Interaction and Coalescence of Drops and Bubbles Rising through a

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Interaction and Coalescence of Drops and Bubbles Rising through a Tube Eisa Almatroushi† and Ali Borhan* Department of Chemical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802

An experimental study of the interaction and coalescence of two drops (of the same fluid) or bubbles translating under the action of buoyancy in a cylindrical tube is performed. The close approach of two Newtonian drops or bubbles of different size in a Newtonian continuous phase is examined using image analysis, and measurements of the coalescence time are reported for various drop size ratios, Bond numbers, and drop to suspending fluid viscosity ratios. The time scale for coalescence in the non-axisymmetric configuration is found to be substantially larger than that for coalescence in the axisymmetric configuration. Experimental measurements of the radius of the liquid film between the two drops are used in conjunction with a simple film-drainage model to predict the dependence of the coalescence time on the drop size ratio. The agreement between the model predictions and the experimental measurements is satisfactory for axisymmetric coalescence in the low viscosity ratio systems. For the systems with O(1) viscosity ratio, on the other hand, model predictions are qualitatively different from experimental observations. Introduction The motion and interaction of drops and bubbles are frequently encountered in many industrial applications such as food processing, polymerization processes, dispersion, extraction, enhanced oil recovery, and production of lubricant oils, paints, detergents, pharmaceuticals, and cosmetics. An associated phenomenon that plays a significant role in many of these operations is drop coalescence. There are instances where it is desirable for coalescence to occur, as in the case of separation processes, while in other cases coalescence can be a highly undesirable process. For example, the coalescence of gas bubbles during a gas-liquid reaction can lower the efficiency of the reactor by reducing the overall interfacial area. An adequate understanding of the hydrodynamic aspects of the motion and interaction of multiple drops is required for useful modeling of the heat and mass transport and/or reactive processes that can accompany drop or bubble motion in such multiphase systems. The interaction and coalescence of two bubbles in a Newtonian fluid has received much attention in the past. A useful review of previous studies of bubble coalescence is given by Chaudhari and Hofmann.1 Coalescence of two drops requires that the separation distance between the two drops eventually become much smaller than the size of either drop. The coalescence process can be characterized by three distinct stages. The first stage is the close approach of the two bubbles or drops to form a thin liquid film between them, with the thickness of the liquid film being on the order of a few microns. The second stage is film thinning or drainage, wherein the liquid film thickness is gradually reduced. Thinning of the film is primarily driven by gravity and capillary forces and can be significantly affected by the physical properties of the bulk phases and the interfacial tension between them. When the thickness of the film is reduced to about 100 nm, van der Waals attraction accelerates the drainage process. Once the film becomes sufficiently thin (less than about 10 nm thick), it eventually ruptures and leads to coalescence. * Corresponding author. Tel: 814-865-7847. Fax: 814-865-7846. E-mail: [email protected]. † Present address: Dept. of Chemical & Petroleum Engineering, United Arab Emirates University, P.O. Box 17555, Al-Ain, United Arab Emirates.

The time scale for film rupture is negligibly small as compared to that for film thinning. Hence, the film-thinning or drainage step is generally believed to be the rate-controlling step in the coalescence process. If the contact time of the two drops is less than the drainage time scale, coalescence will not occur. The rate of film drainage is strongly influenced by the mobility of the bounding interfaces and the physical properties of the bulk fluids. The presence of surface-active species or electrical charges at the interfaces bounding the thin film can dramatically affect the rate of drainage and the coalescence process as a whole. Hence, a major emphasis of most previous studies of bubble coalescence has been on understanding the influence of electrolytes and surfactants on the overall coalescence behavior of drops and bubbles (e.g., Cristini et al.2 and Blawzdziewicz et al.3). Although interest in modeling the coalescence process has motivated a number of theoretical studies of the film-drainage problem, experimental studies of the interaction and coalescence of drops in pure liquids have been limited. A few experimental studies dealing with the buoyancy-driven interaction and coalescence of gas bubbles under low Reynolds number conditions have been reported in the literature. The main focus of these studies has been on understanding the interaction between two bubbles. The in-line interaction of two gas bubbles was examined by Crabtree and Bridgewater,4 Narayanan et al.,5 and Bhaga and Weber.6 The results of these studies showed that the wake of the leading bubble can play a vital role both in capturing nonaligned bubbles and in their subsequent coalescence behavior. In particular, Crabtree and Bridgewater4 found that coalescence between widely separated bubbles could be achieved even when the difference in the free rise velocities of the two bubbles would suggest otherwise. On the other hand, Bhaga and Weber6 concluded that the rise velocity of the trailing bubble must be greater than the free rise velocity of the leading bubble in order for coalescence to occur. They also found that the coalescence behavior of the bubbles was affected by deformations of the trailing bubble in the wake of the leading one. More recently, Manga and Stone7 studied the non-axisymmetric buoyancy-driven interaction of two air bubbles rising in a large container filled with corn syrup. They found that the initial horizontal displacement of the two bubbles determined

