A new model for coalescence efficiency of drops in stirred dispersions

Ckmicaf Engineering Science, Vol. 48. No. It, pp. 202552038, Printed in Great Britain.

A NEW

Department

UXE-Zso9/93 $6.03 + 0.00 (b 1993 Pcrgamon Press Ltd

1993.

MODEL FOR COALESCENCE EFFICIENCY DROPS IN STIRRED DISPERSIONS SANJEEV KUMAR, R. KUMAR+ and K. S. GANDHI of Chemical Engineering, Indian Institute of Science, Bangalore 560012, (Received

12 May

1992; accepted for publication

23 November

OF

India

1992)

model for coalescence efficiency of two drops embedded in an eddy has been developed. Unlike the other models which consider only head-on collisions, the model considers the droplets to

Abstract-A

approach at any arbitrary angle. The drop pair is permitted to undergo rotation while they approach each other. For coalescence to occur, the drops are assumed to approach each other under a squeezing force acting over the life time of eddy but which can vary with time depending upon the angle of approach. The model accounts for the deformation of tip regions of the approaching drops and, describes the rupture of the intervening film, based on stability considerations while film drainage is continuing under the combined influence of the hydrodynamic and van der Waals forces.The coalescenceefficiencyis definedas the ratio of the rangeof angles resulting in coalescence to the total range of all possible approach angles.. The model not only reconciles the contradictory predictions made by the earlier models based on similar framework but also brings out the important role of dispersed-phase viscosity. It further predicts that the dispersions involving pure phases can be stabilized at high rps values. Apart from explaining the hitherto unexplained experimental data of Konno et al. aualitatively, the model also offers an alternate explanation for the interesting observations of Shinnar.

ZNTRODUCTZON Stirred liquid-liquid dispersions of two immiscible phases are often employed in chemical industry to carry out processes such as liquid-liquid extraction, direct contact heat transfer, suspension polymerization and multiphase reactions. In all these situations, the drop size distribution in the vessel is governed by the dynamic equilibrium between breakage and coalescence rates. Breakage of drops due to their interaction with eddies of inertial subrange is relatively well understood and the maximum stable drop size, d,., , in lean dispersions can be evaluated with reasonable confidence. Moreover, mechanistic models of breakage frequency are also available. Coalescence phenomenon, however, is more complex as it involves not only the approach of two drops, but also drainage and eventual rupture of the intervening film of the continuous -phase. The coalescence frequency, required for the solution of the population balance equations, is defined as the product of collision frequency and coalescence efficiency. The present investigation concerns itself with prediction of coalescence efficiency. A number of models have been reported in the literature for predicting coalescence efficiency, t]. These models are based on different ways of visualizing coalescence phenomenon in stirred vessels. Howarth (1964) assumed coalescence to occur immediately if the velocity of approach of two drops along their line of centres exceeded a critical value. He assumed that the velocities of approach of two drops ‘Also at Jawaharlal Nehru Centre for Advanced Scientific

Research, Indian Institute of Science, Bangalore. Author to whom correspondence should be addressed. 2025

follow the three-dimensional Maxwellian distribution. From this he obtained q as the fraction of drops having the approach velocity exceeding the critical value w,,. Thus, rj = exp

- 3w; _ 4u2

( > _

It is seen that the above expression for coalescence efficiency is independent of fluid properties as the model neglects the presence of drainage and rupture of the intervening film. Coulaloglou and Tavlarides (1977) considered both film drainage and rupture in the development of their model. They viewed the influence of the turbulent environment on two drops to be equivalent to thinning of the intervening film under the application of a constant squeezing force for certain contact time t;. Coalescence would occur if the contact time was larger than rE,the time required for the film to thin to critical thickness of film rupture. They assumed the drops to deform due to the squeezing force and form a plane parallel disc. Drop interface was treated as immobile and, thus, the thinning rate expression for the approach of two circular plane parallel rigid plates was employed to compute the coalescence time, t,. They assumed the coalescing droplet pair to be entrained in an eddy of a length scale (d, + d2), where dl and d2 are the diameters of the coalescing drops, and mean contact time G was estimated to be the life time of such an eddy. Their final expression for coalescence efficiency is q=exp[(

-c~kpcE)(*~]

(21

where C, is a parameter that strongly depends on the

2026

SANJEEV KUMAR et al.

critical thickness of film rupture. If C1 is estimated from the now available models for critical thickness of film rupture, it is found that quite low values oft] are obtained for both large and small drops. Muralidhar and Ramkrishna (1986), Das et al. (1987) and Muralidhar et al. (1988) visualized a drop pair to be continuously impinged by small eddies. They assumed drops to approach each other with a mean force due to the asymmetric distribution of pressure fluctuations around the drop, and a fluctuating component of the force superimposed on it arising out of the random nature of these fluctuations. Stochastic nature of the superimposed force thus renders the film drainage process also stochastic in nature. If the time scale of the force fluctuation is much smaller than the time scale of film thinning, the force can be represented as white noise (uncorrelated). If the two time scales are comparable, then the force cab be represented as coloured noise (correlated). The drainage is permitted to continue till the thickness of the separating film either reaches the critical value, when the drops are assumed to coalesce, or goes beyond the original collision thickness, when the drops are assumed to separate. Average probability of the occurrence of the coalescence event was equated to the coalescence efficiency. Das et al. (1987) and Muralidhar et OZ.(1988) assumed the drops to be rigid spheres, whereas Muralidhar and Ramkrishna (1986) considered the drops to be deformable with immobile interfaces. Allowance for drop deformation requires consideration of another time scale. If the deformation time scale is much larger than the time scale of force fluctuation, drops remain deformed under the influence of the mean force. In the other limit, when the time scale of deformation is very small, drops instantaneously respond to the fluctuating force. It is interesting to note that the predictions made in the two extreme limits are opposite to each other. Thus, the value of 9 and the predicted trends depend on the model used. The selection of an appropriate model requires correct description of the parameters representing the noise in the vessel and this is not easily done.’ All these models also require the evaluation of the squeezing forces based only on the distribution of small eddies around the drops, which needs further examination. Table 1 summarizes the predictions regarding the influence of various variables on q made by the various models. It is seen from this table that some models predict q to increase with respect to some variable whereas other models predict exactly the opposite behaviour with respect to the very same variable. Thus, model predictions can go from one extreme to the other even when they are starting from the same premises. Further, the existing models either predict very low values of ‘1 (Coulaloglou and Tavlarides, 1977), or very high values of q when contact time was not limited (Das et al., 1987, Muralidhar et al., 1988). Moreover, all the models treat the critical film thickness to be independent of the system properties, whereas, in reality, it is strongly dependent on them.

Table

1. Dependence of coalescence efficiency on various parameters as predicted by the available models Authors

Howarth (1964) Coulaloglou and Tavlarides (1977) Das et al. (1987) Muralidhar et al. (1988)

(small force fluctuations) Muralidhar et al. (1988) (large force fluctuations) Muralidhar and Ramkrishna (1986) (static deformation) Muralidhar and Ramkrishna (1986) (dynamic deformation)

N

Drop size

fl

I&

+

+

x

x

-

+

-

X

+

-

-

-

-

X

+

+

+

x

-

+

+

-

+

-

-

+

+

+ indicates that q increases with increasing the variable concerned. - means that it decreases. x indicates that q does not depend on this variable.

