Systematic Study on the Coalescence and Breakup Behaviors of

Article pubs.acs.org/IECR

Systematic Study on the Coalescence and Breakup Behaviors of Multiple Parallel Bubbles Rising in Power-law Fluid Jingru Liu, Chunying Zhu, Taotao Fu, and Youguang Ma* State Key Laboratory of Chemical Engineering, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China ABSTRACT: The coalescence and breakup behaviors of multiple parallel bubbles rising in power-law non-Newtonian fluid were investigated numerically by Volume of Fluid (VOF) method. The effects of the number of bubbles, bubble diameter, bubble interval, and flow index of power-law fluid on bubble coalescence and breakup were studied systematically. The dimensionless critical horizontal intervals of bubble coalescence for two, three, and four bubbles with different diameters were attained respectively by simulation under different flow indexes of power-law fluid. A quantitative criterion was developed to compute the critical bubble interval for bubble coalescence based on the critical approach velocity theory. Dramatic deformations of bubbles were found during the coalescence processes. Two different bubble coalescence and breakup behaviors were found and illustrated: (I) all bubbles coalesce into one big bubble; (II) coalescence of partial bubbles among multiple bubbles. For two or three parallel bubbles, only regime (I) occurs. However, for four parallel bubbles, other than the coalescence into one big bubble and subsequent breakup, the collision and coalescence between two neighboring bubbles among the four parallel bubbles could take place more easily. The breakup of the coalescing bubble usually appears in the cases of big bubble size and low flow index of power-law fluid.

1. INTRODUCTION Gas−liquid contact equipment such as bubble column, gas stripper, and biochemical reactor are widely applied in chemical industry because of their good performance of heat and mass transfer and low cost of maintenance and operation.1−3 The widespread phenomena of bubble coalescence and breakup alter bubble size distribution and gas−liquid contact area, which have remarkable effect on mass transfer coefficient and chemical reaction rate. Non-Newtonian fluids are frequently used in the fields of enhanced oil recovery, wastewater treatment, polymerization process, and production of foods and pharmaceuticals.4 Bubble behaviors in non-Newtonian fluid are more complicated compared to that in Newtonian fluid, while the relevant researches are mainly focused on Newtonian fluid.5 Previous studies of in-line bubble coalescence in nonNewtonian fluid try to explore the mechanism of coalescence process and the effect of rheological properties on bubble coalescence. The experiment of in-line bubbles rising in nonNewtonian fluid indicated that the competition between the stress creation by the passage of bubbles and the relaxation due to the fluid’s memory governs the in-line bubble interactions and coalescence.6 The flow field during in-line bubble coalescence revealed that the dragging force and pushing force generated by shear-thinning effect and viscoelastic effect respectively play important role in the course of bubble coalescence in nonNewtonian fluid.7 Recently, more and more researchers8−12 make use of various numerical methods such as Volume of Fluid method (VOF), Level Set method (LS), Lattice Boltzmann method (LB), and Front Tracking method (FT) to investigate bubble dynamics or interactions, in general, their studied results were in reasonable agreement with the existing experimental data. However, it should be pointed out that bubble coalescence occurs numerically as soon as the interfaces come closer than the grid width for numerical technique of VOF, LS, and LB. At © 2014 American Chemical Society

present, no successful model has been developed to determine the separation or coalescence between two bubbles depending on the dynamic conditions,13 while thoroughly understanding of the mechanisms of bubble coalescence and breakup is helpful for the improvement of the models. In this study, we focus mainly on the coalescence process of parallel bubbles and try to reveal the mechanism of bubble coalescence. For parallel bubble, although there are some researches on interactions between two parallel bubbles,14−17 few researches were conducted on coalescence of parallel bubbles. Bouncing and coalescence of a pair of bubbles rising side by side in Newtonian fluid was studied experimentally,18 the critical Reynolds number, and Weber number separating bouncing and coalescence were obtained, and the bubble trajectories and velocity changes before and after coalescence of two parallel bubbles were analyzed. Two bubbles both rising side by side and rising in line were simulated and compared with experimental observations; also, the coalescence dynamics of in-line bubbles were discussed.19 De Kee and Chhabra20 reported the coalescence process of two and three parallel bubbles rising in polyacrylamide solution and carboxymethyl cellulose solution, respectively. In the previous work,17,21 we have studied both interaction and coalescence dynamics of two and three parallel bubbles rising in non-Newtonian fluids. When it comes to bubble coalescence, most of the studies basically concentrated on in-line bubbles, much less is known about parallel bubble coalescence especially for multiple bubbles. It is the most concerned issue that under what conditions the Received: Revised: Accepted: Published: 4850

