- Email: [email protected]
Collective Effects of Gravitational and Brownian Coalescence on Droplet Growth
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
178, 47–52 (1996)
0091
Collective Effects of Gravitational and Brownian Coalescence on Droplet Growth HUA WANG 1
ROBERT H. DAVIS 2
AND
Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424 Received January 17, 1995; accepted July 20, 1995
tional to the drop radius. In contrast, the rate of drop growth increases with time for coalescence due to differential sedimentation, because the settling velocity at a low Reynolds number is proportional to the square of the drop radius. For both cases, the coalescence rate decreases with increasing ratio of the viscosities of the drop and continuous phases; this is because it is more difficult to squeeze the fluid out from between two approaching drops as their internal viscosity increases (4). Brownian motion and gravity sedimentation may play important roles simultaneously under many conditions and drop size ranges. For example, micrometer-sized hydrocarbon droplets in water under normal gravity have a Pe´clet number near unity, and so Brownian and gravitational motions are comparable. The same may be true for supramicrometer droplets in a reduced gravity environment. Even submicrometer droplets, for which Brownian motion is dominant, will begin to sediment after growing due to Brownian coalescence. However, very little work has been done on the simultaneous effects of Brownian motion and gravity sedimentation on drop growth and coalescence. An exception is the study of simultaneous flocculation (coalescence) and creaming (sedimentation) of drops by Reddy et al. (5). However, they used the Brownian collision efficiencies for rigid spheres to account for hydrodynamic and interdroplet interactions for both Brownian and gravitational coalescence. They employed the so-called ‘‘additivity approximation,’’ in which the individual Brownian and gravitational collision rates are simply added together. This ad hoc approximation has also been used to describe the aggregation of micrometer-sized rigid particles (6), but it has no physical basis and, in fact, does not yield the proper asymptotic limits for weak gravitational or Brownian effects (1). Only recently, Zinchenko and Davis (1) developed an efficient technique to compute the collision rates for coupled gravity and Brownian effects with complete hydrodynamic interactions. Unlike previous studies on drop or particle collisions, a numerical solution was presented for arbitrary Pe´clet numbers, thus covering the whole range of drop sizes in typical dispersion systems. This technique was based on an analytical continuation into the plane of a complex Pe´clet
The temporal evolution of drop-size distributions induced by collective gravity sedimentation and Brownian motion and coalescence is studied by population dynamics. For initial distributions with arbitrary Pe´clet numbers representing the ratio of gravitational and Brownian motion, the exact collision efficiencies predicted by Zinchenko and Davis (1) are used to cover the whole range of drop sizes. It is found that coupled Brownian and gravity effects are important for Pe´clet numbers in a wide range from 10 02 to 10 3 , with the distribution shifting to larger sizes by coalescence at a much greater rate than for either mechanism acting alone. Moreover, the additivity approximation, in which the individual Brownian and gravitational coalescence rates are simply added, significantly underestimates the droplet growth rates. q 1996 Academic Press, Inc.
Key Words: coalescence; coagulation; drop growth; population dynamics.
1. INTRODUCTION
Predicting the evolution of the drop-size distribution in a liquid–liquid emulsion is an important goal in many chemical and material processes. The distribution evolves as a result of drop coalescence and possible breakage. In an unstirred emulsion, drops of different sizes will experience relative motion due to gravity, which may lead to their collisions and coalescence. Drops of a few micrometers in size, or smaller, will experience Brownian motion, which may also lead to coalescence. The coalescence rate for two interacting spherical drops induced by gravity motion and Brownian motion independently has been predicted by Zhang and Davis (2). Their results were used as collision kernals in population dynamics equations by Wang and Davis (3) to study the growth of a homogeneous drop distribution with time due to either Brownian or gravitational coalescence acting alone. For Brownian coalescence, the droplet growth slows down with time because the Brownian diffusivity is inversely propor1 Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. 2 To whom correspondence should be addressed.
47
AID
JCIS 3951
/
m4747$$$27
01-30-96 07:19:53
0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
coida
AP: Colloid
48
WANG AND DAVIS
number and a special conformal mapping to represent the solutions as a convergent power series for all real Pe´clet numbers. A key finding is that the synergistic effects of Brownian motion and gravity sedimentation give much higher coalescence rates than either mechanism acting alone and that predicted by the additivity approximation. Since coalescence is slowest for micrometer-sized drops for which neither mechanism dominates, this synergistic effect may play a significant role in applications such as cloud droplet growth where the rate-limiting step is through to occur for the micrometer size range (7). The calculation of Simons et al. (8) for combined Brownian and gravitational coagulation of particles without hydrodynamic interactions also predicts that the collision rate exceeds the sum of the collision rates of the two mechanisms acting alone. The present paper extends our previous work on individual mechanisms (3) by predicting drop growth due to coalescence induced collectively by Brownian motion and gravity sedimentation and by solving the population dynamics equations using the collision rates for arbitrary Pe´clet numbers from Zinchenko and Davis (1). The dispersions are assumed to be sufficiently dilute that pairwise collisions dominant the coalescence process, and the drops are sufficiently small that they remain spherical, have negligible inertia, and do not break apart. 2. THEORETICAL DEVELOPMENT
We consider a dilute, homogeneous dispersion containing spherical drops of viscosity m* and density r *, dispersed in an immiscible fluid of viscosity m and density r. The dispersion is quiescent (no stirring or imposed flow) other than as a result of the motion of individual drops. The growth of the drops due to coalescence is studied by using population dynamics equations. In discretized form, these equations are (3, 7, 9) i Å1
i Å 1, 2, . . . , N,
Jij Å pni n j (ai / aj ) 2V
(0) ij
Eij .
