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Drop coalescence in turbulent dispersions
Chemical Engineering Science 54 (1999) 5667}5675
Drop coalescence in turbulent dispersions Shiping Liu , Dongming Li * Polymer Processing Laboratory, East China University of Science and Technology, Shanghai 200237, People's Republic of China Department of Chemistry, Tongji University, Shanghai 200092, People's Republic of China
Abstract The coalescence between drops in an isotropic turbulent dispersion is examined. With this result, a new model without any adjustable or empirical parameters is developed to predict the minimum drop size stable against coalescence both in surfactant systems and pure systems. The drop size distribution and coalescence e$ciency are also discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Drop coalescence; Turbulent dispersion; Minimum drop size
1. Introduction The dispersion of immiscible liquids is of great importance in many chemical engineering processes. For example, in systems such as extraction, suspension polymerization and multiphase reactions, the interfacial area available for mass transfer is increased by forming liquid}liquid dispersions in stirred tanks or column contactors, as could be determined by drop size distribution and volume fraction (holdup) of the dispersed phase. In a turbulent dispersion, the breakup and coalescence of drops usually take place continuously. These rate processes not only determine the drop size distribution, which is an equilibrium property, but also have a profound e!ect on mass transfer in dispersions. The phenomenological model concerning coalescence and breakage of drops has been extensively used to simulate the drop size distribution, which could be a manifestation of the two rate processes (Tsouris and Tavlarides, 1994). However, in order to understand the dispersive process better, it is important that drop coalescence and breakup can be assessed. In a dispersion, the important parameters closely related to drop coalescence and breakup are the minimum and maximum stable drop sizes d and d . For
example, if a dispersion is maintained by agitation for a su$ciently long time, a local dynamic equilibrium between coalescence and breakage will be established.
* Corresponding author.
Under this condition, d and d can be observed. Any
drop which is smaller than d will coalesce, while those
larger than d will break up. Furthermore, the #ow in
a stirred tank could be divided into regions of dispersion (near the impeller) and coalescence (relatively stagnant zones). In the region of coalescence the equilibrium spectrum of drop size distribution depends on the coalescence process. The Sauter average diameter d is directly pro portional to d . On the other hand, when the breakup
process is dominant, d is proportional to d . There
fore, d and d are important in interpreting drop size
distribution and modeling the related two-phase #ows. In locally isotropic turbulence, theoretical analysis on breakup of drops has received wide attention since the investigations of Taylor (1932) and Hinze (1955). The diameter of the largest drop against breakage, d , is
determined by the process of breakup. The phenomenon in the inertial subrange is relatively well understood so that d in a dilute dispersion can be evaluated with
reasonable con"dence (Hinze, 1955; Shinnar, 1961; Sprow, 1967; McManamey, 1979; Lagisetty et al., 1986; Calabrese, 1986; Clark, 1988). One of the widely used expressions was given by Hinze (1955) d "A (c/o ) e\ (1)
A in which A is a constant. Hinze (1955) found that A "0.725 from the experimental data of Clay (1940) using two coaxial cylinders, of which the inner one rotated. For a dispersed phase with low viscosity in an agitated tank, the above relation becomes d "A (c/o ) (ND)\ ,
A '
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 1 0 0 - 1
(2)
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
where A is a constant. The value of A varied substan tially in previous investigations. For example, Lagisetty et al. (1986) suggested a value of 0.125, similar to the one obtained by Sprow (1967), while the correlation obtained by Calabrese et al. (1986) gave a value of 0.0883. Recently, Sathyagal et al. (1996) found that their experimental results agreed well with the one given by Calabrese et al. (1986), while the model of Lagisetty et al. (1986) predicted much larger values for d . Thus we take A "0.0883
when d is needed.
Coalescence phenomenon, however, is more complex since it involves not only the approach of two drops, but also the drainage and eventual rupture of the intervening liquid "lm, in which the physical properties of the #uids and interfaces will play an important role. For example, coalescence in a dispersion is usually reduced considerably by the combined action of a protective colloid and of turbulent agitation. For a highly viscous dispersed phase, the viscoelastic properties become important (Alvarez et al., 1994). In suspension polymerization, the suspension viscosity may be large enough for breakage and coalescence of drops to cease as conversion increases (Vivaldo-Lima et al., 1997). We deal with the coalescence process in dispersions with Newtonian #uids. Kumar et al. (1993) and Tsouris and Tavlarides (1994) have reviewed the models for the coalescence process in the literature. It is convenient to view the process as the drainage of the continuous-phase "lm entrapped between any two drops. The coalescence time, or the time required for "lm drainage and rupture, is an important parameter characterizing the process. However, these models usually deal with undeformable interfaces (e.g., a "lm between two rigid spheres or a plane parallel "lm) so that the initial and/or critical "lm thicknesses are needed for calculating the coalescence time. For example, Shinnar and Church (1960) have investigated the in#uence of turbulence on coalescence of droplets and obtained d "A e\ o\ A(h ), (3)
A where A is a constant, A(h ) is the energy necessary to separate completely two drops with unit diameter, which are initially separated by a distance h , and is determined by the properties of the liquids. Sprow (1967) derived a similar expression. Thomas (1981) developed a formula for the drops with tangentially immobile interface:
ch d &2.4 , (4)
koe A A where h is the "lm thickness at rupture between the two drops. Both models involve an empirical parameter, A(h ) or h, that has to be assumed or determined experi mentally, implying the dependence of d on the physical
properties is unknown (Thomas, 1981).
For drops with an immobile interface, only the physical properties of the continuous phase appear in the expressions for the models. If the interfaces are partially or fully mobile, the situation may be di!erent. For example, Park and Blair (1975) could not "nd turbulence-stabilized emulsions for a pure system involving a dispersed phase with low viscosity k . Kumar et al. B (1993) predicted that the dispersion with pure phases cannot be stabilized at low stirrer speed, which was exactly opposite to the experimental results of Shinnar for surfactant systems (1961). Therefore, the viscosity of dispersed phase may come into e!ect. In the present paper, we consider the coalescence in a turbulent dispersion with and without surfactant. A model is developed to predict the minimum stable drop size. The drop size distribution and coalescence e$ciency are also discussed. For simplicity, we assume that both phases are Newtonian #uids.
2. The proposed model 2.1. Coalescence time between drops in turbulent yow For coalescence to occur, the intervening liquid "lm between two drops has to drain until the "lm ruptures. A theoretical model has been developed for drainage and rupture of a dimpled liquid "lm between a drop and a surface (Lin and Slattery, 1982a,b). With this model, the coalescence time between two drops or bubbles can be predicted by giving only the drop sizes and the required physical properties of #uids and interfaces. Dispersed systems usually contain at least a trace of surfactant, which may give rise to interfacial tension gradient and/or interfacial viscosities so as to retard interfacial mobility. Based on the analyses on the e!ects of surface tension gradient and surface viscosities on bubble coalescence (Li, 1996) and drop coalescence in pure systems (Liu et al., 1995, 1996; Li and Liu, 1996), Li and Zhao (1996) derived a more general expression for coalescence time between drops when surfactant is present in the continuous phase. For simplicity, we consider two equal-sized drops. The coalescence time t becomes t"0.0272f
p Sk F \Sd >S A , 4 c \SB \S
(5)
where the correlations for u and f are 65a u" , (1#320a )(1#0.66k ) Q 4.0 106.0 f"1# exp ! log a 84.0# 1#k 1#0.09k Q Q
(6)
.