November 6, 2013 February 18, 2014 March 7, 2014 March 7, 2014 dx.doi.org/10.1021/ie4037565 | Ind. Eng. Chem. Res. 2014, 53, 4850−4860

Industrial & Engineering Chemistry Research

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with pure gas, F = 1 indicates a cell with pure liquid, and 0 < F < 1 indicates a cell with a mixture of gas and liquid. The tracking of gas−liquid interface is achieved by the solution of the continuity equation for the volume fraction of gas:

coalescence could take place and how the shape of the bubbles evolves with time if coalescence takes place.22 The knowledge of parallel bubble coalescence and breakup could increase the predictability of gas−liquid equipment design as well as improve the understanding of the mechanisms of bubble coalescence and interaction systematically. The information about interactions and coalescence between parallel bubbles is also required for the multiscale modeling approach.23,24 In this article, coalescence and breakup processes of multiple parallel bubbles in power-law fluid were studied systematically using VOF method, the critical bubble intervals for bubble coalescence were obtained for different bubble numbers and rheological properties, different regimes of multiple bubbles coalescence were classified according to dimensionless numbers, and the mechanisms of parallel bubble coalescence and breakup were analyzed.

∂Fg ∂t

(1)

∂ (ρ(F )u ⃗) + ∇·(ρ(F )uu⃗ ⃗) ∂t = −∇p + ∇·[2μ(F )D⃗ ] + FS + ρ(F )g ⃗

(2)

(8)

Thus, the volume fraction of liquid is calculated based on the following constraint: Fl + Fg = 1

(9)

2.4. Numerical Methods. The numerical methods and detail settings were consistent with our previous study.21 The physical model of the simulation was simplified to a 2D rectangle computational domain with the dimension of 200 mm ×300 mm, as shown in Figure 1a. Based on our previous investigation of grid

2. MATHEMATICAL MODELS AND NUMERICAL METHODS 2.1. Continuity and Momentum Equations. The continuity and momentum equations for incompressible power-law fluid were represented as ∇·u ⃗ = 0

+ ∇·(uF ⃗ g) = 0

where u⃗ is the velocity vector, p is the pressure and D⃗ is the stress tensor. FS stands for source term induced by surface tension. When a computational cell is occupied by gas−liquid two phases, mixture properties in eq 2 are estimated as follows: ρ(F ) = ρl (F ) + ρg (1 − F )

(3)

μ(F ) = μ l (F ) + μg (1 − F )

(4)

2.2. Physical Properties Equations. The viscosities of nonNewtonian fluid are calculated using power-law model: μ = Kγ ṅ − 1

Figure 1. Computational domain and grid partition.

(5)

1 where γ ̇ = (2(D⃗ : D⃗ ))1/2 and D⃗ = 2 (∇u ⃗ + ∇u ⃗T ). According to the Continuum Surface Force (CSF) model proposed by Brackbill et al.,25 only the surface tension force normal to the interface is considered and treated as constant along the surface. The CSF model is incorporated into the calculation and results in a source term in the momentum equation:

Fs = σ

independence, mesh interval size 0.2 mm was chosen to ensure high computing accuracy and low time cost, as shown in Figure 1b. The exit of bubble column was set as pressure outlet and the walls were defined as no-slip boundary condition. In the VOF model, the volume fraction function is solved using geometric reconstruction scheme based on piecewise linear interface calculation (PLIC). The PLIC method is comparatively precise because it takes transport among adjacent interface fluid into consideration in detail.26 The pressure implicit with splitting of operator (PISO) algorithm was used to solve the pressure− velocity coupling in the momentum equation. Pressure staggering option (PRESTO) and first order upwind scheme are used for pressure and momentum discretization, respectively. The under relaxation factor for pressure, momentum, and body force was set as 0.3, 0.5, and 0.7, respectively, in order to improve the stability and convergence rate. The computational domain was filled with stationary liquid after initialization and the operation pressure was set as atmospheric pressure at pressure outlet. Bubbles with different sizes and intervals were patched at bottom of the computational domain, as shown in Figure 1a. Time step was set as 1.0 × 10−5 s and the residual of velocity and continuity less than 1.0 × 10−5 was considered as convergence, as

ρκ ∇F 1 (ρ 2 g

+ ρl )