[2]
ni and n j are the numbers of drops per unit volume in the discrete size categories i and j, respectively, and ai and aj are the large and small drop radii, respectively. V ij( 0 ) is the magnitude of the Hadamard–Rybczinski relative velocity for two isolated drops (10, 11),
AID
2( mP / 1)( r * 0 r )(a 2i 0 a 2j )g Å , 3(3mP / 2) m
JCIS 3951
Pe Å
/
m4747$$$28
[3]
01-30-96 07:19:53
1/2(ai / aj )V D (ij0 )
(0) ij
,
[4]
where the relative Brownian diffusivity of two isolated drops is D (ij0 ) Å
[1]
where Jij is the rate of collisions per unit volume of size i drops with size j drops,
V
where mˆ Å m* / m is the viscosity ratio and g is the gravitational acceleration vector. The collision efficiency, Eij , takes account of simultaneous gravity and Brownian effects, hydrodynamic forces which cause the drops to move around one another, and any interdroplet repulsive and attractive forces. The ratio of gravitational and Brownian effects is given by the Pe´clet number,
N
dni 1 Å ∑ Jj (i0j ) 0 ∑ Jij , dt 2 jÅ1 jÅ1
(0) ij
FIG. 1. The collision efficiency as a function of the relative Pe´clet number for drops with l Å 0.25, mˆ Å 10, and no interdroplet forces. The solid line represents the exact solution of Zinchenko and Davis (1), the dotted line represents the additivity approximation, the dashed line is the Brownian collision efficiency, and the dashed-dotted line is the gravitational collision efficiency.
kT( mP / 1)(1 / l 01 ) , 2pm(3mˆ / 2)ai
[5]
where k Å 1.38 1 10 023 J/K is the Boltzmann constant, T is the absolute temperature, and l Å aj /aj £ 1 is the radius ratio. As Pe r ` , the collision efficiency tends to the gravitational limiting values E `ij calculated by Zhang and Davis (2) and by Zinchenko (12). In the opposite limit of Pe r 0, the Brownian collision efficiency is Cij /Pe, where the coefficient Cij has been computed by Zhang and Davis (2) and by Zinchenko and Davis (1). 3. RESULTS AND DISCUSSION
Figure 1 shows example collision efficiencies for a pair of drops with size ratio l Å 0.25 and viscosity ratio mˆ Å 10. The solid line is the exact collision efficiency for combined
coida
AP: Colloid
TEMPORAL EVOLUTION OF DROP-SIZE DISTRIBUTIONS
Brownian and gravitational motion calculated by Zinchenko and Davis (1), and the dotted line is the collision efficiency calculated using the additivity approximation Eij Å E `ij / Cij /Pe. This figure shows that the combined effects are important over a surprising wide range of Pe´clet numbers, and that the additivity approximation significantly underestimates the collision efficiency over this range. Thus, we expect the predicted dynamics of the drop size evolution in an unstable emulsion to be substantially faster when using the exact collision efficiencies. The initial drop-size distributions employed in this study are normal distributions on a number basis, as discussed in Wang and Davis (3). The governing equations were nondimensionalized using the average radius in the initial distribution, ao , as the characteristic length and tg Å 4ao / (3foVgo Eo ), or tb Å Peotg , the characteristic gravity and Brownian coalescence time scales defined in (3), as the characteristic time for Peo ú 1 or Peo õ 1, respectively. The characteristic Pe´clet number for the initial distribution is defined as Peo Å ao£o /Do , where £o and Do are the sedimentation velocity and Brownian diffusivity of an isolated drop of radius ao . The range of dimensionless droplet radii considered is from 0.08 to 600, although the numerical codes can handle even larger size ranges. A finite-difference method described by Wang and Davis (3) was used to solve the dimensionless population dynamics equations. The dimensionless time step used in the calculations is typically 0.01. The final average radius changes by less than 1% when the dimensionless time step is reduced to 0.001. In all of the calculations, the total volume fraction of the dispersed phase was computed after each time-step and checked against the input value; these values do not vary by more than 1% for the conditions used in this study. The dimensionless parameters which affect the macroscopic behavior of the dispersions we are interested in include the characteristic Pe´clet number of the initial distribution, Peo , the viscosity ratio mˆ Å m* / m, the standard deviation of the radii in the initial distribution, sˆ Å s /ao , and any dimensionless parameters describing the interdroplet forces. Note that the volume fraction of the dispersed phase, fo , is scaled out of the population dynamics equations. For simplicity, we present results only for negligible interdroplet forces; unlike solid particles, drops have finite collision rates even in the absence of attractive forces (1, 2). The effects of interdroplet forces are presented by Wang (13). 3.1. Initial Distributions with Small Pe´clet Numbers First, we consider the evolution of the drop-size distribution when Brownian effects initially dominate over gravity, i.e., Peo õ 1. Figure 2 shows the evolution of the drop-size distribution for an initial dispersion having Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0 (bubbles). The function f is the volume density function defined such that f (a)da is the volume of
AID
JCIS 3951
/
m4747$$$28
01-30-96 07:19:53
49
FIG. 2. Time evolution of the drop-size distribution for collective gravitational and Brownian coalescence of an initial distribution having sˆ Å 0.2, Peo Å 0.01, and mˆ Å 0; the solid curves represent t Å 0, 2, 10, 30, 50, and 60 times tb , respectively, from left to right, and the dashed curves represent the corresponding results for Peo Å 0 (Brownian coalescence only).
