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Interaction of two in-line bubbles of equal size rising in viscous liquid
CJCHE-01515; No of Pages 9 Chinese Journal of Chemical Engineering xxx (xxxx) xxx
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Chinese Journal of Chemical Engineering journal homepage: www.elsevier.com/locate/CJChE
Article
Interaction of two in-line bubbles of equal size rising in viscous liquid☆ Zhen Tian, Xi Li ⁎, Youwei Cheng, Lijun Wang College of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 24 April 2019 Received in revised form 23 May 2019 Accepted 4 June 2019 Available online xxxx Keywords: Bubble column reactor Computational fluid dynamics Bubble interaction Bubble rise velocity Volume-of-fluid method Bubble aspect ratio
a b s t r a c t The interaction of bubbles is the key to understand gas–liquid bubbling flow. Two-dimensional axis-symmetry computational fluid dynamics simulations on the interactive bubbles were performed with VOF method, which was validated by experimental work. It is testified that several different bubble interactive behaviors could be acquired under different conditions. Firstly, for large bubbles (d: 4, 6, 8, 10 mm), the trailing bubble rising velocity and aspect ratio have negative correlations with liquid viscosity and surface tension. The influences of viscosity and surface tension on leading bubble are negligible. Secondly, for smaller bubbles (d: 1, 2 mm), the results are complicated. The two bubbles tend to move together due to the attractive force by the wake and the potential repulsive force. Especially for high viscous or high surface tension liquid, the bubble pairs undergo several times acceleration and deceleration. In addition, bubble deformation plays an important role during bubble interaction which cannot be neglected. © 2019 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.
1. Introduction Bubble dynamics in gas–liquid flow plays a significant role in many industrial and natural processes, such as coal and petroleum processing, chemical engineering and cleaning [1]. A great amount of the early work concentrates on single bubble rising velocity [2,3]. However, in practical applications bubble swarm mostly emerges [4], and hydrodynamics of each bubble would be evidently affected by the wake of the neighboring bubbles, which becomes more apparent in the near wake region [5,6]. It is considerable importance to understand two bubble interactions for developing reliable models suitable for the momentum exchange between two phases when gas volume fractions are high and bubble interaction cannot be ignored [7]. Thus, it is crucial to understand two bubble interactions as the separation distance among them gets smaller. Experimentally, Crabtree [8], Bhaga [9] and Komasawa [10] investigated the spherical cap bubble rising in line and provided one method of calculating the coalescence velocity of the bubbles by the asymptotic wake theory, which neglected the bubble shape and radial position, and then deduced the empirical equation of the average apparent wake velocity contribution. Narayanan [11] studied bubble pairs rising velocity in aqueous glycerin solutions at Reynolds number range from 0.5 up to 80 and indicated that the coalescence of bubbles was followed the analysis of weightless solid spheres when Re b 7. When Re N 7, the coalescence of bubbles was under the influence of two additional velocities produced owing to the wake structure. Katz [12] conducted ☆ Supported by the National Natural Science Foundation of China (91334105). ⁎ Corresponding author. E-mail address: [email protected] (X. Li).
