Analysis of Bubble Coalescence Dynamics and Postrupture

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Cite This: Ind. Eng. Chem. Res. 2017, 56, 14781−14792

Analysis of Bubble Coalescence Dynamics and Postrupture Oscillation of Capillary-Held Bubbles in Water Yesenia Saavedra Moreno, Ghislain Bournival, and Seher Ata* School of Mining Engineering, University of New South Wales, Sydney, New South Wales 2052, Australia S Supporting Information *

ABSTRACT: This paper presents a three-dimensional computational study of coalescence dynamics of two capillary-held air bubbles in water using the volume of fluid (VOF) method. The interface motion of the newly formed bubble indicated that smaller initial separation distances resulted in a slightly faster expansion of the neck. The velocity vectors showed that the inward motion of the air at the contact point of the two bubbles favors the generation of small eddies at the top and bottom part of the neck at the early stage of coalescence and bulges at the edges of the bubble at a later stage. The release of free-surface energy drove the bubble contraction and expansion and consequently the oscillation motion, which created differences in pressure across the bubble interface. The computationally simulated bubbles followed the oscillatory motion observed experimentally, but with a lower damping constant and lower angular frequency. occurred at greater approach speed (150 μm/s). However, they found that high concentration of NaCl electrolyte increased film stability by immobilizing the interface and therefore inhibiting coalescence. For nonionic surfactants, it is believed that the adsorption at the liquid−air interface is mainly driven by the hydrophobic forces.8,9 The amount of surfactant adsorbed affects the surface properties of the interface, causing the Gibbs−Marangoni phenomena, which is commonly used to explain the effect of surface tension on bubble coalescence. However, surface forces have also been recognized as an important contributor in preventing film rupture in some nonionic surfactant systems.10 Bournival et al.11 observed a direct relationship between increasing surfactant concentration and coalescence time until a maximum coalescence time was reached. However, all surfactants investigated had a maximum coalescence time until no change with increasing surfactant concentration was observed. High-speed cameras and laser technologies are major optical technique developments which have been used as tools to improve understanding of bubble coalescence dynamics. Thoroddsen et al.12 investigated the motion of the neck during the merging of two air bubbles of approximately 2.4 mm in average diameter using a high speed imaging technique with a capture rate of 1 × 106 frames per second (fps). The frame sequences taken allowed the authors to analyze and quantify

1. INTRODUCTION The coalescence of air bubbles is an important phenomenon in a number of industrial processes that use bubbling in a liquid flow such as in froth flotation, wastewater treatment, and paper recycling. Bubble coalescence is initiated by the thinning of the liquid separating the two bubbles followed by a rupture of the liquid film.1 Once coalescence occurs, a neck is formed joining the two bubbles. As the neck opens, the surface area and the curvature of the two initial bubbles decrease. The newly formed bubble expands horizontally and contracts vertically and vice versa until a stable spherical shape is achieved. The release of free-surface energy due to this coalescence process is imparted to the surrounding liquid as kinetic energy.2 The effects of inorganic electrolytes and surfactants in inhibiting bubble coalescence have been widely studied experimentally and theoretically. Many dissolved salts have a critical concentration beyond which the likelihood of bubble coalescence is significantly reduced.3−6 Lessard and Zieminski3 introduced a transition concentration for each electrolyte studied where the probability of coalescence was 50%, using 100% as the coalescence of a bubble pair in pure water. They found that electrolytes with valence combinations of 3−1 and 2−2 were the most effective electrolytes with transition concentrations of 0.032 and 0.035 M, respectively. These findings were corroborated by Craig et al.,4 who observed that highly charged salts were more effective as bubble coalescence inhibitors at lower transition concentrations. More recently, Yaminsky et al.7 reported that the speed at which the bubbles were brought together played an important role in bubble coalescence, finding that instantaneous coalescence of bubbles © 2017 American Chemical Society

Received: Revised: Accepted: Published: 14781

August 2, 2017 October 8, 2017 November 21, 2017 November 21, 2017 DOI: 10.1021/acs.iecr.7b03197 Ind. Eng. Chem. Res. 2017, 56, 14781−14792

Article

Industrial & Engineering Chemistry Research

These studies show that no research has numerically investigated the oscillation following the merging of two captive bubbles. The present study provides insight into bubble coalescence dynamics by mapping the flow velocity vectors inside the bubble and in the surrounding liquid, and local dynamic pressure across the bubble interface. This objective was achieved by modeling two air bubbles fixed to adjacent capillaries in three dimensions using the VOF method. The developed model was validated by comparing the overall oscillatory motion of the bubbles with experimental data produced in a well-controlled system. The analysis provided further understanding of postrupture bubble oscillation and the fluid dynamics properties regulating the coalescence process.

