes exhibitspatial variations. At a given location also, owing tocontinuous redispersion and coalescence, a distributionof drop sizes is observed. Therefore, to get an overallpicture of the quality of dispersion it makes sense to talkof a representative drop size of the dispersion. TheSauter mean diameter (d32) and the maximum stabledrop diameter (dmax) are the two choices of therepresentative drop diameter. While the first is theratio of third moment of drop size distribution to secondmoment of drop size distribution, the second representsthe maximum drop size that can be observed for the

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level of turbulence prevailing in the dispersion. TheSauter mean diameter assumes more importancebecause it is directly related to specific interfacial areaa by the following expression:

d32 ¼ 6/Pa

ð1Þ

where ϕ is the holdup of the dispersed phase. In manyinstances, direct proportionality between the two represen-tative drop diameters has been reported (Brown and Pitt,1972; Nishikawa et al., 1987a; Collias and Pruddhornme,1992; Zerfa and Brooks, 1996; Calabrese et al., 1986a).Therefore, correlation for either of the representativediameters can be used to compute the other representativediameter.

Experimental measurements of drop size distribu-tions in stirred tanks are indispensable in view of theexorbitant computational demands of a fully predictivemodel that will solve the discretized population balanceequations along with the flow and turbulence equationson a fine grid in the computational domain. The numberof equations to be solved will be very large as there willbe one equation for each drop class. This approach hasnot been attempted so far. A simplification can beachieved by solving the flow equation on the fine gridfollowed by solution of population balance equations ona very coarse grid. Some studies using this simplifiedapproach have been reported (Alopaeus et al., 1999;Maggioris et al., 2000; Alopaeus et al., 2002). This stilldoes not obviate the need of experimental measurementsentirely as the breakage and coalescence models(Coulaloglou and Tavlarides, 1977) embedded in thepopulation balance equations contain model constantswhich need to be estimated for a given system by usingthe experimental data.

The purpose of the present study is to investigate theeffect of three operating parameters — impeller speed,feed phase ratio and mean residence time, on the dropsize and drop size distributions and to develop suitablesystem specific correlations. The experimental datagenerated in this study will also be used to develop andvalidate the population balance models. This studyconsiders aqueous in organic (A/O) dispersions. Thoughthe A/O dispersions are not preferred in continuousplants as during settling they tend to give thickerdispersion band for the same specific settling rate (Lottet al., 1972) thereby requiring large settlers whichadversely affect the plant economics, still in manyinstances the end stages in a mixer–settler cascade arepreferably operated as A/O system to avoid loss ofcostly organic solvent through entrainment. Since the

study aims at developing suitable correlations for therepresentative drop sizes, it is worthwhile to reviewdifferent correlations for representative drop sizes inliquid–liquid dispersion in stirred tanks reported in theliterature and the semiempirical theory behind thefunctional form used in the majority of the correlations.

2. Literature review

Owing to the immense industrial importance ofliquid–liquid dispersions, several studies on experimen-tal measurements of drop size in liquid–liquid disper-sions in stirred tanks have been reported in literature.Some of these studies are summarized in Table 1.Majority of them (Vermeulen et al., 1955; Rodger et al.,1956; Weinstein and Treybal, 1973; Mlynek andResnick, 1972; Fernandes and Sharma, 1967; Brownand Pitt, 1974; McManamey, 1978; Calabrese et al.,1986a,b; Wang and Calabrese 1986; Nishikaw et al.,1987a,b; Laso et al., 1987; Chatzi et al., 1991; Zhou andKresta, 1998; Pacek et al., 1999; Ruiz et al., 2002;Desnoyer et al., 2003; Giapos et al., 2005; Sechremeliet al., 2006) have been done on batch vessels. Only fewof them have been done on continuous flow stirred tanks(Wienstein and Treybal, 1973; Fernandes and Sharma,1967; Quadros and Baptista, 2003). In most of the cases,the experimental data have been correlated using thefunctional form developed by Hinze (1955) and modi-fied by subsequent researchers (Shinnar and Church,1960; Doulah, 1975). In some cases, altogether differentfunctional forms are reported (Wienstein and Treybal,1973; Quadros and Baptista, 2003). Since model ofHinze (1955) has been used in majority of cases, it isbriefly discussed below for sake of completeness.

The model is essentially based on identification andquantification of the restoring and disrupting forcesacting on a drop of diameter d in a turbulent flow field.The dynamic pressure fluctuations or turbulent pres-sure fluctuations cause a stress τt to act on the surfaceof the drop. Owing to the deformation of drop, aninternal flow within the drop is established giving riseto an internal dynamic pressure. This dynamic pressureis of the same order of magnitude as the external stressand causes flow velocities of the order of

ffiffiffiffiffiffiffiffiffiffiffist=qd

p. The

viscous stresses associated with this flow are of theorder of ld=dð Þ ffiffiffiffiffiffiffiffiffiffiffi

st=qdp

and tend to counteract thedeformation of the drop. Further more, interfacialtension σ also gives rise to a surface stress of order ofmagnitude σ /d to counteract the deformation. Thesethree stresses τt, ld=dð Þ ffiffiffiffiffiffiffiffiffiffiffi

st=qdp

and σ /d govern thedeformation and breakup of the drop. Combination ofthese three stresses gives two dimensionless groups

123

which control drop deformation and breakup. Thesegroups are:

We ¼ dstr

ð2Þ

Vi ¼ ldffiffiffiffiffiffiffiffiffiffidqdr

p ð3Þ

The first group is the called generalizedWeber group,which is the ratio of the disrupting force due toturbulence to the restoring force due to interfacialtension. The second group is termed as viscosity group.Breakup of drop is assumed to occur when generalizedWeber group assumes a critical value Wecrit, expressionfor which can be given as:

