Ground-based modeling and experiments have been performed on the interaction and coalescence of drops leading to macroscopic phase separation. The focus has been on gravity-induced motion, with research also initiated on thermocapillary motion of drops. The drop size distribution initially shifts toward larger drops with time due to coalescence, and then a back towards smaller drops due to the larger preferentially settling out. As a consequence, the phase separation rate initially increases with time and then decreases

Davis, Robert H.

Hawker, Debra

Wang, Hua

English

NASA Technical Reports Server

3 N95- 14537 
PHASE SEGREGATION DUE TO SIMULTANEOUS 
MIGRATION AND COALESCENCE 
Robert H. Davis, Hua Wang, and Debra Hawker 
Department of Chemical Engineering 
University of Colorado 
Boulder, Colorado 80309-0424 
ABSTRACT 
Ground-based modeling and experiments have been performed on the interaction and coalescence of 
drops leading to macroscopic phase separation. The focus has been on gravity-induced motion, with reseqh 
also initiated on thermocapillary motion of drops. The drop size distribution initially shifts toward larger 
drops with time due to coalescence, and then back towards smaller drops due to the larger drops preferentially 
settling out. As a consequence, the phase separation rate initially increases with time and then decreases. 
INTRODUCTION 
Dispersions of drops of one fluid in a second, immiscible fluid are frequently encountered in industrial and 
natural processes such as extraction, rain-drop growth, food and beverage processing, and the formation of 
liquid-phase-miscibility-gap materials. In the absence of stirring or bulk motion, dispersed drops of different 
sizes may undergo relative motion under the action of gravity. The larger drops migrate more rapidly and 
catch up to the smaller drops in their path, leading to possible contact and coalescence. In a finite sample, 
drop migration and coalescence generally result in an inhomogeneous dispersion with phase separation. In 
extraction, this is the desired result; the two phases must separate subsequent to mixing for the partitioned 
component to be recovered. A counter-example is the processing of liquid-phase-miscibility-gap metals, for 
which the desired product is a composite material with fine particles of one metal uniformly dispersed in a 
matrix of the other. Low-gravity environments suppress buoyancy-driven droplet motion, but coalescence 
and phase separation may still occur due to thermocapillary effects [l]. 
Gravity sedimentation of particles in the absence of coalescence or aggregation has been studied exten- 
sively [2]. Similarly, coalescence of drops in spatially uniform dispersions in the absence of phase separation 
has been modeled extensively [3-61. A notable example of simultaneous sedimentation and coalescence is the 
study of Fkddy, Melik & Fogler [?I. They noted that the probability size distribution shifts toward larger 
drops when coalescence dominates, toward smaller particles when sedimentation dominates, and initially 
toward larger drops and subsequently toward small drops when both mechanisms are important. 
In the current work, we predict the macroscopic phase separation and drop-size distributions for 
buoyancy-driven sedimentation and coalescence of immiscible dispersions of drops by solving the general 
population dynamics equations with both timedependence and spatial-dependence retained, and incorpo- 
rating a mass balance on the drops arriving at the moving interface. Preliminary experiments on two-phase 
systems are also presented. Comparison work has been performed on drop coalescence due to Brownian 
diffusion and thermocapillary effects, with the focus to-date on spatially uniform dispersions [5,6,8]. 
THEORY 
We consider a dilute dispersion of spherical drops of viscosity pf and density p' dispersed in an immiscible 
fluid of viscosity p and density p ,  The drops are considered to be nonBrownian, but not so large that inertial 
effects or deformation are important. For common liquids, these conditions are met for drops with diameters 
