Representative drop sizes and drop size distributions in A/O
dispersions in continuous flow stirred tank
K.K. Singh a , S.M. Mahajani a,⁎, K.T. Shenoy b , S.K. Ghosh b
a
b
Department of Chemical Engineering, Indian Institute of Technology, Powai, Mumbai, 400076, India
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 a
continuous flow stirred tank agitated by a four-bladed top shrouded turbine with trapezoidal blades. The organic phase is a mixture
of n-paraffin, tributyl phosphate (TBP) and di-2-ethyl hexyl phosphoric acid (D2EHPA), the aqueous phase is dilute phosphoric
acid. Drop size measurements have been performed for different values of impeller speed, feed phase ratio and mean residence time
at two locations in the tank, near the wall. Surfactant stabilization of the dispersion has been used as the drop size measuring
technique. Log-normal distributions are found to fit the experimental drop size distributions. Experimental results have been used
to 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 stirred
tanks play an important role in hydrometallurgical
plants using mixer–settlers, wherein the objective is to
preferentially extract a valuable component from one
liquid phase into another immiscible liquid phase. The
overall extraction affected by the mixer or the stage
efficiency depends, among other things, on specific
interfacial area available for mass transfer that in turn
depends on sizes of the drops of the dispersed phase.
The sizes of the drops depend on several factors such as
impeller geometry, impeller speed, impeller location in
the tank, feed phase ratio and physical properties of the
phases. For an optimum design, a quantitative description of the effect of all these factors on the drop sizes is
required. In a stirred tank due to inhomogeneous
dissipation of power (Cutter, 1966), drop sizes exhibit
spatial variations. At a given location also, owing to
continuous redispersion and coalescence, a distribution
of drop sizes is observed. Therefore, to get an overall
picture of the quality of dispersion it makes sense to talk
of a representative drop size of the dispersion. The
Sauter mean diameter (d32) and the maximum stable
drop diameter (dmax) are the two choices of the
representative drop diameter. While the first is the
ratio of third moment of drop size distribution to second
moment of drop size distribution, the second represents
the maximum drop size that can be observed for the
122
level of turbulence prevailing in the dispersion. The
Sauter mean diameter assumes more importance
because it is directly related to specific interfacial area
ā by the following expression:
d32 ¼
6/
P
a
ð1Þ
where ϕ is the holdup of the dispersed phase. In many
instances, direct proportionality between the two representative 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 representative
diameters can be used to compute the other representative
diameter.
Experimental measurements of drop size distributions in stirred tanks are indispensable in view of the
exorbitant computational demands of a fully predictive
model that will solve the discretized population balance
equations along with the flow and turbulence equations
on a fine grid in the computational domain. The number
of equations to be solved will be very large as there will
be one equation for each drop class. This approach has
not been attempted so far. A simplification can be
achieved by solving the flow equation on the fine grid
followed by solution of population balance equations on
a very coarse grid. Some studies using this simplified
approach have been reported (Alopaeus et al., 1999;
Maggioris et al., 2000; Alopaeus et al., 2002). This still
does not obviate the need of experimental measurements
entirely as the breakage and coalescence models
(Coulaloglou and Tavlarides, 1977) embedded in the
population balance equations contain model constants
which need to be estimated for a given system by using
the experimental data.
The purpose of the present study is to investigate the
effect of three operating parameters — impeller speed,
feed phase ratio and mean residence time, on the drop
size and drop size distributions and to develop suitable
system specific correlations. The experimental data
generated in this study will also be used to develop and
validate the population balance models. This study
considers aqueous in organic (A/O) dispersions. Though
the A/O dispersions are not preferred in continuous
plants as during settling they tend to give thicker
dispersion band for the same specific settling rate (Lott
et al., 1972) thereby requiring large settlers which
adversely affect the plant economics, still in many
instances the end stages in a mixer–settler cascade are
preferably operated as A/O system to avoid loss of
costly organic solvent through entrainment. Since the
study aims at developing suitable correlations for the
representative drop sizes, it is worthwhile to review
different correlations for representative drop sizes in
liquid–liquid dispersion in stirred tanks reported in the
literature and the semiempirical theory behind the
functional form used in the majority of the correlations.
2. Literature review
Owing to the immense industrial importance of
liquid–liquid dispersions, several studies on experimental measurements of drop size in liquid–liquid dispersions 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 and
Resnick, 1972; Fernandes and Sharma, 1967; Brown
and 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 and
Kresta, 1998; Pacek et al., 1999; Ruiz et al., 2002;
Desnoyer et al., 2003; Giapos et al., 2005; Sechremeli
et al., 2006) have been done on batch vessels. Only few
of 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 the
functional form developed by Hinze (1955) and modified by subsequent researchers (Shinnar and Church,
1960; Doulah, 1975). In some cases, altogether different
functional forms are reported (Wienstein and Treybal,
1973; Quadros and Baptista, 2003). Since model of
Hinze (1955) has been used in majority of cases, it is
briefly discussed below for sake of completeness.
The model is essentially based on identification and
quantification of the restoring and disrupting forces
acting on a drop of diameter d in a turbulent flow field.