experiments about air bubbles rising in line in distilled water of small size (d ≤ 475 μm, Re ≤ 35, Eo ≤ 0.027) and pointed out that the relative velocity of two bubbles was dependent on bubble diameter. For large bubbles (349 μm ≤ d ≤ 475 μm), the relative velocity increased as the distance decreased. For small bubbles (d ≤ 349 μm), the relative velocity decreased as the bubble distance decreased which could be owing to a possible repulsive force caused by pressure gradient. Theoretically, Harper [13], using the irrotational first approximation in larger Reynolds number and inviscid liquid, found that there existed a repulsive force between two bubbles rising in line which was inversely the fourth power of the separation distance between them. Yuan [14] conducted numerical research on two spherical bubbles motion of equal size moving along their line of centers and drew the conclusion that an equilibrium distance was reached when two bubbles rose in line, ignoring shape deformation, at which the force induced by the wake interaction effect equaled to that by the pressure gradients. Watanabe [15] also analytically and numerically studied bubble pairs rising in line for Reynolds number from 5 up to 150 and predicted that a stable equilibrium distance was presented between bubble pairs, which was also owing to the equilibration between viscous attractive forces and the potential repulsive forces. However, their numerical results were not consistent with the experimental results, which was owing to the different initial bubble distance. Fan [16] considered that the viscous drag force played an dominant role and inertial force was negligible, and three mathematical models were developed for calculating the trailing bubble velocity under the action of leading bubble for intermediate Reynolds number range [Re∼ O(100)]. In addition, several similar correlations [17–23] have been put forward using an analytical approximation method, in which a new theoretically reasonable
https://doi.org/10.1016/j.cjche.2019.06.003 1004-9541/© 2019 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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artificial origin has been introduced. For low to moderate Reynolds numbers (Re ≤ 50), Ramírez-Muñoz [22] pointed out that the following three forces (inertial forces, buoyancy and quasi-steady drag) played an important role in pushing the trailing bubble motion in the leading bubble wake region. However, both the history and added-mass body acceleration forces were negligible. For moderated to large Reynolds numbers (50 ≤ Re ≤ 300), Baz-Rodríguez [23], by introducing a theoretically justified artificial origin and adapting the type of analytical solution suitable for far wake region, proposed an equation applying to calculate the axial velocity distribution. In addition, due to an upward repulsion force was generated, induced by the potential flow field ahead of the trailing bubble, the leading bubble was pushed and moved upward together. As a result of the analysis, an equilibrium distance among bubble pairs was existed, which was in accordance with the conclusions of Yuan [14] and Watanabe [15]. In view of the preceding arguments, it is obvious that some uncertainties still exist concerning the interaction of two in-line bubbles. Experiments mainly focus on the spherical or spherical cap bubble, which neglect ellipsoid bubble. Theoretical methods are often based on some assumption and simplification on account of the strong nonlinearity caused by bubble deformations. The correlations put forward till now on the bubble drag force are still restricted to spherical bubbles [6]. While detailed analytical formulas based on some assumptions provide a better understanding of underlying principles, they are not generally suitable for real industrial situations in which liquid viscosities or surface tension might be significant. In this article, therefore, the VOF method was exploited to simulate bubble behavior. To validate the model, numerical results were compared with those of the experiments for trailing bubble. The effects of bubble diameter, liquid viscosity, surface tension on the interactive bubble were systematically researched within the laminar flow regime, in order to improve comprehension of bubble pair interaction rising in line from a numerical standpoint. 2. Numerical Simulation 2.1. Continuity and momentum equations As indicated by previous experimental and simulated work [1, 24–27], the inline configuration is found to be stable and bubble shape is axisymmetric for interaction of two in-line bubbles of equal size rising in viscous liquid, which is also validated by experimental work. A 2D-
axisymmetry model compared with three-dimensional simulation is enough to describe bubble rising trajectory and shape during bubble interactions. Thus in this study, a simplified physical model of simulation process is adopted for minimizing the computational costs. The simulations are conducted in transient and axisymmetric models by using Ansys Fluent, and the axis of symmetry is along the center of bubble column. Both gas and liquid phases are assumed as incompressible fluids, and the flow is Newtonian and laminar. Continuity and momentum equations are as follows [28]: ∇u¼0
ð1Þ
h i ∂ðρuÞ þ ∇ðρuuÞ ¼ −∇p þ ∇ μ ∇u þ ∇T u þ F s þ ρg ∂t
ð2Þ