Thus, until the actual drainage rates as well as stability considerations are taken into account, the quantitative predictions made by these models can at best be approximate. Another restriction in the existing models is that they permit only head-on collisions, whereas that is not the only possible approach of droplets towards each other in a stirred vessel. In the present paper a new model of coalescence efficiency is presented, which not only accounts for arbitrary angle of approach, but also follows drainage of the film right up to its rupture.

THE MODEL

For two drops to coalesce, they must approach each other and the intervening film separating them must drain to a thickness at which it ruptures due to the growth of perturbations. The drainage rate and the growth of perturbations are governed by hydrodynamic, van der Waals and other pertinent forces. The drops do not coalesce if the film does not drain to the required thickness within the available time of contact. The time of contact and the hydrodynamic forces depend on the manner in which the drop-drop interaction is visualized. In a stirred vessel, three modes of interaction and coalescence can be imagined. The drops can get embedded in eddies of inertial range and experience a squeezing force for a finite time. The second mechanism assumes the drops to be drawn together by a random force generated by asymmetric bombardment of drops by small eddies. The third possibility is the shear coalescence in the relatively calmer sections of the vessel or of smaller drops embedded in Kolmogorov scale eddies. At very high dispersed-phase hold-up values, coalescence may be predominantly caused by the velocity gradients. The present model addresses the question of coalescence efficiency when the drops are embedded in eddies of inertial subrange. These eddies are

A new model for coalescenceefficiencyof drops in stirreddispersions

2027

present throughout the vessel, though the energies associated with them can be position-dependent. We consider coalescence events where two drops approach each other while both are embedded in the same eddy and are already in collisional contact. Thus, both the hydrodynamic force and the time of contact have been evaluated by visualizing the drops to be embedded in an eddy of size comparable to the sum of their diameters. It is quite possible that at the end of the contact time, when the external force has reduced to zero, the film may still drain under the influence of attractive van der Waals forces and result in coalescence. Such events have been neglected in the development of the model.

Fig. 1. Two unequal size drops colliding in non-head-on fashion.

Qualitative description ofthe approach of two drops and their subsequent coalescence The approach of two drops and their subsequent coalescence can either be viewed as a single-step process or can be split into collision (when the drops approach very near to each other) and coalescence (where the drainage and the rupture of intervening film occur). Most of the investigators have adopted the latter approach and treated collision frequency and coalescence efficiency separately. Chesters (199 l), however, has considered the approach of two drops and proposed the possibility of their rebounding. He has considered a drop moving towards another one with the velocity of the pertinent turbulent fluctuation. He has neglected viscous dissipation in the continuous phase during the approach and assumed that the kinetic energy of the drop is completely converted into interfacial energy resulting in strong deformation, which can finally result in drop rebound. Such rebounds have indeed been observed when drops collide in air (Jiang et a[., 1992) but the experimental results do not substantiate those expected from Chesters’ analysis. Further, the complete neglect of viscous resistance at small interdrop distances needs justification as drop deformation can occur only when viscous forces have become important. In view of these, the present work retains widely adopted framework of delineating between collision and coalescence, and treats only the processes occurring during coalescence between drops which are already very close to each other. Two drops caught in an eddy experience a squeezing force exerted by the eddy on them. Traditionally, this force has been assumed to bc directed along the line joining their centres. Such encounters can be classified as head-on collisions. However, in an eddy, the drops can be initially caught in such a way that the line joining their centres can make any angle with the direction of the force exerted by the eddy. We need to consider only angles between 0 and n/2, where f3= 0 indicates head-on collision and 0 = 742 indicates a grazing collision. Deviations from head-on collision introduce a hitherto ignored phenomenon: rotation of the colliding doublet. When two drops are subjected to a head-on collision, they do not experience any torque and, therefore, do not rotate. Such a doublet

experiences a constant squeezing force throughout the available contact time. In contrast, the drops colliding with each other at an angle experience a net torque and rotate around an axis passing through their centre of mass as shown in Fig. 1. Only that component of the force directed along the line joining their centres, F cos 0, will make them approach each other, while the component of the force perpendicular to this, Fsin0, will induce rotation. As the doublet rotates, the squeezing component of the force, F cos 6, decreases. If the rotating doublet reaches 0 = x/2 within the life time of the eddy, the drops separate from each other from then on, and the rest of the eddy lifetime available is ineffective in bringing about coalescence. Thus, the contact time available for non-head-on collisions is the smaller of the two time periods: time required to reach 0 = z/2 or the lifetime of the eddy. Hence, in the new model, we have to consider the rotational motion of the doublet. Let us now consider in some detail the effect of the squeezing force. Initially, the drops may approach each other with the velocity given by Hadamard-Rybzynski formula and retain spherical shape (Batchelor, 1967). As h, the distance separating them becomes of the order a, their approach velocity decreases (Haber et al., 1973). When the drops are at a distance h Q a, a lubrication regime is set up between them (Davis et al., 1989) and the approach velocity decreases rapidly while the pressure in the tip region increases. At some stage, this pressure attains a value of the order of 2u/a and the drops begin to flatten. If the drop size is large, pressure gradient at the edge of the flattened region induces faster thinning and a dimple gets formed which vanishes as time progresses. However, the size of the drops encountered in stirred dispersions is small, and dimple formation is not expected to occur. Hence, the drop shape smoothly changes from perfectly spherical to a disc at the tip (Emil1 and Woods, 1973; Ivanov and Dimitrov, 1988). If the force applied on the drop is small, excess pressure as large as 2u/a may not develop in the tip region and the drops may reach very small film thicknesses without deforming significantly. The drainage rates are significantly affected by the presence or absence of the disc, and this also has to be quantitatively modelled.

2028

SANJEEV

KUMAR

At a distance of about 500 A, drops can attract or repel each other depending upon the net effect of van der Wals attractive forces and electrical double-layer repulsive forces. If the repulsive forms are assumed to be absent, the van der Waals forces result in an increased squeezing force. Additional phenomena also come into play. If the interfaces of the discs of the drops are sufficiently close to each other, an increase in the interfacial energy due to a perturbation on the interface can be less than the decrease in the van der Waals attraction energy between the perturbed interfaces. The free energy of the system then decreases due to perturbations. This results in spontaneous growth of the perturbation (Scheludko, 1967) till the perturbed interfaces meet each other at some point and film rupture occurs. At this point, drops are assumed to have coalesced. As coalescence times calculated by ignoring stability phenomena are invariably underestimated, it is necessary to incorporate stability considerations into the model to calculate the critical film thickness of rupture. The above description is valid for all collisions irrespective of their angle of approach. However, some additional features occur during film drainage when the collision is not head-on. While the film drainage continues for a rotating doublet, the decreasing film thickness require an increased pressure buildup in the tip region whereas the squeezing force decreases due to the rotation. Thus, the drop deformation now depends on the net effect of the two opposing factors. Unlike in head-on collisions, here the drops can be deformed at some stage, but attain spherical shape later. Such changes in drop shape are possible only when the deformation time scale is smaller than the film thinning time scale, and that always seems to be the case. Thus, to evaluate coalescence efficiency, it is necessary to obtain expressions for the forces acting on the drops that cause drainage, the time available for coalescence to occur, to determine the state of deformation of the drops and the rate at which drainage occurs, and the final thickness at which film ruptures leading to drop coalescence. These are discussed below. Expression for force For two drops of radii a, and a, embedded in an eddy of length scale equal to the sum of their diameters, Coulaloglou and Tavlarides (1977) have given the following expression for the hydrodynamic force squeezing the two drops towards each other: 2_

PCU2(0 where

I= 2(a, + az).