10.1021/ie0505615 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/08/2005

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the type of bubble interaction that occurred. Small initial horizontal displacements led to bubble alignment such that the smaller leading bubble coated the larger trailing one. When the initial horizontal displacement was sufficiently large, the smaller leading bubble no longer coated the trailing bubble. Instead, the larger bubble passed by the smaller one and its deformation caused the smaller bubble to be sucked in toward it from behind. Zhang et al.8 presented experimental results for the hydrodynamic interaction between two viscous drops undergoing gravity-induced relative motion under low Reynolds number conditions in a large container. For different initial horizontal displacements, they measured the relative velocity of the two drops as well as the time required for the two drops to flow around each other. They found that hydrodynamic interactions significantly reduced the relative velocity of the two drops and caused them to flow around each other with curved trajectories. Their experimental results also confirmed the theoretical prediction that the hydrodynamic interactions increased with decreasing drop separation, decreasing drop size ratio, and increasing viscosity ratio. In the case of the confined motion of drops, the interaction and coalescence of drops and bubbles have received much less attention than the complementary problem of drop breakup. Most experiments on the coalescence of drops in emulsions flowing through porous media have indirectly inferred the effects of various parameters (such as interfacial tension, viscosity, and flow rate) on the overall rate of coalescence by measuring the drop size distribution leaving the porous media under different experimental conditions. However, the parameter values affecting the interaction between drops in such experiments are very difficult to control and cannot be varied systematically. There have been a few experimental studies in which the time scale for thin-film drainage between two confined drops has been examined (e.g., Olbricht and Kung9 and Aul and Olbricht10). Olbricht and Kung9 experimentally studied the coalescence behavior of two neutrally-buoyant drops of unequal size suspended in pressure-driven flow through a horizontal capillary tube. They found that the interaction between drops resulted in their coalescence over a limited range of drop sizes. They did not observe drop coalescence when the trailing drop was smaller than a critical size that depended on the capillary number and the viscosity ratio. In particular, when the capillary number exceeded 0.5, coalescence between drops did not occur for any of the drop sizes considered in their experiments. Aul and Olbricht10 extended the results of Olbricht and Kung9 by examining the influence of drop size on the coalescence process. They measured a coalescence time, defined as the time elapsed from the instant of apparent contact between drops to coalescence, and found that it was an increasing function of the size of the trailing drop. However, their results for the coalescence time were not sensitive to the value of the leading drop size or the capillary number. In this work, we present the results of an experimental study of the interaction and coalescence of drops and bubbles rising under the action of buoyancy in a vertical cylindrical tube where the continuous phase preferentially wets the tube wall and the drop phase is completely nonwetting. We considered this particular experimental configuration to avoid the difficulties encountered in coalescence experiments performed in large containers wherein the lack of control of the lateral location of drops rendered their motion past each other (as opposed to their coalescence) the more likely outcome of their interaction. In the experimental configuration used in this study, the presence of the tube wall prevented the drops from moving past each

Table 1. Two-Phase Systems Used in the Experimentsa suspending fluid

drop fluid

96.2% GW 96.2% GW 96.2% GW 96.2% GW 96.2% GW

air UCON-65 UCON-165 UCON-625 UCON-1145

a

λµ γF σ (N/m) (mPa‚s) (kg/m3) × 103 0 27 100 278 528

1.3 957 976 993 996

66.0 4.0 4.0 4.0 11.5

λ

γ

Bo

0.00 0.06 0.23 0.63 1.19

0.00 0.77 0.78 0.80 0.80

4.1 4.6 4.3 3.6 3.4

GW, glycerol-water.