the motion of the neck and bubble surface shape. More recently, Bournival et al.11 used a high speed camera to study the coalescence dynamics of bubbles in nonionic surfactant solutions. The authors generated two bubbles attached to the tips of a pair of fine capillaries to examine the relationship between coalescence time, defined as the time interval from the moment at which the bubbles are in contact to the rupture of the liquid film separating the two bubbles, and surfactant concentration. Coalescence and oscillation of the entire bubble were captured and analyzed by tracking the newly formed bubble’s change in projected surface area over time. A direct relationship between increasing surfactant concentration and coalescence time was observed and the postrupture dampening of the oscillation of the newly formed bubble increased with surfactant concentration. The study suggested that surface elasticity partly dampened the oscillation of the newly formed bubble while other factors like surface viscosity may contribute to the dissipation of energy. The experimental work provided an analysis of the surface area of the oscillated bubble in two dimensions, quantifying the expansion and contraction of the coalesced bubble as a projected area. However, there are still questions on how fluid dynamics properties, such as flow velocity and dynamic pressure, control the bubble coalescence and postrupture oscillation. The objective of the paper is to explain bubble coalescence dynamics by characterizing the flow velocity field vectors and local dynamic pressure inside the bubble and in the surrounding liquid across the bubble interface using computer simulation. Several numerical investigations have used computational fluid dynamics (CFD) as a tool for studying multiphase flows.13 Most of these studies use multiphase methods such as volume of fluid (VOF) method,14−22 front tracking method,23,24 and level set method.25−27 The VOF method, first introduced by Hirt and Nichols,28 tracks the liquid−air interface by defining a fractional volume of each fluid phase in a specific computational cell. Using this method, Tomiyama et al.14 investigated the effect of fluid properties and surface tension on the motion of a single bubble rising in a laminar flow. The computational results were in reasonable agreement with the experimental data suggesting that VOF may predict the qualitative and quantitative behavior of bubble motion under different conditions. Nevertheless, further development of the numerical model to consider a larger computational domain and a threedimensional model is required for a precise quantitative prediction of the bubble rising motion. The study of coalescence dynamics of a bubble pair rising freely in a liquid phase using the VOF method has been well documented.15,16,18 Hasan and Zakaria18 developed a twodimensional model of two coaxial free bubbles rising in a liquid phase and used the VOF method to track the motion, shape, and velocity of the two bubbles during the interaction and coalescence processes. Although the two-dimensional numerical study did not fully represent the experimental conditions, the findings corroborated the experimental data available in the literature such as Chen et al.15 The study of Hasan and Zakaria18 showed that the VOF method may be considered an accurate tool for modeling bubble coalescence time and the subsequent oscillation behavior. Some authors have carried out qualitative analysis of the interaction of a pair of bubbles rising side by side.17,19,29,30 Such computational studies were intended only to investigate the effect of approach and rise velocity on bubble coalescence, considering liquid film drainage before the film rupture.

2. METHODOLOGY 2.1. Computational Solution Method. The VOF method developed by Hirt and Nichols28 was used to simulate the coalescence of two air bubbles fixed to adjacent capillaries in water. The computational model was implemented in the commercial code ANSYS-Fluent v 17.2. 2.1.1. Governing Equations. In the VOF method, both phases were assumed to be insoluble and all computational cells in the domain were either occupied by the water or air phase or a combination of both.28 The water phase was defined as the primary phase and the air phase as the secondary phase. A VOF function F was defined to track the fractional volume of a particular fluid in any specific cell of the computational domain at time t. If the cell was completely empty of air, F = 0; if the cell was completely full of air phase, F = 1; and if the cell contained the interface between the water and air phase, 0 < F < 1. Assuming that the mass of the fluid was preserved, the VOF method tracked the water−air interface by solving a continuity equation for the secondary phase (air) given by (1)

∇·V = 0

where V is the velocity vector of the air phase in the entire domain. A momentum equation was solved for both water and air phases by the following equation: ∂ρ V + ∇ (ρ V · V ) ∂t = − ∇p + ρ g + ∇·μ(∇V + ∇VT) + Fσ v

(2)

where ρ and t represent the density and time; p, g, μ, Fσv, are pressure, gravity force, viscosity, and surface tension force per unit volume, respectively; and T is the matrix transpose operator. The fluid properties, ρ and μ, were determined by the properties of each phase as ρ = Fairρair + (1 − Fair)ρwater

(3)

μ = Fairμair + (1 − Fair)μwater

(4)

The surface tension force per unit volume (Fσv) was calculated by adopting the continuous surface force model developed by Brackbill et al.31 and assuming a constant surface tension: Fσ v = σk n

(5)

In eq 5, σ is the surface tension, k is the surface curvature, and n is the normal vector to the interface. The surface curvature of the interface, k, was calculated by 14782

DOI: 10.1021/acs.iecr.7b03197 Ind. Eng. Chem. Res. 2017, 56, 14781−14792

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Industrial & Engineering Chemistry Research k=