(6)

where the expression for the curvature κ is obtained from the divergence of the unit normal vector to the interface: κ = −(∇·n)̑ = n = ∇Fg

⎤ ⇀ 1 ⎡ n⃗ ⎢ ⇀ ·∇|n | − (∇·n ⃗)⎥ , ⎦ |n ⃗ | ⎣ | n |

n̑ =

n⃗ ⇀

,

|n | (7)

2.3. Volume Fraction Equation. Volume of fluid (VOF) method employs a volume fraction function to indicate the fractional amount of fluid in the computational cell. For gas− liquid system, the volume fraction function F = 0 indicates a cell 4851

dx.doi.org/10.1021/ie4037565 | Ind. Eng. Chem. Res. 2014, 53, 4850−4860


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and 0.06354 N·m−1, measured by a vibrating tube density meter (DMA 4500 M, Anton Paar, Austria) and an Optical Contact Angle Measuring Device (OCA15EC, Dataphysics), respectively. The rheological properties of the liquid were measured by a DV-III ultraprogrammable rheometer (Brookfield Engineering Laboratories, U.S.A.) and correlated by power-law model, the obtained consistency coefficient K and flow index n are 0.0539 Pa·sn and 0.7262. All experiments were carried out at room temperature (298.15 K) and atmospheric pressure (0.1013 MPa). The physical property parameters used in the simulation were kept consistent with the measured values in the experiment. Figure 4 compared the process of two in-line bubble coalescence between experiment and simulation. In the experiment, the steady shape for bubbles rising in 0.4% carboxymethylcellulose sodium (CMC) aqueous solution is ellipsoidal. In the experiment, t = 0 s was selected when the two in line bubbles evolve to their steady shape after formation. Due to the limitation of Fluent software, setting the initial bubble shape as ellipsoidal was difficult, in the simulation, considering the spherical bubble would rapidly evolve into ellipsoidal one very closed to experimental shape under given physical conditions, therefore, we set the initial bubble shape as spherical. The initial bubble diameter was set as 8 mm and the initial distance between bubbles is 40 mm referring to the equivalent diameter and initial distance in the experiment. As shown in Figure 4, at t = 0.2 s, the bubble shapes have already evolved to ellipsoidal shapes quite similar to experimental shapes before coalescence, and the shapes did not change until the trailing bubble get close to the leading bubble; the trailing bubbles experience pursuing stage and finally catch up with the leading bubble. After pursuing stage, two in line bubbles begin to coalesce. The leading bubble was squeezed and became oblate, while the following bubble was stretched in vertical direction. The slight difference during bubbles coalescence may result from the difference of film drainage rate and bubble deformation rate between experiment and simulation; in the simulation, the film drainage rate and bubble deformation rate were a little faster than that in the experiment. The flow fields around two bubbles interact with each other which result in the deformation of bubbles. The well agreement of experiment and simulation indicates the validity of the computational methods.

shown in Figure 2. The calculations were conducted using commercial software Fluent 6.3 on the platform of high

Figure 2. Variation of residuals of velocity and continuity with iterations.

performance parallel computing system. Postprocessing of simulated data was done using software Tecplot 360.

3. EXPERIMENTAL SECTION In order to validate the accuracy of the numerical methods, at first, the comparison between the simulation and experiment of in line bubble coalescence was made. The experimental setup is shown in Figure 3. The bubble column is made up of transparent

4. RESULTS AND DISCUSSION 4.1. Criterion for Coalescence and Coalescence Regimes. Whether bubbles coalesce or not is closely related to the approach velocity of these bubbles. Kirkpatrick and Lockett27 pointed out the importance of the approach velocity for bubble coalescence in the early days. Their research showed that coalescence takes place rapidly at low approach velocity, while it would be difficult for bubbles to coalesce once the approach velocity exceeds a critical value. According to Chesters and Hofman,28 the critical Weber number for bubble coalescence was defined based on the maximum relative velocity of the bubbles to determine bubble coalescence. The Weber number was defined as follows:

Figure 3. Schematic representation of the experimental apparatus.

plexiglas with dimension of 0.15 m × 0.15 m × 1.50 m. Bubbles were generated through the orifice submerged in the liquid in the center of the bottom section of the column. Air was injected into the column by the syringe pump (Harvard microprocessor multiple syringe pump, U.S.A.), and the accurate control of gas flow by precision syringe pump enable generation of in line bubbles. The images of rising bubbles were captured by high speed camera (Motion Pro Y5, REDLAKE Global, U.S.A.) and transmitted onto the connected computer. The obtained pictures were analyzed quantitatively by processing software-Matlab. Carboxymethylcellulose sodium (CMC) aqueous solution with mass concentrations 0.4% was used as non-Newtonian fluid. The density and surface tension of the liquid was 1003 kg·m−3