of 2-100 pm [9]. 
101 
https://ntrs.nasa.gov/search.jsp?R=19950008123 2020-06-16T10:24:14+00:00Z
The typical problem of interest is illustrated in Fig. 1. The initial condition (Fig. la) is a uniform 
suspension of droplets of one fluid dispersed in a second, immiscible liquid. After mixing is stopped, the 
drops begin to rise due to buoyancy (assuming, for illustrative purposes, that p' c p). As the drops reach the 
top of the container, they coalesce into an overlying, segregated layer of the dispersed-phase fluid (Fig. lb). 
This layer grows with time (Fig. IC) until all of the dispersed drops have coalesced into it (Fig. Id). The 
problem is complicated by the possibility that the drops also collide and coalesce with each other as they 
rise, and, because the larger drops move faster and leave the smaller ones behind, the drop size distribution 
varies with both time and position. 
The temporal and spatial evolutions of the drop size distribution are studied by using population 
dynamics equations: 
where ni is the number of drops per unit volume in the discrete size category i, ui is the Stokes' settling 
velocity of drops of size i, t is time, z is the direction of drop migration (vertical, as defined in Fig. l), 
Jij is the rate of collisions per unit volume of size i drops with size j drops, and N is the total number of 
size categories. The first term on the right-hand-side of Eq. (1) is the rate of formation of size i drops by 
collision of two smaller drops, and the second term is the rate of loss of size i drops due to their collisions to 
form larger drops. The collision rate between drops in size category i (larger drops) and the size category j 
(smaller drops) may be expressed as [9]: 
where ai and aj are the large and small drop radii, respectively, and Eij is the collision efficiency. The 
collision efficiency equals unity when the drops move independently until colliding; values of Eij differing 
from unity are given previously [SI and account for hydrodynamic forces which cause the drops to move 
around one another and the attractive forces which pull them together. 
The rate at which the lighter phase grows at the top of the vessel due to droplets reaching this phase 
is determined by a mass balance: 
where the left-hand-side is the rate of accumulation of the upper phase and the right-hand-side is the flux 
of drops into the upper phase, A is the cross-sectional area of the container, & = $ma is the volume of a 
drop of size i, and 4 = zc, &ni is the total volume fraction of droplets in the suspension just below the 
upper interface at z = h,(t). The initial condition is h, = H at t = 0. 
The population dynamics equations were nondimensionalized and then solved using the Lax-Wendroff 
finite-difference method [lo]. The numerical methods used in the modeling involve logarithmic discretization 
of drop spectra into N categories which have equal spacing in the logarithm of droplet mass or volume, 
and the mass or volume of a droplet within each discrete category doubles every fourth category [3]. The 
initial size distributions, assumed uniform for 0 5 z HI were chosen to be normal distributions on a 
number basis, as specified by the number-averaged drop radius, a, and the standard deviation, CT, of drop 
radii. In dimensionless form, normal distributions are characterized by a single dimensionless parameter, 
B = c/a,. The dimensionless parameters which affect the macroscopic behavior of a dispersion include the 
viscosity ratio, 6 = $ / p ,  the dimensionless standard deviation of radii in the initial distribution, B = u/ao, 
the initial volume fraction, r$,, and the time scale ratio, N7 c To/Tc = 3t#,H/4a0, where ro = H/u ,  is the 
characteristic settling time for drop with velocity uo and radius a, to travel the length of the container, and 
= 4a0/(3q5,uO) is the characteristic coalescence time. 
102 
Figure 2 shows the evolution of the average radius (defined in [5] as the radius of a drop having the 
massaveraged volume) versus time at the container midpoint for a dispersion having B = 0.2, f i  = 0.1, 