⎤ 1 ⎡⎛⎜ n ⎞⎟ ⎢⎝ ∇ ⎠ | n | − ( ∇ n )⎥ ⎦ |n | ⎣ |n |

mesh refinement was implemented by increasing the number of cells per average diameter of the bubbles to improve the resolution of the interface curvature around the bubbles and reduce the computational error of the interface tracking. The mesh refinement was created in ANSYS-Fluent by marking a hexahedron with dimensions of 2.5Db,ave × 2.5Db,ave × 2.5Db,ave, as depicted in Figure 2a. A case with 23 cells per 2 mm was compared against two cases with mesh refinement of 46 and 92 cells per 2 mm. The mesh independence solution was evaluated by extracting the mean velocity profiles at the cross section plane. A small difference of 4.1% was observed in the mean velocity for the finest mesh (92 cells per 2 mm) with respect to the case with no mesh refinement (23 cells per 2 mm), indicating a mesh independent solution. Consequently, for computational efficiency and numerical accuracy, a mesh density of 92 cells per 2 mm was selected for all subsequent analysis in this study. A high level of grid quality throughout the domain was achieved where the aspect ratio of the cells was below 5.2 and the skewness was 0.51.37 Time-step sizes below 1.5 × 10−6 s were selected to satisfy a courant number (CN = |V|Δt/ΔX) below 0.6. This dimensionless number quantifies the velocity magnitude at a specific cell and the ratio of the time-step size to the element size. To compare the computational simulations with the experimental results, a simulation time of 0.025 s was chosen. 2.2. Computational Parameters in ANSYS-Fluent. Two air bubbles, each with 2 mm diameter, were patched into the domain and placed at the tips of the capillary tubes. The other ends of the capillary tubes were closed to the atmosphere. The immersion depth of the capillary tubes in water was 20 mm and the air phase was patched into the capillary tubes. A contact angle (θeq) of 160° between the capillary tube and the water− air interface was defined. It was measured inside the air phase and calculated from the experimental images by using an image processing technique (ImageJ). The rectangular structure was open to the atmosphere by a pressure outlet set at the top of the rectangular structure as shown in Figure 2b. Both water and air phases were assumed to be Newtonian fluids and incompressible flow, which, by definition, assumed the speed of the flow was significantly lower than the speed of sound.38 Previous computational studies on bubble coalescence have shown that these assumptions are valid for simulating the motion of coalesced bubbles.18,27,29,30 The flow was considered laminar and unsteady. The physical properties for the water and air phases specified in the computational simulations are shown in Table 1. 2.2.1. Solution Methods. A pressure-base solver and a time dependent solution were chosen as a solver approach for the model. The pressure implicit with splitting operator (PISO) was applied as the pressure velocity coupling in eq 2. A georeconstruction scheme was selected for tracking the water−air interface as it is the most precise reconstruction scheme available in ANSYS-Fluent.40 This scheme is based on a piecewise linear approximation of the water−air interface at a specific cell by a plane in three dimensions.41 All the parameters used in ANSYS-Fluent are shown in Table 2. 2.2.2. VOF Computational Cases. The liquid film thickness between two bubbles at the rupture could not be measured in the experimental case. Thus, two computational cases with initial separation distances (h) between the two bubbles of 0.05 and 0.02 mm were generated. These separation distances were in the range of 0.1 mm, which is the film thickness of the initial liquid drainage estimated by Kirkpatrick and Lockett,42 and

(6)

The normal vector, n, was evaluated by the following equation, which took into account the three-phase contact perimeter formed between the bubble interface (i.e., water and air) and the capillary tube, known as the wall adhesion effect: n = n wall cos θeq + t wall sin θeq

(7)

where nwall and twall are the unit vectors normal and tangent to the capillary tube respectively, and θeq is the equilibrium contact angle between the water−air interface and the capillary tube. 2.1.2. Geometry and Mesh Design. A rectangular structure of dimensions 18 mm × 30 mm × 8 mm with two cylindrical capillaries of 20 mm long was created in ANSYS-Design modeler. The geometry used was sufficiently large with the ratio of the average diameter of the two initial bubbles (Db,ave) to the domain diameter being less than about 0.25 to eliminate any wall effects on the bubble coalescence dynamics.32 A schematic representation of the geometry used in ANSYSFluent is presented in Figure 1. The two vertical capillaries were

Figure 1. Schematic representation of rectangular structure with two cylindrical capillary tubes for the 3D computational simulations.

chosen to have an outer diameter of 1.07 mm and inner diameter of 0.69 mm to replicate the experimental conditions of Bournival et al.,11 which for experimental purposes offered a well-controlled experimental system, where the neck expansion and the bulging of the extremities of bubbles were not as constrained by the capillary tubes as other configurations. This capillary arrangement also allowed the (almost free) surface motion of the newly formed bubble, minimizing the effect of the capillary tubes on the horizontal expansion of the coalesced bubbles. Practically, the vertical capillaries effectively allowed the coating of the bubbles with hydrophobic particles. Thus, that configuration offered practical advantages in a stirred system and for the packing of particles on the surface of the bubbles.33−36 The domain was discretized in ANSYS-Meshing using a multizone mesh method to generate hexahedral meshes. A local 14783