We =

ρv 2R eq σ

(10)

where v is the approach velocity of the bubbles, Req is the equivalent radii of the bubble, ρ and σ are the density and surface tension of the liquid, respectively. Duineveld29 found that two 4852



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approach velocity theory generally agrees with the film drainage model for bubble coalescence. Bubble coalescence could take place only if the contacting time is greater than the film drainage time; if the approach velocity is too great, the contacting time would become decreased, which is unfavorable for bubble coalescence. According to Kirkpatrick and Lockett,27 the increase of approach velocity leads to the increase of film area, which decreases the rate of film drainage. In addition, the increase of approach velocity induces the increase of film pressure and interaction of vortex around bubbles, which also prevent bubble coalescence. For freely rising parallel bubbles, the approach velocity depends on bubble size, physical properties of the fluid and bubble configuration such as bubble interval and angles between bubble center and perpendicular line. Wijngaarden30 proposed the calculation of approach velocity for parallel bubbles as follows: 2 ⎡ ⎤ d0 3U∞ ⎛ dx ⎞ 2 ⎛ x 0 ⎞3 ⎜ ⎟ − 1⎥ ⎜ ⎟ = k ⎢ ⎝ dt ⎠ x03 ⎣⎝ x ⎠ ⎦

(11)

where k is a constant, x0 is initial horizontal bubble interval between the centers of bubbles, x is distance between centers of bubbles, U∞ is the rise velocity of a single bubble, and d0 is initial bubble diameter. In order to predict the approach velocity for parallel bubbles rising in power-law fluid, an amendment was made as follows: 2 ⎡ ⎤ d0 3U∞ ⎛ dx ⎞ 2 ⎛ x 0 ⎞3 ⎜ ⎟ − 1⎥ ⎜ ⎟ = f (m , μ) ⎢ ⎝ dt ⎠ x03 ⎣⎝ x ⎠ ⎦

(12)

where the modifying factor is f (m , μ) = ambμc

(13)

where m is number of bubbles, viscosity of power-law fluid μ is calculated by eq 5. Critical bubble interval is the minimum initial interval above which the coalescence of parallel bubbles could not take place, and it is an important parameter for gas−liquid equipment design such as sieve plates and sparger. The dimensionless form of critical bubble interval is

xc* = xc/d0

(14)

where xc is the critical initial horizontal maximum bubble interval for coalescence; d0 is the initial diameter of the bubbles. The typical evolution of horizontal interval along the height for two bubbles with critical bubble interval was shown in Figure 5. At t = 0, the dimensionless bubble interval is the critical bubble interval for coalescence, and then, the interval between the two bubbles along the height decreased gradually until coalescence. According to eq 12, the critical approach velocity for bubble coalescence could be evaluated from the following equation: vm2 =

2 ⎡ ⎤ d03U∞ ⎛ dx ⎞ 2 ⎛ xc ⎞3 ⎜ ⎟ − 1⎥ ⎜ ⎟ = f (m , μ) ⎢ ⎝ dt ⎠ xc3 ⎣⎝ x ⎠ ⎦

(15)

where vm is critical approach velocity for bubble coalescence, the parameters a, b, and c in f(m,μ)could be obtained by nonlinear regression using least-squares method:

Figure 4. Comparison of coalescence of two in-line bubbles between experiment and simulation: left, experiment; right, simulation.

f (m , μ) = 1.2308m−0.7526μ−0.4391

equally sized bubbles could coalesce in pure water whenever the Weber number was below a critical value, Wc = 0.18. The

(16)

We assume that the resistance is constant during the approaching of bubbles. The critical point for bubble coalescence is that the 4853



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vmt −

1 2 at = x − d0 2

(17)

where a = dv/dt = d2x/dt2. According to Chesters,31 the coalescence time could be calculated as t=

ρd0 2vm 4σ

(18)

Equations 17 and 18 could be integrated to give x − d0 =

2 ρd02 2 ⎛ 1 d2x ρd0 ⎞ ⎟ vm⎜1 − 4σ ⎝ 8 dt 2 σ ⎠

(19)

Integrating eqs 15 and 19, the relationship between bubble interval and time could be established as x − d0 = f (m , μ) Figure 5. Schematic representation of critical bubble interval for coalescence and typical variation of horizontal bubble interval along the height for two bubbles.