and different values of N r .  It is seen that the average droplet radius initially increases with time due to 
coalescence; after the larger drops sediment out of the dispersion, the average radius began to decrease. 
The variation of the droplet volume fraction with dimensionless position at different times is shown in 
Fig. 3 for a dispersion with ji = 0.1, b = 0.2, and Nr=20. The volume fraction in the lower regions of the 
container decreases rapidly with time as the drops rise. For short times, the volume fraction in the upper 
regk1a8 of the dispersion is the same as the initial volume fraction. This is because droplet6 which rise out of 
a control volume at the top of dispersion are replaced at an equal rate by droplets moving into the control 
volume. With time increasing, however, larger drops rise out of the dispersion, and the volume fraction at 
any height in the dispersion system is significantly less than the initial volume fracction. 
Typical results for the gravitational phaae separation rate are shown in Fig. 4 for a dispersion having 
fi  = 0.1, B = 0.1 (dashed lines), B = 0.2 (solid lines), (p,=0.05, and different Nr.  When N,=O, no coalescence 
occurs. Phase separation then takes place at a constant rate until the largest drops rise out of the dispersion, 
after which the phase separation rate monotonically decreases due to fewer and smaller drops arriving at 
the phase interface. When Ny > 0, the phase separation rate initially increases, because coalescence leads 
to larger drops with faster rise rates. Once these drops rise out of suspension, however, the phase separation 
rate reaches a maximum and then decreases as only smaller drops remain. Note that relatively large values 
of N7 2 O(10) are necessary for coalescence to significantly affect the phase separation process. This is 
because typical collision efficiencies are O(10-l) for dispersions with f i  = 0.1 [9]. 
I 
EXPERIMENTS 
Experiments to observe and measure drop coalescence and phase segregation due to gravity sediment* 
tion have been initiated with two phase systems: lI2-propanediol drops in dibutyl sebacate, and an aqueous 
biphase mixture containing 1% dextran (MW = 500) and 6.5% polyethylene glycol (PEG, MW = 8000) by 
weight. The experiments were carried out in an optical cuvette of 1 cm x 1 cm cross-section, with heights up 
to 20 cm. Before transferring the two phases to the cuvette, vortexing was performed twice for one minute 
each. This resulted in a range of drop diameters of approximately 5-30 ,urn. The height of the segregated 
minority phase was followed as a function of time using a videomicroscope connected to a video recorder and 
a personal computer containing Global Lab Image analysis software by Data Translation. For the aqueous 
biphase system, samples at various heights were taken using inserted syringes and analyzed for the volume 
fraction of the droplet phase. 
The experimental results are in qualitative agreement with the theory; quantitative comparison has 
not yet been made. Figure 5 shows a plot of the height of the segregated droplet phase, normalized by the 
final height it achieves after complete separation, versus time for two values of the total height of the 1,2- 
propanediol/dibutyl sebacate dispersion: H = 10 cm and H = 14.5 em. In both cases, a sigmoidal-shaped 
curve is observed, as expected. The initial phase separation rate is slow because of the small drop sises and 
velocities; it then speeds up as coalescence occurs to yield larger drops, and then it decreases again as the 
large drops settle out of the dispersion. As predicted from Fig. 4, the phase separation is faster for the taller 
dispersion (Nr greater) because there is more opportunity for coalescence. At a given position, the observed 
drop size increased initially due to coalescence and then decreases due to sedimentation of the larger drops, 
as predicted in Fig. 2, but quantitative analysis of the drop size distributions has not yet been performed. 
Similar results have been obtained with the aqueous biphasic system. Figure 6 shows the volume percent 