(7)
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
As k P0, the last expression is not appropriate, and Q a more accurate correlation for f is f"1#7.70 exp[!1.25"log 190a" ],
(8)
The parameter in these equations are
1 c cNF k F , "!0.254 #12.76 B a k cDN k cd A A and
(9)
i #e k "4 . (10) Q kd A In these equations, d is the drop diameter, k and k are A B the viscosities of the continuous and dispersed phases, respectively, c is the equilibrium interfacial tension, c is the change rate of interfacial tension with interfacial concentration, i and e are interfacial shear and dilational viscosities, respectively, cN is the equilibrium interfacial concentration, DN is the interfacial di!usion coe$cient, and F is the force pressing the drops together. Since the retarded London-van der Waals forces are considered, the van der Waals constant B is assumed to be approximately 10\ Jm (Kitchener and Prosser, 1957; Black et al., 1960; Churaev, 1974). Both parameters a and k characterize the mobility of the interfaces. As the Q interfacial mobility decreases, coalescence time increases. As aP0, which results either from large surface tension gradient or large viscosity of the dispersed phase, or as k PR for large interfacial viscosities, the interfaces Q become immobile so that the coalescence process is the slowest. In these cases, u"0 and f"1, so that Eq. (5) reduces to the one obtained previously (Chen, 1985; Li, 1994). For a pure system, the interfaces are mobile, and we may use a simpler expression for coalescence time (Li and Liu, 1996). For simplicity, we therefore consider two limiting cases, in which the interfaces are either tangentially immobile as a result of large interfacial stress or mobile in a pure system. In turbulent #ow, if it is assumed, as in Kolmogorov's theory, that the eddies responsible for coalescence belong to the inertial subrange, the force pressing the drops together may be estimated in the order of magnitude as F&o e d , (11) A where e is the energy dissipation per unit mass, and o is A the density of the continuous phase. Lacking more detailed information, we adopt a coe$cient of unity in Eq. (11) in the following calculations. For immobile interfaces in surfactant systems, Eq. (5) becomes k o e d t"0.0272 A A . c B
(12)
and for a pure system, the interfaces are mobile and circulation prevails inside a drop, and the coalescence
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time t becomes (Liu et al., 1995; Li and Liu, 1996) 1 c k B A "1363.3 t E k o e d B A c B #217.3 , E k o e d A A
(13)
where E is the dimensionless curvature radius of the liquid "lm, which is a function of the interfacial mobility coe$cient M E"12.61#2.166 arctg(2M )
(14)
With the force expressed by Eq. (11), M becomes
pc k M"1.12 A k o e d B A
(15)
in which k is the viscosity of the dispersed phase. Both B Eqs. (12) and (13) indicate that the coalescence time increases with the intensity of turbulence. This means that increase in turbulence intensity will decrease the probability of drop coalescence. 2.2. Minimum drop size in turbulent yow The coalescence and breakup of drops are in dynamic equilibrium in turbulent #ow under steady state. As two drops are brought together in a dispersion, a thin liquid "lm forms between them and drains. During drainage, turbulent #uctuations could separate the drops again. Assuming that the eddies in the inertial subrange are responsible for the motion of drops, Levich (1962) estimated the average contact time ¹ between drops in turbulent #ow as ¹&d e\ .
(16)
Again we adopt a coe$cient of unity in Eq. (16) in the following calculations for lack of more detailed information. A simple criterion for coalescence to occur is t(¹, i.e., the contact time should be long enough for the liquid "lm to drain and rupture before the drops are separated again in turbulence. With Eqs. (12) and (16), we obtain the minimum diameter of a drop which is stable against coalescence in a turbulent dispersion c B d " .
0.0272k o e A A
(17)
This agrees with Eqs. (3) and (4), in which d varies with
e\ in particular. From Eq. (17), the minimum stable diameter can be predicted directly as a function of physical properties of #uids and energy dissipation e. Since the expression for coalescence time in a pure system is more complex, we cannot explicitly express d as a function of the system parameters. With
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Eqs. (13) and (16), we have c k B A E k o e d B A
c B #217.3 "1 (18) E k o e d A A
from which d can be solved by the Newton method.
For a very dilute dispersion, e is equivalent to the energy dissipation for a homogeneous continuous phase. However, in a concentrated dispersion, the two-phase dispersion has to be considered as a whole. The density and viscosity of such a dispersion depend on the volume fraction of the dispersed phase (Taylor, 1932): 1363.3
oH"o #o (1! ), (19) B A k #0.4k A , kH"k 1#2.5 B (20) A k #k B A where superscript H refers to a two-phase dispersion, subscript d and c correspond to the dispersed and continuous phases, respectively. Considering Kolmogorov's microscale, Doulah (1975) showed that
e lH , (21) " eH l A where e and eH are the energy dissipation for the pure continuous phase and the two-phase dispersion, respectively, l and lH are the respective kinematic viscosities. A In the presence of the drops, the viscosity of the dispersion increases (lH'l ) so that the energy dissipation A decreases (eH(e). With Eqs. (19)}(21), we may take account of the e!ect of holdup on d and d in
turbulent #ow. An important di!erence between the two limiting cases, i.e., the drops with immobile interfaces in surfactant systems and those with mobile interfaces in a pure system, is the e!ect of the viscosity of the dispersed phase. For immobile interfaces, d is not a function of k for
B low , though it is a weak function of k through Eqs. B (19)}(21) for higher . For mobile interfaces, the interfacial mobility increases as k"k /k decreases, as inB A dicated by Eq. (5), so that d increases rapidly as
k decreases. If the interfaces are partially mobile, all of the bulk and interfacial properties appeared in Eqs. (5)}(10) will a!ect d . Increase in those properties that retard
the interfacial mobility, such as k , c , i and e , will B decrease d .
2.3. Coalescence ezciency Coulaloglou and Tavlarides (1977) have assumed that once the drops collide with each other the contact lasts for some time, called contact time, and then they either coalesce or bounce away from each other. Therefore, coalescence occurs when the random contact time exceed
the coalescence time. The authors assumed that the contact time is exponentially distributed and they de"ned a coalescence e$ciency function as j"exp(!t/¹).
(22)
With Eqs. (12) and (16), we obtain the coalescence e$ciency as a function of physical properties and energy dissipation for drops with immobile interfaces:
k o e d j"exp !0.0272 A A . c B
(23)
With Eqs. (13) and (16), we obtain an expression for mobile interfaces in a pure system:
c k B A !1363.3 E k o e d B A \ c B !217.3 . (24) E k o e d A A Both equations indicate that coalescence may be prevented by increasing turbulence intensity. For general cases, the bulk and interfacial properties that retard the interfacial mobility will reduce the coalescence e$ciency.
j"exp
2.4. Ewect of d on drop size distribution in turbulent
dispersions Under steady state, the drop size distribution is not a function of time. If a drop signi"cantly larger than d is formed by coalescence, it will have a high prob ability of being broken in a short time. Likewise if a drop is signi"cantly smaller than d , it will coalesce. Thus
dispersion is in dynamic equilibrium and a steady-state drop size distribution is maintained. Combining Eqs. (1) and (17) for immobile interfaces, we have d c k A
" A (25) d 3.19 o e B
A which implies that the drop size distribution becomes narrower as e increases. This equation characterizes the drop size distribution in a turbulent dispersion and provides information for scale up of mixing equipment. Eq. (23) shows that coalescence may be prevented by increasing turbulence intensity. Such a dispersion was considered as turbulence-stabilized (Shinnar, 1961). Stabilization is possible only if the minimum diameter stabilized by turbulent #uctuations is less than the maximum stable diameter, i.e., d (d . Any drop in a turbulent
dispersion must be larger than d and smaller than d .