(4)

We expect 1to lie in the inertial subrange and, substituting for u’(I) from CoulaIoglou and Tavlarides (1977), eq. (3) yields

et al.

F = 20.6~,N~D~‘~ Cal + a2)4,3.

(5)

Equation (5) yields the squeezing force for a head-on collision. If, however, the drops approach each other at an arbitrary angle, only the component Fcos 8 provides the squeezing force. The other component F sin 8 provides the torque responsible for rotation of the doublet. Apart from the hydrodynamic force, the drops experience the van der Waals attractive force and electrostatic repulsive force, which become significant at small distances ( -K 500 A). Miller and Neogi (1985) summarized these for deformed and undeformed spherical drops and small degrees of overlap of electrical double layers. They are given for spherical drops by F vdw F cdl

=

=

Aa,

$

12Sxa,C,,N,kBT K

c2

exp (-

~ch)

(7)

whereas for deformed drops these take the following form:

exp (-

sh).

(9)

These forces act along the line joining the centres of the drop and, hence, the total squeezing force is given by F, = F cos 8 + F+,

- FedI

(10)

and the tangential component F, is F, = F sin 0.

(11)

However, in the present work, only van der Waals attractive forces have been taken into account. Time of contact The maximum time over which the hydrodynamic force is applicable corresponds to the lifetime of the eddy (Coulaloglou and Tavlarides, 1977)

t eddy= 1.562 caiNfo”,;;

2/3

.

This provides an estimate of the contact time for head-on collisions, whereas for collisions that begin at any other angle, BO,the contact time can be different. As discussed earlier, the contact time in such a case will either be the time required for the doublet to reach 7t/2 due to the rotation or the lifetime of the eddy itself, whichever is less. Allan and Mason (1962) have showed that two colliding drops rotate as a rigid dumb-bell without relative tangential motion. In the present case, the tangential component of the force, F sin 0, is assumed to act on the centres of the drops and provide the torque. The resulting torque is balanced by the rate of change of angular momentum.

A new model for

head-on collision, the contact time is equal to teddy_ Thus,

Thus,

I$ =(Fsin@(a,

+ a2)

(13)

OI

d2e F(a, + 4

-=

sin

e.

(14)

I

dt2

Here I, the moment of inertia of doublet around its centre of mass, is given by

where Ml and M2 are the masses of the drops of radii ~1~and a2, respectively. The initial conditions used to solve the above equation are att=O tl=t?

0

and

2029

coalescence efficiencyof drops in stirred dispersions

E=O dt

(21)

Formation of the disc and its dimensions When two drops approach each other, conditions may arise when the approaching surfaces deform into a disc. To evaluate the velocity of approach of the drops, it is necessary not only to know the point of formation of disc but also its size. Due to different internal pressures, unequal drops do not yield a plane parallel disc, but a curved one. However, in most of the cases, the disc formed is of much smaller size than the size of the smaller drop and, therefore, its curvature can be neglected to obtain the following simplified expression for the radius of the disc (Coulaloglou and Tavlarides, 1977): Fcose

’

adisc

(16)

Equation (14) can be integrated once with the initial conditions to obtain the following expression for angular velocity:

=

J--

-

K(T

4

(22)

where u,, mean radius, is given by a, = ~

ala2

(23)

a1+a2

dt? t =

~J%$cos

8,

-

bogey.

(17)

Equation (17), however, does not admit an analytic solution. To facilitate a simple solution, we, therefore, use the following approximate relationships for cos 8: case =

1b202, x/2 - 0,

8 0 n/3 > 2 x/2 2 e b n/3

(18)

where h2 = -$ (1 - x/6). The maximum deviation between the values of cos 0 and those given by the above equations does not exceed 5% in the entire range of 6. Substituting for cos B,, and cos r3from the above expressions in eq. (17) and integrating, the following final expressions are obtained for 8 depending upon the value of B,, and t:

and F cos I3 is the squeezing component of the force exerted by turbulent pressure fluctuations. Drainage rates Coalescence can occur only if the time available for contact is larger than the time required for the drainage to occur to such an extent that the intervening liquid film ruptures. During this period, the expressions for drainage rates may undergo changes as the approaching drops may deform. We assume that the system is free of any surfaceactive impurities and interaction of the two approaching drops with other drops in the vessel can be neglected. In such a situation, drainage between two drops can be approximated to that for two drops approaching each other in an unbounded continuous phase. When the drops are sufficiently apart, they can

I

B,, cash (,,&I&), + ,/%(r

- L/3)(1 - b2@ - n/6)“‘,

60 Q n/3, t 2 tn,3 (20) e, z rrj3.

Here tn,3 corresponds to the time taken to reach x/3 when B. is smaller than x//3. For an initial angle &, the pertinent equations of the set (20) are solved to obtain the value of tnlZ, the time required to reach 8 = n/2. If t-,2 < t&,y, it indicates that rotating doublet reaches 0 = x/2 within the eddy lifetime and, therefore, the contact time is reduced to tr,2. If, however, tn12> t.ddy, it indicates that inspite of a non-

be assumed to be spherical. However, as they appreach each other, a disc is formed in the tip region of the drops. Drainage for undeformed drops. IIaber et al. (1973) have presented exact solutions for calculating the velocities of two spherical drops of unequal size, and

2030

SANIEEV KUMAR

of different viscosities, approaching each other due to a force acting along the line joining their centres in a third unbounded liquid. The solution is given in terms of an implicit infinite series. At small distances of separation, their solution requires a large number of terms to obtain convergence. Davis et al. (1989) have recently calculated the approach velocity between two unequal drops by using a combination of lubrication approximation in the intervening film and boundary integral theory for flow inside the drop. They showed that, at small film thicknesses, the predictions are same as those obtained from the exact analysis of Haber et al. (1973). They have obtained the following expression for approach velocity:

v, = -

F,k

1 + 1.69m + 0.43m2 1 + 0.39m

67~&

(24)

Drops with plane parallel disc in the tip region. Ivanov and Traykov (1976) have modelled the drainage process between two deformed drops as the approach of two plane parallel fully mobile interfaces. They considered full Navier-Stokes equations inside the drop and lubrication equations for the film region, and obtained the following expression for the film thinning velocity:

VdiSf=

where

1

“=I

$

(26) Unlike other expressions for film thinning, eq. (26) shows that intervening film can drain down to zero thickness without the aid of van der Waals attractive force. When the drops are far away from each other, the approach velocity is given by the HadamardRybzynski equation (Batchelor 1967): _ F,(3 -I- 3A) SF- 6~&(2 + 32)’

2F’h3

3wc,k4i,c Y

[I + (;)‘I””

(29)

where adisc is given by eq. (22). The mobility ratio, k,/k, is given by

a,

?;’

In the limit m < 1, drops behave as rigid spheres and Taylor’s equation (Ivanov and Dimitrov, 1988) is recovered. In the other limit, m B 1, the drop surface is fully mobile and the following expression results:

v

et al.

this work were of the order unity. In such situation, the inertial forces would also he important. However, creeping flow analysis has been used because an analysis incorporating inertial forces is not available in literature at present.