DOI: 10.1021/acs.iecr.7b03197 Ind. Eng. Chem. Res. 2017, 56, 14781−14792

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Figure 2. (a) Cross-sectional view of the region prior to mesh refinement and after the first and second mesh refinement. (b) Boundary conditions used in the computational domain.

all experiments to evaluate cleanliness of the system. The bubbles were left aging for 90 s in Milli-Q water (resistivity of 18.2 MΩ cm) before being brought in contact. The system was deemed cleaned if the bubbles coalesced within one frame interval (i.e., between two frames). The details of the cleaning procedure may be found in the work of Bournival et al.11 All the experiments were conducted at a temperature of 20 ± 2 °C, which was maintained during all experiments. Five independent trials were run for replicating the same experimental conditions. The authors used a high speed camera (Phantom 5, Vision Research Inc., USA) to study the coalescence dynamics of the bubbles with a capture rate of 6024 fps. The analysis of the oscillation was performed as described in section 3.1.2.

Table 1. Physical Properties of Water and Air Phases Used in the Computational Simulations physical property

water phase

air phase

fluid density (ρ), kg/m3 dynamic viscosity (μ), mPa s surface tension (σ), mN/m temperature (T), °C

998.239 1.00239 72.811 2011

1.20439 1.805 × 10−2 39 2011

Table 2. Parameters Used in ANSYS-Fluent parameter solver solution method discretization

setting pressure-base transient pressure velocity coupling: PISO momentum: second-order upwind volume fraction: geo-reconstruction

3. RESULTS AND DISCUSSION 3.1. Analysis of Bubble Coalescence Dynamics. Two capillary-held bubbles were simulated using the VOF method in ANSYS-Fluent to evaluate the bubble coalescence dynamics, similar to the work of Bournival et al.11 The results of the computational study were first used to evaluate the effect of the initial separation distance between the two bubbles on the neck expansion by calculating and comparing the Weber (We), Reynolds (Re), and Ohnesorge (Oh) numbers for both computational cases and the experimental trial. The postrupture oscillation behavior of coalesced bubbles was assessed by measuring the projected surface area of the coalesced bubbles and comparing it to the experimental case. 3.1.1. Neck Expansion during Coalescence. The neck expansion during coalescence can be characterized by the Weber (We), Reynolds (Re), and Ohnesorge (Oh) numbers. The We number quantifies the relative magnitude of the inertial force with the surface tension, as follows:

10−5 mm which is the critical film thickness in water.43 It was not possible to simulate smaller separation distances, as the cell size was 0.02 mm. 2.3. Experimental Data. The experimental work of Bournival et al.11 was used for validating the computational results of the present study. These authors studied the coalescence dynamics of capillary bubbles in nonionic surfactant solutions by evaluating the oscillation of the resultant bubble. They generated two bubbles attached to the tips of a pair of fine, vertical capillaries. The two air bubbles were brought together in a controlled environment to allow coalescence using an electronic linear actuator (T-LA28A, Zaber Technologies Inc.). A resolution time of 3.33 ms, which was the time taken for the two bubbles to coalesce, was set for 14784

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Figure 3. (a) Neck expansion contours as a function of time and (b) comparison of the shape of the two bubbles (black continuous line) prior to coalescence with the bubble shape at t = 0.8 ms (blue dashed line), for the experimental trial conducted by Bournival et al.11

We =

ρ Vr 2λ inertial force = water surface tension force σ

(τ = ρwater R b,ave 3/σ ) which compared the surface tension force per unit area (σ/Rb,ave) with the inertial force per unit area.47 A prefactor, C, was calculated through the best fit of the data to the power-law scaling:

(8)

The Weber number takes into account the velocity of the neck radius (Vr), the density of the water phase (ρwater), the horizontal width of the two coalesced bubbles (λ), which is determined by the neck radius (r) and the surface tension (σ). The Reynold number (Re) describes the ratio of inertial force to viscous force, and it is given by the following equation: ρ Vrλ inertial force = water Re = μwater viscous force

r R b,ave

⎛ t ⎞A = C⎜ ⎟ ⎝τ⎠

(11)

The growth of the neck for the experimental and simulated data was fitted to eq 11 and is plotted in Figure 4. All three

(9)

where μwater is the dynamic viscosity of the water phase. The importance of the surface tension, viscous and inertial forces on the motion of the neck expansion can be measured by Ohnesorge number (Oh), defined as Oh =

We = Re

μwater ρwater σλ

(10)