drops with radii in the range a { (1/2)da, per unit volume of dispersion (3). The evolution of the drop-size distribution for the same initial distribution but with Peo Å 0 (Brownianinduced coalescence only) is shown by the dashed lines. For short times (t/ tb £ 2), the predicted drop-size distributions (solid lines) overlap with those predicted by assuming Peo Å 0 (dashed lines). As time increases, however, the drop sizes increase due to coalescence. Since the gravitational velocity is proportional to the square of the drop radius, whereas the Brownian diffusivity is inversely proportional to the drop radius, the relative importance of gravity-induced coalescence increases with time. After t É 10tb , the dropsize distributions predicted using the exact collision efficiency shift toward larger drop sizes much faster than those predicted by assuming Peo Å 0, because the gravity effects then dominate over Brownian effects. The effect of the viscosity ratio on the evolution of average drop radius, » a … , defined as the radius of a drop having the volume-averaged drop volume (3), is shown in Fig. 3 for dispersions with Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0, 0.5, 2, and 10. It is seen that, with an increase in the viscosity ratio from 0 to 10, the growth rate of the average drop radius slows down. This is because the hydrodynamic resistance to close approach between colliding drops increases with increasing drop viscosity due to the greater difficulty in squeezing fluid out from the gap separating the two drops (4). Figure 3 also shows that the additivity approximation may considerably underestimate the drop growth rate. For short times, the additivity and exact results agree, because both are dominated by Brownian effects. At longer times, gravity effects are also important, and the additivity approximation underpredicts the average radius by 50% or more.
coida
AP: Colloid
50
WANG AND DAVIS
FIG. 3. The average droplet radius versus time for collective gravitational and Brownian coalescence of initial distributions having Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0, 0.5, 2, and 10. The dashed lines represent the corresponding results using the additivity approximation. The dotted line is the Brownian result (Peo Å 0) for mˆ Å 0.
Note that the curves switch from the concave-down, characteristic of Brownian collisions, to the concave-up, characteristic of gravity collisions (see (3)) as time progresses. The influence of the initial average drop size, as measured by the characteristic Pe´ clet number, on the evolution of average drop size is shown in Fig. 4 for a dispersion having sˆ Å 0.2 and mˆ Å 10. It is shown that, with an increase of Peo from 10 04 to 0.1, the average radius deviates significantly from that of Peo Å 0. Even when Peo Å 10 02 , significant errors are introduced if the results for Peo Å 0 ( Brownian coalescence only ) are used. The time needed for the average drop size to increase to » a … Å 10ao
FIG. 5. The average drop radius at t Å 50tb as a function of Peo for a dispersion with an initial distribution having sˆ Å 0.2 and mˆ Å 10. The upper curve is the result using exact collision efficiencies, and the lower curve is the result using the additivity approximation.
is 34, 87, 132, 151, and 160 times tb , respectively, for Peo Å 0.1, 10 02 , 10 03 , 10 04 , and 0. A plot of the average radius at t Å 50tb as a function of the initial Pe´ clet number is shown in Fig. 5. When Peo r 0 ( Brownian effects alone ) , the calculated value is » a … Å 6.8ao , whereas the calculated values using the exact collision efficiencies are » a … Å 17.1ao , 7.7ao , and 7.0ao , respectively, for Peo Å 10 01 , 10 02 , and 10 03 . The calculated values using the additivity approximation are » a … Å 9.1ao , 7.0ao , and 6.8ao , respectively, for Peo Å 10 01 , 10 02 , and 10 03 . Thus, neglecting gravitational effects, or inaccurately accounting for them using the additivity approximation, leads to a significant underprediction of the droplet growth for Peo § 0 ( 10 02 ) . 3.2. Initial Distributions with Moderate and Large Pe´clet Numbers
FIG. 4. Time evolution of the average radius for collective gravitational and Brownian coalescence of initial distributions having mˆ Å 10, sˆ Å 0.2, and different Peo ; the dashed line is the Brownian result for Peo Å 0.
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
The results presented in Figs. 2–5 are for dispersion systems with Peo õ 1 for which coalescence is initially dominated by Brownian effects. A transition in the dominant collision mechanism to gravitational coalescence with increasing time was found. We now consider the droplet growth behavior for dispersion systems with Peo § 1. For such systems, gravity-induced coalescence always dominates. Attention here is focused on how the effects of Brownian motion and coalescence enhance the gravitational drop growth rates, by comparing the results predicted using exact collision efficiencies for coupled gravity and Brownian induced coalescence with those predicted using the collision efficiencies for Peo Å ` (gravity-induced coalescence alone). Figure 6 presents the drop-size distribution at t/ tg Å 0, 10, and 20, evolved from an initial dispersion having mˆ Å
coida
AP: Colloid
TEMPORAL EVOLUTION OF DROP-SIZE DISTRIBUTIONS
FIG. 6. Time evolution of the drop-size distribution for collective gravitational and Brownian coalescence of initial distributions having mˆ Å 10 and sˆ Å 0.2. The solid lines represent the results for t/ tg Å 0, 10, and 20, respectively, from left to right, for a system with Peo Å 10; the dotted lines represent the results for Peo Å ` (gravity-induced coalescence alone) at t/ tg Å 10, 20, 30, and 50, respectively, from left to right.