The dynamic pressure fluctuations or turbulent pressure fluctuations cause a stress τt to act on the surface
of the drop. Owing to the deformation of drop, an
internal flow within the drop is established giving rise
to an internal dynamic pressure. This dynamic pressure
is of the same order of magnitude as the external
pffiffiffiffiffiffiffiffiffiffiffistress
and causes flow velocities of the order of st =qd . The
viscous stresses p
associated
with this flow are of the
ffiffiffiffiffiffiffiffiffiffiffi
order of ðld =d Þ st =qd and tend to counteract the
deformation of the drop. Further more, interfacial
tension σ also gives rise to a surface stress of order of
magnitude σ / d to counteract
the
ffi deformation. These
pffiffiffiffiffiffiffiffiffiffi
three stresses τt, ðld =d Þ st =qd and σ / d govern the
deformation and breakup of the drop. Combination of
these three stresses gives two dimensionless groups
123
which control drop deformation and breakup. These
groups are:
We ¼
dst
r
ð2Þ
ld
Vi ¼ pffiffiffiffiffiffiffiffiffiffi
dqd r
ð3Þ
where e is specific turbulence energy dissipation rate. If
Vi → 0, then combining Eqs. (4), (5) and (6), maximum
stable drop diameter can be expressed as:
q 3=5
c
0
e2=5 ¼ c2 :
ð7Þ
dmax
r
For stirred tanks having agitator of diameter D,
rotating at speed of N with fully established turbulence:
e~N 3 D2 :
The first group is the called generalized Weber group,
which is the ratio of the disrupting force due to
turbulence to the restoring force due to interfacial
tension. The second group is termed as viscosity group.
Breakup of drop is assumed to occur when generalized
Weber group assumes a critical value Wecrit, expression
for which can be given as:
Wecrit ¼ c1 ½1 þ uðViÞ
ð4Þ
where φ is a function of Vi and reduces to zero when
Vi → 0. The effect of dispersed phase viscosity is thus
to increase the critical Weber number. The dynamic
turbulent pressure fluctuations as seen by a drop are
caused by changes in velocity over a distance equal to
the diameter of the drop. If these fluctuations are
assumed to be responsible for the breakup of drops then
P
substitution of qc u2 for τt on dimensional ground in
Eq. (2) leads to:
P
Wecrit ¼
0
qc dmax
u2
r
ð5Þ
0
where dmax
is the maximum stable drop diameter i.e. the
drop diameter for which Weber number is equal to the
0
critical Weber number. For drops larger than dmax
, the
Weber number will be more than the critical number and
hence they will not be stable and will under go breakage.
P
u2 is the average value of the square of velocity
differences over a distance equal to maximum stable
0
drop diameter, dmax
. It is assumed that the drops are
broken by eddies having sizes of the order of the drop
diameter. While the smaller eddies collide with the
drops but fail to break them, the larger eddies convey the
drops rather than breaking them. Therefore, if it is
assumed that the drop sizes are of the order of the length
scale
of the inertial subrange of the turbulence spectrum,
P
u2 of Eq. (5) will be independent of viscosity and can be
expressed as
2=3
P 0
u2 ~ dmax
e
ð6Þ
This reduces Eq. (7) to
3=5
r
0
dmax
¼ c3
D0:8 N 1:2
qc
ð8Þ
ð9Þ
which can be rearranged further to
0
dmax
¼ c3 We0:6
I
D
ð10Þ
where, WeI is a dimensionless group called the impeller
Weber number and is defined as
WeI ¼
N 2 D3 qc
:
r
ð11Þ
Eq. (10) is the most popular functional form to
correlate maximum stable drop diameter in dilute
dispersions. Frame work of Hinze (1955) has been
extended to account for the drops that are smaller than
the characteristic length scale of eddies in inertial
subrange of turbulence
spectrum (Shinnar and Church,
P
1960). In this case u2 depends on viscosity also and can
be expressed, instead of Eq. (6), as follows:
2
P e 0
u2 ~
:
ð12Þ
dmax
m
Using Eqs. (4), (5), (8) and (12)
0
dmax
1=3
1=3
¼ c4 ReI WeI
D
ð13Þ
where
ReI ¼
ND2
m
ð14Þ
where, ReI is a dimensionless number called impeller
Reynolds number. Another important parameter, effects
of which can be significant in dispersions encountered
in practice is the dispersed phase holdup. A high value
of dispersed phase hold up is likely to increase the
maximum stable drop diameter both by damping of
124
Table 1
Summary of some of the studies on liquid–liquid dispersions in mechanically agitated contactors
Reference
Experimental
Correlations
Vermeulen et al. Several systems both A/O and O/A were studied in N 2 d 5=3 D4=3 q
m
¼ 0:016
(1955)
batch experiments. Paddle impeller was used. Hold up
5=3
rf/
values were between 0.1 to 0.4. Drop size measurement
d
was done using light transmission technique.
f/ ¼
d/¼0:1
qm ¼ 0:6qd þ 0:4qc
Rodger et al.
(1956)
Fernandes
and Sharma
(1967)
Mlynek
and Resnick
(1972)
Wienstein
and Treybal
(1973)
Brown and Pitt
(1974)
McManamey
(1978)
0:36 k 15 16
Seventeen different O/A dispersions were studied in
K D3 N 2 qc
D
md
t
Dq
batch experiments. Turbine impeller was used in the P
a¼
exp 3:6
W
D
T
to
qc
r
mc
study. Holdup was equal to 0.5 in all experiments.
Direct photography and light transmission were used
as drop size measurement techniques.
Dispersions of several esters in caustic soda were P
aaNDT 1=2 /
studied. Batch and a few continuous experiments were P
aaND/
done to study the overall specific interfacial area by
following a fast pseudo first order reaction. Different
impellers — disk turbines, paddle impellers and
propellers were studied. Experiments in tanks of
different sizes were also done to study the effect of
scale. Holdup was varied between 0.1–0.5.