Letting d "d , the critical energy dissipation is de
"ned as c k A , e "A oB A
(26)
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
where A "(A /3.19) , e.g., A "1.40;10\ for A "0.725 (Hinze, 1955). For mobile interfaces in a pure system, we can also de"ne e with Eqs. (1) and (18), so that 1363.3 B k o c e A A A E k B 217.3 B o e A "1. # (27) A E k c A The value of e can be obtained by Newton's method. If e(e (i.e., d (d ), the dispersion is turbulence-sta
bilized. If e'e (i.e., d 'd for calculated values),
the dispersion will behave like an unstabilized dispersion in which Eqs. (17) and (18) do not apply (Shinnar, 1961). In this case, turbulence alone cannot prevent coalescence in the dispersion, since the drops coalesce and breakup rapidly.
3. Comparison and discussion Several investigators studied experimentally the behavior of liquid}liquid dispersions maintained by agitation and measured the drop size distributions. Shinnar (1961), Shinnar and Church (1960) dispersed a molten microcrystalline wax in hot water (system 1). Chatzi et al. (1991) used a system of 1% styrene in water under 603C (system 2). System 3 (Chatzi and Kiparissides, 1995) was one of 50% n-butyl chloride in water under 203C. In these systems, surfactant polyvinylalcohol (PVA) was added, so we compare our model for immobile interfaces with these results. The physical properties of the three systems are listed in Table 1. To calculate d , e is
needed. Chatzi et al. (1991) suggested e"100DN/p for ' their experiments, which is used in our prediction of d for system 2. Lacking more detailed information for
the other two systems, we use e&62DN (Thomas, ' 1981). Moreover, it is necessary to take account of the damping e!ects of the suspension on the local turbulence intensities at high holdup. From Eqs. (19)}(21), we obtain e/eH"6.15 for system 3.
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Part of the drop size distributions at di!erent impeller speed measured by Shinnar (1961), the minimum stable diameters d and d , predicted by Eq. (4) (Thomas,
1981) and Eq. (17), respectively, are listed in Table 2, in which d is the average diameter in a size group, X is the number of drops in the size group, > is the volume percentage of the drops. Shinnar calculated both d (90)
and d (10), which were de"ned so that 90% of the
cumulative volume of all drops had a diameter smaller than d (90) and 10% were smaller than d (10). Table
2 shows that the minimum stable size d , in which the
rupture "lm thickness was assumed to be 0.1 lm, is closer to d (10) and much larger than the observed minimum
diameters, while the values predicted by this analysis, d , are in good agreement with the actual minimum
stable diameters. The comparison with other two systems (Chatzi et al., 1991; Chatzi and Kiparissides, 1995) is shown in Table 3. The surface tension c at di!erent concentration of PVA, C, and the measured minimum stable diameters are also listed. If the interfaces are partially or fully mobile, the situation may be di!erent. An important di!erence between mobile and immobile interfaces is the e!ect of k on d . B
Park and Blair (1975) could not "nd turbulence-stabilized emulsions for a pure system involving a dispersed phase with low k . Kumar et al. (1993) predicted that the B dispersion with pure phases cannot be stabilized at low stirrer speeds. This is exactly opposite to the observation of Shinnar (1961), in which PVA added to the continuous phase may make the drop interface immobile. We discuss the phenomena according to our analysis for mobile interfaces. Park and Blair (1975) conducted some experiments for a pure system, in which the interface of a drop may be assumed to be mobile. The system consisted of mutually saturated water (continuous phase), and methlisobutylketone (MIBK). The experiments were carried out in a 6 in diameter mixing vessel with four ba%es. Liquid depth H was maintained at 5.625 in. The holdup was
"0.005 and 0.1. The turbine blade agitator was a standard mix #at blade design. Its diameter D was 3 in with ' a blade width = of 0.375 in. The impeller speed was '
Table 1 Physical properties and impeller diameter System
k (mPas) A
k (mPa s) B
o (kg/m) A
o (kg/m) B
c (mN/m)
D (m) '
1 2 3 4
0.36 0.466 1.0 0.89
17.8 0.475 0.45 0.59
964 983.2 988 997
791.4 870.4 887 795
38.5
0.127 0.075 0.127 0.076
1: Shinnar (1961), molten microcrystalline wax in hot water, with surfactant PVA; 2: Chatzi et al. (1991), 1% styrene in water under 603C, with surfactant PVA; 3: Chatzi and Kiparissides (1995), 50% n-butyl chloride in water under 203C, with surfactant PVA; 4: Park and Blair (1975), methlisobutylketone (MIBK) in water, no surfactant.
10.5
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Table 2 Comparison of predicted minimum stable drop diameters d by Eq. (17) with experimental data (Shinnar, 1961)
N (r.p.m.) d l
X
>
X
15 30 45 60 75 90 105 120 135 150
* 61 * 67 * 83 * 127 * 171
* 0.07 * 0.81 * 2.43 * 8.8 * 23
10 25 47 79 115 165 210 177 2 2
156
d d (90)
d (10)
d (Eq. (17))
d (Eq. (4))
177 265 128 95 48
220 >
X
0.002 0.038 0.240 0.97 2.77 6.85 13.1 17.3 2 2
14 75 87 171 258 281 2 2
127 174 96 73 36
312 > 0.005 0.205 0.81 3.83 11.35 21.2 2 2
97 135 75 56 26
Table 3 Comparison of predicted d by Eq. (17) with experimental data
(Chatzi et al., 1991 and Chatzi and Kiparissides, 1995) System
2 2 3 3 3 3
C (g/l)
0.1 3.0 0.5 0.5 1.0 1.0
c (mN/m)
15.3 3.8 3.2 3.2 2.8 2.8
N (r.p.m.)
300 300 400 700 400 700
d (lm)
Experimental
Eq. (17)
22.5 12.5 10.7 8.7 7.0 6.3
27.1 14.6 8.5 5.3 8.0 5.0
maintained at N"375 r.p.m. For turbine or #at-bladed impellers, the average energy dissipation e is the power input per unit mass in the volume swept by the impeller (McManamey, 1979) 4P e" , (28) no D= A ' ' where P is the power input to the impeller, which can be calculated by P"P o ND, (29) A ' where P is the power coe$cient. For the tank and impeller (D /= "8) used, since D /D O1/3 and ' ' ' 2 H/D O1, the energy dissipation should be corrected 2 to
32P D H 2 e" ND, (30) ' p 3D D ' 2 where D is the diameter of the tank. Under the condi2 tion, P "2.9 for Re'10 (Chen, 1985).