2⎡ 2 ⎤⎛ ρd05U∞ ⎛ xc ⎞3 1 ρ d0 d2x ⎞ ⎜ ⎟ − 1⎥⎜1 − ⎢ ⎟ 8 σ dt 2 ⎠ 4σxc3 ⎣⎝ x ⎠ ⎦⎝

(20)

Bubble interval and time could be normalized as follow:

coalescence time or film drainage time equals to bubble contact time. The process of bubble approaching could be represented as

x* =

x , d0

t* =

tU∞ d0

(21)

Figure 6. Variation of critical bubble interval for coalescence with initial bubble diameter, flow index of power-law fluid and number of bubbles: dot, obtained from simulation; line, predicted values by eq 22; (a) two parallel bubbles; (b) three parallel bubbles; (c) four parallel bubbles; (d) flow index n = 0.6. 4854



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Therefore, the dimensionless form of eq 18 would be ⎤ 2 ⎡⎛ * ⎞ 3 ρd0U∞ xc ⎢ ⎜ ⎟ − 1⎥ x * − 1 = f (m , μ) ⎥⎦ 4σxc*3 ⎢⎣⎝ x* ⎠ 2 ⎛ 1 ρd0U∞ d2x * ⎞ ⎟ ⎜1 − 8 σ dt *2 ⎠ ⎝

(22)

The above partial differential equations could be solved with self-developed Matlab integral program. Figure 6 compared the values of critical bubble interval obtained from simulation and predicted by eq 22. The solid dots represent values obtained from simulation, and solid line was predicted by eq 22. It can be seen that the predicted values are in good agreement with the simulation data, with a maximum relative deviation of 7%. Bubble shapes evolve generally from spherical to ellipsoidal and finally to spherical cap with the increase of bubble diameter in this simulation. The diversity of bubble shapes may result in the difference of flow field structure, big bubble may suffer from a greater resistance to coalesce, and the critical bubble interval decreases with the increase of bubble diameter, as shown in Figure 6a, b, and c. By far, the research on the effect of bubble size on coalescence process is scarce. With the increase of bubble diameter, the film thickness and the volume of the liquid film needing to be drained out would increase, leading to an increase of film drainage time, which is unfavorable for coalescence. It could be found that for big diameter bubbles the dimensionless critical interval xc* tends toward 1. The decrease of flow index enhances the shear-thinning effect, leading to the reducing of local viscosity, this facilitates the bubbles approaching and contacting with each other; thus, the critical bubble interval increases remarkably with the decrease of the flow index. While once flow index n > 0.6, the shear-thinning effect of nonNewtonian fluid has nearly no influence on critical bubble interval. Figure 6d shows the effect of bubble number on critical bubble interval taking flow index n = 0.6 for example. Interactions between bubbles were intensified as bubble number increased, which result in the increase of critical bubble interval and coalescence of bubbles become more easily. Bubble size, rising velocities, and rheological properties of power-law fluid have significant impact on bubble coalescence and breakup, these variables could be evaluated by the following dimensionless numbers: Eo =

ρgde2 σ

(23)

Re =

ρu 2 − nden K

(24)

Figure 7. Regimes of coalescence and breakup of multiple parallel bubbles in power-law fluid: ■ two bubbles coalesce into one; □ two bubbles coalesce into one and breakup; ▼ three bubbles coalesce into one; ▽ three bubbles coalesce into one and breakup; ▲ four bubbles coalesce into one; △ four bubbles coalesce into one and breakup; ◆ the middle two of four bubbles coalesce into one; ◇ the middle two of four bubbles coalesce into one and breakup; ● two pairs of four bubbles coalesce into one; ○ two pairs of four bubbles coalesce into one and breakup.