other. We report measurements of the relative mobility of the drops as a function of their separation distance as well as the coalescence time in cases where drop collisions led to coalescence. Experimental Procedures The experimental setup was similar to that used by Borhan and Pallinti11 in their study of the breakup of drops and bubbles translating through cylindrical tubes. It consisted of a precision bore glass tube positioned vertically within a Plexiglas chamber of square cross-section containing an aqueous solution of sodium iodide. The refractive index of the sodium iodide solution was matched to that of the glass tube in order to minimize optical distortions due to the refraction of light at the outer wall of the tube. Two tubes, both of length 120 cm, with different inside diameters of 0.347 and 0.796 cm were used in the experiments. The exterior and interior bulk phases used in the experiments were Newtonian fluids with densities F and γF and viscosities µ and λµ, respectively. The interface between the two fluids was characterized by uniform interfacial tension σ. Specifically, a 96.2% glycerol-water (GW) mixture was used as the suspending fluid, and air as well as a variety of UCON oils were used as the drop phase. The physical properties and the dimensionless parameters characterizing the various two-phase systems used in the experiments are listed in Table 1. All of the physical properties shown in this table were measured at a temperature of 25 °C. The viscosities of all liquids were measured using a Brookfield cone and plate rheometer, and the interfacial tensions were determined using a Fisher ring tensiometer. The leading and trailing drops were characterized by equivalent-spherical radii κ1R and κ2R, respectively, where R denotes the tube radius. The Reynolds number for the motion of the drops remained small in all experiments. The dimensionless parameters governing the dynamics of the drops include dimensionless size of the leading drop (κ1), leading to trailing drop size ratio (R ) κ1/κ2), drop to suspending fluid viscosity ratio (λ), drop to suspending fluid density ratio (γ), and the Bond number (Bo ≡ (1 - γ)FgR2/σ) representing the relative strength of the gravitational force as compared to interfacial tension, where g is the magnitude of the gravitational acceleration. For all two-phase systems considered, the density of the drop phase was less than that of the continuous phase so that the drop moved from the bottom of the tube to the top, and the Bond number based on the tube radius varied in the range 3 < Bo < 5. To begin each experiment, the tube was filled with the suspending fluid from a liquid reservoir that was maintained at a constant temperature of 25 °C. The temperatures at the inlet and outlet of the tube were monitored during the experiment using thermocouples connected to a digital thermometer. The drop injection section near the inlet region (at the bottom) of the tube consisted of two injection ports vertically offset by a small distance to allow simultaneous injection of the leading and trailing drops. Using a micrometer syringe, the desired

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Figure 1. Steady shapes of drops rising in a vertical tube; (a) air bubbles with Bo ) 4.1 and (i) κ ) 0.3, (ii) κ ) 0.5, (iii) κ ) 0.7, (iv) κ ) 0.8, (v) κ ) 1.0, (vi) κ ) 1.2 and (b) viscous drops with λ ) 0.23, Bo ) 4.2, and (i) κ ) 0.3, (ii) κ ) 0.5, (iii) κ ) 0.6, (iv) κ ) 0.7, (v) κ ) 0.8, (vi) κ ) 0.9.

volume of the drop fluid was injected at the symmetry axis of the tube to generate the leading drop of dimensionless size (κ1). A second drop of dimensionless size (κ2) was then injected from the injection port once the first drop had translated several tube diameters. The initial axial spacing between the two drops was made sufficiently large for the drops to attain equilibrium shapes. The size of the trailing drop was selected such that the trailing drop had a larger single-drop velocity than the leading one, thereby causing the two drops to come together. A video camera mounted on a motor-driven moving platform with a variable speed controller was used to monitor the drops as they passed through the entire length of the tube. The motion of the drops was recorded and indexed using a video recorder capable of single frame playback. The recorded images were played back frame by frame, and the Image-Pro software (Media Cybernetics, Silver Spring, MD) was used to analyze the digitized images. The axisymmetric drop profiles obtained from image analysis were used to obtain more accurate measurements of drop volume. The single-drop terminal velocities were determined by measuring the time required for the drop to travel a specified vertical distance designated by markers on the tube wall, while the relative velocity of the two interacting drops was determined from the time evolution of the interparticle distance measured using Image-Pro. Each experiment was repeated to ensure reproducibility of the measurements. In all cases, the reported values of drop velocity represent the average of three velocity measurements, with each measurement having a variation of less than 5% from the reported mean value. The tube and the syringe were thoroughly cleaned with distilled water, benzene, and acetone and then dried in air before each new set of experiments. Results and Discussion We begin with a brief description of the motion of single drops within the tube before proceeding to present the results of the coalescence experiments. As will become clear later in this section, results of the single-drop experiments will allow

us to explain some of the observations in the coalescence experiments. In addition, experimental measurements in the single-drop experiments were used as a guide in selecting appropriate sizes for the leading and trailing drops in the coalescence experiments. The shape of a single drop or bubble injected at the tube centerline remained axisymmetric as the drop passed through the tube. Typical profiles (in the meridional plane) of the steady shapes observed experimentally for air bubbles and viscous drops are shown in Figure 1 for various drop sizes. The location of the tube wall coincides with the edges of the images in this figure. The steady shape of an air bubble approached an elongated ellipsoid as the bubble size increased. Increasing the viscosity of the interior phase relative to that of the suspending fluid led to a more tapered drop shape without significantly affecting the qualitative features of the leading end of the drop. For fixed Bond number, drop deformation was only slightly affected by large variations in the viscosity ratio. Increasing the Bond number produced larger deformations in the form of axial elongation of the drop, particularly for larger drops. Once the transition to a cylindrical shape occurred, increases in the Bond number led to faster rates of elongation with increasing drop size and thicker liquid films between the cylindrical drops and the tube wall. However, as the Bond number exceeded a limiting value, the film thickness reached a plateau value that remained unchanged with further increases in the Bond number. The typical dependence of the steady rise velocity of single drops on drop size is shown in Figure 2, where the terminal velocity U is made dimensionless with characteristic velocity (1 - γ)FgR2/µ. The curves in this figure represent best fits to the experimental data, taking into account the fact that U must vanish as the drop size tends to zero. The terminal velocity initially increased with drop size and achieved a local maximum at an intermediate value of κ ) 0.5, as the larger retarding effect of the tube wall began to overwhelm the increase in the buoyancy force with increasing drop size. The terminal velocity eventually reached a plateau and became independent of drop