440 >
X 65 135 175 195 164
0.035 0.605 2.51 6.9 11.2
91 130 68 44 20
X 95 183 179 255
440 > 0.10 1.55 5.15 17.4
79 115 57 44 20
627
627
X
>
X
>
410 297 210
1.90 11.30 27.0
334 310 199
1.6 12.0 26.0
53 98 34 33 14
51 92 35 33 14
The e!ect of the bulk viscosities, expressed as k"k /k , on d and d for "0 and 0.2 is shown in B A
Fig. 1. The energy dissipation e is calculated from Eq. (30), d for immobile and mobile interfaces are cal culated from Eqs. (17) and (18), respectively, d is deter mined by Eq. (2) with A "0.0883 (Calabrese et al., 1986), and the other parameters are from the experiments of Park and Blair (1975). The drop with immobile interface behaves like a rigid sphere, so that d is not
a function of k for "0, though it is a weak function of B k by Eqs. (19)}(21) for higher holdup . For drops with B mobile interface, d decreases rapidly as k increases.
When k'30, d as computed is close to that for immo bile interfaces. When k is small, the interfacial mobility is large so that liquid drops coalesce more rapidly. When k'1.1, d (d and the system is stable. However,
when k is small, calculated d 'd so that the system
is unstable. In the experiments of Park and Blair (1975), k"0.663, so they could not "nd turbulence-stabilized emulsions. For their experiment on "0.005, predicted d 'd when N'2 r.p.s. It implies that the MIBK
Water system (Park and Blair, 1975) is unlikely to be stabilized by increasing turbulence intensity alone, unless the interfacial mobility of drops is retarded, which agrees with the experimental observation. The above comparisons indicate that the model gives an appropriate prediction for the minimum stable diameter under di!erent experimental conditions. Since d is proportional to d when coalescence process
is dominant, it shows also the e!ect of physical properties as well as turbulence intensity on drop size distribution. Fig. 2 shows the coalescence e$ciency j for drops with immobile interfaces, determined by Eq. (23), as a function of drop size d for various values of hold up and energy
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Fig. 1. E!ect of bulk viscosity k"k /k on computed drop size. B A
dissipation e. Here d and d are calculated from Eqs.
(1) and (17), in which the physical properties of Shinnar's experiments (1961) are used. The coalescence e$ciency j increases with but decreases as e increases. As drop size increase, coalescence may be prevented. Fig. 3 shows the coalescence e$ciency j as a function of drop size d for di!erent parameters, in which the physical properties are from the experiments of Park and Blair (1975). For drops with mobile interfaces in a pure system, the coalescence e$ciency, determined by Eq. (24), is much higher than that for immobile interface (Eq. (23)). Fig. 3a shows that j increases rapidly with holdup , a similar result to that for immobile interfaces. For drops with mobile interfaces, an important feature is that j increases rapidly as k decreases, as shown in Fig. 3b, as B compared to immobile interfaces. From the comparison and discussion, we may draw a few conclusions. The prediction of d is important to understand the
phenomena in a dispersion system. The relation between d and physical parameters tells us how to reach the
required condition and how to correlate experimental data.
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Fig. 2. Coalescence e$ciency as a function of drop diameter for immobile interfaces (Eq. (23)).
In a pure system without any surfactant, interfacial mobility is relatively large when k is small. Coalescence B time is short and d is large, so that the interfacial area
decreases rapidly because of coalescence. The interfacial area may be increased by suppressing the interfacial mobility by adding surface-active agent and by increasing the energy dissipation. When a surfactant is present, the interfacial mobility may be retarded. When the interfaces are partially mobile, all of the bulk and interfacial properties appearing in Eqs. (5)}(10) will a!ect coalescence time t, minimum stable diameter d , coalescence e$ciency j, and drop
size distribution. The increase in those properties that retard interfacial mobility, such as k , c , i and e , will B reduce d and j. If the interfaces are immobile, coales cence time is large so that d is small. At low holdup ,
coalescence is usually negligible. However, when or e is larger, coalescence may become controlling step (Nishikawa, 1987). When e'e , the drop sizes are so small that the interfacial area is large. Moreover, coalescence and break up of drops occur rapidly, so the rate of mass transfer increases greatly.
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
N P r r G t ¹ = ' X >
agitation speed, s\ power input to the impeller, kg ) m/s "lm radius, m radius of drop i, m coalescence time, s contact time, s impeller blade width, m number of drops in a size group volume percentage of drops
Greek letters c c e e e i j k l o
interfacial tension, N/m change rate of interfacial tension with interfacial concentration, Nm/mol interfacial shear viscosity, Ns/m energy dissipation, m/s critical energy dissipation, m/s interfacial dilational viscosity, Ns/m coalescence e$ciency viscosity, Pa ) s kinematic viscosity, m/s density, kg/m volume fraction of the dispersed phase (holdup)
Subscript c d
continuous phase dispersed phase
Superscript Fig. 3. Coalescence e$ciency as a function of drop diameter for immobile (dashed line, Eq. (23)) and mobile interfaces (Eq. (24)): (a) e!ect of holdup (k "0.59 mPa s); (b) e!ect of k ( "0). B B
H
dispersion
Acknowledgements Notation A constant determined by experiment B London-van der Waals constant, Jm cN equilibrium interfacial concentration, mol/m C concentration of solution d drop diameter, m d average diameter in a size group, m d maximum stable drop size, m
d minimum stable drop size, m
D impeller diameter, m ' D tank diameter, m 2 DN interfacial di!usion coe$cient, m/s E dimensionless curvature radius F force, N h "lm thickness at rupture, m k viscosity ratio of dispersed and continuous phases, k /k B A l length scale, m M interfacial mobility coe$cient
The authors are grateful for the "nancial support by the National Natural Science Foundation of China, grant no. 29573114.
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675 Clark, M. M. (1988). Chemical Engineering Science, 43, 671}679. Clay, P. H. (1940). Proceedings of the Royal academic Science (Amsterdam), 43, 852, 979. Coulaloglou, C. A., & Tavlarides, T. T. (1977). Chemical Engineering Science, 32, 1289}1297. Doulah, M. S. (1975). Industrial and Engineering Chemistry Fundamentals, 14, 137. Hinze, J. O. (1955). A.I.Ch.E. Journal, 1, 289}295. Kitchener, J. A., & Prosser, A. P. (1957). Proceedings of the Royal Society of London A, 242, 403}409. Kumar, S., Kumar, R., & Gandhi, K. S. (1993). Chemical Engineering Science, 48, 2025}2038. Lagisetty, J. S., Das, P. K., & Kumar, R. (1986). Chemical Engineering Science, 41, 65}72. Levich, V. G. (1962). Physicochemical Hydrodynamics. Englewood Cli!s, NJ: Prentice-Hall. Li, D. (1994). Journal of Colloid Interface Science, 163, 108}119. Li, D. (1996). Journal of Colloid Interface Science, 181, 34}44. Li, D., & Liu, S. (1996). Langmuir, 12, 5216}5220. Li, D., & Zhao, L. (1996). Wuli Huaxue Xuebao. Acta Physica et.Chemica, 12, 709}715.