one big bubble and the subsequent possible breakup (regime I) takes place usually when 0 < Re < 20. The coalescing bubbles for great initial bubble diameter were unstable and tend to deform, therefore the coalescing bubbles finally breakup at high Eo numbers (5 < Eo < 15). As bubble number increases from two to four, the rising of bubbles were hindered by intensified interactions, and the Re numbers for bubble coalescence and breakup were decreased. For two or three parallel bubbles, only one coalescence behavior could take place: all bubbles coalesce into one big bubble. For four bubbles, the coalescence of partial bubbles among multiple bubbles and breakup happen when 20 < Re < 100 corresponding to regimes II (a) and II (b) as shown in Figure 6. Compared with the coalescence of the two middle bubbles (0 < Eo < 6), coalescence of two pair of four bubbles take place in a wide range of Eo number. Breakup of coalescing bubbles in regimes II occurs usually when Re > 60, this may be induced by the fluctuation of bubble rise and strong shear stress of fluid due to high bubble rising velocity. The effect of number of bubbles, initial bubble diameter and flow index on coalescence process will be discussed in the follow section. 4.2. All Bubbles Coalesce into One and the Subsequent Breakup. According to film drainage theory,32−34 in line bubble coalescence could be divided into three stages: first, a liquid film is formed because of bubble collision; second, bubbles keep in touch with each other until film drains out to a critical thickness; finally, liquid film ruptures and coalescence takes place. As shown in Figure 8, parallel bubbles merge into a long bubble if they get close enough to each other. The process of parallel bubble coalescence could also be divided into three stages: first, parallel bubbles approach and contact with each other due to bubble attraction and collision; second, parallel bubbles are connected through the gas bridges formed between bubbles; finally, the gas bridges extend in vertical direction rapidly and the bubbles merge into one. The merged bubble is unstable and could breakup easily due to flow field interaction. Figure 8 shows the typical coalescence process of parallel bubbles into one for different bubble numbers. The coalescing bubble tends to breakup after coalescence but finally recover to steady shape without breakup, as shown in Figure 8a and d. The bubble itself could accommodate and dissipate the excess kinetic energy of external flow field through surface distortion and internal motion to

where ρ, σ, K and n are density, surface tension, consistency coefficient, and flow index of power-law fluid. Vertical velocity u was chosen as the characteristic velocity in the definition of Re.18 de is equivalent volume diameter and calculated as follows: de =

3

6Vb/π

(25)

According to Re and Eo numbers, coalescence and breakup behaviors of multiple parallel bubbles rising in power-law fluid could be divided into several regimes, as shown in Figure 7. In general, two different regimes of bubble coalescence and breakup (regime I and regime II) were identified as displayed in Figure 7. The lines in Figure 7 refer to transition line between different bubble behavior regimes. All bubbles coalescing into 4855



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Figure 8. All bubbles coalesce into one and the possible breakup process for different bubble numbers: (I) two parallel bubbles: (a) d0 = 6 mm, n = 0.4, x* =1.1833, (b) d0 = 8 mm, n = 0.4, x* =1.1125, (c) d0 = 6 mm, n = 0.2, x* = 1.1833; (II) three parallel bubbles: (d) d0 = 4 mm, n = 0.4, x* = 1.3000, (e) d0 =6 mm, n = 0.4, x* = 1.1833; (III) four parallel bubbles: (f) d0 = 6 mm, n = 0.4, x* =1.1833, (g) d0 = 10 mm, n = 0.4, x* = 1.0800, (h) d0 = 10 mm, n = 0.8, x* = 1.0800.

coalescing bubble finally breakup into two daughter bubbles, this may be because the surface tension and viscous dissipation could

maintain the bubble shape and avoid breakup.35 While with the bubble diameter increase to 8 mm as shown in Figure 8b, the 4856



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Figure 9. Middle two of four parallel bubbles coalescence and breakup process: (a) d0 = 2 mm, n = 0.4, x* = 1.5000; (b) d0 = 6 mm, n = 0.4, x* = 1.2500; (c) d0 = 6 mm, n = 0.2, x* = 1.2500; (d) d0 = 10 mm, n = 0.4, x* = 1.0900.

not hold the large amount of kinetic energy imposed by surrounding flow. Also, the enhancement of the shear-thinning effect due to decrease of flow index of power-law fluid speeds up the breakup process of the coalescing bubble, as shown in Figure 8c. With the increase of bubble number, it would be difficult for all four bubbles to merge thoroughly in horizontal direction and coalesce into one big bubble, breakup of coalescing bubble become frequent as shown in Figure 8e and f. When the initial bubble size increases to 10 mm as in Figure 8g, there are bulges and hollows formed on the surface of coalescing bubble. The irregularity of bubble shape results in the instability of gas bridges;36 in addition, the gas bridges suffer from a great external force from the around fluid due to the great buoyancy during big bubble rising, therefore, the big coalescing bubble breaks up into two daughter bubbles with the generation of small satellite bubble. Figure 8g and h show the effect of rheological property on coalescence and breakup processes of four big bubbles. Low flow index of power-law fluid contributes to the stability of the gas bridge, while high flow index reduces the violence of bubble collision and deformation. The relative high viscous resistance results in the collapse of gas bridges at early time, the generated bubble in the middle was ruptured into two small bubbles and sucked into the wake of big bubbles on two sides. The coalescence processes of big bubble and small bubble agree well with in line coalescence of two-unequal bubbles in nonNewtonian fluid.5 Little satellite bubbles could be formed due to the collapse of gas bridges under the effect of bubble interactions and fluid stress. As shown in Figure 8g, at t = 0.100 s, the gas bridges ruptured and the satellite bubble formed. The cracked piece of gas bridges tried to recover to its stable shape in the