Wecrit ¼ c1 1þ u Við Þ½ � ð4Þ

where φ is a function of Vi and reduces to zero whenVi→0. The effect of dispersed phase viscosity is thusto increase the critical Weber number. The dynamicturbulent pressure fluctuations as seen by a drop arecaused by changes in velocity over a distance equal tothe diameter of the drop. If these fluctuations areassumed to be responsible for the breakup of drops thensubstitution of qc

Pu2 for τt on dimensional ground in

Eq. (2) leads to:

Wecrit ¼ qcd0max

Pu2

rð5Þ

where dmax0 is the maximum stable drop diameter i.e. the

drop diameter for which Weber number is equal to thecritical Weber number. For drops larger than dmax

0 , theWeber number will be more than the critical number andhence they will not be stable and will under go breakage.Pu2 is the average value of the square of velocitydifferences over a distance equal to maximum stabledrop diameter, dmax

0 . It is assumed that the drops arebroken by eddies having sizes of the order of the dropdiameter. While the smaller eddies collide with thedrops but fail to break them, the larger eddies convey thedrops rather than breaking them. Therefore, if it isassumed that the drop sizes are of the order of the lengthscale of the inertial subrange of the turbulence spectrum,Pu2 of Eq. (5) will be independent of viscosity and can beexpressed as

Pu2~ d0maxe

� �2=3 ð6Þ

where e is specific turbulence energy dissipation rate. IfVi→0, then combining Eqs. (4), (5) and (6), maximumstable drop diameter can be expressed as:

d0maxqcr

� �3=5e2=5 ¼ c2: ð7Þ

For stirred tanks having agitator of diameter D,rotating at speed of N with fully established turbulence:

e~N 3D2: ð8ÞThis reduces Eq. (7) to

d0max ¼ c3rqc

� �3=5

D�0:8N�1:2 ð9Þ

which can be rearranged further to

d0max

D¼ c3We�0:6

I ð10Þ

where, WeI is a dimensionless group called the impellerWeber number and is defined as

WeI ¼ N2D3qcr

: ð11Þ

Eq. (10) is the most popular functional form tocorrelate maximum stable drop diameter in dilutedispersions. Frame work of Hinze (1955) has beenextended to account for the drops that are smaller thanthe characteristic length scale of eddies in inertialsubrange of turbulence spectrum (Shinnar and Church,1960). In this case

Pu2 depends on viscosity also and can

be expressed, instead of Eq. (6), as follows:

Pu2~

em

� �d0max

� �2: ð12Þ

Using Eqs. (4), (5), (8) and (12)

d0max

D¼ c4Re

�1=3I We�1=3

I ð13Þ

where

ReI ¼ ND2

mð14Þ

where, ReI is a dimensionless number called impellerReynolds number. Another important parameter, effectsof which can be significant in dispersions encounteredin practice is the dispersed phase holdup. A high valueof dispersed phase hold up is likely to increase themaximum stable drop diameter both by damping of

Table 1Summary of some of the studies on liquid–liquid dispersions in mechanically agitated contactors

Reference Experimental Correlations Remarks

Vermeulen et al.(1955)

Several systems both A/O and O/A were studied inbatch experiments. Paddle impeller was used. Hold upvalues were between 0.1 to 0.4. Drop size measurementwas done using light transmission technique.

N2d5=3D4=3qmrf 5=3/

¼ 0:016

f/ ¼ dd/¼0:1

qm ¼ 0:6qd þ 0:4qc

Effect of holdup on drop size was accounted for through afunction fϕ. Plot of fϕ versus ϕ was reported.

Rodger et al.(1956)

Seventeen different O/A dispersions were studied inbatch experiments. Turbine impeller was used in thestudy. Holdup was equal to 0.5 in all experiments.Direct photography and light transmission were usedas drop size measurement techniques.

Pa ¼ KD

D3N2qcr

0:36DT

� �k mdmc

� �15 tto

� �16

exp 3:6Dqqc

W

Correlation for specific interfacial area was obtained. Settlingtimes were also measured and included in the correlation toobtain good fit. t0 is the reference settling time equal to1 min.

Fernandesand Sharma(1967)

Dispersions of several esters in caustic soda werestudied. Batch and a few continuous experiments weredone to study the overall specific interfacial area byfollowing a fast pseudo first order reaction. Differentimpellers — disk turbines, paddle impellers andpropellers were studied. Experiments in tanks ofdifferent sizes were also done to study the effect ofscale. Holdup was varied between 0.1–0.5.

PaaNDT1=2/PaaND/

No practical difference between batch and continuousoperation drop sizes was observed. First correlation is forturbines, paddles and propellers of Tb40 cm, the secondcorrelation is for turbines with TN40 cm.

Mlynekand Resnick(1972)

Batch experiments were done with mixture of CCl4and iso-octane as dispersed and distilled water ascontinuous phase. Rushton turbine impeller was used.Holdup varied between 0.025 and 0.25. Measurementof drop size was done by encapsulation of drops in apolymeric film using a specially designed trap.

d32D

¼ 0:058We�0:6I 1þ 5:4/ð Þ Variation in local drop sizes was found to be small.

Wiensteinand Treybal(1973)

Eight different systems, O/A and A/O were studied.Both batch and continuous experiments were done.Holdup varied between 0.08–0.6. Light transmissionmethod was used for drop size measurement. Turbineimpeller was used.

d32 ¼ 10 �2:316þ0:672/ð Þm0:0722c e�0:194 rqc

� �0:196

d32 ¼ 10 �2:066þ0:732/ð Þm0:047c e�0:204 rqc

� �0:274

First equation is for batch vessels. The second equation is forcontinuous vessels. In drop size correlations for continuoussystems, residence time was not included, to account for itseffect a separate correlation for hold up in terms of residencetime was proposed.