of minority phase versus distance from the bottom of the cuvette for a total minority phase volume percent 
of 6.5% and total height of the dispersion of H = 17 cm. Note that the minority, dextran-rich phase is more 
dense than the continuous, PEG-rich phase, and so the droplets settle downward. Similar to the behavior 
predicted in Fig. 3, the droplet volume fraction is uniform and equal to its initial value for short times and 
103 
then decreases with time. The decrease occurs first in the upper portion of the container, due to droplets 
settling downward. I 
CONCLUSIONS 
Quantitative predictions of the temporal and spatial evolutions of the drop size distributions and macro- 
scopic phase separation rates in droplet dispersions due to buoyancy-driven motion are presented. Droplet 
coalescence significantly increases the phase separation rate initially, and then the phase separation rate 
decreases after the larger drops are removed from the dispersion. The predicted trends are verified by 
experiments. 
ACKNOWLEDGMENTS 
This work was supported by NASA grant NAG3-1389. The authors thank two undergraduate assistants, 
Tyler Kinkade and Nathan Reader, for assistance with the experiments. 
REFERENCES 
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2. Davis, R.H. & Acrivos, A. Sedimentation of noncolloidal particles at low Reynolds numbers. Ann. 
3. Rogers, J.R. & Davis, R.H. Modeling of collision and coalescence of droplets in microgravity processing 
4. Satrape, J.V. Interactions and collisions of bubbles in thermocapillary motion. Phys. Fluids A 4 (1992) 
5. Wang, H. & Davis, R.H. Droplet growth due to Brownian, gravitational, or thermocapillary motion 
6. Zhang, X., Wang, H. & Davis, R.H. Collective effects of temperature and gravity on droplet coalescence 
7. Reddy, S.R., Melik, D.H. & Fogler, H.S. Emulsion stability-theoretical studies on simultaneous floccu- 
precipitation in immiscible alloys. J. Mat. Sci. 23 (1988) 1573-1579. 
Rev. Fluid Mech. 17 (1985) 91-118. 
of Zn-Bi immiscible alloys. Metallurgical Dans. 21A (1990) 59-98. 
1883-1900. 
and coalescence in dilute dispersions J. Colloid Interface. Sci. 159 (1993) 108-118. 
Physics of Fluids A 5 (1993) 1602-1613. 
lation and creaming. J. Colloid Interface Sci. 82 (1981) 116-127. 
8. Zhang, X. & Davis, R.H. The collision rate of small drops due to thermocapillary migration J. Colloid 
9. Zhang, X .  & Davis, R.H. The collisions of small drops due to Brownian and gravitational motion J. 
10. Lax, P.D. & Wendroff, B. 1960 Systems of conservation laws. Comm. Pure Appl. Meth. 13 (1960) 
Interface Sci. 152 (1992) 548-561. I 
Fluid Mech. 230 (1991) 479-504. 
217-237. 
10.4 
==3 
Fig. 1-Schematic of the time evolution of the phase separation process due 
to the simultaneous migration and coalescence of rising drops or bubbles. 
0 
0 
-2 2.0 ~. 
0 
V 
1 .o 
8 
Fig. 2-Predicted time 
evolution of the average 
drop radius at z / H  = 0.5 
for a dispersion having 
j k0 . l  and h 0 . 2 .  
0.0 0.2 0.4 0.6 0.8 
t/r, 
Fig. 3-Predicted varill- 
tion of volume fraction 
with position at differ- 
ent times for a dispersion 
having jl = 0.1, 6 = 0.2, 
and NT = 20. 
0.0 0.2 0.4 0.6 0.8 1.0 
z/H 
105 
10 
0.0 0.2 0.4 0.6 0.8 1 .o 
t/T. 
4. 
2 6,1 
0. 
Fig. 4-Predicted rate of 
phase separation versus 
time for a dispersion hav- 
ing fi  = 0.1, b 5 0.2 
(solid linea), B = 0.1 
(dashed limes), q60 = 0.05 
and different N+ in a con- 
tainer of finite depth. 
Fig. &Measured height 
of segregated minority 
phase (normalized by fi- 
nal height) versus time 
for 1,2-propanediol drops 
at q60 = 0.034 in dibutyl 
sebacate with H=10 cm 
(open squares) and H = 
14.5 cm (closed squares). 
Fig. 6-Volume percent- 
age of dispersed phase 
versus distance from the 
bottom of the curvette 
for dextran-rich drops at 
do = 0.065 in PEG-rich 
continuous phase with 
H = 17 cm at t = 15, 
30, 45, 60, 120, and 240 
min (top to bottom). 
106 


Phase segregation due to simultaneous migration and coalescence

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