Frames of the VOF computational cases and experimental trial were produced and then analyzed using ImageJ. The neck expansion of the two coalesced bubbles is illustrated in Figure 3a. The neck grows rapidly in both vertical directions until a maximum neck radius is achieved, at 2.0 ms. Figure 3b illustrates the contour of the bubbles prior to coalescence compared with the bubble shape at t = 0.8 ms. The initial contact point between the two bubbles was established as the origin of the Cartesian coordinates system. The neck radius (r) was measured from the initial contact in a downward vertical direction to eliminate the neck growth restriction due to the capillaries, as shown in Figure 3b. The initial shape of the two bubbles at t = 0 ms was used to measure the horizontal width of the coalesced bubbles (λ). The motion of the neck radius (r) for two liquid drops in an inviscid system has been described as a power-law dependence on time, r ∝ tA, where A is the power exponent.44−46 Studies on bubble coalescence have also reported that the growth of the neck radius follows a power-law dependence on time during the early stage of coalescence.2,12,47 In this study, the neck radius (r) was normalized by the average radius of the two initial bubbles (Rb,ave = (Rb1 + Rb2)/ 2). The time (t) was normalized by the capillary-inertial time

Figure 4. Neck radius (r) normalized by average radius of the two initial bubbles (Rb,ave) as a function of t/τ for (solid red triangle) experimental trial with A = 0.55 and C = 1.10, (solid green square) VOF computational case with h = 0.02 mm, A = 0.40, and C = 1.23, and (solid blue circle) VOF computational case with h = 0.05 mm, A = 0.46, and C = 1.25. The solid lines show the best fit to eq 11. The initial radii of the bubbles for the experimental trial were Rb1 = 0.971 mm and Rb2 = 0.982 mm. Both computational cases had two bubbles of equal radii of 1 mm.

cases followed the power-law scaling with exponents in a range between 0.40 and 0.55 for r/Rb,ave < 0.85. The observed initial neck radius in all three cases deviated slightly, leading to different power-law exponents. However, these exponents were relatively close to 0.5, which is the theoretical power exponent suggested by Eggers et al.44 for bubbles in water. Duchemin et al.46 pointed out that the growth of the neck radius resulting from the coalescence of two identical drops in an inviscid flow had a power-law exponent of 0.5. The experimental and computational results were consistent with the experiments of Thoroddsen et al.12 who also showed an overall agreement with a power-law exponent of 0.5 for the expansion of the neck 14785

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Industrial & Engineering Chemistry Research radius of two bubbles in ethyl alcohol. The prefactors C in the present study were slightly smaller, i.e., 1.10−1.25, than the prefactor observed by Thoroddsen et al.12 who found a value of 1.39 ± 0.03. Therefore, the results presented in Figure 4 not only represented well the motion of the neck radius as a powerlaw dependence but were also in good agreement with the other theoretical and experimental findings. Figure 5 illustrates the horizontal width of the two coalesced bubbles as a function of the normalized neck radius for all three

width between the coalesced bubbles increased and therefore more surrounding liquid was pushed outward. This motion created a surface wave across the newly formed bubble surface. The VOF case captured the bubble surface deformation very well as illustrated in Figure 6. However, the experiments had wider horizontal widths than the computational case as shown graphically in Figure 5. The deviation could be expected due to the piecewise linear scheme used for calculating the interface, which may have compromised the reconstruction of the sharp edges during the neck expansion.48 The neck growth velocity during the early stages of bubble coalescence was calculated from the derivative of the neck expansion equation, Vr = dr/dt, and used to calculate the Weber and Reynolds numbers (eqs 8 and 9). The time at which the rupture of thin liquid film occurred was chosen as time zero. For both computational cases and the experimental trial, the Weber number was compared with the Reynolds number (Figure 7). It is important to note that the experiments had a

Figure 5. Horizontal width of the two coalesced bubbles (λ) as a function of r/Rb,ave for (red triangle) experimental trial, (green square) VOF computational case with h = 0.02 mm, and (blue circle) VOF computational case with h = 0.05 mm.

cases. Although the initial horizontal width observed in the computational case with an initial separation distance of 0.05 mm was smaller than that for the computational case with a separation distance of 0.02 mm, the two cases had nearly the same horizontal width-neck radius curves. Based on this result, the computational case with h = 0.05 mm was selected for further analysis. Snapshots of the experimental trial and contours of air volume fraction for the VOF case with h = 0.05 mm are presented in Figure 6. Three representative r/Rb,ave values were selected for comparison. As the neck radius grew, the horizontal

Figure 7. Comparison of (closed markers) Weber number and (open markers) Reynold number as a function of time for (red triangle) experimental trial, (green square) VOF computational case with h = 0.02 mm, and (blue circle) VOF computational case with h = 0.05 mm.