Brown and Pitt(1974)

Three O/A systems with MIBK, kerosene, n-butanolas dispersed phase and water as continuous phasewere investigated. Holdup was equal to 0.05.Photoelectric probe was used for drop sizemeasurement. Disk turbine impeller was used.

d5=332

qre2=3

� � e1=3tcT2=3

� �¼ c

NtcWT

� �DT

� �8=3

¼ 0:0122

Drop sizes were measured at impeller tip. tc is the circulationtime given by second equation. The term in the secondbracket of left hand side of first equation was include toaccount for the effect of geometrical parameters in thecorrelation for drop size.

McManamey(1978)

Used experimental data of other authors.d32 ¼ c

rq

� �0:6

e�0:4i

Showed that the whole power dissipation should beassumed to occur in the impeller swept volume only andthis value should be use to correlate the drop size.

124

Calabrese et al.(1986a)

Study was aimed at finding effect of dispersed phaseviscosity on drop size. Silicone oils with viscosity lessthan 0.5 Pas were calledmoderately viscous, with 1 Pasintermediately viscous and more than 4 Pas as highlyviscous. Five different grades of silicone oil in waterwere used to obtain dispersed phases of varied viscosity.Hold upwas 0.0015. Batch experiments were donewithdirect photography as drop size measuring technique.Rushton turbine impeller was used.

d32d0

¼ 1þ 11:5qcqd

� �1=2lde1=3d1=332

r

" #5=3

d32D

¼ 2:1ldlc

� �3=8 qcND2

lc

� ��3=4

d32 ¼ 0:6dmax

d32 ¼ 0:5dmax

Data for intermediately viscous oils showed a lot of scatterand could not be correlated. First and third equations are formoderately viscous oils. The second and fourth equations arefor highly viscous oil. Dependence of d32 on μd for highlyviscous oils was different from as expected from asemiempirical model. Log-normal drop size distributionswere obtained. d0 is the diameter of inviscid drop.

Wang andCalabrese(1986)

Objective of the study was to establish relativeimportance of dispersed phase viscosity andinterfacial tension on drop size. Silicone oils weredispersed in water, methanol and their solution. Batchexperiments were done with Rushton turbine impeller.Holdup was less than 0.002. Direct photography wasused for drop size measurement.

d32D

¼ 0:066We�0:66I 1þ 13:8V 0:82

i

d32D

� �0:33" #0:59

Vi ¼ ldNDr

qcqd

� �1=2

μd varied between 0.001-1 Pas and σ varied between 0.001–0.045 N/m. Transition from low to moderate viscositybehavior to high viscosity behavior was found to shifttoward high viscosity as σ reduced. The equation is valid forμdb0.5 Pas. For μd=1 Pas, a lot of scatter was observed.

Calabrese et al.(1986b)

Total 349 data from published studies, including thatof previous two studies, were used to obtain thecorrelation of broader utility.

d32D

¼0:054 1þ3/ð ÞWe�0:6I 1þ 4:42 1�2:5/ð ÞVi

d32D

� �13

" #35

Unlike two previous studies, correlation here accounts for theeffect of high hold up.

Nishikawa et al.(1987a)

Honey bee's wax was used as dispersed phase in hightemperature batch mixing experiments with Rushtonturbine impeller. Distilled water or millet jelly wasused as the continuous phase. Holdup was variedbetween 0.005–0.36. Measurement technique usedwas stabilization of dispersion by siphoning it intochilled water followed by imaging under themicroscope.

d32 ¼ 0:105e�2=5 D

T

� �65 1þ 2:5/2=3� � ld

lc

� �15

d

ldlc

� �18

c

rq

� �35

d32 ¼ 0:0371e�1=4 d

D

� �34 1þ 3:5/3=4� � ld

lc

� �15

d

ldlc

� �18

c

rq

� �38

d32 ¼ 0:5dmax

d32 ¼ 0:45dmax

It was argued that depending on value of specific power inputdispersions can be coalescence or breakup controlled. Asspecific power input increases transition from breakupcontrolled to coalescence controlled takes place. First andthird equations are for breakup controlled and second andfourth for coalescence controlled regime. Note thatcorrelation for coalescence controlled regime is notdimensionless. Correlation for transition value of specificpower input was also given.

Nishikawa et al.(1987b)

Honey bee's wax was used as dispersed phase in thedistilled water as continuous phase. Batch experimentswere done with disk turbine in tanks of different sizes tostudy the effect of scale. Measurement technique usedwas stabilization of dispersion by siphoning it intochilled water followed by imaging under themicroscope.

d32 ¼ 0:105e�25

DT

� �65 T

T0

� ��25 1þ 2:5

TT0

� �12/2=3

0B@

1CA ld

lc

� �15

d

ldlc

� �18

c

rq

� �35

d32 ¼ 0:0371e�14

dD

� �34 T

T0

� ��144 1þ 3:5

TT0

� �12/3=4

0B@

1CA ld

lc

� �15

d

ldlc

� �18

c

rq

� �38

The correlations of the previous study were modified toincorporate the terms for the effect of scale. The first equationis for breakup controlled regime, the second equation is forcoalescence controlled regime. T0 is the reference tankdiameter equal to 25 cm.

Laso et al.(1987)

Dispersed phase was a mixture of CCl4 with n-heptaneor 1-octanol or MIBK. Continuous phase was water.Batch studies with flat blade turbine in a baffled tank.Drop size measurement was done by siphoning thedispersion into a capillary, photographing it andsending it back to the tank.

d32D

¼ 0:118We�0:4I /0:27 ld

lc

� ��0:056 Holdup values not mentioned clearly, however from one ofthe presented data it seems to be 0.09.