10, sˆ Å 0.2, and Peo Å 10. The evolution of the drop-size distribution due to gravity-induced coalescence alone (Peo Å ` ) is also presented in Fig. 6 as dashed lines for t/ tg Å 10, 20, 30, and 50. For Peo Å ` , the size-distribution evolves into a bimodal distribution with large spread; a large concentration of small drops remains even for large times. This behavior occurs because the gravitational collision efficiency is very small for l ! 1, since small drops flow around large ones (3). For Peo Å 10, however, the size distribution evolves into a unimodal distribution, with fewer small drops. This is because weak Brownian effects, which are more important for smaller drops, promote the coalescence of the small drops. Once these small drops coalescence into larger drops, the drop-size evolution accelerates because the gravitational velocity increases in proportion to the square of the drop radius. As a result, the evolution of drop-size distribution is much faster for dispersions with moderately large Pe´clet numbers when the effects of Brownian motion are included in the collision efficiencies employed. The influence of the initial Pe´clet number on the growth of average drop radius is shown in Fig. 7 for initial dispersions having mˆ Å 10 and sˆ Å 0.2. The times for the average drop radius » a … to reach 25ao are 12.2, 22.2, 34.4, 44.1, 49.8, and 53.3 times tg , respectively, for Peo Å 1, 10, 10 2 , 10 3 , 10 4 , and ` . The influence of Brownian motion on the drop growth is significant even for Peo as high as 10 3 . Figure 8 is a plot of the average radius at t Å 10tg as a function of the initial Pe´clet number. The solid line represents the results using the exact collision efficiencies (1), whereas the dashed line represents the results using the additivity approximation. When Peo r ` (gravity-induced coalescence alone), the calculated value is » a … Å 1.3ao ,
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
51
FIG. 7. Time evolution of the average radius for drops with initial distributions having sˆ Å 0.2, mˆ Å 0, and different Peo ; the dashed line is the result for Peo Å ` .
whereas the calculated values using the exact collision efficiencies are » a … Å 12.8ao , 3.5ao , 1.9ao , and 1.5ao , respectively, for Peo Å 1, 10, 10 2 , and 10 3 . The calculated values using the additivity approximation are » a … Å 5.9ao , 2.3ao , 1.5ao , and 1.3ao , respectively, for Peo Å 1, 10, 10 2 , and 10 3 . Neglecting Brownian effects, or inaccurately accounting for them by the additivity approximation, significantly underpredicts the average radius for Peo £ 0(10 3 ). 4. CONCLUDING REMARKS
Quantitative predictions of the temporal evolution of drop size distributions in droplet dispersions due to collisions and
FIG. 8. The average drop radius at t Å 10tg as a function of Peo for a dispersion with an initial distribution having sˆ Å 0.2 and mˆ Å 10. The upper curve is the result using exact collision efficiencies, and the lower curve is the result using the additivity approximation.
coida
AP: Colloid
52
WANG AND DAVIS
coalescence induced collectively by gravity sedimentation and Brownian motion are presented in this paper. Complete hydrodynamic interactions are included in the analysis by using the exact collision efficiencies computed by Zinchenko and Davis (1). A dimensionless Pe´ clet number, Peo , is defined as the characteristic ratio of gravity-induced motion and Brownian motion for the initial drop size distribution. For Peo õ 1, Brownian effects initially predominate. With increasing time, however, the drop sizes increase due to coalescence and gravity effects eventually dominate, even for Peo Å 0 ( 10 02 ) . For the case of Peo ú 1, Brownian motion promotes coalescence significantly even for values of Peo Å 0 ( 10 3 ) . Also, due to the effects of Brownian coalescence on the smaller drops, the drop-size distribution evolves into a unimodal distribution instead of a bimodal distribution. Coupled Brownian and gravity effects are therefore important for a wide range, approximately 10 02 £ Peo £ 10 3 . It is also found that the additivity approximation significantly underestimates the droplet growth rate for a similar range of Peo .
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
ACKNOWLEDGMENTS This paper is based upon work supported by NASA Grants NAG3-993 and NAG3-1389, and by NSF Grant CTS-8914236. The authors also thank Dr. Alexander Zinchenko for many discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Zinchenko, A. Z., and Davis, R. H., J. Fluid Mech. 280, 119 (1994). Zhang, X., and Davis, R. H., J. Fluid Mech. 230, 479 (1991). Wang, H., and Davis, R. H., J. Colloid Interface. Sci. 159, 108 (1993). Davis, R. H., Schonberg, J. A., and Rallison, J. M., Phys. Fluids A 1, 77 (1989). Reddy, S. R., Melik, D. H., and Fogler, H. S., J. Colloid Interface Sci. 82, 116 (1981). Prieve, D. C., and Ruckenstein, E., AIChE J. 20, 1178 (1974). Rogers, J. R., and Davis, R. H., J. Atmos. Sci. 47, 1075. Simons, S., Williams, M. M. R., and Cassell, J. S., J. Aerosol Sci. 17, 789 (1986). Berry, E. X., and Reinhardt, R. L., J. Atmos. Sci. 31, 1814 (1974). Hadamard, J. S., C. R. Acad. Sci. (Paris) 152, 1735 (1911). Rybczynski, W., Bull. Acad. Sci. Cracovie A, 40 (1911). Zinchenko, A. Z., Prikl. Mat. Mekh. 46, 58 (1982). Wang, H., Ph.D. dissertation, University of Colorado, Boulder, 1994.