Figure 2. Dependence of the dimensionless rise velocity of single drops on their dimensionless size (κ) for systems with Bo = 4.

size for κ > 0.8. The mobility of large drops was enhanced with increasing Bond number due to the formation of more elongated drop shapes. The resulting increase in the thickness of the liquid film between the drop and the tube wall reduced the retarding effect of the wall. The mobility of viscous drops was reduced as the viscosity ratio was increased. However, there was a limiting value of viscosity ratio beyond which drop mobility became independent of the viscosity ratio. The relative reduction in the terminal velocity from the maximum to the plateau value also became more pronounced as the viscosity ratio increased (see Figure 2). This had a significant effect on the coalescence behavior of drops, as will be discussed later. A more detailed description of single-particle motion in a tube is provided by Borhan and Pallinti.12,13 To examine the coalescence behavior of two drops or bubbles, various combinations of leading and trailing drop sizes had to be specifically chosen to ensure drop-drop interaction within the test section. In all of the experiments described here, the size of the leading drop was fixed, and trailing drops of different sizes were used to vary the drop size ratio (R). When the separation distance between the leading and trailing drops was greater than a tube diameter, the two drops translated and deformed independent of each other. In this configuration, each drop achieved a steady axisymmetric shape which was the same as that observed for a single drop of the same size. The measured velocities of the leading and trailing drops (denoted by U1 and U2, respectively) were also identical to those measured in the single-drop experiments. Therefore, the initial conditions associated with the injection of drops into the tube did not play a significant role in the coalescence experiments. In addition, the results of single-drop experiments could be used to select the trailing drop size (κ2) in each coalescence experiment such that the trailing drop had a larger rise velocity than the leading one, thereby allowing it to catch up with the leading drop. To maximize the range of values of R that could be studied experimentally, the size of the leading drop was fixed at a value of κ1 ) 0.85. This value of κ1 corresponded to the point at which the plateau region in the single-drop velocity first appeared (see Figure 2). As such, contact between the two drops could not be achieved in most cases where the trailing drop was larger than the leading one (i.e., for R < 1). When the size of the trailing drop was smaller than that of the leading one, but large enough for the single-drop velocity

to remain above the plateau value in Figure 2, the trailing drop approached the leading drop and appeared to collide with it in an axisymmetric configuration to form a doublet. In this configuration, the two drops were separated from each other by a thin film of the suspending fluid. There was a slight change in the shape of the trailing drop as the separation distance between the two drops decreased below a tube diameter and hydrodynamic interactions between them became significant. In most cases, this was accompanied by a measurable change in the translational velocity of the trailing drop. On the other hand, the shape and velocity of the leading drop remained nearly unchanged throughout the interaction process, all the way to the point of apparent contact between drops. This is not a surprising result considering that the migration velocity is independent of drop size for single drops larger than the leading drop, because of the plateau in the migration velocity of large single drops beyond κ ) 0.8. Similar observations have been reported by Olbricht and Kung9 in the case of flow-driven interaction of two drops. The hydrodynamic interaction between the two drops was characterized experimentally by monitoring the separation distance between them as a function of time, starting from an arbitrary instant at which the separation distance was larger than a tube diameter to the instant of apparent contact between drops. The velocity of the trailing drop was found to increase as it approached the leading drop, due to the influence of the disturbance flow behind the leading drop. In some cases, when the size of the trailing drop became very small, the trailing drop migrated from the tube centerline toward the wall as it came into apparent contact with the leading drop, thereby leading to the formation of a non-axisymmetric doublet. The two drops remained in a doublet configuration with no significant change in their shapes until they either coalesced into a single drop as a result of film rupture or left the test section of the tube, depending on the rate of film drainage. The coalescence behavior of drops was characterized by measuring a coalescence time (τ) defined as the elapsed time between the instants of apparent contact and coalescence. Following Olbricht and Kung,9 the instant of apparent contact between drops was used as the reference point in time for measuring the characteristic time scale for coalescence because it could be identified unambiguously in the experiments. In all of the experiments reported here, the time between apparent contact and coalescence was much larger than the time required for the interparticle distance to decrease from one tube diameter to its value at the instant of apparent contact. Hence, the choice of apparent contact as the reference configuration for determining the value of τ did not have a significant effect on measurements of the coalescence time scale. Experimental measurements of the coalescence time as a function of drop size ratio R are shown in Figures 3-7. The coalescence times reported in these figures are made dimensionless with the characteristic time R/U1. Hence, the value of τ represents the approximate number of tube radii the doublet translated before coalescence occurred. The coalescence time for two air bubbles (shown in Figure 3) exhibits a dramatic increase as the size ratio R exceeds a value of 2. When the volume of the trailing bubble was made sufficiently small, the hydrodynamic interaction between the bubbles caused the trailing bubble to migrate from the tube centerline toward the tube wall. The doublet then maintained a non-axisymmetric configuration as it traveled through the tube until coalescence occurred. The rapid increase in the coalescence time for R > 2 in Figure 3 is due to the onset of coalescence in the non-