(continued on next page) 125

Table 1 (continued)

Reference Experimental Correlations Remarks

Chatzi et al.(1991)

Batch experiments with styrene as the dispersed phase,water with 0.1 g/l PVA as the continuous phase.Holdup was equal to 0.01. Impeller used was Rushtonturbine. Drop sizes were measured with particle sizeanalyzer.

d32D

¼ 0:045 F0:003ð ÞWe�0:6I

Drop size distributions observed were bimodal.

Zhou and Kresta(1998)

Batch experiments with water as the continuous andsilicone oil as the dispersed phase. Four differentimpellers — A310, HE3, PBT and RT were used.Holdup was 0.0003. PDPA (Phase Doppler ParticleAnalyzer) was used for drop size measurement.

d32 ¼ 118:6 emaxND2

� ��0:270 Objective of the study was to compare different scale-upcriteria such that data for different impellers can berepresented by a single correlation. Relation between d32and dmax was not found to be strictly linear.

Pacek et al.(1999)

Batch experiments were done with water as thecontinuous phase and chlorobenzene or sunflower oilas the dispersed phase. Different impellers — highshear (RT), high flow (Chemineer HE3) and ultra highshear (Chemineer CS) were used. Measurements weredone by direct photography near the tank wall.Dispersed phase holdup ranged between 0.01to 0.05.

d32aeb Values of the exponent, ranging between -0.47 to -0.72, were

tabulated as a function of impeller type, dispersed phase anddispersed phase holdup.

Ruiz et al.(2002)

Batch experiments were done with 0.25 M sodiumsulfonate solution as continuous phase and a 1:1mixture of salicylaldoxime (LIX-860 N-IC) and aketoxime (LIX84-IC) in an aliphatic diluent (ESCAID103) as the dispersed phase. Dispersed phase holdupranged between 0.006–0.018. Pump-mix doubleshrouded impeller with eight curved blades wasused in the study. Single point measurement wasdone by siphoning dispersion in a gelatin solutionfollowed by quick cooling in an ice bath to freeze thedrops.

dmax

D¼ 0:353We�0:6

I

Number volume distributions were found to be log normal.Effects of temperature, extractant concentration and pH werealso investigated. Increase in temperature caused drop size toreduce. Extractant concentration in the investigated range of 7-20% (w/w) did not affect the drop size. Reduction in pHresulted in smaller drops.

126

Desnoyer et al.(2003)

Batch experiments with a mixture of TBP andSobesso 150 as the continuous phase and 20 g/lNiCl2 (a fast coalescing system) or 3 M HCl (a slowcoalescing system) solution as the dispersed phase.Hold up values ranged between 0.1–0.6. Impellerused was PBTU. Drop size measurements were doneusing laser granulometer.

d32D

¼ 0:28We�0:6I 1þ 0:92/ð Þ

d32D

¼ 0:14We�0:6I 1þ 0:48/ð Þ

d32D

¼ f /ð ÞWe�n /ð Þ=2I

Though the Sauter mean diameters could be correlated usingthe frame work of Hinze (1955) and Doulah (1975), theycould be better correlated with the functional form given inthe third equation with exponent of Weber number showingdependence on holdup value. The first equation is for NiCl2as the dispersed phase and second for HCl as the dispersedphase.

Quadrosand Baptista(2003)

Experiments to determine interfacial area in continuousflow stirred tank with di-isobutylene diluted withbenzene as the dispersed and sulfuric acid as theaqueous phase. Impellers used were straight bladepaddles with two or four number of blades. A chemicalmethod was used to determine overall interfacial areaunder different operating conditions. Hold up valuesranged between 0.061–0.166. Impeller speeds werevaried to cover a large range of Weber number.

d32 ¼ 6/ 1þ c1WeI/

� �2( )

c2/2 þ c3/

� �d32D

¼ 0:0336We�0:6I 1þ 13:76/ð Þ

d32D

¼ 0:0286We�0:6I 1þ 13:24/ð Þ

The first functional form was found to represent the Sautermean diameters over a wide range of Weber number.However, for high Weber numbers (WeN1900), Sautermean diameter could be correlated using the frame work ofHinze and Doulah. The second correlation is for two bladepaddle while the third correlation is for four blade paddle.

Giapos et al.(2005)

Batch experiments were done to understand the effectof number of blades on drop size for kerosene indistilled water dispersions. Direct photography wasemployed to measured drop size which was renderedpossible due to low values of holdups rangingbetween 0.01–0.07. Disk turbines with 2, 4, 6 and8 number of blades were used in this study.

d32~N�a The exponent a was found to vary between 0.62 to 1.23depending upon the value of holdup. As expected, animpeller with more number of blades gave finer dropscompared to an impeller having less number of blade androtating at the same speed.

Sechremeli et al.(2006)

Batch were done to compare drop sizes produced by adisk turbine and an open impeller having samediameter and blade width. Distilled water was usedas continuous and kerosene as the dispersed phase.Direct photography was used to measure drop size.Holdup values ranged between 0.01–0.1.

d32~N�a Values of exponent were found to range between 0.61 - 1.03.For open impeller (power number=4), the Sauter meandiameters were larger by 6-82% than the closed impeller(power number=5). Proportionality between dmax and d32was not observed.