coida
AP: Colloid
178, 47–52 (1996)
0091
Collective Effects of Gravitational and Brownian Coalescence on Droplet Growth HUA WANG 1
ROBERT H. DAVIS 2
AND
Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424 Received January 17, 1995; accepted July 20, 1995
tional to the drop radius. In contrast, the rate of drop growth increases with time for coalescence due to differential sedimentation, because the settling velocity at a low Reynolds number is proportional to the square of the drop radius. For both cases, the coalescence rate decreases with increasing ratio of the viscosities of the drop and continuous phases; this is because it is more difficult to squeeze the fluid out from between two approaching drops as their internal viscosity increases (4). Brownian motion and gravity sedimentation may play important roles simultaneously under many conditions and drop size ranges. For example, micrometer-sized hydrocarbon droplets in water under normal gravity have a Pe´clet number near unity, and so Brownian and gravitational motions are comparable. The same may be true for supramicrometer droplets in a reduced gravity environment. Even submicrometer droplets, for which Brownian motion is dominant, will begin to sediment after growing due to Brownian coalescence. However, very little work has been done on the simultaneous effects of Brownian motion and gravity sedimentation on drop growth and coalescence. An exception is the study of simultaneous flocculation (coalescence) and creaming (sedimentation) of drops by Reddy et al. (5). However, they used the Brownian collision efficiencies for rigid spheres to account for hydrodynamic and interdroplet interactions for both Brownian and gravitational coalescence. They employed the so-called ‘‘additivity approximation,’’ in which the individual Brownian and gravitational collision rates are simply added together. This ad hoc approximation has also been used to describe the aggregation of micrometer-sized rigid particles (6), but it has no physical basis and, in fact, does not yield the proper asymptotic limits for weak gravitational or Brownian effects (1). Only recently, Zinchenko and Davis (1) developed an efficient technique to compute the collision rates for coupled gravity and Brownian effects with complete hydrodynamic interactions. Unlike previous studies on drop or particle collisions, a numerical solution was presented for arbitrary Pe´clet numbers, thus covering the whole range of drop sizes in typical dispersion systems. This technique was based on an analytical continuation into the plane of a complex Pe´clet
The temporal evolution of drop-size distributions induced by collective gravity sedimentation and Brownian motion and coalescence is studied by population dynamics. For initial distributions with arbitrary Pe´clet numbers representing the ratio of gravitational and Brownian motion, the exact collision efficiencies predicted by Zinchenko and Davis (1) are used to cover the whole range of drop sizes. It is found that coupled Brownian and gravity effects are important for Pe´clet numbers in a wide range from 10 02 to 10 3 , with the distribution shifting to larger sizes by coalescence at a much greater rate than for either mechanism acting alone. Moreover, the additivity approximation, in which the individual Brownian and gravitational coalescence rates are simply added, significantly underestimates the droplet growth rates. q 1996 Academic Press, Inc.
Key Words: coalescence; coagulation; drop growth; population dynamics.
1. INTRODUCTION
Predicting the evolution of the drop-size distribution in a liquid–liquid emulsion is an important goal in many chemical and material processes. The distribution evolves as a result of drop coalescence and possible breakage. In an unstirred emulsion, drops of different sizes will experience relative motion due to gravity, which may lead to their collisions and coalescence. Drops of a few micrometers in size, or smaller, will experience Brownian motion, which may also lead to coalescence. The coalescence rate for two interacting spherical drops induced by gravity motion and Brownian motion independently has been predicted by Zhang and Davis (2). Their results were used as collision kernals in population dynamics equations by Wang and Davis (3) to study the growth of a homogeneous drop distribution with time due to either Brownian or gravitational coalescence acting alone. For Brownian coalescence, the droplet growth slows down with time because the Brownian diffusivity is inversely propor1 Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. 2 To whom correspondence should be addressed.
47
AID
JCIS 3951
/
m4747$$$27
01-30-96 07:19:53
0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
coida
AP: Colloid
48
WANG AND DAVIS
number and a special conformal mapping to represent the solutions as a convergent power series for all real Pe´clet numbers. A key finding is that the synergistic effects of Brownian motion and gravity sedimentation give much higher coalescence rates than either mechanism acting alone and that predicted by the additivity approximation. Since coalescence is slowest for micrometer-sized drops for which neither mechanism dominates, this synergistic effect may play a significant role in applications such as cloud droplet growth where the rate-limiting step is through to occur for the micrometer size range (7). The calculation of Simons et al. (8) for combined Brownian and gravitational coagulation of particles without hydrodynamic interactions also predicts that the collision rate exceeds the sum of the collision rates of the two mechanisms acting alone. The present paper extends our previous work on individual mechanisms (3) by predicting drop growth due to coalescence induced collectively by Brownian motion and gravity sedimentation and by solving the population dynamics equations using the collision rates for arbitrary Pe´clet numbers from Zinchenko and Davis (1). The dispersions are assumed to be sufficiently dilute that pairwise collisions dominant the coalescence process, and the drops are sufficiently small that they remain spherical, have negligible inertia, and do not break apart. 2. THEORETICAL DEVELOPMENT
We consider a dilute, homogeneous dispersion containing spherical drops of viscosity m* and density r *, dispersed in an immiscible fluid of viscosity m and density r. The dispersion is quiescent (no stirring or imposed flow) other than as a result of the motion of individual drops. The growth of the drops due to coalescence is studied by using population dynamics equations. In discretized form, these equations are (3, 7, 9) i Å1
i Å 1, 2, . . . , N,
Jij Å pni n j (ai / aj ) 2V
(0) ij
Eij .