402


Figure 3. Dependence of the dimensionless coalescence time (τ) and impact velocity (∆Ui) on drop size ratio for two air bubbles with Bo ) 4.1. Solid circles correspond to coalescence in the non-axisymmetric configuration.

Figure 4. Dependence of the dimensionless coalescence time (τ) and impact velocity (∆Ui) on drop size ratio for two viscous drops with λ ) 0.06 and Bo ) 4.6.

axisymmetric configuration, as denoted by the solid circles in this figure. Although a collision between the two bubbles was always observed in the experiments with R > 1, such collisions did not always lead to bubble coalescence. When the trailing bubble size was reduced below a threshold value, the two bubbles travelled the entire length of the tube in contact with each other without coalescing. The corresponding results for buoyancy-driven coalescence of viscous drops are shown in Figures 4-7 for different viscosity ratios. The dependence of the coalescence time on drop size ratio was qualitatively similar to that observed for air bubbles. Namely, as the volume of the trailing drop was reduced (for fixed leading drop size), the coalescence time developed a minimum in the approximate range 1.5 < R < 2.5. Since the development of the non-axisymmetric doublet is mainly due to the influence of the disturbance flow behind the leading drop, and the strength of the disturbance flow is dependent on the viscosity of the drop phase, the viscosity ratio can be expected to affect the onset of non-axisymmetric coalescence. Comparison of the solid symbols in Figures 3-7 shows that the onset of coalescence in the non-axisymmetric configuration did indeed

Figure 5. Dependence of the dimensionless coalescence time (τ) and impact velocity (∆Ui) on drop size ratio for two viscous drops with λ ) 0.23 and Bo ) 4.3. Solid circles correspond to coalescence in the non-axisymmetric configuration.

Figure 6. Dependence of the dimensionless coalescence time (τ) and impact velocity (∆Ui) on drop size ratio for two viscous drops with λ ) 0.63 and Bo ) 3.6. Solid circles correspond to coalescence in the non-axisymmetric configuration.

occur at larger values of R for viscous drops compared to air bubbles. For example, in the experiments with the UCON-65 drops (Figure 4), the doublet maintained an axisymmetric configuration for all values of R < 2.6 considered in the experiments. Similarly, non-axisymmetric coalescence was first observed for R ) 2.6 in experiments with the UCON-625 drops (Figure 7), as compared to a value of R ) 2.0 for air bubbles. The experimental data in Figures 4-7 show a dramatic increase in the coalescence time as the volume of the trailing drop approaches that of the leading drop (i.e., as R f 1). Although the experimental data shown in Figure 3 for air bubbles seem to approach a plateau as the value of R is reduced to about 1.1, the coalescence time did in fact exhibit a rapid increase as the value of R was further reduced toward 1. This is indicated by the dashed line, which represents the best fit to all of the experimental data, including those not shown on the scale of Figure 3. In the limit R f 1, the difference in the singledrop velocities of the leading and trailing drops (∆U ≡ U2 U1) tended to zero. Since the interaction between drops was


interfaces, the attractive van der Waals force produces a negative disjoining pressure (P*v) in the thin film, which can be written as14

P*v )

Figure 7. Dependence of the dimensionless coalescence time (τ) and impact velocity (∆Ui) on drop size ratio for two viscous drops with λ ) 1.19 and Bo ) 3.4. Solid circles correspond to coalescence in the nonaxisymmetric configuration.