Batch experiments were done with mixture of CCl4 d32
0:6
and iso-octane as dispersed and distilled water as D ¼ 0:058WeI ð1 þ 5:4/Þ
continuous phase. Rushton turbine impeller was used.
Holdup varied between 0.025 and 0.25. Measurement
of drop size was done by encapsulation of drops in a
polymeric film using a specially designed trap.
0:196
Eight different systems, O/A and A/O were studied.
ð2:316þ0:672/Þ 0:0722 0:194 r
mc
e
Both batch and continuous experiments were done. d32 ¼ 10
q
c 0:274
Holdup varied between 0.08–0.6. Light transmission
r
method was used for drop size measurement. Turbine d32 ¼ 10ð2:066þ0:732/Þ mc0:047 e0:204
qc
impeller was used.
e1=3 t Three O/A systems with MIBK, kerosene, n-butanol c
5=3 q
d32 e2=3
¼c
as dispersed phase and water as continuous phase
r
T 2=3
8=3
were investigated. Holdup was equal to 0.05.
W
D
Photoelectric probe was used for drop size Ntc
¼ 0:0122
T
T
measurement. Disk turbine impeller was used.
Used experimental data of other authors.
0:6
r
e0:4
d32 ¼ c
i
q
Remarks
Effect of holdup on drop size was accounted for through a
function fϕ. Plot of fϕ versus ϕ was reported.
Correlation for specific interfacial area was obtained. Settling
times were also measured and included in the correlation to
obtain good fit. t0 is the reference settling time equal to
1 min.
No practical difference between batch and continuous
operation drop sizes was observed. First correlation is for
turbines, paddles and propellers of T b 40 cm, the second
correlation is for turbines with T N 40 cm.
Variation in local drop sizes was found to be small.
First equation is for batch vessels. The second equation is for
continuous vessels. In drop size correlations for continuous
systems, residence time was not included, to account for its
effect a separate correlation for hold up in terms of residence
time was proposed.
Drop sizes were measured at impeller tip. tc is the circulation
time given by second equation. The term in the second
bracket of left hand side of first equation was include to
account for the effect of geometrical parameters in the
correlation for drop size.
Showed that the whole power dissipation should be
assumed to occur in the impeller swept volume only and
this value should be use to correlate the drop size.
Calabrese et al. Study was aimed at finding effect of dispersed phase
(1986a)
viscosity on drop size. Silicone oils with viscosity less
than 0.5 Pas were called moderately viscous, with 1 Pas
intermediately viscous and more than 4 Pas as highly
viscous. Five different grades of silicone oil in water
were used to obtain dispersed phases of varied viscosity.
Hold up was 0.0015. Batch experiments were done with
direct photography as drop size measuring technique.
Rushton turbine impeller was used.
Wang and
Calabrese
(1986)
Calabrese et al.
(1986b)
"
Objective of the study was to establish relative
0:33 #0:59
importance of dispersed phase viscosity and d32 ¼ 0:066WeI0:66 1 þ 13:8Vi0:82 d32
D
interfacial tension on drop size. Silicone oils were D
dispersed in water, methanol and their solution. Batch
ld ND qc 1=2
Vi ¼
experiments were done with Rushton turbine impeller.
r
qd
Holdup was less than 0.002. Direct photography was
used for drop size measurement.
"
Total 349 data from published studies, including that
13 #35
of previous two studies, were used to obtain the d32 ¼0:054ð1þ3/ÞWe0:6 1þ 4:42ð12:5/ÞVi d32
I
D
D
correlation of broader utility.
Nishikawa et al. Honey bee's wax was used as dispersed phase in high
(1987a)
temperature batch mixing experiments with Rushton
turbine impeller. Distilled water or millet jelly was
used as the continuous phase. Holdup was varied
between 0.005–0.36. Measurement technique used
was stabilization of dispersion by siphoning it into
chilled water followed by imaging under the
microscope.
Nishikawa et al. Honey bee's wax was used as dispersed phase in the
(1987b)
distilled water as continuous phase. Batch experiments
were done with disk turbine in tanks of different sizes to
study the effect of scale. Measurement technique used
was stabilization of dispersion by siphoning it into
chilled water followed by imaging under the
microscope.
Laso et al.
(1987)
"
1=2 1=3 1=3 #5=3
d32
q
ld e d32
¼ 1 þ 11:5 c
d0
qd
r
3=8 3=4
d32
ld
qc ND2
¼ 2:1
D
lc
lc
d32 ¼ 0:6dmax
d32 ¼ 0:5dmax
d32
d32
d32
d32
1
1
3
6 l l r
D 5
5
d 5
d 8
2=3
¼ 0:105e
1 þ 2:5/
T
lc d lc c q
1
1
3
3 l l r
d 4
8
d 5
d 8
¼ 0:0371e1=4
1 þ 3:5/3=4
D
lc d lc c q
¼ 0:5dmax
¼ 0:45dmax
2=5
0
1
2 6 2
1
1 1 3
D 5 T
5 B1 þ 2:5 T 2 /2=3 C ld 5 ld 8 r 5
d32 ¼ 0:105e 5
@
A
T
T0
T0
lc d lc c q
0
1
3
14
1
1 1 1 3
d 4 T
4 B1 þ 3:5 T 2 /3=4 C ld 5 ld 8 r 8
d32 ¼ 0:0371e 4
@
A
D
T0
T0
lc d lc c q
0:056
Dispersed phase was a mixture of CCl4 with n-heptane d
l
32
¼ 0:118WeI0:4 /0:27 d
or 1-octanol or MIBK. Continuous phase was water.