where p represents the pressure. Fs stands for the source term that represents surface tension contribution. The density and viscosity in Eq. (2) are estimated supposing linear contributions of two phases when a computational cell is occupied by two phases. ρ ¼ ρl þ ρg ð1−F Þ
ð3Þ
μ ¼ μ l þ μ g ð1−F Þ
ð4Þ
2.2. Volume fraction equation The Volume Fraction Equation (VOF) [29] is applied to simulate interaction of the two in-line bubbles of equal size rising in Newtonian viscous liquids. In the VOF model, a single momentum equation is established to express the two fluids, and each phase volume fraction in all computational cells is marked by the volume ratio function F which is utilized to distinguish each phase. When the indicator function F equals to 0 or 1, it means that the cell is occupied by pure gas or pure liquid, respectively, and when F is within 0-1, it means that the cell is occupied by both gas and liquid [30]. The transport equation of fluid volume function is as follows: DF ∂F ¼ þ ðu ∇ÞF ¼ 0 Dt ∂t
ð5Þ
Fig. 1. Illustration of the hydrodynamic features of bubble depicted in the present study.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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8 <0 F ¼ 0bFb1 : 1
in bubbles in interface in liquid fluid
ð6Þ
2.3. Continuum surface tension equation The continuum surface force model [31] is applied to compute the surface tension force for cells existing gas–liquid interface. The piecewise-linear approach put forward by Youngs [32] is applied as the geometric reconstruction scheme to represent the phases interface. Fs ¼ σ
ρκ∇F l 0:5 ρg þ ρl
3
Table 1 Parameters selected for the current study Case
T /K
ρl /kg·m−3
μ /mPa·s
σ /mN·m−1
d /mm
Mo
1 2 3 4 5 6 7 8 9
293 – – 323 373 – 423 – 473
970 970 970 951 921 921 859 859 829
300.3 99.4 39.8 19.2 99.4 99.4 59.6 59.6 39.8
20.3 20.3 20.3 19.2 16.5 82.5 14.4 72.0 12.6
1, 2, 4, 6, 8, 10 1, 2, 6 1, 2, 6 1, 2, 4, 6, 8, 10 1, 2, 4, 6, 8, 10 1, 2, 6, 8 1, 2, 4, 6, 8 2, 6 1, 2, 4, 6
9.8 0.12 2.9×10-3 2.0 0.23 1.9×10-3 4.8×10-2 3.9×10-4 1.4×10-2
ð7Þ
where κ = ∇ ⋅ n, n = (n/|n|), n = ∇ Fq. 2.4. Model geometry and solution method The rectangle computational domain is with a width of 12 radii and a height 180 radii, which is enough to eliminate the boundary effect [25]. In addition, the geometry structured grids are used with typical grid size of R/70 equably, which is enough to rule out the mesh effect and ensure high computing accuracy, especially for bubble shape during interaction. The pressure-based Navier–Stokes solution algorithm is used as the solution method, where a finite volume methodology is applied to discretize the governing equations. The Pressure-Implicit with splitting of Operators (PISO) pressure–velocity coupling scheme is adopted in the momentum equation, which is remarkable for transient calculations. The discretization scheme applied for pressure equation is pressure staggering option (PRESTO!). The QUICK scheme is used as discretization scheme for momentum equation. The Euler explicit formation is adopted to solve the volume fraction equation where the maximum Courant number is specified as 0.25. Implicit body force is taken into account to avoid poorly segregated algorithms convergence, which may be caused by large body forces existing in multiphase flow, such as surface tension forces and buoyancy force induced by high density differences. Variable time stepping is applied to automatically adjust the time step by specifying the global courant number as 0.25. The value of absolute criteria is 10−4 for velocity and continuity residual as a convergence criterion. As reported by Komasawa [10], the bubble behavior, such as rising velocity, bubble geometry and liquid velocity profile around a bubble, for free rising bubble had no significant differences with that of bubble held stationary. Considering that fixed frames needed large computational domains necessarily, which then consumed much time, the leading bubble position was set as moving reference frames. This method implied that a velocity-inlet boundary condition was adopted at the top of the region with a uniform velocity. The magnitude of velocity was equivalent to the velocity of the isolated bubble, and the inlet was consisted of liquid phase. In addition, a downward moving wall velocity of the same value as the isolated bubble and a no slip wall condition were applied in the wall boundary condition. Symmetry boundary condition was adopted at the axis. A pressure outlet boundary condition was specified at the bottom of the computational domain. Firstly, the single leading bubble was simulated to acquire flow distribution, such as axial velocity, radial velocity and volume fraction of two phases, inside and around the bubble by using above assumption, which indicated appropriate results on account of the position of the leading bubble undergoing only very small transformation. Then the flow distribution around the leading bubble was copied and pasted into a new position under the leading bubble with a separation distance equivalent to 75% of a minimum distance between bubbles (Lmin), above which there was no interaction between bubbles [33,34]. The stabilization length of the liquid flow field under the leading bubble (Lmin) was defined to highlight the influence of the bubble in its surroundings.