127

128

turbulence and enhancement of frequency of dropcoalescence by increasing drop population density inthe continuous phase. Considering damping of turbu-lence in concentrated dispersions, Doulah (1975)proposed the following expression to account for theincrease in drop size with increase in hold up of dispersedphase.

dmax

d0max

¼ 1þ 2:5/ð Þ qcqm

� �6=5

ð15Þ

which can be generalized to the following functionalform considering terms of ϕ2 onward negligible

dmax

d0max

¼ 1þ c5/þ c6/2� �: ð16Þ

Using Eq. (10) for dmax0 and Eq. (16), the functional

form of the maximum stable drop diameter for highholdup of dispersed phase assumes the following form

dmax

D¼ c3 1þ c5/þ c6/

2� �We�0:6

I : ð17Þ

This functional form has been the most extensivelyused by different researchers to correlate their experi-mental data for concentrated dispersions. Functionalform of Eq. (17) assumes that the dispersed phase is notviscous or that the viscosity of the dispersed phase doesnot contribute to the stabilization of the drop. Severalstudies, however, have focused on stabilization due todispersed phase viscosity (Calabrese et al., 1986a,b;Wang and Calabrese 1986). One of such studies(Calabrese et al. 1986b), based on a large number ofexperimental data, proposes the following expressionfor the Sauter mean diameter:

d32D

¼0:054 1þ3/ð ÞWe�0:6I 1þ4:42 1� 2:5/ð ÞVi

d32D

� �13

" #35

ð18Þwhere

Vi ¼ ldNDr

qcqd

� �1=2

: ð19Þ

Note that, the first part of the right hand side ofEq. (18) uses the frame work of Hinze (1955) as modi-fied by Doulah (1975), the second part, the term in thebracket, accounts for the stabilization effect of thedispersed phase viscosity.

Apart from the models discussed above, there existsanother class of models that views the drop deformation

and breakage process as Vigot element in which thestabilization effect of interfacial tension is representedby a spring element and stabilization effect due todispersed phase viscosity as the dash pot. Both of themact in parallel to resist the deforming stress due toturbulence (Arai et al., 1977; Lagisetty et al., 1986).These models have been extended to account for effecton drop size of dispersed phase rheologies (Lagisetty etal., 1986; Koshy et al., 1988b, Gandhi and Kumar1990), presence of surfactants (Koshy et al., 1988a) andcirculation patterns (Kumar et al., 1992). Some modelsaccounting for breakage mechanisms different from thatdue to turbulent pressure fluctuations have also beenproposed (Kumar et al., 1991; Wichterle, 1995; Kumaret al., 1998). A few studies focusing on the fine detailslike the effect of intermittency of turbulence onrepresentative drop diameter have also been presented(Baldyga and Bourne, 1993). However, as far thefunctional form used to fit the experimental data isconcerned, the model of Hinze (1955) as extended byDoulah (1975) continues to be the most popular one.

3. Experimental

Fig. 1 shows the schematic diagram of the experi-mental setup used in this study. The experimental setupconsists of a cylindrical tank of 240 mm diameter and240 mm height. Four baffles, each of 220 mm height,having width equal to 10% of tank diameter, areprovided to prevent vortex formation and enhancemixing. At the bottom plate of the tank is a suctionorifice with diameter equal to 1/4 of the tank diameter.The suction orifice provides connectivity between thetank and a cylindrical chamber called suction box(65 mm diameter and equal height), which in turn isconnected with two feed tanks. One of the feed tankscontains organic phase and the other aqueous phase. Afour-bladed top shrouded turbine with trapezoidalblades, a pump-mix impeller, is used in the study.Diameter of the turbine is 148 mm, blade width is31 mm and blade length is 37 mm. Shaft diameter is12 mm, hub height is 13 mm and hub outer diameter is20 mm. Thickness of disk and blade is 2 mm. The off-bottom clearance of the impeller is equal to half of themixer height. In this position, the impeller has a powernumber of 3.0. The mixer has two ports for samplewithdrawal with rubber-clip arrangement — one in theplane of the impeller disk and the other in a planehalfway between the impeller disk and bottom of themixer. Organic phase used in the experiments is amixture of n-paraffin, D2EHPA and TBP. The aqueousphase is 30% phosphoric acid. The physical properties

Table 2Physical properties of the phases (at 25 °C) and range of variablesstudied

Phase ρ (kg/m3) μ (cP) σs (N/m)

Organic 859 4.01 0.0267Aqueous 1328 4.28 0.0194

N (rpm) τ (min) ϕf

Range of variables 100–150 0.5–2.0 0.2–0.5

Fig. 1. Schematic diagram of the experimental setup.

129

of the phases are given in Table 2. At steady state, thedispersion created in the mixer overflows from thecentral opening (diameter equal to half of the tankdiameter) in the top baffle to a horizontal settler througha full width launder. In the settler, separation of phasesoccurs and the clear organic flows from the top to theorganic storage tank, the aqueous phase flows frombottom to the aqueous storage tank through an interfacecontroller. From the storage tanks the phases arepumped back to the respective feed tanks from wherethey are again sucked into the mixer by the pumpingaction of the pump-mix impeller. The flow rates aremaintained at desired values using the valves andflowmeters located downstream of the pumps. Thus thesystem is operated in a closed loop.

Before starting the actual experiments, phases wereequilibrated by running the system for 24 h. All theexperiments were conducted with aqueous as thedispersed phase. Filling the mixer fully with organic atthe time of startup ensured organic continuity. Total 27experiments were conducted for different impellerspeeds, feed phase ratios and mean residence times ofthe phases in the mixer. The range over which thesevariables were varied is given in Table 2. Unlike thebatch mixing studies where impeller speed can be variedover a wide range (between the speed required for visualhomogenization and speed at which air entrapment takesplace), the upper limit of the speed in the continuous

mixing studies is limited by the phase separationcharacteristics of the dispersion in the settler. Owingto a low interfacial tension, the systems studied here iseasy to disperse and difficult to separate and even at lowimpeller speeds the dispersion band tends to flood thesettler. This effectively limits the upper limit of speedvariation and hence the relatively narrow range ofimpeller speed as given in Table 2.