D
lc
Batch studies with flat blade turbine in a baffled tank.
Drop size measurement was done by siphoning the
dispersion into a capillary, photographing it and
sending it back to the tank.
Data for intermediately viscous oils showed a lot of scatter
and could not be correlated. First and third equations are for
moderately viscous oils. The second and fourth equations are
for highly viscous oil. Dependence of d32 on μd for highly
viscous oils was different from as expected from a
semiempirical model. Log-normal drop size distributions
were obtained. d0 is the diameter of inviscid drop.
μd varied between 0.001-1 Pas and σ varied between 0.001–
0.045 N/m. Transition from low to moderate viscosity
behavior to high viscosity behavior was found to shift
toward high viscosity as σ reduced. The equation is valid for
μd b 0.5 Pas. For μd = 1 Pas, a lot of scatter was observed.
Unlike two previous studies, correlation here accounts for the
effect of high hold up.
It was argued that depending on value of specific power input
dispersions can be coalescence or breakup controlled. As
specific power input increases transition from breakup
controlled to coalescence controlled takes place. First and
third equations are for breakup controlled and second and
fourth for coalescence controlled regime. Note that
correlation for coalescence controlled regime is not
dimensionless. Correlation for transition value of specific
power input was also given.
The correlations of the previous study were modified to
incorporate the terms for the effect of scale. The first equation
is for breakup controlled regime, the second equation is for
coalescence controlled regime. T0 is the reference tank
diameter equal to 25 cm.
Holdup values not mentioned clearly, however from one of
the presented data it seems to be 0.09.
125
(continued on next page)
126
Table 1 (continued)
Reference
Chatzi et al.
(1991)
Experimental
Correlations
Batch experiments with styrene as the dispersed phase, d32
0:6
water with 0.1 g/l PVA as the continuous phase. D ¼ 0:045ðF0:003ÞWeI
Holdup was equal to 0.01. Impeller used was Rushton
turbine. Drop sizes were measured with particle size
analyzer.
0:270
Zhou and Kresta Batch experiments with water as the continuous and
d32 ¼ 118:6 emax ND2
(1998)
silicone oil as the dispersed phase. Four different
impellers — A310, HE3, PBT and RT were used.
Holdup was 0.0003. PDPA (Phase Doppler Particle
Analyzer) was used for drop size measurement.
Pacek et al.
Batch experiments were done with water as the d aeb
32
(1999)
continuous phase and chlorobenzene or sunflower oil
as the dispersed phase. Different impellers — high
shear (RT), high flow (Chemineer HE3) and ultra high
shear (Chemineer CS) were used. Measurements were
done by direct photography near the tank wall.
Dispersed phase holdup ranged between 0.01to 0.05.
Ruiz et al.
Batch experiments were done with 0.25 M sodium dmax
0:6
(2002)
sulfonate solution as continuous phase and a 1:1 D ¼ 0:353WeI
mixture of salicylaldoxime (LIX-860 N-IC) and a
ketoxime (LIX84-IC) in an aliphatic diluent (ESCAID
103) as the dispersed phase. Dispersed phase holdup
ranged between 0.006–0.018. Pump-mix double
shrouded impeller with eight curved blades was
used in the study. Single point measurement was
done by siphoning dispersion in a gelatin solution
followed by quick cooling in an ice bath to freeze the
drops.
Remarks
Drop size distributions observed were bimodal.
Objective of the study was to compare different scale-up
criteria such that data for different impellers can be
represented by a single correlation. Relation between d32
and dmax was not found to be strictly linear.
Values of the exponent, ranging between -0.47 to -0.72, were
tabulated as a function of impeller type, dispersed phase and
dispersed phase holdup.
Number volume distributions were found to be log normal.
Effects of temperature, extractant concentration and pH were
also investigated. Increase in temperature caused drop size to
reduce. Extractant concentration in the investigated range of 720% (w/w) did not affect the drop size. Reduction in pH
resulted in smaller drops.
Desnoyer et al.
(2003)
Quadros
and Baptista
(2003)
Batch experiments with a mixture of TBP and d32
¼ 0:28We0:6
ð1 þ 0:92/Þ
I
Sobesso 150 as the continuous phase and 20 g/l
D
d
32
0:6
NiCl2 (a fast coalescing system) or 3 M HCl (a slow
¼ 0:14WeI ð1 þ 0:48/Þ
coalescing system) solution as the dispersed phase. dD
32
nð/Þ=2
¼ f ð/ÞWeI
Hold up values ranged between 0.1–0.6. Impeller
D
used was PBTU. Drop size measurements were done
using laser granulometer.
Experiments to determine interfacial area in continuous
flow stirred tank with di-isobutylene diluted with
benzene as the dispersed and sulfuric acid as the
aqueous phase. Impellers used were straight blade
paddles with two or four number of blades. A chemical
method was used to determine overall interfacial area
under different operating conditions. Hold up values
ranged between 0.061–0.166. Impeller speeds were
varied to cover a large range of Weber number.
Giapos et al.
Batch experiments were done to understand the effect
(2005)
of number of blades on drop size for kerosene in
distilled water dispersions. Direct photography was
employed to measured drop size which was rendered
possible due to low values of holdups ranging
between 0.01–0.07. Disk turbines with 2, 4, 6 and
8 number of blades were used in this study.