The standard used to assess the values of Lmin was based on determining the position where the absolute value of the normalized difference between the inlet velocity Uin and the column centerline velocity Uz was equivalent to 0.01 (|Uz-Uin |/Uin = 0.01). By this means, the initial flow field for the simulation of two in-line bubbles, with identical bubble shape, was created, divided by a requested distance, where the two bubble interaction was roughly ignored. For ease of understanding, the features depicted are schematically illustrated in Fig. 1. Fig. 1 (a) gives a global view of the following hydrodynamic features: the bubble velocity, minimum interaction distances below (Lmin) the leading bubble. Fig. 1(b) gives a global view of two interaction bubble at initial condition. 2.5. Experiment In addition, some experimental work has been carried out to validate the numerical method. The experimental setup contained two parts: bubble column and high speed camera system. The bubble column consisted of a cylindrical stainless-steel column with dimensions 50mm diameter and 600-mm height. On both the front and rear sides of the column three pairs of high strength compression quartz windows, with a viewing area of 20 mm × 100 mm, were uniformly installed along the test section. A thermostat was installed outside of the column, heated by temperature controlling systems. The experiment was carried out at temperature from 293 K up to 473 K. Nitrogen from the highpressure cylinders was introduced into the bottom of column through various diameter orifice nozzle (do: 0.5 mm, 1.12 mm, 2.5 mm). A pair of bubbles was released in succession from a single nozzle through quick-closing valve, and shared the same axis of symmetry during interaction. Silicone oil was used as the liquid phase. The initial liquid level exceeded the nozzle about 500 mm. The transient bubble rising velocity and bubble shape were obtained by a high-speed camera (Mikrotron CUBE7). The orifice was used to calibrate the image pixel resolution. Image resolution was 0.03 mm·pixel−1, and errors caused by distinguishing bubble edges was under 3 pixel. The relative error was
Fig. 2. Simulated drag coefficient of single bubble compared with experimental data, Bhaga and Turton model.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
under ± 4.5% due to the minimum diameter of the bubbles that was about 2 mm. In addition, with increase of bubble diameter, the relative error decreased. Liquid density was computed by measuring mass and volume separately. Liquid viscosity and surface tension was measured by rotational rheometer (HAAKE RS6000) and
video-based, contact angle device (OCA 20) separately. Uncertainties estimated at 95% confidence were 2.0% and 1.0% for viscosity and surface tension, respectively. Table 1 shows the physical properties of the silicone oil at different temperature. The simulation conditions are also presented in Table 1.