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Lin, C. -Y., & Slattery, J. C. (1982a). A.I.Ch.E. Journal, 28, 147}156. Lin, C. -Y., & Slattery, J. C. (1982b). A.I.Ch.E. Journal, 28, 786}792. Liu, S., Li, D., & Jia, S. (1995). Wuli Huaxue Xuebao. Acta Physica et.-Chemica, 11, 997}1003. McManamey, W. J. (1979). Chemical Engineering Science, 34, 432}434. Nishikawa, M., Mori, F., & Fujieda, S. (1987). Journal of Chemical Engineering of Japan, 20, 82}88. Park, J. Y., & Blair, L. M. (1975). Chemical Engineering Science, 30, 1057}1064. Sathyagal, A. N., & Ramkrishna, D. (1996). Chemical Engineering Science, 51, 1377}1391. Shinnar, R. (1961). Journal of Fluid Mechanics, 10, 259}275. Shinnar, R., & Church, J. M. (1960). Industrial and Engineering Chemistry, 52, 253}256. Sprow, F. B. (1967). A.I.Ch.E. Journal, 13, 995}998. Taylor, G. I. (1932). Proceedings of the Royal Society A, 138, 41. Thomas, R. M. (1981). Journal of Multiphase Flow, 7, 709}717. Tsouris, C., & Tavlarides, L. L. (1994). A.I.Ch.E. Journal, 40, 395}406. Vivaldo-Lima, E., Wood, P. E., Hamielec, A. E., & Penlidis, A. (1997). Industrial and Engineering Chemistry Research, 36, 939}965.
Drop coalescence in turbulent dispersions Shiping Liu , Dongming Li * Polymer Processing Laboratory, East China University of Science and Technology, Shanghai 200237, People's Republic of China Department of Chemistry, Tongji University, Shanghai 200092, People's Republic of China
Abstract The coalescence between drops in an isotropic turbulent dispersion is examined. With this result, a new model without any adjustable or empirical parameters is developed to predict the minimum drop size stable against coalescence both in surfactant systems and pure systems. The drop size distribution and coalescence e$ciency are also discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Drop coalescence; Turbulent dispersion; Minimum drop size
1. Introduction The dispersion of immiscible liquids is of great importance in many chemical engineering processes. For example, in systems such as extraction, suspension polymerization and multiphase reactions, the interfacial area available for mass transfer is increased by forming liquid}liquid dispersions in stirred tanks or column contactors, as could be determined by drop size distribution and volume fraction (holdup) of the dispersed phase. In a turbulent dispersion, the breakup and coalescence of drops usually take place continuously. These rate processes not only determine the drop size distribution, which is an equilibrium property, but also have a profound e!ect on mass transfer in dispersions. The phenomenological model concerning coalescence and breakage of drops has been extensively used to simulate the drop size distribution, which could be a manifestation of the two rate processes (Tsouris and Tavlarides, 1994). However, in order to understand the dispersive process better, it is important that drop coalescence and breakup can be assessed. In a dispersion, the important parameters closely related to drop coalescence and breakup are the minimum and maximum stable drop sizes d and d . For
example, if a dispersion is maintained by agitation for a su$ciently long time, a local dynamic equilibrium between coalescence and breakage will be established.
* Corresponding author.
Under this condition, d and d can be observed. Any
drop which is smaller than d will coalesce, while those
larger than d will break up. Furthermore, the #ow in
a stirred tank could be divided into regions of dispersion (near the impeller) and coalescence (relatively stagnant zones). In the region of coalescence the equilibrium spectrum of drop size distribution depends on the coalescence process. The Sauter average diameter d is directly pro portional to d . On the other hand, when the breakup
process is dominant, d is proportional to d . There
fore, d and d are important in interpreting drop size
distribution and modeling the related two-phase #ows. In locally isotropic turbulence, theoretical analysis on breakup of drops has received wide attention since the investigations of Taylor (1932) and Hinze (1955). The diameter of the largest drop against breakage, d , is
determined by the process of breakup. The phenomenon in the inertial subrange is relatively well understood so that d in a dilute dispersion can be evaluated with
reasonable con"dence (Hinze, 1955; Shinnar, 1961; Sprow, 1967; McManamey, 1979; Lagisetty et al., 1986; Calabrese, 1986; Clark, 1988). One of the widely used expressions was given by Hinze (1955) d "A (c/o ) e\ (1)
A in which A is a constant. Hinze (1955) found that A "0.725 from the experimental data of Clay (1940) using two coaxial cylinders, of which the inner one rotated. For a dispersed phase with low viscosity in an agitated tank, the above relation becomes d "A (c/o ) (ND)\ ,
A '
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 1 0 0 - 1
(2)
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
where A is a constant. The value of A varied substan tially in previous investigations. For example, Lagisetty et al. (1986) suggested a value of 0.125, similar to the one obtained by Sprow (1967), while the correlation obtained by Calabrese et al. (1986) gave a value of 0.0883. Recently, Sathyagal et al. (1996) found that their experimental results agreed well with the one given by Calabrese et al. (1986), while the model of Lagisetty et al. (1986) predicted much larger values for d . Thus we take A "0.0883
when d is needed.
Coalescence phenomenon, however, is more complex since it involves not only the approach of two drops, but also the drainage and eventual rupture of the intervening liquid "lm, in which the physical properties of the #uids and interfaces will play an important role. For example, coalescence in a dispersion is usually reduced considerably by the combined action of a protective colloid and of turbulent agitation. For a highly viscous dispersed phase, the viscoelastic properties become important (Alvarez et al., 1994). In suspension polymerization, the suspension viscosity may be large enough for breakage and coalescence of drops to cease as conversion increases (Vivaldo-Lima et al., 1997). We deal with the coalescence process in dispersions with Newtonian #uids. Kumar et al. (1993) and Tsouris and Tavlarides (1994) have reviewed the models for the coalescence process in the literature. It is convenient to view the process as the drainage of the continuous-phase "lm entrapped between any two drops. The coalescence time, or the time required for "lm drainage and rupture, is an important parameter characterizing the process. However, these models usually deal with undeformable interfaces (e.g., a "lm between two rigid spheres or a plane parallel "lm) so that the initial and/or critical "lm thicknesses are needed for calculating the coalescence time. For example, Shinnar and Church (1960) have investigated the in#uence of turbulence on coalescence of droplets and obtained d "A e\ o\ A(h ), (3)
A where A is a constant, A(h ) is the energy necessary to separate completely two drops with unit diameter, which are initially separated by a distance h , and is determined by the properties of the liquids. Sprow (1967) derived a similar expression. Thomas (1981) developed a formula for the drops with tangentially immobile interface:
ch d &2.4 , (4)
koe A A where h is the "lm thickness at rupture between the two drops. Both models involve an empirical parameter, A(h ) or h, that has to be assumed or determined experi mentally, implying the dependence of d on the physical
properties is unknown (Thomas, 1981).
For drops with an immobile interface, only the physical properties of the continuous phase appear in the expressions for the models. If the interfaces are partially or fully mobile, the situation may be di!erent. For example, Park and Blair (1975) could not "nd turbulence-stabilized emulsions for a pure system involving a dispersed phase with low viscosity k . Kumar et al. B (1993) predicted that the dispersion with pure phases cannot be stabilized at low stirrer speed, which was exactly opposite to the experimental results of Shinnar for surfactant systems (1961). Therefore, the viscosity of dispersed phase may come into e!ect. In the present paper, we consider the coalescence in a turbulent dispersion with and without surfactant. A model is developed to predict the minimum stable drop size. The drop size distribution and coalescence e$ciency are also discussed. For simplicity, we assume that both phases are Newtonian #uids.