driven by the difference in their velocities, it is not surprising that the coalescence time increased significantly as ∆U tended to zero. Consequently, the finite length of the tube imposed a lower bound on the value of ∆U below which coalescence did not occur during the finite time it took the doublet to reach the tube exit. Also shown in Figures 3-7 are measurements of the impact velocity (∆Ui), defined as the relative velocity between the two drops at the instant of apparent contact. The impact velocity was obtained from the time evolution of the interparticle distance captured through image analysis and was made dimensionless with the single-drop rise velocity of the leading drop. The impact velocity exhibited a local maximum in the approximate range 1.5 < R < 2.5, with the location of the maximum shifted to larger values of R for viscous drops. For the system with the highest viscosity ratio (λ ) 1.19), the coalescence time formed a broad minimum as the impact velocity between the two drops reached a maximum. For lower viscosity ratio systems, however, an inverse correlation between the impact velocity and coalescence time was only observed when the leading and trailing drops were of comparable size, in the sense that a significant increase in the impact velocity led to a much smaller coalescence time scale for R < 1.6. Experimental measurements of coalescence time can be compared with the predictions of a simple film-drainage model similar to that used by Olbricht and Kung.9 The model is based on lubrication flow between two parallel disks squeezed together by an applied force. Using lubrication analysis, the rate of thinning of the liquid film between the two interfaces can be written in dimensionless form as

( )

dh 2h3 F )dt 3πRf4

(1)

where h(t) and Rf represent the film thickness and the film radius (both made dimensionless with R), respectively, and the driving force (F) is made dimensionless with ∆FgR3. For small drop separations, the total applied force can be viewed as the sum of a contribution arising from the hydrodynamic interaction between drops and the van der Waals force that is attractive between two drops of the same fluid. For planar fluid-fluid

A* h*m

(2)

where the asterisk denotes dimensional quantities. When the liquid film thickness is less than about 12-15 nm, the powerlaw index m and the Hamaker constant A* take on values of 3 and 10-20 to 10-21 J, respectively. When the separation distance between the interacting interfaces exceeds a few hundred angstroms, a change in the nature of van der Waals forces leads to the new constant values m ) 4 and A* ) 10-27 to 10-28 Nm2.15 The transition in the van der Waals force involves a complicated dependence of the force on the film thickness, which is asymptotic to h-4 at large separations and to h-3 at relatively small separations. Since the film thickness at the point of rupture is expected to be on the order of 10 nm, we use the value of m ) 4 for the entire film-drainage process. Accounting for the negative disjoining pressure in the liquid film, the total force exerted on the film can be approximated as

F ) 6πκ2 f (κ2)∆U + πARf2h-4

(3)

where all quantities are dimensionless, and the dimensionless Hamaker constant is defined as A ) A*/∆FgR5. The first term on the right-hand side of eq 3 represents the hydrodynamic force acting to squeeze the thin film, which is estimated to be the force required to change the velocity of the trailing drop from U2 (its value before interacting with the leading drop) to the measured velocity of the doublet. The latter was found to be the same as the single-drop rise velocity of the leading drop (U1) in all cases. The function f (κ) is the wall correction factor for the buoyancy-driven motion of a single drop of dimensionless size κ in the tube. It can be determined from the following expression:

f (κ) )

2(λ + 1)κ2 3(3λ + 2)U(κ)

(4)

where U(κ) is given by the experimental measurements reported in Figure 2 for the mobility of single drops in the tube. The typical dependence of the wall correction factor on the dimensionless drop size is shown in Figure 8 for drops with O(1) viscosity ratio. The dashed line in this figure represents the theoretical prediction of Haberman and Sayre16 for buoyancydriven motion of spherical drops in a tube. There is good agreement between the experimental measurements and the theoretical predictions for κ e 0.8. For larger values of κ, however, significant deformations of the drop shape lead to large deviations of the experimental measurements from theoretical predictions (which do not account for shape deformations). Substituting eq 3 into eq 1, assuming constant film radius throughout the drainage process, and integrating from some initial film thickness (hi) to the film thickness at rupture (hr) leads to the following estimate for the dimensionless coalescence time:

3 τ ) [6Aκ2 f (κ2)∆U]-1/2Rf3 4

{

tan-1 [6Aκ2 f (κ2)∆U]-1/2ARf

(

)}

1 1 - 2 2 hr hi

(5)

While the value of the film thickness at rupture can be estimated

404


Figure 8. Wall correction factor for the rise of a single viscous drop of dimensionless size (κ) with λ ) 1.19 and Bo ) 3.4. The dashed line corresponds to the theoretical predictions of Haberman and Sayre16 for the motion of spherical drops.