Fig. 3. Numerical data of the bubbles interface, velocity field and streamlines in the liquid phase, as a function of L/R compared with experimental image at different temperature.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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3. Result and Discussion 3.1. Single bubble rising The first case considered is that an isolated bubble rising in liquid. The accuracy and validity of the numerical simulation are checked firstly, by carrying out some comparisons of the isolated bubble rising simulation results with experimental results and correlations or theoretical equations available in the literature. Single bubble rising in a viscous Newtonian fluid has been researched both numerically and experimentally [1,6,26]. Bhaga [35] and Turton [36] studied the bubble rising velocity in viscous liquid and proposed the CD-Re relationship. As shown in Fig. 2, the simulated gas bubble drag coefficient is in accordance with experimental data and the calculated results by Eqs. (8), (9). In general, the CFD-VOF model used in this paper is practicable. h i1=0:9 C D ¼ ð2:67Þ0:9 þ ð16= ReÞ0:9 CD ¼
ð8Þ
24 0:413 1 þ 0:173 Re0:657 þ Re 1 þ 16300 Re−1:09
ð9Þ
3.2. Sequential images of two in-line bubbles When two bubbles rise in-line, the trailing bubble hydrodynamic behavior are evidently influenced by the leading bubble by changing the
1.8
U/U0
1.6
Experiment d: 2mm-UB/U0 d: 4mm-UB/U0 d: 8mm-UB/U0 Fan-2003 Rodriguez-2013
velocity field and pressure field, which becomes more apparent in the leading bubble wake neighboring region. In this connection, it is specifically significant to obtain bubble pairs interaction behavior along with the two bubble separation distance becoming shorter. In the following sections, in order to access detailed knowledge of the bubble interaction behavior during the approaching process, the interactions of two in-line bubbles rising under buoyancy in a viscous fluid at various conditions are simulated. For convenience, label the leading bubble A and the trailing bubble B. Firstly, it is significant to present a qualitative description on the bubble shape and velocity change process of the two bubbles as the trailing bubble approaches the leading one. Fig. 3 shows simulated sequential images about the description of bubbles shape, vector velocity and axial velocity streamlines as the dimensionless separation distance (L/R) among them becoming shorter, compared with experimental images. The conditions of Fig. 3 are at temperature 293, 423 and 473 K with the bubble diameter of 2 mm and 6 mm. It is demonstrated that the inline configuration is stable, and bubble shape is axisymmetric during bubble interaction from experimental observation. From the observation of Fig. 3, a very close qualitative accordance is observed among the calculated images and photographs. The leading bubble shape remains mostly unchanged, and just the bottom is progressively flattened as the trailing bubble approaches. In addition, elongation of the trailing bubble can be found in both the simulation and experiment image. The reason for this phenomenon could be that the leading bubble wake generates high velocity area at the bottom, which pushes the trailing bubble becomes narrow. Furthermore, as temperature increases, the initial Simulation Experiment d: 2 mm-EB/E0 d: 2 mm-EB/E0 d: 2 mm-EA/E0 d: 4 mm-EB/E0 d: 4 mm-EB/E0 d: 8 mm-EB/E0 d: 4 mm-EA/E0 d: 8 mm-EB/E0 d: 8 mm-EA/E0
1.8 1.6 1.4
E/E0
Simulation d: 2mm-UB/U0 d: 2mm-UA/U0 d: 4mm-UB/U0 d: 4mm-UA/U0 d: 8mm-UB/U0 d: 8mm-UA/U0
1.2
1.4
5
1.0 1.2
0.8 0.6
1.0 1
10
1
10
L/R
L/R
(a) case 1, T :293 K
U/U0
1.8 1.6 1.4
1.8
Simulation Experiment d: 2 mm-EB/E0 d: 2 mm-EB/E0
1.6
d: 2 mm-EA/E0
d: 4 mm-EB/E0
d: 4 mm-EB/E0
1.4
d: 4 mm-EA/E0 d: 6 mm-EB/E0
E/E0