2. The proposed model 2.1. Coalescence time between drops in turbulent yow For coalescence to occur, the intervening liquid "lm between two drops has to drain until the "lm ruptures. A theoretical model has been developed for drainage and rupture of a dimpled liquid "lm between a drop and a surface (Lin and Slattery, 1982a,b). With this model, the coalescence time between two drops or bubbles can be predicted by giving only the drop sizes and the required physical properties of #uids and interfaces. Dispersed systems usually contain at least a trace of surfactant, which may give rise to interfacial tension gradient and/or interfacial viscosities so as to retard interfacial mobility. Based on the analyses on the e!ects of surface tension gradient and surface viscosities on bubble coalescence (Li, 1996) and drop coalescence in pure systems (Liu et al., 1995, 1996; Li and Liu, 1996), Li and Zhao (1996) derived a more general expression for coalescence time between drops when surfactant is present in the continuous phase. For simplicity, we consider two equal-sized drops. The coalescence time t becomes t"0.0272f
p Sk F \Sd >S A , 4 c \SB \S
(5)
where the correlations for u and f are 65a u" , (1#320a )(1#0.66k ) Q 4.0 106.0 f"1# exp ! log a 84.0# 1#k 1#0.09k Q Q
(6)
.
(7)
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
As k P0, the last expression is not appropriate, and Q a more accurate correlation for f is f"1#7.70 exp[!1.25"log 190a" ],
(8)
The parameter in these equations are
1 c cNF k F , "!0.254 #12.76 B a k cDN k cd A A and
(9)
i #e k "4 . (10) Q kd A In these equations, d is the drop diameter, k and k are A B the viscosities of the continuous and dispersed phases, respectively, c is the equilibrium interfacial tension, c is the change rate of interfacial tension with interfacial concentration, i and e are interfacial shear and dilational viscosities, respectively, cN is the equilibrium interfacial concentration, DN is the interfacial di!usion coe$cient, and F is the force pressing the drops together. Since the retarded London-van der Waals forces are considered, the van der Waals constant B is assumed to be approximately 10\ Jm (Kitchener and Prosser, 1957; Black et al., 1960; Churaev, 1974). Both parameters a and k characterize the mobility of the interfaces. As the Q interfacial mobility decreases, coalescence time increases. As aP0, which results either from large surface tension gradient or large viscosity of the dispersed phase, or as k PR for large interfacial viscosities, the interfaces Q become immobile so that the coalescence process is the slowest. In these cases, u"0 and f"1, so that Eq. (5) reduces to the one obtained previously (Chen, 1985; Li, 1994). For a pure system, the interfaces are mobile, and we may use a simpler expression for coalescence time (Li and Liu, 1996). For simplicity, we therefore consider two limiting cases, in which the interfaces are either tangentially immobile as a result of large interfacial stress or mobile in a pure system. In turbulent #ow, if it is assumed, as in Kolmogorov's theory, that the eddies responsible for coalescence belong to the inertial subrange, the force pressing the drops together may be estimated in the order of magnitude as F&o e d , (11) A where e is the energy dissipation per unit mass, and o is A the density of the continuous phase. Lacking more detailed information, we adopt a coe$cient of unity in Eq. (11) in the following calculations. For immobile interfaces in surfactant systems, Eq. (5) becomes k o e d t"0.0272 A A . c B
(12)
and for a pure system, the interfaces are mobile and circulation prevails inside a drop, and the coalescence
5669
time t becomes (Liu et al., 1995; Li and Liu, 1996) 1 c k B A "1363.3 t E k o e d B A c B #217.3 , E k o e d A A
(13)
where E is the dimensionless curvature radius of the liquid "lm, which is a function of the interfacial mobility coe$cient M E"12.61#2.166 arctg(2M )
(14)
With the force expressed by Eq. (11), M becomes
pc k M"1.12 A k o e d B A
(15)
in which k is the viscosity of the dispersed phase. Both B Eqs. (12) and (13) indicate that the coalescence time increases with the intensity of turbulence. This means that increase in turbulence intensity will decrease the probability of drop coalescence. 2.2. Minimum drop size in turbulent yow The coalescence and breakup of drops are in dynamic equilibrium in turbulent #ow under steady state. As two drops are brought together in a dispersion, a thin liquid "lm forms between them and drains. During drainage, turbulent #uctuations could separate the drops again. Assuming that the eddies in the inertial subrange are responsible for the motion of drops, Levich (1962) estimated the average contact time ¹ between drops in turbulent #ow as ¹&d e\ .
(16)
Again we adopt a coe$cient of unity in Eq. (16) in the following calculations for lack of more detailed information. A simple criterion for coalescence to occur is t(¹, i.e., the contact time should be long enough for the liquid "lm to drain and rupture before the drops are separated again in turbulence. With Eqs. (12) and (16), we obtain the minimum diameter of a drop which is stable against coalescence in a turbulent dispersion c B d " .
0.0272k o e A A
(17)
This agrees with Eqs. (3) and (4), in which d varies with
e\ in particular. From Eq. (17), the minimum stable diameter can be predicted directly as a function of physical properties of #uids and energy dissipation e. Since the expression for coalescence time in a pure system is more complex, we cannot explicitly express d as a function of the system parameters. With
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Eqs. (13) and (16), we have c k B A E k o e d B A
c B #217.3 "1 (18) E k o e d A A
from which d can be solved by the Newton method.
For a very dilute dispersion, e is equivalent to the energy dissipation for a homogeneous continuous phase. However, in a concentrated dispersion, the two-phase dispersion has to be considered as a whole. The density and viscosity of such a dispersion depend on the volume fraction of the dispersed phase (Taylor, 1932): 1363.3
oH"o #o (1! ), (19) B A k #0.4k A , kH"k 1#2.5 B (20) A k #k B A where superscript H refers to a two-phase dispersion, subscript d and c correspond to the dispersed and continuous phases, respectively. Considering Kolmogorov's microscale, Doulah (1975) showed that
e lH , (21) " eH l A where e and eH are the energy dissipation for the pure continuous phase and the two-phase dispersion, respectively, l and lH are the respective kinematic viscosities. A In the presence of the drops, the viscosity of the dispersion increases (lH'l ) so that the energy dissipation A decreases (eH(e). With Eqs. (19)}(21), we may take account of the e!ect of holdup on d and d in
turbulent #ow. An important di!erence between the two limiting cases, i.e., the drops with immobile interfaces in surfactant systems and those with mobile interfaces in a pure system, is the e!ect of the viscosity of the dispersed phase. For immobile interfaces, d is not a function of k for
B low , though it is a weak function of k through Eqs. B (19)}(21) for higher . For mobile interfaces, the interfacial mobility increases as k"k /k decreases, as inB A dicated by Eq. (5), so that d increases rapidly as
k decreases. If the interfaces are partially mobile, all of the bulk and interfacial properties appeared in Eqs. (5)}(10) will a!ect d . Increase in those properties that retard
the interfacial mobility, such as k , c , i and e , will B decrease d .