time resolution of 0.2 ms from the rupture of the thin liquid film. For the experimental data, the motion of the neck in the early stages of expansion was not fully counterbalanced by the inertial force, although the value was close to unity as stressed by the Reynolds number (Re ≤ 0.24). This finding is corroborated by the results of Stover et al.2 who found that the motion of the resultant bubbles is the result of the reduction in total surface area, which causes a decrease in the surface energy of the system. As such the motion of the interface is driven by the surface tension, which determines the amount of surface energy to be released. The excess energy is passed to the surrounding liquid where it is opposed by the viscosity and the inertia of the liquid. On the other hand, the Weber number for the computational cases differed slightly from the experimental results, showing the inertial force as superior in magnitude to the driving force (i.e., surface tension) followed by surface tension for longer times although all results are close to unity. Since the motion is driven by the surface tension, the small discrepancy could be attributed to the poor capability of VOF to simulate the initial complex phenomenon of the liquid film rupture. Previous numerical studies have pointed out the limitation of VOF to accurately calculate the local curvature near the interface, which is then used for calculating the surface tension force per unit volume.49,50 Moreover, water, in the absence of any surfactant, could have small variations in surface tension,51 which means

Figure 6. Comparison of the neck radius for (a) experimental trial conducted by Bournival et al.11 and (b) computational case with h = 0.05 mm. The dashed red lines represent the contour of the bubbles before coalescence while the green arrows indicate the horizontal width between the two coalesced bubbles. The capillary tubes are 1.07 mm in diameter and act as a scale. 14786

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simulations and the experimental data. The integration constant is generally very close to zero and therefore not relevant in the present analysis. The calculation assumed equilibrium surface tension at the start of the simulation, which did not change after coalescence. As such, no surface tension gradient caused by a change in surface area was generated, limiting any Gibbs elasticity effect although relaxation may in fact occur.51 High-frame-rate animations of the VOF numerical cases were produced and then analyzed using ImageJ to track the changes in the projected surface area over time. The five parameters in eq 12 were fitted using the solver command in Microsoft Excel and minimizing the sum of the square error. An oscillation period of 1.5 was selected for fitting the simulation data to eq 12. A previous study determined that 1.5−2 periods are sufficient to detect small deviations for the damping constant and angular frequency.53 The projected area curves for both computational case and experimental trials are illustrated in Figure 9. In the VOF case,

that the assumption of a constant surface tension in the computational cases could also lead to small deviation in the results specifically at the early stage of the neck expansion. Figure 8 shows that the Ohnesorge number was between 15 and 3. This indicates that the neck motion driven by surface

Figure 8. Ohnesorge number as a function of time for (red triangle) experimental trial, (green square) VOF computational case with h = 0.02 mm, and (blue circle) VOF computational case with h = 0.05 mm.

tension was initially restrained by a greater magnitude of the fluid viscosity than the inertial force, followed by an exponential decay of viscous force to inertial force ratio. Therefore, it appears that the simulation of the coalescence dynamics by VOF is relatively accurate since the Weber numbers are relatively close to unity and the Ohnesorge numbers are consistent. However, the early stage of coalescence (first 1 ms) represented by the initial expansion of the neck may lack in computational accuracy. 3.1.2. Oscillation Behavior of Coalesced Bubbles. After obtaining the VOF results, the oscillating behavior was quantified following the approach which has its basis in the equation proposed by Schulze.52 Bournival et al.11 approximated the surface oscillation of a coalescing bubble using the following equation: A′ = A 0′ × e−δt sin(ω0t + φ) + B

Figure 9. Comparison of modeled data (eq 12) for (red triangle) experimental data and (blue circle) VOF computational case with h = 0.05 mm.

the average projected surface area toward an infinite time was calculated by assuming that the total volume of the bubbles was preserved during the postrupture oscillation. From these two curves, it can be seen that the computational case followed the oscillatory motion of the bubble after coalescence. The VOF results showed a higher initial amplitude peak of the bubble oscillation compared to the experimental trial and indicated that the coalesced bubbles in the VOF case were initially further from the final equilibrium state at which a stable shape was reached. In the experiments, the damping surface oscillation was rapid suggesting that the coalesced bubbles expanded and contracted quickly. The VOF computational case showed a lower resistance of the bubble surface on the neck expansion after the film rupture, leading to a lower damping constant (error of 13.3%) and slower angular frequency (error of 18.6%) as shown in Table 3.

(12)

where A′ and A0′ denote the normalized relative projected area and initial amplitude, respectively, δ is the damping constant in meters per second, φ is phase shift, t is time in milliseconds, ω0 is the angular frequency in meters per second, and B is an integration constant. It is important to state that the normalized relative projected area A′ in eq 12 (which was calculated by the instantaneous projected area minus the average projected area toward an infinite time divided by the average projected area toward an infinite time) is characterized by five parameters, with three highly significant parameters. The initial amplitude describes the magnitude of the deviation of the initial projected area of the bubble from the final equilibrium state. The damping constant, which depends on the fluid and interfacial properties, shows if the oscillation is slowly (low damping constant) or rapidly (high dampening constant) dampening. The angular frequency gives an indication of how fast the oscillation completes one period and is important for regulating the violent nature of the process at extreme points (peaks and troughs) of the oscillation. The phase shift describes only the starting position of the bubble’s oscillation. It is not a very critical parameter in this study and therefore not meaningful in comparing the