[2]
ni and n j are the numbers of drops per unit volume in the discrete size categories i and j, respectively, and ai and aj are the large and small drop radii, respectively. V ij( 0 ) is the magnitude of the Hadamard–Rybczinski relative velocity for two isolated drops (10, 11),
AID
2( mP / 1)( r * 0 r )(a 2i 0 a 2j )g Å , 3(3mP / 2) m
JCIS 3951
Pe Å
/
m4747$$$28
[3]
01-30-96 07:19:53
1/2(ai / aj )V D (ij0 )
(0) ij
,
[4]
where the relative Brownian diffusivity of two isolated drops is D (ij0 ) Å
[1]
where Jij is the rate of collisions per unit volume of size i drops with size j drops,
V
where mˆ Å m* / m is the viscosity ratio and g is the gravitational acceleration vector. The collision efficiency, Eij , takes account of simultaneous gravity and Brownian effects, hydrodynamic forces which cause the drops to move around one another, and any interdroplet repulsive and attractive forces. The ratio of gravitational and Brownian effects is given by the Pe´clet number,
N
dni 1 Å ∑ Jj (i0j ) 0 ∑ Jij , dt 2 jÅ1 jÅ1
(0) ij
FIG. 1. The collision efficiency as a function of the relative Pe´clet number for drops with l Å 0.25, mˆ Å 10, and no interdroplet forces. The solid line represents the exact solution of Zinchenko and Davis (1), the dotted line represents the additivity approximation, the dashed line is the Brownian collision efficiency, and the dashed-dotted line is the gravitational collision efficiency.
kT( mP / 1)(1 / l 01 ) , 2pm(3mˆ / 2)ai
[5]
where k Å 1.38 1 10 023 J/K is the Boltzmann constant, T is the absolute temperature, and l Å aj /aj £ 1 is the radius ratio. As Pe r ` , the collision efficiency tends to the gravitational limiting values E `ij calculated by Zhang and Davis (2) and by Zinchenko (12). In the opposite limit of Pe r 0, the Brownian collision efficiency is Cij /Pe, where the coefficient Cij has been computed by Zhang and Davis (2) and by Zinchenko and Davis (1). 3. RESULTS AND DISCUSSION
Figure 1 shows example collision efficiencies for a pair of drops with size ratio l Å 0.25 and viscosity ratio mˆ Å 10. The solid line is the exact collision efficiency for combined
coida
AP: Colloid
TEMPORAL EVOLUTION OF DROP-SIZE DISTRIBUTIONS
Brownian and gravitational motion calculated by Zinchenko and Davis (1), and the dotted line is the collision efficiency calculated using the additivity approximation Eij Å E `ij / Cij /Pe. This figure shows that the combined effects are important over a surprising wide range of Pe´clet numbers, and that the additivity approximation significantly underestimates the collision efficiency over this range. Thus, we expect the predicted dynamics of the drop size evolution in an unstable emulsion to be substantially faster when using the exact collision efficiencies. The initial drop-size distributions employed in this study are normal distributions on a number basis, as discussed in Wang and Davis (3). The governing equations were nondimensionalized using the average radius in the initial distribution, ao , as the characteristic length and tg Å 4ao / (3foVgo Eo ), or tb Å Peotg , the characteristic gravity and Brownian coalescence time scales defined in (3), as the characteristic time for Peo ú 1 or Peo õ 1, respectively. The characteristic Pe´clet number for the initial distribution is defined as Peo Å ao£o /Do , where £o and Do are the sedimentation velocity and Brownian diffusivity of an isolated drop of radius ao . The range of dimensionless droplet radii considered is from 0.08 to 600, although the numerical codes can handle even larger size ranges. A finite-difference method described by Wang and Davis (3) was used to solve the dimensionless population dynamics equations. The dimensionless time step used in the calculations is typically 0.01. The final average radius changes by less than 1% when the dimensionless time step is reduced to 0.001. In all of the calculations, the total volume fraction of the dispersed phase was computed after each time-step and checked against the input value; these values do not vary by more than 1% for the conditions used in this study. The dimensionless parameters which affect the macroscopic behavior of the dispersions we are interested in include the characteristic Pe´clet number of the initial distribution, Peo , the viscosity ratio mˆ Å m* / m, the standard deviation of the radii in the initial distribution, sˆ Å s /ao , and any dimensionless parameters describing the interdroplet forces. Note that the volume fraction of the dispersed phase, fo , is scaled out of the population dynamics equations. For simplicity, we present results only for negligible interdroplet forces; unlike solid particles, drops have finite collision rates even in the absence of attractive forces (1, 2). The effects of interdroplet forces are presented by Wang (13). 3.1. Initial Distributions with Small Pe´clet Numbers First, we consider the evolution of the drop-size distribution when Brownian effects initially dominate over gravity, i.e., Peo õ 1. Figure 2 shows the evolution of the drop-size distribution for an initial dispersion having Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0 (bubbles). The function f is the volume density function defined such that f (a)da is the volume of
AID
JCIS 3951
/
m4747$$$28
01-30-96 07:19:53
49
FIG. 2. Time evolution of the drop-size distribution for collective gravitational and Brownian coalescence of an initial distribution having sˆ Å 0.2, Peo Å 0.01, and mˆ Å 0; the solid curves represent t Å 0, 2, 10, 30, 50, and 60 times tb , respectively, from left to right, and the dashed curves represent the corresponding results for Peo Å 0 (Brownian coalescence only).