(27)

Following the procedure suggested by Dimitrov et al. (1984) for the approach velocity of two rigid spheres in the entire range of film thicknesses, the above two limiting expressions for film thinning, eqs (24) and (27), have been added in parallel to obtain a simple approximate relationship for the approach velocity for two drops for the entire range of film thicknesses (co 2 h 2 0), to yield F,(3 + 31) 6n/@l,(2 + 31) K, = l+S 1 + 0.38m 3 + 31 @) k 1 + 1.69~~~+ 0.43m2 2 + 31 where F, is the total force pushing drops together. In the limit k/a, % 1 and h/a, 4 1, the above equation reduces to eqs (27) and (24), respectively. Dimitrov et al. (1984) found that the approximation made above predicts the thinning rates for solid spheres with an accuracy better than 7% in the entire range of the film thicknesses. The above analysis assumes creeping flow, whereas the maximum drop Reynolds numbers encountered in

h _ p

1

10W~,,,c)4~c”1’4 1

k-

PdhFS

h -*

In the limit k,/k % 1, eq. (29) simplifies to 113 32F,2 k5/3 v. d’sc X2P,P&dd I

m-4

(31)

which shows that, at small film thicknesses, the approach rate becomes independent of continuousphase viscosity. Neglecting inertial forces in the drop phase, Chesters (1991) proposed the following expression for drainage of the intervening film between deformed drops: vdise =

The above expression, however, does not reduce to the Reynolds thinning equation in the limit /.L,,--t co. Hence, the results of Traykov and Ivanov (1976) were used in this work. However, calculations were also made using the expression of Chesters (1991). A comparison of the predictions made by these two approaches showed that, while qualitatively they were similar, there were quantitative differences.

Shape transition thickness At present a comprehensive expression for predicting approach rates of drops undergoing shape transition is not available. Though boundary element technique can bc used to make some calculations, it requires large computational time and even then it fails to give results at small thicknesses. In the present model, therefore, we assume that at some film thickness, called the shape transition thickness, the drop changes its shape from spherical to plane parallel disc at the tip. The transition thickness is assumed to occur at a film thickness where the film thinning rates corresponding to spherical drops and the plane parallel

A new model for coalescence efficiency of drops in stirred dispersions disc are equal. Thus, we assume KZJ(LII*) = Vdi,&,“,)

v = v*,, (33) v = vai.,. v=

2.

It is interesting to note that when the above approach is applied to two drops with immobile interfaces, the shape transition thickness is found to be F/27ca which is in good agreement with Fj2scr required for the onset of flattening (Ivanov and Dimitrov, 1988). For mobile surfaces also, the idea of shape transition could be tested, though only for creeping flow, where exact analysis of Yiantsios and Davis (1990) is available. The agreement was found to be good for the purpose of calculating the drainage times, giving confidence in the proposed approximation for calculating the shape transition thickness even for mobile interfaces. Critical thickness of film rupture As the film thins down, instabilities set in and the film ruptures. Various investigators (Scheludko, 1967; Jain and Ivanov, 1980; Maldarelli et al, 1980) have shown that plane parallel and unbounded films become unstable due to perturbations of wave number k once the transition thickness corresponding to it is crossed. During the coalescence of two approaching drops, the film drainage and growth of perturbation occur simultaneously. Out of the many growing waves, the one that leads to the largest thickness of film rupture is called critical thickness of film rupture. This thickness can be calculated by maximizing the value of h er,kT as obtained from the following expression (Jain and Ivanov, 1980), with respect to wave number k subject to the condition that the minimum value of k is 27r/adisc (Ruckenstein and Sharma, 1987): h L - cr, _ = .zOexp ( L

where jf(k, at which number k have given ables:

\

hc,,, B(!,h) Jw) Y r

dh‘l /

203 1

the doublet experiences a constant external force for the available contact time and, therefore, the size of the disc remains constant. In contrast, non-head-on collisions result in a varying squeezing force and, therefore, the disc diameter also keeps changing continually. Jn the absence of any analysis of growth of perturbation in such a situation, we use in eq. (34) constant values of the force and disc radius, corresponding to the magnitude of force at the time of shape transition. It is quite possible that under some situations, e.g. for very small drops, the drops remain spherical throughout. No analysis is available for stability of the thin liquid region separating two spherical drops. However, it is clear from eq. (26) that the film can actually drain tq zero thickness. We have chosen a thickness of 25 A as rupture thickness merely to stop the computation at some stage, as the time required for further drainage is negligible. If drops undergo double transition (spherical to deformed one and uice uersa), we assume that the perturbations that began to grow on/plane parallel interfaces continue to grow even when drops have become spherical and yield a, values of the same order that would have been obtained had the plane parallel interface persisted all through. Expression for coalescence eficiency It is assumed that drops collide with each other at all possible angles between 0 and nJ2 with equal probability. Once &, is known, it is possible to predict, as will be discussed in the next section, whether coalescence will occur or not. The coalescence efficiency for two drops of any size can then be defined as range

of 0, yielding

11=

coalescence

x/2

_

(37)

It will be seen that above definition of coalescence efficiency includes the earlier concept that coalescence times must be smaller than the time of contact, and is broader in scope since it allows different angles of approach as well.