For each experiment, attainment of steady state wasobserved by following the thickness of the dispersionband in the settler which was measured every 15 minand steady state was deemed to have been achievedwhen three consecutive measurements were almostsame. Typical time required to attain the steady state wasabout 3 h. After attainment of steady state, samples ofdispersion were withdrawn from sample ports intomeasuring cylinders. From the volumes of the settledphases, local values of the hold up were computed. To

Fig. 2. A typical image of the stabilized dispersion.

130

measure the drop sizes, the sample of dispersion waswithdrawn into petri dish containing the organic phaseladen with a surfactant (sorbitan monooleate) presenceof which prevented the coalescence of drops and thusstabilized the dispersion. The dish containing stabilizeddispersion was kept under a microscope mounted on acamera which in turn was connected to a personalcomputer. Since the field of view of this imaging systemwas small, the full image of the stabilized dispersion wasnot obtained in a single frame. Therefore, by system-atically moving the dish under the microscope severalimages, sufficient to give adequate numbers of drops, ofthe stabilized dispersion were captured. Utmost care wastaken to avoid imaging the same area twice. The imageswere analyzed using an image processing software.

Fig. 3. Comparison of Sauter mean diameters at upper and lowe

For each case, at least three hundred drops weremeasured to ensure good statistical accuracy. Fig. 2shows a typical image of the stabilized dispersion. Asimilar method of measuring the drop size in emulsionsproduced in simple shear flow has been reportedrecently (Nandi et al., 2006). At the end of eachexperiment, impeller was stopped and the valvesconnecting the feed tanks to the mixer were simulta-neously closed. Following the settling of the dispersion,the volume occupied by the separated phases weremeasured to compute the average holdup in the mixer.

From the measurements of the counted drops,maximum drop diameter, Sauter mean diameter anddrop size distribution were obtained. The Sauter meandiameter was obtained using the following expression:

d32 ¼P

d3Pd2

: ð20Þ

4. Results and discussion

4.1. Characteristic drop diameter

Most of the studies on drop size distribution in agitatedtanks have been performed in batch mode. In these studies thedispersed phase holdup is an independent variable. However,in continuous flow agitated tanks, the hold up in the tank is notan independent variable as it may change with, apart from feedphase ratio, impeller speed and mean residence time. Using theholdup in tank as a correlating variable for continuous flowagitated tanks will therefore not indicate the true dependenceof drop diameter on speed and mean residence time.Considering this, the feed phase fraction, instead of holdupin the tank, is used to correlate the drop diameter. To decipherthe functional form of the relationship between the Sauter

r sampling locations, as predicted by Eqs. (21) and (22).

131

mean diameter and the three independent variables, theexperimental data were used to train a neural network basedon back propagation algorithm. The response of the trainedneural network indicated that the Sauter mean diameter couldbe correlated as a quadratic function of the feed holdup, apower law function of the impeller speed and an exponentialfunction of mean residence time. This led to the selection ofthe functional form for the correlation of the Sauter meandiameter, as given in (21)–(23). Note that this functional formresembles the form suggested by Eq. (17) albeit with vesselholdup replaced with the feed holdup and an additionalmultiplicative function to account for the effect of meanresidence time on Sauter mean diameter. The values ofcoefficients and exponents were obtained by using Gauss–Newton algorithm. The final correlations are given below:

d32D

¼ 1:2825� 10�3N�1:681

� 1þ 2:6539/fþ1:3986/2f

� �exp 0:4457sð Þ

ð21Þ

for the upper sampling location and

d32D

¼ 2:451�10�3N�1:723

� 1�0:8785/f þ 4:2848/2f

� �exp 0:4148sð Þ

ð22Þ

for the lower sampling location.Fig. 3 shows the comparison of Sauter mean diameters as

predicted by Eqs. (21) and (22) with±10% error bars aroundthe diameters observed at the upper sampling location. As canbe seen the diameters at the upper sampling location, owing toit being in the plane of the impeller discharge stream, aremarginally smaller than the diameters at the lower samplinglocation. However, difference being small, it can be assumedthat the quality of dispersion is more or less homogeneous and

Fig. 4. Parity plot for the Sauter mean

it is better to have a single correlation for Sauter mean diameterfor practical utility. This correlation was found to be:

d32D

¼ 1:849�10�3N�1:7025

� 1þ0:392/f þ3:2435/2f

� �exp 0:4302sð Þ: ð23Þ

Fig. 4 shows the parity plot for Eq. (23). The quantitativeestimate of the goodness of the fit of correlation given byEq. (23) is given in Table 3. The ±95% confidence intervals(Draper and Smith, 1966) of the regressed model constants aregiven in Table 4. As can be seen that for most of the constants theconfidence intervals are reasonably narrow thereby indicatinggood statistical accuracy of the correlation given by Eq. (23).

Since in the present study effects of impeller diameter andphysical properties on drop size have not been investigated,Weber number will not be a proper correlating variable.However, in order to be in line with the correlation reported inliterature, it would be interesting to recast Eq. (23) in terms ofWeber number, as follows

d32D

¼ 2:946�10−6We�0:85125I

� 1þ0:392/fþ3:2435/2f

� �exp 0:4302sð Þ: ð24Þ

In the above correlations τ is the mean residence time, inminutes, of the phases in the mixer. The above correlationsshow that the characteristic drop size reduces with increase inspeed and increases with an increase in feed holdup. This is inconformity with the trend of semiempirical model representedby Eq. (17). The exponent on N is also close to the value of−1.2 as suggested by the semiempirical model. The correla-tions show that, for the range of the experiments, the drop sizeincreases with increase in mean residence time. This appears tobe contradictory to the expectation that increasing meanresidence time should ensure longer interaction between theimpeller and the droplets and hence should result into a smaller

diameter predicted by Eq. (23).