2.3. Coalescence ezciency Coulaloglou and Tavlarides (1977) have assumed that once the drops collide with each other the contact lasts for some time, called contact time, and then they either coalesce or bounce away from each other. Therefore, coalescence occurs when the random contact time exceed
the coalescence time. The authors assumed that the contact time is exponentially distributed and they de"ned a coalescence e$ciency function as j"exp(!t/¹).
(22)
With Eqs. (12) and (16), we obtain the coalescence e$ciency as a function of physical properties and energy dissipation for drops with immobile interfaces:
k o e d j"exp !0.0272 A A . c B
(23)
With Eqs. (13) and (16), we obtain an expression for mobile interfaces in a pure system:
c k B A !1363.3 E k o e d B A \ c B !217.3 . (24) E k o e d A A Both equations indicate that coalescence may be prevented by increasing turbulence intensity. For general cases, the bulk and interfacial properties that retard the interfacial mobility will reduce the coalescence e$ciency.
j"exp
2.4. Ewect of d on drop size distribution in turbulent
dispersions Under steady state, the drop size distribution is not a function of time. If a drop signi"cantly larger than d is formed by coalescence, it will have a high prob ability of being broken in a short time. Likewise if a drop is signi"cantly smaller than d , it will coalesce. Thus
dispersion is in dynamic equilibrium and a steady-state drop size distribution is maintained. Combining Eqs. (1) and (17) for immobile interfaces, we have d c k A
" A (25) d 3.19 o e B
A which implies that the drop size distribution becomes narrower as e increases. This equation characterizes the drop size distribution in a turbulent dispersion and provides information for scale up of mixing equipment. Eq. (23) shows that coalescence may be prevented by increasing turbulence intensity. Such a dispersion was considered as turbulence-stabilized (Shinnar, 1961). Stabilization is possible only if the minimum diameter stabilized by turbulent #uctuations is less than the maximum stable diameter, i.e., d (d . Any drop in a turbulent
dispersion must be larger than d and smaller than d .
Letting d "d , the critical energy dissipation is de
"ned as c k A , e "A oB A
(26)
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
where A "(A /3.19) , e.g., A "1.40;10\ for A "0.725 (Hinze, 1955). For mobile interfaces in a pure system, we can also de"ne e with Eqs. (1) and (18), so that 1363.3 B k o c e A A A E k B 217.3 B o e A "1. # (27) A E k c A The value of e can be obtained by Newton's method. If e(e (i.e., d (d ), the dispersion is turbulence-sta
bilized. If e'e (i.e., d 'd for calculated values),
the dispersion will behave like an unstabilized dispersion in which Eqs. (17) and (18) do not apply (Shinnar, 1961). In this case, turbulence alone cannot prevent coalescence in the dispersion, since the drops coalesce and breakup rapidly.
3. Comparison and discussion Several investigators studied experimentally the behavior of liquid}liquid dispersions maintained by agitation and measured the drop size distributions. Shinnar (1961), Shinnar and Church (1960) dispersed a molten microcrystalline wax in hot water (system 1). Chatzi et al. (1991) used a system of 1% styrene in water under 603C (system 2). System 3 (Chatzi and Kiparissides, 1995) was one of 50% n-butyl chloride in water under 203C. In these systems, surfactant polyvinylalcohol (PVA) was added, so we compare our model for immobile interfaces with these results. The physical properties of the three systems are listed in Table 1. To calculate d , e is
needed. Chatzi et al. (1991) suggested e"100DN/p for ' their experiments, which is used in our prediction of d for system 2. Lacking more detailed information for
the other two systems, we use e&62DN (Thomas, ' 1981). Moreover, it is necessary to take account of the damping e!ects of the suspension on the local turbulence intensities at high holdup. From Eqs. (19)}(21), we obtain e/eH"6.15 for system 3.
5671
Part of the drop size distributions at di!erent impeller speed measured by Shinnar (1961), the minimum stable diameters d and d , predicted by Eq. (4) (Thomas,
1981) and Eq. (17), respectively, are listed in Table 2, in which d is the average diameter in a size group, X is the number of drops in the size group, > is the volume percentage of the drops. Shinnar calculated both d (90)
and d (10), which were de"ned so that 90% of the
cumulative volume of all drops had a diameter smaller than d (90) and 10% were smaller than d (10). Table
2 shows that the minimum stable size d , in which the
rupture "lm thickness was assumed to be 0.1 lm, is closer to d (10) and much larger than the observed minimum
diameters, while the values predicted by this analysis, d , are in good agreement with the actual minimum
stable diameters. The comparison with other two systems (Chatzi et al., 1991; Chatzi and Kiparissides, 1995) is shown in Table 3. The surface tension c at di!erent concentration of PVA, C, and the measured minimum stable diameters are also listed. If the interfaces are partially or fully mobile, the situation may be di!erent. An important di!erence between mobile and immobile interfaces is the e!ect of k on d . B
Park and Blair (1975) could not "nd turbulence-stabilized emulsions for a pure system involving a dispersed phase with low k . Kumar et al. (1993) predicted that the B dispersion with pure phases cannot be stabilized at low stirrer speeds. This is exactly opposite to the observation of Shinnar (1961), in which PVA added to the continuous phase may make the drop interface immobile. We discuss the phenomena according to our analysis for mobile interfaces. Park and Blair (1975) conducted some experiments for a pure system, in which the interface of a drop may be assumed to be mobile. The system consisted of mutually saturated water (continuous phase), and methlisobutylketone (MIBK). The experiments were carried out in a 6 in diameter mixing vessel with four ba%es. Liquid depth H was maintained at 5.625 in. The holdup was
"0.005 and 0.1. The turbine blade agitator was a standard mix #at blade design. Its diameter D was 3 in with ' a blade width = of 0.375 in. The impeller speed was '
Table 1 Physical properties and impeller diameter System
k (mPas) A
k (mPa s) B
o (kg/m) A
o (kg/m) B
c (mN/m)
D (m) '
1 2 3 4
0.36 0.466 1.0 0.89
17.8 0.475 0.45 0.59
964 983.2 988 997
791.4 870.4 887 795
38.5
0.127 0.075 0.127 0.076
1: Shinnar (1961), molten microcrystalline wax in hot water, with surfactant PVA; 2: Chatzi et al. (1991), 1% styrene in water under 603C, with surfactant PVA; 3: Chatzi and Kiparissides (1995), 50% n-butyl chloride in water under 203C, with surfactant PVA; 4: Park and Blair (1975), methlisobutylketone (MIBK) in water, no surfactant.
10.5
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Table 2 Comparison of predicted minimum stable drop diameters d by Eq. (17) with experimental data (Shinnar, 1961)
N (r.p.m.) d l
X
>
X
15 30 45 60 75 90 105 120 135 150
* 61 * 67 * 83 * 127 * 171
* 0.07 * 0.81 * 2.43 * 8.8 * 23
10 25 47 79 115 165 210 177 2 2
156
d d (90)
d (10)
d (Eq. (17))
d (Eq. (4))
177 265 128 95 48
220 >
X
0.002 0.038 0.240 0.97 2.77 6.85 13.1 17.3 2 2
14 75 87 171 258 281 2 2
127 174 96 73 36
312 > 0.005 0.205 0.81 3.83 11.35 21.2 2 2
97 135 75 56 26
Table 3 Comparison of predicted d by Eq. (17) with experimental data
(Chatzi et al., 1991 and Chatzi and Kiparissides, 1995) System
2 2 3 3 3 3
C (g/l)
0.1 3.0 0.5 0.5 1.0 1.0
c (mN/m)
15.3 3.8 3.2 3.2 2.8 2.8
N (r.p.m.)