(34) METHOD

h) is growth rate and h,(k) is the thickness the perturbation corresponding to wave becomes unstable. Maldarelli et al. (1980) the following expressions for these vari-

(35)

kh + Iz + k3h3A2 (36) Equation (34) also requires the value of V, which can be obtained through eq. (29). For head-on collisions,

OF SOLUTION

The coalescence efficiency can be estimated by identifying the range of 13that results in drop coalescence. Conceptually, the range is found as follows. For a given droplet pair, the force exerted by the eddy and the eddy lifetime are computed from eqs (5) and (12). For any given angle of approach, B,,, the contact time is found from eq. (21). The droplet pair is assumed to be initially separated by a distance h,. The film thickness is computed as a function of time from eq. (33), and, if a disc forms, its diameter is noted. Then the stability calculations represented by eqs (34)-(36) are carried out to determine the critical film thickness. Now the calculation for film thickness as a function of time are continued to check if h, is reached within tl or not. Since there are a variety of ways in which coalescence can occur, a general computational pro-

2032

SANJEEVKUMAR

cedure accounting for all such possibilities is required. For head-on collisions, coalescence can occur while the drops remain deformed throughout or a part of the drainage process. Drops can also coalesce without deforming at all. Non-head-on col. lisions can yield coalescence not only in this way but also when the drops deform and again come back to spherical shape, and when the contact time becomes less than cedardue to the rotation of the doublet. The details of the computational procedure accounting for all these possibilities have been presented by Gupta (1992). The range of 8 for which coalescence occurs is found iteratively with an accuracy of 0.1”. The accuracy of the standard numerical techniques employed was checked by changing the acceptable levels of accuracy. RESULTS

AND

DISCUSSION

etal.

t! *z

60-

3 ‘;j

40

REGION

-

OF

COALESCENCE

Lo. Drop radius

(pm)

Fig. 2 Influence of angle of approach on coalescence for different drou sizes CN = 2 s.‘. uA= 0.5 mPas_’ and

The coalescence efficiency as predicted from the present model depends on a large number of parameters: a,, a2, pc. pd, cl=, pp, CT,A, IV, D, &o and h,. In this section, we have first discussed the effect of some of the important variables on coalescence efficiency of equal sized drops, and then the implications of these results for the behaviour of dispersions. The values of the parameters influencing r), unless otherwise stated explicitly, have been taken as pc = p,, = 1000 kg m- 3, ~~,=p.,= lmPas_‘, A= 1~10-~~J, $o=O, Ed= 1x10-10m,D=7x10-2mandh,,=2x10-6m. The range of angles of approach resulting in coalescence For each set of parameters, there is a range of approach angles which results in coalescence. Figure 2 shows a typical plot of the range of angles of approach that leads to coalescence as a function of drop size. The plot yields a coalescence region whose shape reveals several interesting features of coalescence. It is seen that there is a size at which drops coalesce with each other for angles of approach as high as 80”, indicating that such optimally sized drops can coalesce even while virtually grazing past each other. Below this optimal drop size, collisions become increasingly less efficient and, at very low drop sizes, there is no coalescence even if the drops meet head-on. As the drop size is increased beyond the optimal size, both non-head-on collisions with large angles of approach and head-on collisions become ineffective, till a drop size is reached for which coalescence can occur only at a specific non-zero angle of approach. Drop sizes larger than this do not coalesce at all. Thus, in contrast with the notion that head-on collisions are the most efficient in leading to coalescence, the model predicts that in certain situations, where a head-on collision fails, a non-head-on collision can succeed in bringing about coalescence. A change in values of parameters that influence drop coalescence can shift the coalescence region in 0 vs drop size space, and can also alter its shape because of the interaction among various system parameters.

0

100

200

200

400

Drop

radius

(pm)

500

Fig. 3. Effect of drop size on coalescence efficiency.

Efict of drop size on coalescence eficiency The effect of drop size on q for three sets of variable values has been presented in Fig. 3. Curve a shows that as the drop size increases, q first increases, passes through a maximum, and reduces to zero. As shown by eqs (5) and (12), an increase in drop size results in increased squeezing force applied for increased contact times. This has a tendency to increase drainage and hence ‘I. The drainage rate is also influenced by the presence or absence of disc, it being greater in the absence of disc. For smaller drops, the excess pressures, 26/a, required in the film region to deform the drop is quite high and drops do not deform significantly. However, as the drop size increases, the drop deformation becomes significant leading to the forrnation of a disc, and steeply decreased approach rates. Even though the contact time increases, the decrease in drainage rates is more dominant and tends to reduce n. These two opposing effects of increased drop sizes on drainage rates cause the observed maximum in efficiency of coalescence. It is possible that, for certain combinations of other values of parameters,

2033

A new model for coalescence efficiencyof drops in stirred dispersions only a part of the general behaviour is observed. This is shown by curves b and c, where the sets of parameters are such that only one of the two effects is manifested. These curves indicate that q can increase as well as decrease with drop size; thus, reconciling the opposite predictions made by earlier models based on similar premises.

Eflect of stirrer speed The rps influences q predominantly through the variation of squeezing force and this is shown in Fig. 4. The most general behaviour is indicated by curve a. As the rps increases, according to eqs (5) and (12), squeezing force increases as N2 whereas time of contact decreases as l/N. At low values of rps, when the squeezing force acting on the drops is small, drops are not deformed significantly. In this situation, an increase in rps results in increased drainage on account of steeply increased squeezing force and, thus, TVincreases. Still larger values of rps result in very large squeezing force and drops get significantly deformed and form a disc. In this range of rps, an increase in rps results not only in decreased contact times but also decreased thinning rates due to increased drop deformation. Hence, coalescence is reduced. Thus, q at first increases with rps, passes through a maximum and decreases to zero at some stirrer speed. There are situations which exhibit behaviour corresponding to only a part of the general curve and these are shown by curves b and c. Like the effect of drop size, the present model predicts that q can either increase or decrease with rps, reconciling the contradictory predictions of earlier models based on similar premises.

1

0.0 0

4

6

Stirrer

ufo= (1 4(d) + 4&)2’ The expression for u;(d) in the stirred vessel is u;(d) = C,N2D4’3d2’3.

(39)

Substitution of eq. (39) into eq. (3X) yields u;(d) = C,N$D4/3dZ’3

(40)

where N

N+=(l

+44)’

A comparison of eqs (39) and (40) shows that the effect of increasing 4 on turbulence intensity, and therefore on II, is analogous to the reduction of rps from N to N,. Thus, effect of I# can be predicted from the effect of rps itself.

16

speed

(a-‘)

20

Fig. 4. Effect of stirrer speed on coalescence efficiency (a=25mNm-I).

‘-O/

0.0

____.

1’ 2

12

Interfacial Effect of dispersed-phase hold-up Presence of another phase has been shown to dampen the intensity of turbulent fluctuations. Laats and Frishman (1974) have proposed the following correlation between mean square velocity fluctuation in turbulent jets in the presence and absence of dispersed phase:

12

22

32

tension

42

(mN/m)

Fig. 5. Effect of interfacial tension on coalescence (N=lOs-‘,~,=5mPas-‘).

efficiency

Eflect of interfacial tension The effect of u on u has been presented in Fig. 5. Curve a shows that as CTincreases, q increases relatively steeply from a very small value to a large value and then becomes constant. This behaviour is mainly due to the influence of interfacial tension on the deformation of the drop. The excess pressure that is required in the tip region for drop deformation to occur is of the order 2g/a. Increased cr reduces the deformability of the drop; consequently the thinning rates increase, and q increases. However, in the limit of negligible drop deformation at larger u values, a further increase in D does not alter thinning rates as the drop remains undeformed, and q becomes independent of G. The above discussion also shows that, as drop size increases, large values of u are required for q to become independent of IT.Curve a also reveals a small maximum. Low values of cr increase the critical thickness of film rupture and increased van der Waals attraction due to large disc area. These effects promote coalescence. However, only in a very small