Table 3Goodness of fit of different correlations presented in this work

Equation For Emax Eavg Beyond ±15%

21 Sauter mean diameter at the upper sampling location 21.7 7.5 5/2722 Sauter mean diameter at the lower sampling location 29.0 8.8 6/2723 Average Sauter mean diameter 27.6 8.5 10/5425 Local holdup at the upper sampling location 8.9 2.4 0/2726 Local holdup at the lower sampling location 11.1 2.7 0/2727 Vessel average holdup 6.2 2.0 0/2728 Relationship between Sauter mean diameter and dmax 22.2 7.3 3/5430 Variance of the log-normal drop size distributions 39.6 9.3 9/5431 Mean of the log-normal drop size distributions 5.3 1.9 0/54

132

characteristic drop size. This will be explained in the nextsection.

It would be interesting to compare the correlation of Eq. (23)with some of the correlations compiled in Table 1 for theireffectiveness to predict the experimental data reported in thiswork. Fig. 5 shows this comparison.

As can be seen from Fig. 5, whereas the correlation of Eq. (23)gives predictions closer to the experimental data, the othercorrelations reported earlier exhibit large deviations. This is notsurprising considering that the drop size distributions in stirredliquid–liquid dispersions are highly system specific due to theirbeing dependent on the physical properties of the phases, phasecontinuity, impeller type, mode of operation and geometricconfiguration of the mixer. Among the compared correlations,the correlation obtained with TBP containing organic as thecontinuous phase and an acidic phase as the dispersed phase(Desnoyer et al., 2003) exhibits the least deviation form thecorrelation given by Eq. (23). This may be attributed to theidentical phase continuity and similar nature of the phases.

4.2. Holdup

To have an estimate of the specific interfacial area, inaddition to the correlation for the Sauter mean diameter, thecorrelation for dispersed phase hold up in the mixer is alsorequired. Whereas, in a batch mixer, hold up of the dispersedphase varies from one location to another (Wang and Mao,2005; Wang et al., 2006), for a continuous flow stirred tankboth local and vessel average values, in general, can bedifferent from the feed holdup. The local and vessel average

Table 4Confidence intervals for model constants of Eqs. (23) and (27)

Equation Model constant

23 Multiplicative constant on right hand side23 Exponent on N23 Coefficient of ϕf

23 Coefficient of ϕf2

23 Coefficient of τ27 Multiplicative constant on right hand side27 Exponent on ϕf

27 Exponent on N27 Exponent on τ

values will depend on the impeller speed, mean residence timeand feed holdup. The local and vessel average holdup in thepresent study could be correlated as a power law function ofthe independent variables. These correlations are as follows:

/ ¼ 2:0274/0:3569f N�1:1288s�0:1153 ð25Þ

for the upper sampling location and

/ ¼ 2:3173/0:3583f N�1:2389s�0:1389 ð26Þ

for the lower sampling location and

P/ ¼ 2:1214/0:4012

f N�1:097s�0:0631 ð27Þ

for vessel average holdup values.Apart from being useful in predicting the specific

interfacial area, these correlations are important in decidingthe intrastage recycle flow rate in order to have a desired phaseratio in the mixer. As expected, the local and vessel averageholdup increase with increasing feed holdup. In the presentcase, the heavy phase being the dispersed phase and the outletof dispersion being from the top of the mixer, the holdupvalues in the tank were observed to be larger than the feedholdup. For any system, increasing influence of impellershould tend to bridge the gap between the holdup in the vesseland the feed holdup. The increasing influence of the impellercan be due to increasing impeller speed or increasing meanresidence time. For the case of heavy phase as dispersed phaseand outlet from the top of the mixer, the hold up in the vessel

Regressed value ±95% confidence interval

0.001849 (0.001758, 0.001039)−1.7025 (−1.7779, −1.6450)0.3920 (0.1851, 0.5970)3.2435 (2.7371, 3.7451)0.4302 (0.3824, 0.4683)2.1214 (2.1091, 2.1743)0.4012 (0.3876, 0.4147)

−1.0970 (−1.1169, −1.0770)−0.0631 (−0.1190, −0.0069)

Fig. 5. Comparison of correlation of Eq. (23) with some of the correlations reported earlier.

133

should, therefore, reduce with increasing impeller speed andincreasing mean residence time. This explains the negativeexponents on N and τ in Eqs. (25)–(27). Now the dependenceof the Sauter mean diameter on mean residence time, as seenearlier, can be explained. The increasing residence timereduces the dispersed phase holdup in the mixer. This willtend to reduce the drop size. However, the heavy phase beingthe dispersed phase, the effective density of the dispersion alsoreduces and for the same impeller speed power introduced inthe system also reduces. This reduction in the power inputtends to increase the drop size. The eventual trend ofdependence of drop size on mean residence time will,therefore, depend on the relative importance of these twoopposing effects and can be either positive, as seen in thepresent case, or negative. At high impeller speeds, the holdupin the tank will be very close to the feed holdup irrespective ofthe mean residence time and in that case it is quite likely thatthe mean residence time will not have an effect on the drop

Fig. 6. Relationship between the maximum drop

size. This probably explains the similar drop sizes observed forboth batch and continuous mixers in an earlier study(Fernandes and Sharma, 1967). Eqs. (21), (22) and (23)when used with Eqs. (25), (26) and (27), respectively, give thelocal and vessel average specific interfacial area for specifiedvalues of the independent variable (impeller speed, meanresidence time and feed holdup). Quantitative estimate ofgoodness of fit of Eqs. (25)–(27) is given in Table 3. The±95% confidence intervals of the evaluated model constants ofEq. (27) are given in Table 4.