Sechremeli et al. Batch were done to compare drop sizes produced by a
(2006)
disk turbine and an open impeller having same
diameter and blade width. Distilled water was used
as continuous and kerosene as the dispersed phase.
Direct photography was used to measure drop size.
Holdup values ranged between 0.01–0.1.
(
d32
)
c1 2 ¼ 6/ 1 þ
c 2 /2 þ c 3 /
WeI /
d32
¼ 0:0336We0:6
ð1 þ 13:76/Þ
I
D
d32
¼ 0:0286We0:6
ð1 þ 13:24/Þ
I
D
Though the Sauter mean diameters could be correlated using
the frame work of Hinze (1955) and Doulah (1975), they
could be better correlated with the functional form given in
the third equation with exponent of Weber number showing
dependence on holdup value. The first equation is for NiCl2
as the dispersed phase and second for HCl as the dispersed
phase.
The first functional form was found to represent the Sauter
mean diameters over a wide range of Weber number.
However, for high Weber numbers (We N 1900), Sauter
mean diameter could be correlated using the frame work of
Hinze and Doulah. The second correlation is for two blade
paddle while the third correlation is for four blade paddle.
d32 ~N a
The exponent a was found to vary between 0.62 to 1.23
depending upon the value of holdup. As expected, an
impeller with more number of blades gave finer drops
compared to an impeller having less number of blade and
rotating at the same speed.
d32 ~N a
Values of exponent were found to range between 0.61 - 1.03.
For open impeller (power number = 4), the Sauter mean
diameters were larger by 6-82% than the closed impeller
(power number = 5). Proportionality between dmax and d32
was not observed.
127
128
turbulence and enhancement of frequency of drop
coalescence by increasing drop population density in
the continuous phase. Considering damping of turbulence in concentrated dispersions, Doulah (1975)
proposed the following expression to account for the
increase in drop size with increase in hold up of dispersed
phase.
dmax
qc 6=5
¼
ð
1
þ
2:5/
Þ
ð15Þ
0
dmax
qm
which can be generalized to the following functional
form considering terms of ϕ2 onward negligible
dmax ¼ 1 þ c5 / þ c6 / 2 :
0
dmax
ð16Þ
0
and Eq. (16), the functional
Using Eq. (10) for dmax
form of the maximum stable drop diameter for high
holdup of dispersed phase assumes the following form
dmax
¼ c3 1 þ c5 / þ c6 /2 We0:6
:
I
D
ð17Þ
This functional form has been the most extensively
used by different researchers to correlate their experimental data for concentrated dispersions. Functional
form of Eq. (17) assumes that the dispersed phase is not
viscous or that the viscosity of the dispersed phase does
not contribute to the stabilization of the drop. Several
studies, however, have focused on stabilization due to
dispersed phase viscosity (Calabrese et al., 1986a,b;
Wang and Calabrese 1986). One of such studies
(Calabrese et al. 1986b), based on a large number of
experimental data, proposes the following expression
for the Sauter mean diameter:
"
13 #35
d32
d32
0:6
1þ4:42ð1 2:5/ÞVi
¼0:054ð1þ3/ÞWeI
D
D
ð18Þ
where
ld ND qc 1=2
Vi ¼
:
r
qd
ð19Þ
Note that, the first part of the right hand side of
Eq. (18) uses the frame work of Hinze (1955) as modified by Doulah (1975), the second part, the term in the
bracket, accounts for the stabilization effect of the
dispersed phase viscosity.
Apart from the models discussed above, there exists
another class of models that views the drop deformation
and breakage process as Vigot element in which the
stabilization effect of interfacial tension is represented
by a spring element and stabilization effect due to
dispersed phase viscosity as the dash pot. Both of them
act in parallel to resist the deforming stress due to
turbulence (Arai et al., 1977; Lagisetty et al., 1986).
These models have been extended to account for effect
on drop size of dispersed phase rheologies (Lagisetty et
al., 1986; Koshy et al., 1988b, Gandhi and Kumar
1990), presence of surfactants (Koshy et al., 1988a) and
circulation patterns (Kumar et al., 1992). Some models
accounting for breakage mechanisms different from that
due to turbulent pressure fluctuations have also been
proposed (Kumar et al., 1991; Wichterle, 1995; Kumar
et al., 1998). A few studies focusing on the fine details
like the effect of intermittency of turbulence on
representative drop diameter have also been presented
(Baldyga and Bourne, 1993). However, as far the
functional form used to fit the experimental data is
concerned, the model of Hinze (1955) as extended by
Doulah (1975) continues to be the most popular one.
3. Experimental
Fig. 1 shows the schematic diagram of the experimental setup used in this study. The experimental setup
consists of a cylindrical tank of 240 mm diameter and
240 mm height. Four baffles, each of 220 mm height,
having width equal to 10% of tank diameter, are
provided to prevent vortex formation and enhance
mixing. At the bottom plate of the tank is a suction
orifice with diameter equal to 1/4 of the tank diameter.
The suction orifice provides connectivity between the
tank and a cylindrical chamber called suction box
(65 mm diameter and equal height), which in turn is
connected with two feed tanks. One of the feed tanks
contains organic phase and the other aqueous phase. A
four-bladed top shrouded turbine with trapezoidal
blades, a pump-mix impeller, is used in the study.