S.~NJEEV KUMAR

2034

range of u can these factors overcome the retarding effect of u through decreased thinning rates due to formation of large discs and, hence, only a very small maximum comes into picture. Thus, the present model predicts that, in general, as (r increases from low values, q first increases steeply, passes through a very small maximum and then becomes independent of it. This reconciles earlier model predictions based on similar premises that q increases with u or does not depend on it. Effect of dispersed-phase viscosity All the earlier published models on coalescence efficiency considered drop interface to be immobile and, hence, p,, was not considered a pertinent variable. However, the above assumption applies only for either pd --t ~13 or when large concentrations of stabilizers are present. For pure systems, interfaces cannot support any tangential stresses and circulation prevails inside the drop. This results in the dependence of approach rate on F(.,as well. Further, growth rates for perturbations are also affected by pd. The effect of ,uLd on 4 due to both of these factors is shown in Fig. 6. It is seen that, as p* increases, ‘1 invariably decreases. Curve a corresponds to large and slightly deformed drops, while curve b corresponds to small and undeformed drops. The stronger retarding effect c(d has on approach rates for undeformed drops [eq. (28)] than for deformed drops [eq. (29)] would imply stronger retarding effect of p,, on q for small drops as opposed to large drops. The curve b confirms this and shows that the dependence of 9 on pd is steeper for small drops. In situations where o is quite low, significantly increased drop deformation can be the result. Thus, in curve c, while the driving force for coalescence remains the same as for curve a, the resistance to drainage is greater because of formation of a larger disc. Hence, & is much more effective in decreasing q here than when compared to curve a. From the above discussion it is clear that pLd,which has not been considered to influence q in earlier published models,

1.0

-----“;

-----_

et al.

is a very important parameter for coalescence in pure systems. The effect of yd on the coalescence efficiency of spherical drops has also been considered by Muralidhar (1988). He neglected the effect of pd on film stability and reached conclusions opposite to the ones obtained here. Effect of continuous-phase viscosity The effect of flc on 4 has heen presented parametrically in Fig. 7 where q is shown vs drop diameter for three he values. It is seen that as p(5increases, PJ decreases. Equations (25) and (30) show that smaller the drop, lesser is the likelihood of interface being fully mobile. Then, in such situations, eq. (28) or eq. (29) show that an increase in fl= reduces the approach rates. Increased fl= also reduces the perturbation growth rates and, therefore, h, is decreased. Hence, the effect of increased p, becomes significant for small drop sizes. The same equations referred to earlier show that for large drops, interfaces become mobile, and thinning rates are not affected by p=,. A small decrease in v for these drops results mainly due to the decreased values of h, at increased pc. In comparison with the present results, earlier models predict stronger dependence of q on pr_ This is mainly because they assume that the drainage rates always correspond to immobile interfaces. In contrast, the present model predicts that pL,can influence q in two ways, through drainage rates and h,. Due to the mobile nature of the interface for pure systems, the dependence of rl on pc is predicted to be weaker by the present model. This suggests that for a given system, coalescence is more effectively suppressed when the more viscous phase is the dispersed phase as the dependence of r] of peais stronger than that on P.. In the limit pd --, co, and for p, as low as 1 mPa s-l, the model predicts that q approaches zero for all the drops. Effect of starting film thickness All the earlier models for 1 predict that the dependence of q on starting film thickness h, is very weak

1.0I-

_.________ ------________fl

4 8

---_____

\

‘G

se

4

.$ a

0.6

E

Q, 0.6

0.6

8

8

B g 0.4

p

B

3 u”

0.6

i ”

0.2

0.4

0.2

0.0

0.0

10 Dispersed

phase

Fig. 6. Effect of dispersed-phase 6Z&k.nCy.

viscosity

viscosity

(mPa.s)

on coalescence

’

6

Drop

100’

radius

Fig. I. Effect of continuous-phase viscosity e~ciency(N=5s-‘,~,=lmPas~‘ando=25mNm-’).

(em)

*

on coalescence

A new model for coalescence efficiency of drops in stirred dispersions and, hence, assume that it need not be specified in a precise way. However, the present model predictions as shown in Fig. 8 clearly indicate that this claim holds only for large drops. For small drops, the dependence of q on h, is quite significant. This is because the large drops deform substantially and most of the contact time is spent in the final stages of film thinning. In comparison, small drops do not deform significantly. In such a case, a substantial portion of the total contact time is spent also on the earlier stages of film thinning. Thus, in contrast with the earlier models, the present model predicts that h,, is an important parameter for small drops and, therefore, should be correctly incorporated in the calculation of collision frequency. Effect of Hamaker constant Hamaker constant determines the magnitude of van der Waals attractive force between two bodies at small film thicknesses and, for most of the liquid-liquid systems, its value lies in the range of 10-2’-10-20 J (Lsraelachvili, 1985). Figure 9 shows the variation of q with drop size for the extremal values of A quoted. It shows that a tenfold increase in A does not alter q for small drops; however, for large drops, it increases 11considerably. Figure 9 shows that the drops lying in the range 190-240 pm, which do not coalesce at a low value of A, do coalesce at a large value of A. Big drops deform substantially and the approach rates for such drops are small at small film thicknesses. Increased A not only increases the approach velocities in this slow film thinning regime but it also increases the critical film thickness of rupture. Since both these factors decrease the times required for coalescence to occur, q is increased. Small drops do not def6i-m to a significant extent. Decreased drop deformation results in large thinning rates and less attractive force, and even a tenfold increase of A does not influence coalescence process significantly and u remains relatively unaffected. Thus, in addition to

the hitherto known parameters affecting 9, the present model predicts that the Hamaker constant for the system has an important role to play in coalescence process through the effect of van der Waalls forces on drainage at small film thicknesses and the dependence of h,, on it. Coalescence eficiency for unequal size drops So far coalescence between two equal sized drops was considered. The qualitative effects described till now were found to hold true for coalescence of unequal-sized drops also. Thus, the present section only shows the effect of changing the ratio of the radii of the two drops. Figure 10 shows q vs the ratio of radii of colliding drops, for a set of fixed parameters. It is seen that coalescence between unequal sized drops is more efficient than when the sizes are equal. For example, a 5OOpm size drop does not coalesce with B drop of its own or greater size, but does so if the other drop is smaller than about 450pm. This happens because of the reduced size of the disc dimension for interaction between unequal-sized drops.

*

*

100=

Drop radius

(cun)

.

Fig. 8. Effect of startingfilm thicknesson coalescence efficiency (N=5s-l, ~~=lmPas-‘, ~~=lOmPas-’ and ,ZI= 25mNm-I). CES48:ll-H

-. Q

1.0 %

2d

0.8

$

x

0.6

ii

al

z 0.4 0

3 u

A

0.2

_

(Joule)

\

‘!\

lo-l*-.l

_._.I_.~

0.0 0

200

100

Drop

radius

300

(pm)

Fig. 9. EtTectof Hamaker constant on coalescenceefficiency (N=5s-‘,&,=lmPas-‘andu=5mNm-1).