4.3. Relationship between d32 and dmax

Several studies report on direct proportionality between theSauter mean diameter and maximum stable drop diameter(Brown and Pitt, 1972; Nishikawa et al.,1987a; Collias andPruddhornme, 1992; Zerfa and Brooks, 1996; Calabrese et al.,1986a). However, a few studies contradict this view (Zhou and

diameter and the Sauter mean diameter.

Fig. 7. Comparison of experimental and log-normal drop size distributions for Qo=300 lph, Qa=200 lph, N=120 rpm, class interval equal to 20 μm.

134

Kresta, 1998; Sechremeli et al., 2006). The system studied here,however, tends to exhibit the direct proportionality between thetwo representative diameters, as shown inFig. 6. Considering theinherent uncertainty associated with measurement of dmax (thereis only one drop with diameter equal to dmax and hence there arefair chances that it can escape imaging), the linear fit of Fig. 6 isreasonably good. The two representative diameters can becorrelated by the following equation:

d32 ¼ 0:5446dmax: ð28ÞThe constant of Eq. (28) compares well with the values

reported in the literature, as reported for some of the previousstudies summarized in the Table 1.

4.4. Drop size distributions

Several distributions were tried but log-normal distributionswere found to give the best fit to the experimentally observeddrop size distributions.

Fig. 8. Comparison of experimental and log-normal drop size distributions for

The number probability density of this distribution is givenby:

f dð Þ ¼ 1

dsffiffiffiffiffiffi2p

p exp � lnd � lPð Þ22s2

:

" #ð29Þ

Figs. 7 and 8 show the graphical comparison of experi-mental and log-normal drop size distributions for two of thecases. A good agreement between the number probabilitydensity as predicted by log-normal distribution and experimen-tal distribution can be observed.

s and μ– of Eq. (29) could be correlated as a power lawfunction of impeller speed, feed holdup and mean residencetime as follows:

s ¼ 0:806 N�0:838/�0:0578f s−0:0788 ð30Þ

lP ¼ 6:3873 N�0:2078/0:1406f s0:1508: ð31Þ

Goodness of fit of Eqs. (30) and (31) is quantified in Table 3.

Qo=400 lph, Qa=150 lph, N=120 rpm, class interval equal to 20 μm.

135

5. Conclusions

Measurements of drop size distributions have beenperformed for aqueous in organic dispersions incontinuous flow stirred tank with a mixture of n-paraffin,D2EHPA and TBP as the organic phase and dilutephosphoric acid as the aqueous phase. Surfactantstabilization has been used as drop size measuringtechnique. The experimentally measured drop sizedistributions exhibit log-normal behavior in dropletnumber density. The characteristic drop sizes exhibitinsignificant spatial variations. The Sauter mean dia-meters have been correlated with impeller speed,dispersed phase feed fraction and mean residence time.Local and vessel average values of hold up have alsobeen correlated. Proportionality between the Sautermean diameter and the maximum stable diameter isobserved.

Nomenclaturea Specific interfacial area [L−1]ci Constant [−]D Impeller diameter [L]d Droplet diameter [L]d0 Diameter of inviscid drop in correlation of

Calabrese et al. (1986a) [L]d32 Sauter mean diameter [L]dmax Maximum stable drop diameter [L]dmax0 Maximum stable drop diameter in dilute

dispersions [L]Eavg Average percentage error [−]Emax Maximum percentage error [−]f (d) Number probability density distribution [L−1]f (ϕ) A function of hold up in correlation of

Desnoyer et al. (2003) [−]K A constant in correlation of Rodger et al.

(1956) [−]n(ϕ) A function of hold up in correlation of

Desnoyer et al. (2003) [−]N Impeller speed [T−1]Qa Aqueous phase flow rate [L3T−1]Qo Organic phase flow rate [L3T−1]ReI Impeller Reynolds number [−]t Settling time in correlation of Rodger et al.

(1956) [T]t0 Reference settling time (equal to 1 min) in

correlation of Rodger et al. (1956) [T]T Tank diameter [L]T0 Reference tank diameter (25 cm) in correlation

of Nishikawa et al. (1987b) [L]tc Circulation time in correlation of Brown et al.

(1974) [T]

Pu2 Average relative velocity between two points

separated by a distance d in turbulent field[L2 T−2]

Vi Viscosity group [−]W Width of the impeller blade [L]We Generalized Weber group [−]Wecrit Critical Weber number [−]WeI Impeller Weber number [−]

Greek letterse Specific turbulent energy dissipation rate

[L2 T− 3]ei Specific turbulent energy dissipation rate in

impeller region [L2 T−3]emax Maximum specific turbulent energy dissipation

rate [L2 T−3]μ– Mean of log-normal drop size distributions [−]μd Viscosity of dispersed phase [ML−1T−1]μc Viscosity of continuous phase [ML−1T−1]ν Kinematic viscosity [L2 T−1]ϕ Local dispersed phase holdup [−]ϕ Vessel average dispersed phase holdup [−]ϕf Dispersed phase feed holdup [−]φ Function of Vi [−]ψ A scale-up function in correlation of Rodger

et al. (1956) [−]ρc Density of continuous phase [ML−3]ρd Density of dispersed phase [ML−3]ρm Effective density of the dispersion [ML−3]σ Interfacial tension [MT−2]σs Surface tension [MT−2]τ Mean residence time [T]τt Stress on droplet surface due to turbulent flow

field [ML−1T−2]

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