Diameter of the turbine is 148 mm, blade width is
31 mm and blade length is 37 mm. Shaft diameter is
12 mm, hub height is 13 mm and hub outer diameter is
20 mm. Thickness of disk and blade is 2 mm. The offbottom clearance of the impeller is equal to half of the
mixer height. In this position, the impeller has a power
number of 3.0. The mixer has two ports for sample
withdrawal with rubber-clip arrangement — one in the
plane of the impeller disk and the other in a plane
halfway between the impeller disk and bottom of the
mixer. Organic phase used in the experiments is a
mixture of n-paraffin, D2EHPA and TBP. The aqueous
phase is 30% phosphoric acid. The physical properties
129
Fig. 1. Schematic diagram of the experimental setup.
of the phases are given in Table 2. At steady state, the
dispersion created in the mixer overflows from the
central opening (diameter equal to half of the tank
diameter) in the top baffle to a horizontal settler through
a full width launder. In the settler, separation of phases
occurs and the clear organic flows from the top to the
organic storage tank, the aqueous phase flows from
bottom to the aqueous storage tank through an interface
controller. From the storage tanks the phases are
pumped back to the respective feed tanks from where
they are again sucked into the mixer by the pumping
action of the pump-mix impeller. The flow rates are
maintained at desired values using the valves and
flowmeters located downstream of the pumps. Thus the
system is operated in a closed loop.
Before starting the actual experiments, phases were
equilibrated by running the system for 24 h. All the
experiments were conducted with aqueous as the
dispersed phase. Filling the mixer fully with organic at
the time of startup ensured organic continuity. Total 27
experiments were conducted for different impeller
speeds, feed phase ratios and mean residence times of
the phases in the mixer. The range over which these
variables were varied is given in Table 2. Unlike the
batch mixing studies where impeller speed can be varied
over a wide range (between the speed required for visual
homogenization and speed at which air entrapment takes
place), the upper limit of the speed in the continuous
mixing studies is limited by the phase separation
characteristics of the dispersion in the settler. Owing
to a low interfacial tension, the systems studied here is
easy to disperse and difficult to separate and even at low
impeller speeds the dispersion band tends to flood the
settler. This effectively limits the upper limit of speed
variation and hence the relatively narrow range of
impeller speed as given in Table 2.
For each experiment, attainment of steady state was
observed by following the thickness of the dispersion
band in the settler which was measured every 15 min
and steady state was deemed to have been achieved
when three consecutive measurements were almost
same. Typical time required to attain the steady state was
about 3 h. After attainment of steady state, samples of
dispersion were withdrawn from sample ports into
measuring cylinders. From the volumes of the settled
phases, local values of the hold up were computed. To
Table 2
Physical properties of the phases (at 25 °C) and range of variables
studied
Phase
ρ (kg/m3)
μ (cP)
σs (N/m)
Organic
Aqueous
859
1328
4.01
4.28
0.0267
0.0194
Range of variables
N (rpm)
100–150
τ (min)
0.5–2.0
ϕf
0.2–0.5
130
For each case, at least three hundred drops were
measured to ensure good statistical accuracy. Fig. 2
shows a typical image of the stabilized dispersion. A
similar method of measuring the drop size in emulsions
produced in simple shear flow has been reported
recently (Nandi et al., 2006). At the end of each
experiment, impeller was stopped and the valves
connecting the feed tanks to the mixer were simultaneously closed. Following the settling of the dispersion,
the volume occupied by the separated phases were
measured to compute the average holdup in the mixer.
From the measurements of the counted drops,
maximum drop diameter, Sauter mean diameter and
drop size distribution were obtained. The Sauter mean
diameter was obtained using the following expression:
Fig. 2. A typical image of the stabilized dispersion.
d32
measure the drop sizes, the sample of dispersion was
withdrawn into petri dish containing the organic phase
laden with a surfactant (sorbitan monooleate) presence
of which prevented the coalescence of drops and thus
stabilized the dispersion. The dish containing stabilized
dispersion was kept under a microscope mounted on a
camera which in turn was connected to a personal
computer. Since the field of view of this imaging system
was small, the full image of the stabilized dispersion was
not obtained in a single frame. Therefore, by systematically moving the dish under the microscope several
images, sufficient to give adequate numbers of drops, of
the stabilized dispersion were captured. Utmost care was
taken to avoid imaging the same area twice. The images
were analyzed using an image processing software.
P 3
d
¼P 2:
d
ð20Þ
4. Results and discussion
4.1. Characteristic drop diameter
Most of the studies on drop size distribution in agitated
tanks have been performed in batch mode. In these studies the
dispersed phase holdup is an independent variable. However,
in continuous flow agitated tanks, the hold up in the tank is not
an independent variable as it may change with, apart from feed
phase ratio, impeller speed and mean residence time. Using the
holdup in tank as a correlating variable for continuous flow
agitated tanks will therefore not indicate the true dependence
of drop diameter on speed and mean residence time.
Considering this, the feed phase fraction, instead of holdup
in the tank, is used to correlate the drop diameter. To decipher
the functional form of the relationship between the Sauter
Fig. 3. Comparison of Sauter mean diameters at upper and lower sampling locations, as predicted by Eqs. (21) and (22).