1.0

10

2035

0.0

. II-

Fig. 10. Effectof unequal drop size on coalescenceefficiency (N=2s-‘,~~=5mPas-‘anda=$mNm-‘).

2036

SANJEEVKUMAR et al.

When the size of the reference drop is considerably reduced, Fig. 10 shows that exactly the opposite behaviour will be observed. Here a drop of size 25 nm does not coalesce with another drop of equal or smaller diameter, but does so if the other drop is larger than size of 35 pm. This happens because of the increased squeezing force available. If small drops collide with larger drops at even very large angles of approach, coalescence occurs. Thus, the present model predicts that, if a dispersion has small as well as large drops, small drops should vanish rapidly due to coalescence with the larger ones. This prediction is in agreement with the observations made by Sprow (1967), Park and Blair (1975) and Shinnar (1961) on coalescing systems. Sprow (1967) and Park and Blair (1975) observed that as drops move away from the impeller region, small drops are lost very quickly and drop size distribution shifts to large drop sizes. Shinnar (1961) has reported that, at steady state, drop size distribution is narrower (ratio of maximum to minimum drop size is nearly two) for stable dispersions than that for unstable dispersions. Since drops do not break at all in stable dispersions, new drops of small size are not formed. The small drops that were present before steady state was reached would have been lost due to coalescence with larger drops and the drop size disribution, hence, is narrow. In the case of unstable dispersions, new small drops are continuously formed due to drop break-up and drop size distribution remains wide. It can be seen that the two sizes selected above for discussion, viz, 500 and 25 pm, exhibit behaviour similar to that shown by 450 and 35 pm drops of Fig. 2. Calculated values of 4 for a reference drop of 150 vrn, a size between two extremes, are also shown in Fig. 10. Here also it is seen that q increases for coalescence with bigger drops. However, as the size of the other drop of the pair decreases, r~ decreases till a&, reaches about 0.5, and, from then on, yl once again increases. This is a result of the complex interaction between various parameters. However, it can be still stated that rl shows a minimum when a2/al - 1. Turbulence stabilized dispersions of pure phases A stable dispersion can result only if drops neither break nor coalesce in the stirred dispersion. The breakage is effectively prevented if none of the drops existing in the vessel is greater than d,.., the largest size of the drop that does not break. It has been shown in Fig. 2 that equal drops above a certain size, called dmio by Shinnar (1961), do not coalesce in the dispersion. If d,, is smaller than d,i,, the drops can continuously break and coalesce yielding unstable dispersion. However, if d,,, is larger than dmi,, then, at steady state, no drop break-up and coalescence can occur and the dispersion gets stabilized (Shinnar, 1961). Thus, depending upon the relative values of d,., and d,,,, the dispersion can be stable or unstable. Such a situation can arise in dispersions without the use of any surface active agents, mainly because of the effect of p,+ on coalescence. This has been shown in

-

100

d,

.

1

1

Stirrer

I

,

I

10 speed

(8-l)

Fig. 11. State of mixing in dispersed phase at steady state (~d=124mPas-1anda=25mNm-‘).

Fig. 11 which presents the variation of d,,, and d,i. with rps for a pLdvalue of 125 mPas- ‘. At rps smaller than about 7 s-r, dmin is larger than d,,, and the dispersion is unstable since drops continuously coalesce and break. For larger rps, d,, is smaller than d,., and, therefore, at steady state, dispersion is stabilized against the coalescence and break-up. As the value of p,, is decreased, d,,, decreases whereas dmin increases. As a result, at low values of p,,, dmin is usually larger than d,,, under the usually studied operating stirrer speeds and the drops should always coalesce and break. This is in agreement with the findings of Park and Blair (1975) who did not observe stabilized dispersions for pure systems involving low values of pa. As pLdincreases, it is always possible to have dmi. smaller than d,,, for all stirrer speeds. This results in no coalescence at steady state. Thus, in a system where p(d increases with time, drops are expected to coalesce and yield bigger drops only as long as d,,, remains smaller than dmin. Once d,., becomes larger than dmin, drops do not coalesce and the drop size does not increase even if FL,,continues to increase. This is in agreement with the studies of Konno et al. (1982) who found that, as pd increases during the course of polymerization reaction, drop size increases initially but soon reaches a constant value, whereas the value of nLdcontinues to increase with time. It is interesting that the model prediction indicating the possibility of stabilized dispersions at increased rps is exactly opposite to the experimental data of Shinnar (1961), who observed that the dispersion was stable at low rps, but became unstable at high rps. However, he used polyvinyl alcohol in the continuous phase which would adsorb at the liquid-liquid interface. Hence, his system did not involve pure components. The adsorbed polymer layer severely retards the mobility of interface, thereby reducing the circulation inside the drop. This effect is similar to the one obtained by increasing 1~~giving lower dmin value. The d max is relatively unaffected by the presence of the

A new model

for coalescence

efficiency of drops in stirreddispersions

adsorbed layer, which is decided by the actual c(d value. With increasing rps, the value of both d,,,.. and d,, decreases. However, d,. falls with rps faster than dmi,,, resulting in a situation where d,, is higher than dmi, at low rps values but becomes smaller than d,, at high rps. Thus, if the presence of stabilizer can be considered as equivalent to increasing pd for coalescence, the lowering of rps can stabilize the dispersion, as observed by Shinnar (1961).

FS

F, F s-31 F vdr h

CONCLUSIONS hm A new model for drop coalescence in stirred vessels hcr.Ir is proposed. The model considers, in contrast to earlier work, drop collisions at all possible angles of he approach. The collisions are caused by eddy drop interaction and, hence, coalescence must occur within h,(k) the eddy lifetime or less, depending upon the angle of approach. The model accounts for the change in the hst Z shape of two approaching drops and the dependence of critical thickness of film rupture on various interk acting parameters. It predicts that the dependence of q on some variables can go through a maximum, thus ks 1 reconciling the opposite sets of predictions for the m effect of these variables available in literature through models based on similar premises. The model also A4 brings out the role of some important, but hitherto N unemphasized variables, p(d,h,, and A, in determining v. It shows that, for pure systems, compared with p=, N.4 an increase in p., more effectively reduces r/.The start4 ing film thickness, h,, which earlier models predicted as inconsequential in affecting ‘I, is found to influence L/3 q for small drops significantly. An increase in L/2 Hamaker constant increases coalescence efficiency for tc Iarge drops quite appreciably. The model makes sevt1 eral new predictions as well. It predicts that, under bddy some conditions, large drops prefer to coalesce in T non-head-on fashion. The coalescence between unequal drops is shown to be generally more efficient than that between equal drops. It also predicts that, depending on the value of pd, there is a drop size, dmin, V beyond which coalescence is not possible and that Vw a dispersion will be stable if d,+ -=zd,... vdioc