drops with radii in the range a { (1/2)da, per unit volume of dispersion (3). The evolution of the drop-size distribution for the same initial distribution but with Peo Å 0 (Brownianinduced coalescence only) is shown by the dashed lines. For short times (t/ tb £ 2), the predicted drop-size distributions (solid lines) overlap with those predicted by assuming Peo Å 0 (dashed lines). As time increases, however, the drop sizes increase due to coalescence. Since the gravitational velocity is proportional to the square of the drop radius, whereas the Brownian diffusivity is inversely proportional to the drop radius, the relative importance of gravity-induced coalescence increases with time. After t É 10tb , the dropsize distributions predicted using the exact collision efficiency shift toward larger drop sizes much faster than those predicted by assuming Peo Å 0, because the gravity effects then dominate over Brownian effects. The effect of the viscosity ratio on the evolution of average drop radius, » a … , defined as the radius of a drop having the volume-averaged drop volume (3), is shown in Fig. 3 for dispersions with Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0, 0.5, 2, and 10. It is seen that, with an increase in the viscosity ratio from 0 to 10, the growth rate of the average drop radius slows down. This is because the hydrodynamic resistance to close approach between colliding drops increases with increasing drop viscosity due to the greater difficulty in squeezing fluid out from the gap separating the two drops (4). Figure 3 also shows that the additivity approximation may considerably underestimate the drop growth rate. For short times, the additivity and exact results agree, because both are dominated by Brownian effects. At longer times, gravity effects are also important, and the additivity approximation underpredicts the average radius by 50% or more.
coida
AP: Colloid
50
WANG AND DAVIS
FIG. 3. The average droplet radius versus time for collective gravitational and Brownian coalescence of initial distributions having Peo Å 0.01, sˆ Å 0.2, and mˆ Å 0, 0.5, 2, and 10. The dashed lines represent the corresponding results using the additivity approximation. The dotted line is the Brownian result (Peo Å 0) for mˆ Å 0.
Note that the curves switch from the concave-down, characteristic of Brownian collisions, to the concave-up, characteristic of gravity collisions (see (3)) as time progresses. The influence of the initial average drop size, as measured by the characteristic Pe´ clet number, on the evolution of average drop size is shown in Fig. 4 for a dispersion having sˆ Å 0.2 and mˆ Å 10. It is shown that, with an increase of Peo from 10 04 to 0.1, the average radius deviates significantly from that of Peo Å 0. Even when Peo Å 10 02 , significant errors are introduced if the results for Peo Å 0 ( Brownian coalescence only ) are used. The time needed for the average drop size to increase to » a … Å 10ao
FIG. 5. The average drop radius at t Å 50tb as a function of Peo for a dispersion with an initial distribution having sˆ Å 0.2 and mˆ Å 10. The upper curve is the result using exact collision efficiencies, and the lower curve is the result using the additivity approximation.
is 34, 87, 132, 151, and 160 times tb , respectively, for Peo Å 0.1, 10 02 , 10 03 , 10 04 , and 0. A plot of the average radius at t Å 50tb as a function of the initial Pe´ clet number is shown in Fig. 5. When Peo r 0 ( Brownian effects alone ) , the calculated value is » a … Å 6.8ao , whereas the calculated values using the exact collision efficiencies are » a … Å 17.1ao , 7.7ao , and 7.0ao , respectively, for Peo Å 10 01 , 10 02 , and 10 03 . The calculated values using the additivity approximation are » a … Å 9.1ao , 7.0ao , and 6.8ao , respectively, for Peo Å 10 01 , 10 02 , and 10 03 . Thus, neglecting gravitational effects, or inaccurately accounting for them using the additivity approximation, leads to a significant underprediction of the droplet growth for Peo § 0 ( 10 02 ) . 3.2. Initial Distributions with Moderate and Large Pe´clet Numbers
FIG. 4. Time evolution of the average radius for collective gravitational and Brownian coalescence of initial distributions having mˆ Å 10, sˆ Å 0.2, and different Peo ; the dashed line is the Brownian result for Peo Å 0.
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
The results presented in Figs. 2–5 are for dispersion systems with Peo õ 1 for which coalescence is initially dominated by Brownian effects. A transition in the dominant collision mechanism to gravitational coalescence with increasing time was found. We now consider the droplet growth behavior for dispersion systems with Peo § 1. For such systems, gravity-induced coalescence always dominates. Attention here is focused on how the effects of Brownian motion and coalescence enhance the gravitational drop growth rates, by comparing the results predicted using exact collision efficiencies for coupled gravity and Brownian induced coalescence with those predicted using the collision efficiencies for Peo Å ` (gravity-induced coalescence alone). Figure 6 presents the drop-size distribution at t/ tg Å 0, 10, and 20, evolved from an initial dispersion having mˆ Å
coida
AP: Colloid
TEMPORAL EVOLUTION OF DROP-SIZE DISTRIBUTIONS
FIG. 6. Time evolution of the drop-size distribution for collective gravitational and Brownian coalescence of initial distributions having mˆ Å 10 and sˆ Å 0.2. The solid lines represent the results for t/ tg Å 0, 10, and 20, respectively, from left to right, for a system with Peo Å 10; the dotted lines represent the results for Peo Å ` (gravity-induced coalescence alone) at t/ tg Å 10, 20, 30, and 50, respectively, from left to right.