131
mean diameter and the three independent variables, the
experimental data were used to train a neural network based
on back propagation algorithm. The response of the trained
neural network indicated that the Sauter mean diameter could
be correlated as a quadratic function of the feed holdup, a
power law function of the impeller speed and an exponential
function of mean residence time. This led to the selection of
the functional form for the correlation of the Sauter mean
diameter, as given in (21)–(23). Note that this functional form
resembles the form suggested by Eq. (17) albeit with vessel
holdup replaced with the feed holdup and an additional
multiplicative function to account for the effect of mean
residence time on Sauter mean diameter. The values of
coefficients and exponents were obtained by using Gauss–
Newton algorithm. The final correlations are given below:
d32
¼ 1:2825 103 N 1:681
D
1 þ 2:6539/f þ1:3986/2f exp ð0:4457sÞ
ð21Þ
for the upper sampling location and
d32
¼ 2:451 103 N 1:723
D
10:8785/f þ 4:2848/2f exp ð0:4148sÞ
it is better to have a single correlation for Sauter mean diameter
for practical utility. This correlation was found to be:
d32
¼ 1:849 103 N 1:7025
D
1þ0:392/f þ 3:2435/2f exp ð0:4302sÞ:
Fig. 4 shows the parity plot for Eq. (23). The quantitative
estimate of the goodness of the fit of correlation given by
Eq. (23) is given in Table 3. The ±95% confidence intervals
(Draper and Smith, 1966) of the regressed model constants are
given in Table 4. As can be seen that for most of the constants the
confidence intervals are reasonably narrow thereby indicating
good statistical accuracy of the correlation given by Eq. (23).
Since in the present study effects of impeller diameter and
physical 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 in
literature, it would be interesting to recast Eq. (23) in terms of
Weber number, as follows
d32
¼ 2:946 10−6 We0:85125
I
D
1þ0:392/f þ3:2435/2f expð0:4302sÞ:
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 around
the diameters observed at the upper sampling location. As can
be seen the diameters at the upper sampling location, owing to
it being in the plane of the impeller discharge stream, are
marginally smaller than the diameters at the lower sampling
location. However, difference being small, it can be assumed
that the quality of dispersion is more or less homogeneous and
ð23Þ
ð24Þ
In the above correlations τ is the mean residence time, in
minutes, of the phases in the mixer. The above correlations
show that the characteristic drop size reduces with increase in
speed and increases with an increase in feed holdup. This is in
conformity with the trend of semiempirical model represented
by Eq. (17). The exponent on N is also close to the value of
−1.2 as suggested by the semiempirical model. The correlations show that, for the range of the experiments, the drop size
increases with increase in mean residence time. This appears to
be contradictory to the expectation that increasing mean
residence time should ensure longer interaction between the
impeller and the droplets and hence should result into a smaller
Fig. 4. Parity plot for the Sauter mean diameter predicted by Eq. (23).
132
Table 3
Goodness of fit of different correlations presented in this work
Equation
For
Emax
Eavg
Beyond ±15%
21
22
23
25
26
27
28
30
31
Sauter mean diameter at the upper sampling location
Sauter mean diameter at the lower sampling location
Average Sauter mean diameter
Local holdup at the upper sampling location
Local holdup at the lower sampling location
Vessel average holdup
Relationship between Sauter mean diameter and dmax
Variance of the log-normal drop size distributions
Mean of the log-normal drop size distributions
21.7
29.0
27.6
8.9
11.1
6.2
22.2
39.6
5.3
7.5
8.8
8.5
2.4
2.7
2.0
7.3
9.3
1.9
5/27
6/27
10/54
0/27
0/27
0/27
3/54
9/54
0/54
characteristic drop size. This will be explained in the next
section.
It would be interesting to compare the correlation of Eq. (23)
with some of the correlations compiled in Table 1 for their
effectiveness to predict the experimental data reported in this
work. 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 other
correlations reported earlier exhibit large deviations. This is not
surprising considering that the drop size distributions in stirred
liquid–liquid dispersions are highly system specific due to their
being dependent on the physical properties of the phases, phase
continuity, impeller type, mode of operation and geometric
configuration of the mixer. Among the compared correlations,
the correlation obtained with TBP containing organic as the
continuous phase and an acidic phase as the dispersed phase
(Desnoyer et al., 2003) exhibits the least deviation form the
correlation given by Eq. (23). This may be attributed to the
identical phase continuity and similar nature of the phases.
4.2. Holdup
To have an estimate of the specific interfacial area, in
addition to the correlation for the Sauter mean diameter, the
correlation for dispersed phase hold up in the mixer is also
required. Whereas, in a batch mixer, hold up of the dispersed
phase varies from one location to another (Wang and Mao,
2005; Wang et al., 2006), for a continuous flow stirred tank
both local and vessel average values, in general, can be
different from the feed holdup. The local and vessel average
values will depend on the impeller speed, mean residence time
and feed holdup. The local and vessel average holdup in the
present study could be correlated as a power law function of
the independent variables. These correlations are as follows:
/ ¼ 2:0274/0:3569
N 1:1288 s0:1153
f
ð25Þ
for the upper sampling location and
N 1:2389 s0:1389
/ ¼ 2:3173/0:3583
f
ð26Þ
for the lower sampling location and
P
/ ¼ 2:1214/0:4012
N 1:097 s0:0631
f
ð27Þ
for vessel average holdup values.