Table 3. Calculated Parameters from Equation 12 for Experimental and VOF Cases with an Initial Separation Distance of 0.05 mm

a

14787

parameter

experimentsa

VOF case

initial amplitude, A0′ damping constant, δ (m/s) angular frequency, ω0 (m/s) phase shift, φ integration constant, B

0.259 0.060 0.586 0.938 1.5 × 10−3

0.346 0.052 0.477 0.798 5.8 × 10−3

Obtained from averaging 5 trials in the work of Bournival et al.11 DOI: 10.1021/acs.iecr.7b03197 Ind. Eng. Chem. Res. 2017, 56, 14781−14792

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Industrial & Engineering Chemistry Research

Figure 10. (a) Velocity vectors colored by velocity magnitude and (b) contour of dynamic pressure gradients for VOF computational case with h = 0.05 mm. A logarithmic scale was selected for the color map of the dynamic pressure contours. The capillary tubes are 1.07 mm outer diameter and 0.69 mm inner diameter as indicated by the black contour lines. The capillaries act as a scale.

separating the two bubbles, the air phase flowed toward the bubbles’ initial contact point and then it moved in the vertical opposite direction, as shown during the first 1.55 ms.54 This airflow pushed the surrounding liquid, producing the neck expansion. Small eddies with velocity magnitudes between 0.25 and 0.75 m/s were observed at the top and bottom part of the neck as the curvature of the bubble surface decreased. The neck expansion generated a surface wave that propagated horizontally in the opposite direction, creating bulges at the edges of the bubble, as illustrated at 3.16 ms. The air phase in the bulging area moved at much higher velocity, in the range of 1.75 to 2.5 m/s, than the rest of the bubble surface. The bubble contracted vertically as the surface wave recoiled and the air phase flowed perpendicular toward the top of the neck creating

3.2. Flow Dynamic Analysis. Flow dynamic analysis provides an insight into the coalescence dynamic experienced by two bubbles during the first cycle of postrupture oscillation. Analysis of velocity vectors and dynamic pressure gradients are presented below. 3.2.1. Velocity Field Analysis. Simulation snapshots of velocity vectors predicted at a vertical plane along the central axis of the capillary tubes are illustrated in Figure 10a. The motion of the interface was captured in the change of the projected surface area over time. Nine points in time during the initial stage of bubble coalescence were chosen. As can be seen in Figure 10a, the velocity vectors indicate the direction and the velocity magnitude of the airflow inside the coalesced bubbles and the surrounding liquid. After the rupture of the liquid film 14788

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Figure 11. Velocity vectors and contours of dynamic pressure gradients for coalescence dynamics of two uncoated bubbles, using the VOF computational case with h = 0.05 mm, compared with an experimental study of bubble coalescence dynamics of two bubbles coated with hydrophobized silica particles carried out by Bournival et al.35 The capillary tubes may be used to scale the images where capillary tubes of 1.04 mm in diameter were used in the experiments and 0.69 mm inner diameter as indicated by the black contour lines on the simulation.

expansion created dynamic pressure gradients at the interface where the motion was constrained by the surrounding liquid. There were contours with higher dynamic pressure gradients in the bubble interface, where the air phase flowed with high velocity, creating eddies as shown in Figure 10a. Consequently, vertical bubble contraction was rapidly driven by the highpressure gradients around the bulged area that pushed the air phase inward, as shown at 3.16 ms. The high dynamic pressure could be useful in understanding the detachment of particles when two bubbles coated with hydrophobic particles coalesce. Bournival et al.35 studied the coalescence dynamics of two bubbles coated with silica particles in water. Some snapshots of the experimental study were compared against the VOF computational case to obtain an insight into the velocity fields and dynamic pressure gradients. The comparison was based on the oscillation time after coalescence. The selected time of the surface contours for the two uncoated bubbles presented in Figure 11 differed slightly from the coated

eddies with small velocity magnitude at the bottom of the neck and leaving a stagnant void in the center of the bubbles. It can be seen from 4.76 to 5.57 ms that the air phase near the bubble interface−capillary tube flowed toward the top of the neck as the bubble continued to vertically contract. 3.2.2. Dynamic Pressure Gradient Analysis. In addition to the velocity field analysis, the dynamic pressure gradients of fluid were also assessed in ANSYS-Fluent to quantify the pressure associated with the difference of velocities at the bubble interface. Dynamic pressure (pd = 0.5 × ρ|V|2), which represents the difference between the total and static pressure in the system, is proportional to the density and square of the velocity vector magnitude of the fluid. Snapshots of dynamic pressure contours along a cross section plane during the first 5.57 ms are shown in Figure 10b. The dynamic pressure scale was set between 50 and 800 Pa for better comparison, and the white color in the snapshots represents values in the range of 0 to 50 Pa. It is evident from the snapshot contours that the neck 14789