Simulation Experiment d: 2 mm-UB/U0 d:2 mm-UB/U0 d: 2 mm-UA/U0 d:4 mm-UB/U0 d: 4 mm-UB/U0 Fan-2003 d: 4 mm-UA/U0 Rodriguez-2013 d: 6 mm-UB/U0 d: 6 mm-UA/U0
2.0
1.2
d: 6 mm-EA/E0
1.0
1.2
0.8
1.0
0.6
1
10
100
1
10
100
L/R
L/R
(b) case 9, T :473 K Fig. 4. Influence of bubble diameter on U/U0 and E/E0 as a function of L/R.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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Table 2 Correlations for trailing bubble rising velocity Investigator
Correlations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !#) ( " Z dU B U B C D0 Re0 d 12 ν t dU B =dτ pffiffiffiffiffiffiffiffiffiffi ddτ − ¼ 2g 1− þ 1− exp − 16ðL þ dÞ d π t0 dt U0 2 t−τ ! !2 ReA Re W 1 1 ReA 1 ð1−U B Þ dU B ð1−WÞ 1− A −1 ¼ 0 þ 1þ βðU B −WÞ þ x þ x0 16 16 W 0 x þ x0 48 x þ x0 W 0 dx " !# C Re 1 x = L/d x0 = 0.32621 + 0.01689ReA UB∗ = UB/U0 W ¼ DA 1− exp − A 2 16 x þ x0
Fan [16] Rodríguez [22]
−1
C DA ReA 1 1 þ ½8= ReB þ 1=2ð1 þ 3:315= ReB 0:5 Þ β¼ −1 2 16 x þ x0 1 þ ½8= ReA þ 1=2ð1 þ 3:315= ReA 0:5 Þ " !# ! 2 ∂W 2 ∂U C Re 1 1‐2:21 Re‐0:5 B U B B −1 ¼ 0 W ¼ 1− DA 1− exp − A β¼ þ U B β− W βþ C D0 ∂x 3C DA 2 16 x þ x0 ∂x 1‐2:21 Re‐0:5 A
W0 ¼ Rodríguez [23]
bubble shape of two bubbles varies from ellipsoidal to spherical cap and a recirculation zone generally appears under the leading bubbles bottom, and a conspicuous bubble interaction appears at larger separation distances, which reflecting that the minimum stabilization length (Lmin) increases with temperature increase. In addition, when bubble diameter is 2 mm, the leading bubble is evidently pushed by the trailing one for the entire range of distances between bubbles. However, for bubble diameter equals to 6 mm, the location of the leading bubble remains roughly unaffected. 3.3. Bubble diameter influence As shown in Fig. 4, the transmutation of the experimental and numerical results about U/U0 (U0: the isolated bubble velocity. UA: leading bubble velocity. UB: trailing bubble velocity) and E/E0 (E0: isolated bubble aspect ratio. EA: leading bubble aspect ratio. EB: trailing bubble aspect ratio) with the dimensionless separation distance (L/R), are presented respectively with various diameters. The relative error between numerical and experimental results is below 8%. The difference may originate from the difference of the initial bubble distance between simulation and experiment. Several trends could be obtained from the results. For separations exceeds 2 radii, both the leading and trailing bubble rising velocities increase with decreasing distance between bubbles. The leading bubble aspect ratio is gradually decreasing with the decreasing distance, whereas the trailing bubble aspect ratio is gradually increasing. For all diameter bubbles, the bubble diameter has a positive effect on the trailing bubble velocity, consistent with the trailing bubble aspect ratio. However, different trends related to bubble size are observed for the behavior of the leading bubble. With regard to larger bubbles (d ≥ 4 mm), the leading bubble was little affected by the existence of the trailing bubble when the two bubble separation distance exceeds 10 radii. When separation distance under 10 radii, there is slightly increase in bubble velocity and bubble aspect ratio gets smaller. With case 1:UB/U0 case 1:UA/U0 case 4:UB/U0 case 4:UA/U0 case 5:UB/U0 case 5:UA/U0 case 7:UB/U0 case 7:UA/U0 case 9:UB/U0 case 9:UA/U0
1.8
U/U0
1.6
U B ¼ U B =U 0
x ¼ L=d