surrounding hydrodynamic conditions. During the recovering process, the deformation of the little bubble could result in the continual change of the flow field. In addition, the little satellite bubble suffered the wake effect from the big leading bubble. The complex flow field interaction and bubble deformation may lead to the asymmetric motion of the little satellite bubble. 4.3. Coalescence of Part of Four Bubbles. 4.3.1. Coalescence of the Two Middle Bubbles. For four parallel bubbles, the frequent happened situation is that two neighboring bubbles interact and coalesce with each other. The reason of two middle bubbles coalescence may be that the bubbles are not close enough for four bubbles merging into one completely, and the two middle bubbles may suffer from more intensive interaction from the bubbles on two sides. Parts a, b, and d of Figure 9 show the effect of initial bubble diameter on coalescence and breakup processes when flow index n = 0.4. The coalescence process of two middle bubbles agrees well with two parallel bubbles coalescence process obtained from experiment and simulation reported in the literature.18,19 The contraction of coalescing bubble in horizontal direction after coalescence for small bubbles was obvious as shown in Figure 9a. However, with the increase of bubble size, the contraction becomes not obvious due to the enhanced wake interaction of the side bubbles. As bubble diameter increases to d0 = 10 mm, as shown in Figure 9d, the coalescing bubble separates before merging into one completely and one of the separated bubbles breaks up asymmetrically. The rising velocity was increased and the wake effect was intensified because of the increase of bubble diameter, turbulence of flow field was induced by the big rising bubble as well. The coalescing bubble was separated and sucked into the wake of bubbles on two sides due to strong wake effect. The asymmetric breakup and 4857



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Figure 10. Coalescence and breakup of two bubble pairs: (a) d0 = 2 mm, n = 0.2, x* = 1.5000; (b) d0 = 8 mm, n = 0.2, x* = 1.2000; (c) d0 = 8 mm, n = 0.4, x* = 1.2000; (d) d0 = 10 mm, n = 0.2, x* = 1.1300; (e) bubbles coalescence process in 1% CMC solution from reference 20. Bubble volume, 7.5 × 10−6 m3; initial horizontal separation between bubbles, 3 mm.

bubble for flow index n = 0.2 breaks up symmetrically into two daughter bubbles and coalesce with the bubbles on two sides. Bubble breakup depends on local hydrodynamics and particularly the effects of wake shear and bubble instability.38 The coalescing bubble in the middle was unstable after coalescence; the neck formed in the middle of the coalescing bubble thinned and finally ruptured due to the attractive interaction of the bubbles on two sides. The two daughter bubbles were sucked into the wake of bubbles on two sides and finally coalesced. This coalescence process is similar with oblique coalescence of two

coalescence situation take place only for big bubble (d0 = 10 mm) in the simulation. Compared to small spherical and ellipsoidal bubble, for spherical cap bubble (Figure 9d), the coalescence and breakup processes were dominated by the wake shear of the leading bubble. The microscale of turbulence and wake vortices could lead to the asymmetrical bubble behavior,37 since tiny deformation of bubble shape would result in the variations of bubble velocity and flow field structure. The effect of flow index on coalescence and breakup of two middle bubbles was compared in Figure 9b and c. The coalescing 4858