Apart from being useful in predicting the specific
interfacial area, these correlations are important in deciding
the intrastage recycle flow rate in order to have a desired phase
ratio in the mixer. As expected, the local and vessel average
holdup increase with increasing feed holdup. In the present
case, the heavy phase being the dispersed phase and the outlet
of dispersion being from the top of the mixer, the holdup
values in the tank were observed to be larger than the feed
holdup. For any system, increasing influence of impeller
should tend to bridge the gap between the holdup in the vessel
and the feed holdup. The increasing influence of the impeller
can be due to increasing impeller speed or increasing mean
residence time. For the case of heavy phase as dispersed phase
and outlet from the top of the mixer, the hold up in the vessel
Table 4
Confidence intervals for model constants of Eqs. (23) and (27)
Equation
Model constant
Regressed value
±95% confidence interval
23
23
23
23
23
27
27
27
27
Multiplicative constant on right hand side
Exponent on N
Coefficient of ϕf
Coefficient of ϕ2f
Coefficient of τ
Multiplicative constant on right hand side
Exponent on ϕf
Exponent on N
Exponent on τ
0.001849
− 1.7025
0.3920
3.2435
0.4302
2.1214
0.4012
− 1.0970
− 0.0631
(0.001758, 0.001039)
(− 1.7779, −1.6450)
(0.1851, 0.5970)
(2.7371, 3.7451)
(0.3824, 0.4683)
(2.1091, 2.1743)
(0.3876, 0.4147)
(− 1.1169, − 1.0770)
(− 0.1190, − 0.0069)
133
Fig. 5. Comparison of correlation of Eq. (23) with some of the correlations reported earlier.
should, therefore, reduce with increasing impeller speed and
increasing mean residence time. This explains the negative
exponents on N and τ in Eqs. (25)–(27). Now the dependence
of the Sauter mean diameter on mean residence time, as seen
earlier, can be explained. The increasing residence time
reduces the dispersed phase holdup in the mixer. This will
tend to reduce the drop size. However, the heavy phase being
the dispersed phase, the effective density of the dispersion also
reduces and for the same impeller speed power introduced in
the system also reduces. This reduction in the power input
tends to increase the drop size. The eventual trend of
dependence of drop size on mean residence time will,
therefore, depend on the relative importance of these two
opposing effects and can be either positive, as seen in the
present case, or negative. At high impeller speeds, the holdup
in the tank will be very close to the feed holdup irrespective of
the mean residence time and in that case it is quite likely that
the mean residence time will not have an effect on the drop
size. This probably explains the similar drop sizes observed for
both 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 the
local and vessel average specific interfacial area for specified
values of the independent variable (impeller speed, mean
residence time and feed holdup). Quantitative estimate of
goodness of fit of Eqs. (25)–(27) is given in Table 3. The
± 95% confidence intervals of the evaluated model constants of
Eq. (27) are given in Table 4.
4.3. Relationship between d32 and dmax
Several studies report on direct proportionality between the
Sauter mean diameter and maximum stable drop diameter
(Brown and Pitt, 1972; Nishikawa et al.,1987a; Collias and
Pruddhornme, 1992; Zerfa and Brooks, 1996; Calabrese et al.,
1986a). However, a few studies contradict this view (Zhou and
Fig. 6. Relationship between the maximum drop diameter and the Sauter mean diameter.
134
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.
Kresta, 1998; Sechremeli et al., 2006). The system studied here,
however, tends to exhibit the direct proportionality between the
two representative diameters, as shown in Fig. 6. Considering the
inherent uncertainty associated with measurement of dmax (there
is only one drop with diameter equal to dmax and hence there are
fair chances that it can escape imaging), the linear fit of Fig. 6 is
reasonably good. The two representative diameters can be
correlated by the following equation:
The number probability density of this distribution is given
by:
"
#
P 2
1
ð lnd l
Þ
f ðd Þ ¼ pffiffiffiffiffiffi exp :
2s2
ds 2p
ð29Þ
The constant of Eq. (28) compares well with the values
reported in the literature, as reported for some of the previous
studies summarized in the Table 1.
Figs. 7 and 8 show the graphical comparison of experimental and log-normal drop size distributions for two of the
cases. A good agreement between the number probability
density as predicted by log-normal distribution and experimental distribution can be observed.
– of Eq. (29) could be correlated as a power law
s and μ
function of impeller speed, feed holdup and mean residence
time as follows:
4.4. Drop size distributions
s ¼ 0:806 N 0:838 /0:0578
s−0:0788
f
ð30Þ
Several distributions were tried but log-normal distributions
were found to give the best fit to the experimentally observed
drop size distributions.
l ¼ 6:3873 N 0:2078 /0:1406
s0:1508 :
f
ð31Þ
d32 ¼ 0:5446dmax :
ð28Þ
P
Goodness of fit of Eqs. (30) and (31) is quantified in Table 3.
Fig. 8. Comparison of experimental and log-normal drop size distributions for Qo = 400 lph, Qa = 150 lph, N = 120 rpm, class interval equal to 20 μm.
135
5. Conclusions
Measurements of drop size distributions have been
performed for aqueous in organic dispersions in
continuous flow stirred tank with a mixture of n-paraffin,
D2EHPA and TBP as the organic phase and dilute
phosphoric acid as the aqueous phase. Surfactant
stabilization has been used as drop size measuring
technique. The experimentally measured drop size
distributions exhibit log-normal behavior in droplet
number density. The characteristic drop sizes exhibit
insignificant spatial variations. The Sauter mean diameters have been correlated with impeller speed,
dispersed phase feed fraction and mean residence time.
Local and vessel average values of hold up have also
been correlated. Proportionality between the Sauter
mean diameter and the maximum stable diameter is
observed.
Nomenclature
ā
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]
0
dmax
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]
P
u2
Vi
W
We
Wecrit
WeI
Average relative velocity between two points
separated by a distance d in turbulent field
[L2 T − 2 ]
Viscosity group [−]
Width of the impeller blade [L]
Generalized Weber group [−]
Critical Weber number [−]
Impeller Weber number [−]
Greek letters
e
Specific turbulent energy dissipation rate
[L 2 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 [−]
ϕ̄
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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