10, sˆ Å 0.2, and Peo Å 10. The evolution of the drop-size distribution due to gravity-induced coalescence alone (Peo Å ` ) is also presented in Fig. 6 as dashed lines for t/ tg Å 10, 20, 30, and 50. For Peo Å ` , the size-distribution evolves into a bimodal distribution with large spread; a large concentration of small drops remains even for large times. This behavior occurs because the gravitational collision efficiency is very small for l ! 1, since small drops flow around large ones (3). For Peo Å 10, however, the size distribution evolves into a unimodal distribution, with fewer small drops. This is because weak Brownian effects, which are more important for smaller drops, promote the coalescence of the small drops. Once these small drops coalescence into larger drops, the drop-size evolution accelerates because the gravitational velocity increases in proportion to the square of the drop radius. As a result, the evolution of drop-size distribution is much faster for dispersions with moderately large Pe´clet numbers when the effects of Brownian motion are included in the collision efficiencies employed. The influence of the initial Pe´clet number on the growth of average drop radius is shown in Fig. 7 for initial dispersions having mˆ Å 10 and sˆ Å 0.2. The times for the average drop radius » a … to reach 25ao are 12.2, 22.2, 34.4, 44.1, 49.8, and 53.3 times tg , respectively, for Peo Å 1, 10, 10 2 , 10 3 , 10 4 , and ` . The influence of Brownian motion on the drop growth is significant even for Peo as high as 10 3 . Figure 8 is a plot of the average radius at t Å 10tg as a function of the initial Pe´clet number. The solid line represents the results using the exact collision efficiencies (1), whereas the dashed line represents the results using the additivity approximation. When Peo r ` (gravity-induced coalescence alone), the calculated value is » a … Å 1.3ao ,
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
51
FIG. 7. Time evolution of the average radius for drops with initial distributions having sˆ Å 0.2, mˆ Å 0, and different Peo ; the dashed line is the result for Peo Å ` .
whereas the calculated values using the exact collision efficiencies are » a … Å 12.8ao , 3.5ao , 1.9ao , and 1.5ao , respectively, for Peo Å 1, 10, 10 2 , and 10 3 . The calculated values using the additivity approximation are » a … Å 5.9ao , 2.3ao , 1.5ao , and 1.3ao , respectively, for Peo Å 1, 10, 10 2 , and 10 3 . Neglecting Brownian effects, or inaccurately accounting for them by the additivity approximation, significantly underpredicts the average radius for Peo £ 0(10 3 ). 4. CONCLUDING REMARKS
Quantitative predictions of the temporal evolution of drop size distributions in droplet dispersions due to collisions and
FIG. 8. The average drop radius at t Å 10tg as a function of Peo for a dispersion with an initial distribution having sˆ Å 0.2 and mˆ Å 10. The upper curve is the result using exact collision efficiencies, and the lower curve is the result using the additivity approximation.
coida
AP: Colloid
52
WANG AND DAVIS
coalescence induced collectively by gravity sedimentation and Brownian motion are presented in this paper. Complete hydrodynamic interactions are included in the analysis by using the exact collision efficiencies computed by Zinchenko and Davis (1). A dimensionless Pe´ clet number, Peo , is defined as the characteristic ratio of gravity-induced motion and Brownian motion for the initial drop size distribution. For Peo õ 1, Brownian effects initially predominate. With increasing time, however, the drop sizes increase due to coalescence and gravity effects eventually dominate, even for Peo Å 0 ( 10 02 ) . For the case of Peo ú 1, Brownian motion promotes coalescence significantly even for values of Peo Å 0 ( 10 3 ) . Also, due to the effects of Brownian coalescence on the smaller drops, the drop-size distribution evolves into a unimodal distribution instead of a bimodal distribution. Coupled Brownian and gravity effects are therefore important for a wide range, approximately 10 02 £ Peo £ 10 3 . It is also found that the additivity approximation significantly underestimates the droplet growth rate for a similar range of Peo .
AID
JCIS 3951
/
m4747$$$29
01-30-96 07:19:53
ACKNOWLEDGMENTS This paper is based upon work supported by NASA Grants NAG3-993 and NAG3-1389, and by NSF Grant CTS-8914236. The authors also thank Dr. Alexander Zinchenko for many discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Zinchenko, A. Z., and Davis, R. H., J. Fluid Mech. 280, 119 (1994). Zhang, X., and Davis, R. H., J. Fluid Mech. 230, 479 (1991). Wang, H., and Davis, R. H., J. Colloid Interface. Sci. 159, 108 (1993). Davis, R. H., Schonberg, J. A., and Rallison, J. M., Phys. Fluids A 1, 77 (1989). Reddy, S. R., Melik, D. H., and Fogler, H. S., J. Colloid Interface Sci. 82, 116 (1981). Prieve, D. C., and Ruckenstein, E., AIChE J. 20, 1178 (1974). Rogers, J. R., and Davis, R. H., J. Atmos. Sci. 47, 1075. Simons, S., Williams, M. M. R., and Cassell, J. S., J. Aerosol Sci. 17, 789 (1986). Berry, E. X., and Reinhardt, R. L., J. Atmos. Sci. 31, 1814 (1974). Hadamard, J. S., C. R. Acad. Sci. (Paris) 152, 1735 (1911). Rybczynski, W., Bull. Acad. Sci. Cracovie A, 40 (1911). Zinchenko, A. Z., Prikl. Mat. Mekh. 46, 58 (1982). Wang, H., Ph.D. dissertation, University of Colorado, Boulder, 1994.
coida
AP: Colloid