NOTATION

a

a. adisc

A b ci CO

d d max &in D eo F

drop radius, m mean drop radius, defined by eq. (23), m radius of the disc, m Hamaker constant, J as defined by eq. (19) constants bulk concentration of electrolyte, .mol l- ’ drop diameter, m maximum size of the drop stable against break-up, m drops larger than this size do not undergo equal drop coalescence, m stirrer diameter, m electronic charge total force exerted by eddy, as shown in Fig. 1, N

w*

2037

component of force along the direction joining the centres of the drops, N component of the force resulting in the rotation of the doublet, N repulsive force due to electrical double layer, N attractive force due to van der Waal interactions, N minimum distance between the surfaces of two drops, m critical film thickness at which rupture occurs, m critical film thickness at which rupture would occur for wavelength k, m as defined by eq. (30), m film thickness at which for a perturbation of wave number k is neutrally stable, m starting thickness for film thinning, m moment of inertia around the centre of the mass of the system, kg m2 wave number for a perturbation, m-l Boltzmann’s constant size of an eddy, m mobility parameter for spherical drops, given by eq. (25) mass of a drop, kg stirrer speed, s- 1 Avogadro’s number defined as *(a, + a,)/I time, s time required for f3 to reach n/3, s time required for 0 to reach ~r/2,s film thinning time for drop coalescence, s available contact time, s eddy lifetime, s temperature mean square velocity fluctuation across length 1, subscript refers to hold-up in the tank, m2se2 approach velocity, m s-l approach velocity for spherical drops defined by eq. (28), ms-’ approach velocity for deformed drops, disc at the tip defined by eq. (29), m s- ’ critical approach velocity in eq. (l), ms- ’

Greek letters growth coefficient for perturbation, s- ’ B dielectric constant for continuous phase & initial amplitude of the perturbation, m &o defined as tanh (veo+o/4kBT) 3 coalescence.efficiency tt angle between the direction of force and the 8 line joining centres, rad initial value of 8 00 l/Debye length (= 8negv2 Co N&kCBT) 1 viscosity ratio (= j+/fl,) viscosity, Pa s- 1 p V magnitude of ion balance density, kg m- ’ P

2038

SANJEEV

interfacial tension, N rn-’ dispersed-phase hold-tip zeta potential angular velocity of doublet,

; JlO

w

K.UMAR

rad s- ’

Subscripts : 192

continuous phase dispersed phase refer to the drops in the colliding

pair

REFERENCES Allan, R. S. and Mason, S. G., 1962, Particle motion in sheared suspensions: XIV. Coalescence of liquid drops in electric and shear fields. J. Colloid Znterface Sci. 17, 383-408. Batchelor, G. K., 1967, An Introduction to Fluid Dynamics. Cambridrce Universitv Press. Cambridee. Burill, K. i. and Woods, D: R., 1973: Film shapes for deformable drops at liquid-liquid interfaces: II. The mechanism of film drainage. J. Colloid Interface Sci. 42, 15-34. Chesters, A. K., 1991, The modelling of coalescence processes in fluid-liquid dispersion: a review of current understandinn Trans. Znstn them. Enars 69. 259-270. Couialoglou, C. A. and Taviarides, L. L., 1977, Description of interaction process in agitated liquid-liquid dispersions. Chem. Engng Sci 32, 1289-1297. Das, P. K., Kumar, R. and Ramkrishna, D., 1987, Coalescence of drops in stirred dispersion: a white noise model. Chem. Engng Se1 42, 213-220. Davis, R. H., Schonberg, J. A. and Rallison, J. M., 1989, The lubrication force between two viscous drops. Phys. Fluids Al, 77-81. Dimitrov, D. S., Stoicheva, N. and Stefanova, D., 1984, J. Colloid Interface Sci. 98, 269. Gupta, Sanjeev Kumar, 1992, Breakage and coalescence of droos in turbulent stirred disuersions. Ph.D. thesis. Indian Ins&e of Science, Bangalire. Haber, S., Hetsroni, G. and Solan, A., 1973, On the low Reynolds number motion of two- droplets. .fnt. J. Multiphase Flow 1, 57-71. Howarth, W. J., 1964, Coalescence of drops in a turbulent flow field. Chem. Engng Sci. 19, 33-38. Israelachvili, J. N., 1985, Inter-molecular and Surface Forces. Academic Press, London. Ivanov, I. B. and Dimitrov, D. S., 1988, Thin film drainage, in

et al.

Thin Liquid Films (Edited by I. B. Ivanov). Marcel Dekker, New York. Ivanov, I. B. and Travkov. T. T.. 1976. Hvdrodvnamics of thin liquid films: raie of’thinning of km;lsion*films from pure liquids. Inc. J. Multiphase Flow 2, 397410. Jain. R. K. and lvanov. I. B.. 1980. Thinning and runture of ring shaped films. Farada; Trans. II 76, 250-266: Jiang, Y. J., Umemura, A. and Law, C. K., 1992, An experimental investigation on the collision behavior of hidrocarbon droplets. J. Fluid Mech. 234, 171-190. Konno, M., Arai, K. and Saito, S., 1982, The effect of stabilizer on coalescence of drops in suspension polymerization of styrene. J. them. Engng Japan 15, 131-135. Laats, M. K. and Frishman, F. A., 1974, Development of technique and investigation of turbulent energy at the axis of a two-Dhase turbulent iet. Fluid Dvnamics 8. 304-307 (translates from Russian).Maldarelli, C., Jain, R. K., Ivanov, I. B. and Ruckenstein, E., 1980, Stability of symmetric and unsymmetric thin film to short and long wave length perturbations. J. Colloid Interface Sci. 78, 118-143. Miller, C. A. an& Neogi, P., 1985, Interfacial Phenomena, Equilibrium and Dynamic Efiects. Marcel Dekker. New York. Muralidhar, R., 1988, Ph.D. thesis, Purdue University. Muralidhar. R. and Ramakrishna. D.. 1986. Analvsis of droplet coalescence in turbulent Gquih-liquih disp&sions. Znd. Engng _ _ Chem. Fundam. 25, 554-560. Muralidhar. R.. Ramkrishna. D.. Das. P. K. and Kumar. R.. 1988, Coale&nce of rigib d;ople& in a stirred di&.r: sion-II. Band-limited foroe fluctuations. Chem. Enana _ Sci. 43, 1559-1568. Park, J. Y. and Blair, L. M., 1975, The effect of coalescence on drop size distribution in an agitated liquid-liquid dispersion. Chem. Engng Sci. 30, 1057-1064. Ruckenstein, E. and Sharma, A., 1987, A new mechanism of film thinning: enhancement of Reynolds’ velocity by surface waves. J. Colloid Znterfkce Sci. 119, 1-13. Scheludko, A., 1967, Thin liquid films. Adv. Colloid Interface sci. 1,391464. Shinnar, R., 1961, On the behaviour of liquid dispersions in mixing vessels. J. Fluid Me& 10,259-275. Sprow, F. B., 1967, Drop size distribution in strongly coalescing agitated liquid-liquid systems. A.I.Ch.E. J. 13, 995-998. Yiantsios, S. G. and Davis, R. H., 1990, On buoyancy driven motion of a drop towards a rigid or a deformable interface. J. Fluid Mech. 217. 547-573.