300 300 400 700 400 700
d (lm)
Experimental
Eq. (17)
22.5 12.5 10.7 8.7 7.0 6.3
27.1 14.6 8.5 5.3 8.0 5.0
maintained at N"375 r.p.m. For turbine or #at-bladed impellers, the average energy dissipation e is the power input per unit mass in the volume swept by the impeller (McManamey, 1979) 4P e" , (28) no D= A ' ' where P is the power input to the impeller, which can be calculated by P"P o ND, (29) A ' where P is the power coe$cient. For the tank and impeller (D /= "8) used, since D /D O1/3 and ' ' ' 2 H/D O1, the energy dissipation should be corrected 2 to
32P D H 2 e" ND, (30) ' p 3D D ' 2 where D is the diameter of the tank. Under the condi2 tion, P "2.9 for Re'10 (Chen, 1985).
440 >
X 65 135 175 195 164
0.035 0.605 2.51 6.9 11.2
91 130 68 44 20
X 95 183 179 255
440 > 0.10 1.55 5.15 17.4
79 115 57 44 20
627
627
X
>
X
>
410 297 210
1.90 11.30 27.0
334 310 199
1.6 12.0 26.0
53 98 34 33 14
51 92 35 33 14
The e!ect of the bulk viscosities, expressed as k"k /k , on d and d for "0 and 0.2 is shown in B A
Fig. 1. The energy dissipation e is calculated from Eq. (30), d for immobile and mobile interfaces are cal culated from Eqs. (17) and (18), respectively, d is deter mined by Eq. (2) with A "0.0883 (Calabrese et al., 1986), and the other parameters are from the experiments of Park and Blair (1975). The drop with immobile interface behaves like a rigid sphere, so that d is not
a function of k for "0, though it is a weak function of B k by Eqs. (19)}(21) for higher holdup . For drops with B mobile interface, d decreases rapidly as k increases.
When k'30, d as computed is close to that for immo bile interfaces. When k is small, the interfacial mobility is large so that liquid drops coalesce more rapidly. When k'1.1, d (d and the system is stable. However,
when k is small, calculated d 'd so that the system
is unstable. In the experiments of Park and Blair (1975), k"0.663, so they could not "nd turbulence-stabilized emulsions. For their experiment on "0.005, predicted d 'd when N'2 r.p.s. It implies that the MIBK
Water system (Park and Blair, 1975) is unlikely to be stabilized by increasing turbulence intensity alone, unless the interfacial mobility of drops is retarded, which agrees with the experimental observation. The above comparisons indicate that the model gives an appropriate prediction for the minimum stable diameter under di!erent experimental conditions. Since d is proportional to d when coalescence process
is dominant, it shows also the e!ect of physical properties as well as turbulence intensity on drop size distribution. Fig. 2 shows the coalescence e$ciency j for drops with immobile interfaces, determined by Eq. (23), as a function of drop size d for various values of hold up and energy
S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
Fig. 1. E!ect of bulk viscosity k"k /k on computed drop size. B A
dissipation e. Here d and d are calculated from Eqs.
(1) and (17), in which the physical properties of Shinnar's experiments (1961) are used. The coalescence e$ciency j increases with but decreases as e increases. As drop size increase, coalescence may be prevented. Fig. 3 shows the coalescence e$ciency j as a function of drop size d for di!erent parameters, in which the physical properties are from the experiments of Park and Blair (1975). For drops with mobile interfaces in a pure system, the coalescence e$ciency, determined by Eq. (24), is much higher than that for immobile interface (Eq. (23)). Fig. 3a shows that j increases rapidly with holdup , a similar result to that for immobile interfaces. For drops with mobile interfaces, an important feature is that j increases rapidly as k decreases, as shown in Fig. 3b, as B compared to immobile interfaces. From the comparison and discussion, we may draw a few conclusions. The prediction of d is important to understand the
phenomena in a dispersion system. The relation between d and physical parameters tells us how to reach the
required condition and how to correlate experimental data.
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Fig. 2. Coalescence e$ciency as a function of drop diameter for immobile interfaces (Eq. (23)).
In a pure system without any surfactant, interfacial mobility is relatively large when k is small. Coalescence B time is short and d is large, so that the interfacial area
decreases rapidly because of coalescence. The interfacial area may be increased by suppressing the interfacial mobility by adding surface-active agent and by increasing the energy dissipation. When a surfactant is present, the interfacial mobility may be retarded. When the interfaces are partially mobile, all of the bulk and interfacial properties appearing in Eqs. (5)}(10) will a!ect coalescence time t, minimum stable diameter d , coalescence e$ciency j, and drop
size distribution. The increase in those properties that retard interfacial mobility, such as k , c , i and e , will B reduce d and j. If the interfaces are immobile, coales cence time is large so that d is small. At low holdup ,
coalescence is usually negligible. However, when or e is larger, coalescence may become controlling step (Nishikawa, 1987). When e'e , the drop sizes are so small that the interfacial area is large. Moreover, coalescence and break up of drops occur rapidly, so the rate of mass transfer increases greatly.
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S. Liu, D. Li / Chemical Engineering Science 54 (1999) 5667}5675
N P r r G t ¹ = ' X >
agitation speed, s\ power input to the impeller, kg ) m/s "lm radius, m radius of drop i, m coalescence time, s contact time, s impeller blade width, m number of drops in a size group volume percentage of drops
Greek letters c c e e e i j k l o
interfacial tension, N/m change rate of interfacial tension with interfacial concentration, Nm/mol interfacial shear viscosity, Ns/m energy dissipation, m/s critical energy dissipation, m/s interfacial dilational viscosity, Ns/m coalescence e$ciency viscosity, Pa ) s kinematic viscosity, m/s density, kg/m volume fraction of the dispersed phase (holdup)
Subscript c d
continuous phase dispersed phase
Superscript Fig. 3. Coalescence e$ciency as a function of drop diameter for immobile (dashed line, Eq. (23)) and mobile interfaces (Eq. (24)): (a) e!ect of holdup (k "0.59 mPa s); (b) e!ect of k ( "0). B B
H
dispersion
Acknowledgements Notation A constant determined by experiment B London-van der Waals constant, Jm cN equilibrium interfacial concentration, mol/m C concentration of solution d drop diameter, m d average diameter in a size group, m d maximum stable drop size, m
d minimum stable drop size, m
D impeller diameter, m ' D tank diameter, m 2 DN interfacial di!usion coe$cient, m/s E dimensionless curvature radius F force, N h "lm thickness at rupture, m k viscosity ratio of dispersed and continuous phases, k /k B A l length scale, m M interfacial mobility coe$cient
The authors are grateful for the "nancial support by the National Natural Science Foundation of China, grant no. 29573114.
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