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Industrial & Engineering Chemistry Research bubbles. Apart from the numerical error observed in the VOF results, one reason is the particle layer around the loaded region, which may restrict the bubble surface deformation.33 Figure 11 illustrates the detachment of a large number of particles from the bubble surface as the bulging area contracted. At 3.36 ms, the bulged area moved at much higher velocity magnitude, in the range of 1.75 to 2.5 m/s, than the rest of the bubble surface. Therefore, a dynamic pressure greater than 800 Pa is observed at the bulging area. The vertical contraction also created a difference in dynamic pressure at the interface, which may explain the results by Bournival et al.,36 who pointed out that the difference in pressure force between the air and the surrounding liquid exerts a force on the particles, leading to particle detachment. Therefore, it could be argued that the pressure gradients generated during the postrupture oscillation of the coalesced bubble affect the stability of particles attached at the interface. However, other factors such as particle size, density, and hydrophobicity may also help quantify the detachment of particles.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Seher Ata: 0000-0002-3705-6441 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to thank the following people: Mr. Collin Turner of McGill University, Canada, for his contribution to the setting of the preliminary VOF computational case; Dr. Yuqing Feng of CSIRO Mineral Resources (Australia) for helpful discussion on the model settings and validation; and Dr. Francisco Trujillo of the School of Chemical Engineering, University of New South Wales (Australia), for facilitating access to a computer for running preliminary models and vital discussion on the model settings. Y.S.M. would also like to thank the Australian government and University of New South Wales (Australia) for financial support through the Australian Government Research Training Program (RTP) scholarship.

4. CONCLUSIONS The flow field arising from the merging of two equal-size bubbles was numerically investigated for two separation distances using a finite volume methodology and was validated against experimental data. The use of the VOF method allowed the detailed investigation of the flow field and pressure gradients in the interface regions of the coalescing bubble. The results indicated that there was a small deviation of the neck expansion motion showing a faster neck growth at the smaller separation distance. The dynamics of the coalesced bubbles were studied by evaluating the velocity fields and the dynamic pressure gradients. The velocity vector showed that as the neck grew the surrounding liquid was pushed outward. The chaotic motion created a surface wave across the newly formed bubble surface, which propagated in the opposite direction toward the edges of the bubble. The neck growth created high dynamic pressure gradients at the top and bottom of the bubble interface, where the air phase flowed with higher velocity than the surrounding liquid, which created eddies. Overall, it was found that the computational case followed the oscillatory motion very well, as observed in experiments, but with lower damping constant and lower angular frequency. Further studies of the velocity fields and pressure gradients during the merging of two bubbles with different types of surfactants are needed. These studies should be complemented by investigation of the changes in surface energy during postrupture oscillation to further elucidate the oscillation mechanism. The distribution of surfactants along the interface during bubble coalescence should also be studied numerically. The variation in the distribution of surfactant may be resolved by using a nonconstant surface tension force, which would take into account surface elasticity. A similar approach may be needed to reconcile the initial growth of the neck, which computer simulations considered to be inertia driven as opposed to surface tension driven. Overall, the VOF method successfully tracked the interface motion of the water−air phases in the current system.

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Video of the VOF simulation with an initial separation distance of 0.05 mm (AVI)

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b03197. 14790

ABBREVIATIONS 3D = three-dimensional A = power exponent A0′ = initial amplitude A′ = normalized projected area B = integration constant C = prefactor CFD = computational fluid dynamics CN = courant number Db,ave = average diameter of the two initial bubbles, mm F = function of volume fraction Fair = volume fraction of air phase Fσv = surface tension force per unit volume, N/m3 g = gravity, m/s2 h = initial separation distance between the two bubbles, mm n = normal vector to the interface nwall = vector normal to the capillary tube Oh = Ohnesorge number p = pressure, Pa pd = dynamic pressure, Pa PISO = pressure implicit with splitting operator r = neck radius, mm Rb,ave = average radius of the two bubbles, mm Rb1 = radius of bubble 1, mm Rb2 = radius of bubble 2, mm Re = Reynolds number t = time, ms twall = vector tangent to the capillary tube T = matrix transpose operator T = temperature, °C V = velocity vector |V| = velocity vector magnitude, m/s Vr = velocity of the neck radius, m/s VOF = volume of fluid We = Weber number DOI: 10.1021/acs.iecr.7b03197 Ind. Eng. Chem. Res. 2017, 56, 14781−14792

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GREEK LETTERS Δt = time-step size, s ΔX = element size, m δ = damping constant, m/s θeq = equilibrium contact angle, deg k = surface curvature of the interface λ = horizontal width of the two coalesced bubbles, mm μ = dynamic viscosity, mPa s μair = dynamic viscosity of air phase, mPa s μwater = dynamic viscosity of water phase, mPa s ρ = density, kg/m3 ρair = density of air phase, kg/m3 ρwater = density of water phase, kg/m3 σ = surface tension, mN/m τ = capillary-inertial time, ms φ = phase shift ω0 = angular frequency, m/s ∇ = gradient operator

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