Figure 9. Dependence of the dimensionless film radius on drop size ratio for κ1 = 0.85.

as being on the order of 10 nm, the value of hi remains unknown since we have no way of estimating the thickness of the liquid film at the instant of apparent contact between drops. However, the only term in eq 5 that depends on hi is the one inside the curly brackets on the right-hand side of this equation. Assuming the value of hi to be independent of drop size ratio (R), it turns out that this term is relatively insensitive to the size of the trailing drop; it varies by less than 10% over the entire range of R values considered in the experiments. Treating this term as a constant for experiments in a fixed two-phase system, the dimensionless drainage time is predicted to vary with the size of the trailing drop according to [κ2 f (κ2)∆U]-1/2Rf 3. The dependence of the relative velocity ∆U on the size of the trailing drop is known from the single-drop experiments (see Figure 9). Hence, it will be possible to compare the experimentally observed dependence of the coalescence time on drop size ratio to that predicted by the film-drainage model, provided that the dependence of the film radius Rf on R can be identified from experimental measurements. The film radius depends on the size of the two drops, their deformability, and the strength of the hydrodynamic force pressing them together. For each experiment, the film radius

Figure 10. Effect of viscosity ratio on the relative mobility of the trailing drop for κ1 = 0.85.

was estimated by analyzing digitized images of the doublet. Measurements of the film radius at various times during the film-drainage process indicated that the film radius remained nearly unchanged from the instant of apparent contact to coalescence. Measured values of the dimensionless film radius are shown in Figure 10 for both air bubbles and viscous drops. In the case of air bubbles, the film radius was a monotonically increasing function of trailing bubble size and approached a constant value as the value of R was reduced to about 1.5. For viscous drops, on the other hand, the film radius achieved a local maximum in the vicinity of R = 1.4 (corresponding to a trailing drop size of κ2 = 0.6). Increasing the trailing drop size beyond this value led to a significant decrease in the film radius, with the reduction being most pronounced for larger viscosity ratios. The observed decrease in the film radius was due to the larger deformation of the trailing drop in the wake of the leading drop as its size became comparable to the tube diameter. As was described earlier in the discussion of single drop motion, deformed air bubbles tended to maintain their fore and aft symmetry, whereas viscous drops achieved a more tapered steady shape with a larger curvature at the leading end. As the trailing drop size increased beyond κ2 = 0.6, the curvature of the leading end of the deformed drop increased, leading to the development of a smaller film as it came into apparent contact with the leading drop. A similar reduction in the film radius was not observed in the case of air bubbles because large trailing bubbles maintained their fore and aft symmetry as they deformed. The eventual reduction in the film radius with increasing size of the trailing drop became more pronounced as the viscosity ratio increased (see Figure 10). This is consistent with our earlier observation in the single-drop experiments that drops with larger viscosity ratios exhibited greater deformations. Figure 10 also shows that in general the observed film radius was larger in systems with higher viscosity ratios. This was the case even for small trailing drops which exhibited negligibly small shape deformations in the single-drop experiments. Increasing the viscosity ratio led to a larger relative velocity between the small trailing drops and the leading drop, as shown in Figure 9. The larger relative velocity produced a stronger hydrodynamic force pressing the two drops together, thereby resulting in the flattening of the leading end of the trailing drop and the development of a larger liquid film between the two drops. For the O(1) viscosity-ratio systems considered, the large relative velocity between the leading and trailing drops also


Figure 11. Planar and nonplanar thin films formed between two coalescing drops with small and O(1) viscosity ratios, respectively; (a) λ ) 0.06 and (b) λ ) 1.19.

Figure 12. Comparison of the experimental measurements of coalescence time with theoretical predictions based on the film-drainage model for viscous drops. The solid and dotted lines represent the model predictions for λ ) 1.19 and λ ) 0.06, respectively.

caused the trailing drop to penetrate the trailing end of the leading drop at the instant of initial contact (see Figure 11b). This led to the development of thin liquid films that were not planar. Model predictions for the R-dependence of the drainage time (for fixed leading drop size) are compared to the experimental measurements of the coalescence time in Figure 12, for the lowest and highest viscosity ratio drops considered in the experiments. The film-drainage model predicts a reduction in the coalescence time as the size of the trailing drop decreases (i.e., with increasing values of R). For R > 1.4, this is mainly due to the reduction in the film radius with decreasing size of the trailing drop. Although the magnitude of the hydrodynamic force squeezing the two drops together also decreases with increasing values of R (and that retards the drainage process), the reduction in the film radius has a more dominant effect on the drainage time for R > 1.4. For 1.0 < R < 1.4, the dependence of the film radius on R is reversed. However,