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initially spherical bubbles reported.39 It would be inferred that the decrease of flow index enhances the shear-thinning effect of non-Newtonian fluid, which make the breakup of the unstable coalescing bubble easier. 4.3.2. Coalescence of Two Pair of Four Bubbles. Coalescence and breakup of two bubble pairs could also take place as shown in Figure 10. Smaller bubble often has a larger capillary pressure to resist deformation, so bubble with smaller diameter usually takes spherical shape when rising in power-law fluid as shown in Figure 10a. Two bubble pairs attract and get close to each other in horizontal direction with bubble rising, the coalescence process is consistent with two bubbles coalescence, as stated above. However, it is notable that two bubble pairs coalesce asynchronously due to enhanced bubble attraction and collision. Compared to a small rigid bubble, the big bubble is easily deformable induced by flow field around bubble and the bubble interactions, and the coalescence process is much different from that of small bubble as shown in Figure 10d. The two middle bubbles formed long tails because of bubble interaction and wake effect and were sucked into the wake of the bubbles at two sides. Two bubble pairs experienced oblique bubble interaction and evolved to in line bubble coalescence. At the same time, satellite bubbles were formed during bubble coalescence. The satellite bubble is generated when the bubbles are large and more susceptible to distortion due to their increased flexibility.38 Parts b and c of Figure 10 compare the shear-thinning effect of power-law fluid on coalescence and breakup processes of two bubble pairs with initial diameter d0 = 8 mm. The flow index has a significant impact on the bubble interactions due to the variation of local viscosity distribution. For flow index n = 0.2, the strong shear thinning effect pinched off the two middle bubbles; the ruptured bubbles then were sucked into the wake of the bubble on two sides and coalesced with them respectively. While for flow index n = 0.4, the shear-thinning effect is not as intensive as that of n = 0.2; the middle two bubbles were only sucked into the bubble wake but not pinched off. De Kee and Chhabra20 reported the coalescence process of multiple parallel bubbles rising in CMC aqueous solution, the process of bubble deformation and entering into the wake of another bubble agree well with this simulation, as shown in Figure 10e.

theory, a general correlation predicting critical bubble interval for bubble coalescence was proposed, which would be helpful for gas−liquid contact equipment design and application.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 21076139), the State Key Laboratory of Chemical Engineering (SKL-ChE08B03), and the Programs of Introducing Talents of Discipline to Universities (B06006). The present study is supported by High Performance Computing Center of Tianjin University, China.

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5. CONCLUSION On basis of the previous studies of coalescence and interaction of two and three parallel bubbles rising in non-Newtonian fluid, VOF-CSF method was applied to study coalescence and breakup of multiple parallel bubbles in non-Newtonian fluid systematically. The effects of the bubble number, initial bubble diameter, initial bubble interval and rheological properties of nonNewtonian fluid on coalescence and breakup processes were investigated. Different regimes of bubble coalescence and breakup were analyzed according to Re and Eo numbers. Apart from all bubble coalescence into one and the subsequent breakup, two of four bubbles tend to interact and coalesce with each other. The coalescence and breakup processes of two bubbles agree well with the results in the literature. The study shows that the initial closeness of parallel bubbles is a key factor in order to favor the drainage of the liquid film and bubble coalescence, the increase of bubble size and decrease of flow index of non-Newtonian fluid could enhance the wake effect, rising velocities, and turbulence of flow field, which facilitate bubble coalescence and breakup. Breakup or not of the coalescing bubbles depends mainly on hydrodynamics and physical properties of fluid. By means of critical approach velocity

NOTATION ⃗ D = stress tensor, N·m−1 de = volume equivalent bubble diameter, mm d0 = initial bubble diameter, mm Eo = Eötvös number F = volume fraction function Fs = body force caused by surface tension, N·m−3 g = gravitational acceleration, m·s−2 K = consistency coefficient, Pa·sn m = number of bubbles n = flow index n̑ = normal vector n⃗ = unit normal vector p = pressure, Pa Re = Reynolds number Req = the equivalent radii of the bubble t = time, s t* = dimensionless time u⃗ = velocity vector, m·s−1 U∞ = the rise velocity of a single bubble v = the approach velocity of the bubbles, m·s−1 vm = critical approach velocity for bubble coalescence, m·s−1 VB = volume of the bubble, mm3 We = Weber number Wc = critical Weber number x = horizontal bubble interval between the centers of bubble, mm x* = dimensionless horizontal bubble interval between the centers of bubble x0 = initial horizontal bubble interval between the centers of bubble, mm xc = critical initial horizontal interval of bubble coalescence, mm xc* = dimensionless critical initial interval of bubble coalescence

Greek Letters

γ̇ = shear rate, s−1 μ = viscosity, Pa·s μ(F) = kinematic viscosity coefficient, m2·s−1 ρ(F) = density, kg·m−3 κ = interfacial curvature σ = surface tension, N·m−1

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g = gas phase l = liquid phase

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