drainage time still remains a decreasing function of R because of the more dominant influence of the hydrodynamic force in this range as it tends to zero for R f 1. The qualitative agreement between model predictions and experimental measurements is satisfactory for axisymmetric coalescence in the low viscosity ratio system. For the system with λ ) 1.19, on the other hand, model predictions are qualitatively different from experimental measurements, which indicate the existence of a minimum in the coalescence time. This could be partly due to the qualitatively different configuration of the liquid film in the O(1) viscosity ratio systems as compared to that observed in the low viscosity ratio systems. For the system with λ ) 1.19, the trailing drop penetrated the trailing end of the leading drop to form a nonplanar liquid film, as shown in Figure 11b. Hence, a parallel disk model for film drainage does not provide a good representation of the true geometry of the liquid film in this case. In the low viscosity ratio system, on the other hand, the reduced relative velocity of the trailing drop resulted in the formation of a more or less planar liquid film similar to the one used in the model. Conclusions Buoyancy-driven interaction of two drops or bubbles rising in an axisymmetric configuration within a vertical tube leads to the formation of a doublet, followed by coalescence of the two drops in the axisymmetric configuration in most cases. When the trailing drop is very small as compared to the leading drop, it migrates toward the tube wall and coalescence occurs in a non-axisymmetric configuration. The time scale for coalescence in the non-axisymmetric configuration is typically much larger than that for coalescence in the axisymmetric configuration. In the case of two interacting air bubbles, the coalescence time for the non-axisymmetric configuration becomes so large (as the trailing bubble size is reduced) that coalescence does not occur before the doublet leaves the tube. For axisymmetric coalescence of low viscosity ratio drops, the coalescence time is found to be a monotonically increasing function of trailing drop size. The observed trend is qualitatively similar to that reported by Aul and Olbricht10 for flow-driven coalescence of drops in microscale capillary tubes, but opposite to the findings of Olbricht and Kung.9 For systems with O(1) viscosity ratio, the coalescence time appears to form a broad minimum as the impact velocity between the two drops reaches a maximum. For low viscosity ratio systems, the dependence of the coalescence time on trailing drop size is qualitatively similar to that predicted by a planar film-drainage model. For O(1) viscosity ratios, however, the liquid film between drops is not planar, and the planar film-drainage model fails to capture the dependence of the coalescence time on trailing drop size. Acknowledgment This work was partially supported by NASA Grant NAG31916. Note Added after ASAP Publication. The captions for Figures 8-10 of the version published on the Web 12/8/05 were incorrect. The correct captions are shown in the version published 12/13/05. Literature Cited (1) Chaudhari, R. V.; Hofmann, H. Coalescence of gas bubbles in liquids. ReV. Chem. Eng. 1994, 2, 131. (2) Cristini, V.; Blawzdziewicz, J.; Loewenberg, M. Near-contact motion of surfactant-covered spherical drops. J. Fluid Mech. 1998, 366, 259.

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(3) Blawzdziewicz, J.; Wajnryb, E.; Loewenberg, M. Hydrodynamic interactions and collision efficiencies of spherical drops covered with an incompressible surfactant film. J. Fluid Mech. 1999, 395, 29. (4) Crabtree, J. R.; Bridgewater, J. Bubble coalescence in viscous liquids. Chem. Eng. Sci. 1971, 26, 839. (5) Narayanan, S.; Goossens, L. H. J.; Kossen, N. W. F. Coalescence of two bubbles rising in line at low Reynolds numbers. Chem. Eng. Sci. 1974, 29, 2071. (6) Bhaga, D.; Weber, M. E. In-line interaction of a pair of bubbles in a viscous liquid. Chem. Eng. Sci. 1980, 35, 2467. (7) Manga, M.; Stone, H. A. Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 1993, 256, 647. (8) Zhang, X.; Davis, R. H.; Ruth, M. F. Experimental study of two interacting drops in an immiscible fluid. J. Fluid Mech. 1993, 249, 227. (9) Olbricht, W. L.; Kung, D. M. The interaction and coalescence of liquid drops in flow through a capillary tube. J. Colloid Interface Sci. 1987, 120, 229. (10) Aul, R. W.; Olbricht, W. L. Coalescence of freely suspended liquid drops in flow through a small pore. J. Colloid Interface Sci. 1991, 145, 2. (11) Borhan, A.; Pallinti, J. Breakup of drops and bubbles translating through cylindrical capillaries. Phys. Fluids 1999, 11 (10), 2846.

(12) Borhan, A.; Pallinti, J. Buoyancy-driven motion of viscous drops through cylindrical capillaries at small Reynolds number. Ind. Eng. Chem. Res. 1995, 34, 2750. (13) Borhan, A.; Pallinti, J. Pressure-driven motion of drops and bubbles through cylindrical capillaries: effect of buoyancy. Ind. Eng. Chem. Res. 1996, 37, 3748. (14) Ruckenstein, E.; Jain, R. K. Spontaneous rupture of thin liquid film. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132. (15) Black, W.; de Jongh, J. G. V.; Overbeek, J. Th. G.; Sparnaay, M. J. Measurements of retarded van der Waals forces. Trans. Faraday Soc. 1960, 56, 1597. (16) Haberman, W. L.; Sayre, R. M. Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David W. Taylor Model Basin Report 1143; U.S. Navy Department: 1958.

ReceiVed for reView May 13, 2005 ReVised manuscript receiVed September 25, 2005 Accepted October 12, 2005 IE0505615

Interaction and Coalescence of Drops and Bubbles Rising through a

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