regard to the smaller bubbles (d ≤ 2 mm), the leading bubble is pushed by the trailing bubble and bubble velocity increases significantly with the decreasing separation distance. The two bubbles tend to rise together, and the separation distance between bubble pairs is slowly reduced, until bubble coalescence takes place. This phenomenon is also experimentally observed by Katz [12] and numerically by Gumulya [24] that for smaller bubbles the leading bubble rising velocity increased considerately (up to 30% of the single bubble velocity), and it was explained by the pressure gradients contribution, which became more dominant for small bubbles. However, for larger bubbles, the wakeinduced relative motion was dominant over the pressure gradients. When separations under 2 radii, for large bubbles, there exists a certain deceleration effect, which may be owing to that an accessional force is generated to move away intervening liquid between the two bubbles, and the bubble aspect ratio of the trailing bubble is also becoming small. Some comparisons about the increase of the trailing bubble velocity are also made between simulated results and the wake correlations proposed by Fan [16] and Rodríguez [22,23] which are presented in Table 2. As shown in Fig. 4, the two correlations also shows that bubble diameter has a positive correlation with the increase of the trailing bubble rising velocity. However, the predicted value underestimates the results. Both the Fan and Rodríguez correlations are based on the assumption that bubble shape is spherical. However, the trailing bubble becomes narrow during actual bubble interaction, which causes the drag force of trailing bubble to become small. Thus the two correlations underestimate the trailing bubble velocity. When the two bubbles distance at 2.5 radii, the bubble rising velocity and aspect ratio, ratio to isolated bubble value, are presented in Fig. 5 respectively with the various bubble diameters. As presented in Fig. 5, the trailing bubble rising velocity has a positive correlation with the diameter, consistent with bubble aspect ratio. However, the leading bubble rising velocity decreases firstly with the diameter and then tends to be roughly constant (UA/U0 = 1.07), and the bubble aspect ratio presents same tread. case 1:EB/E0 case 1:EA/E0 case 4:EB/E0 case 4:EA/E0 case 5:EB/E0 case 5:EA/E0 case 7:EB/E0 case 7:EA/E0 case 9:EB/E0 case 9:EA/E0
1.8
1.6
1.4
E/E0
2.0
x0 ¼ 0:1859 Re0:5 A
1.2
1.4 1.0
1.2 0.8
1.0 0
2
4
6
d /mm
8
10
0
2
4
6
8
10
d /mm
Fig. 5. Numerical result of U/U0 and E/E0 when bubble distance at 2.5 radii.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
U/U0
1.3 1.2
1.10
Mo: 9.8-EB/E0 Mo: 9.8-EA/E0 Mo: 0.12-EB/E0 Mo: 0.12-EA/E0 Mo: 2.9×10-3-EB/E0 Mo: 2.9×10-3-EA/E0
1.08 1.06 1.04
E/E0
Mo: 9.8-UB/U0 Mo: 9.8-UA/U0 Mo: 0.12-UB/U0 Mo: 0.12-UA/U0 Mo: 2.9×10-3 UB/U0 Mo : 2.9×10-3 UA/U0 Fan-2003 Rodriguez-2013
1.4
7
1.02 1.00 0.98
1.1
0.96 0.94
1.0
0.92
10
10
L/R
L/R
(a) d: 2 mm Mo: 9.8-UB/U0 Mo: 9.8-UA/U0 Mo: 0.12-UB/U0 Mo: 0.12-UA/U0 Mo: 2.9×10-3-UB/U0 Mo: 2.9×10-3-UA/U0 Fan-2003 Rodriguez-2013
1.8
U/U0
1.6 1.4
Mo: 9.8-EB/E0 Mo: 9.8-EA/E0 Mo: 0.12-EB/E0 Mo: 0.12-EA/E0 Mo: 2.9×10-3-EB/E0 Mo: 2.9×10-3-EA/E0
1.8 1.6 1.4
E/E0
2.0
1.2 1.0
1.2
0.8 1.0
0.6 1
10
1
100
10
100
L/R
L/R
(b) d: 6 mm Fig. 6. Influence of liquid viscosity on U/U0 and E/E0 as a function of L/R.
As presented in Fig. 6, Morton number replaces liquid viscosity to describe its influence on bubble interactions. For small bubble of diameter 2 mm, trailing bubble rising velocity increases slightly with the decreasing viscosity under the same bubble distance, while leading bubble velocity has minor decrease. In addition, the influence of viscosity on bubble aspect ratio is small both for the leading and trailing bubbles, for large bubble of 6-mm diameter, trailing bubble velocity increases significantly with the decreasing viscosity, while almost no difference was made on leading bubble velocity. Meanwhile, bubble aspect ratio has the corresponding change with bubble rising velocity. Trailing bubble aspect ratio seems to be very sensitive to the two bubble separation distance, with this sensitivity increase as the viscosity decreases. The results of Fan and Rodríguez correlations are also presented on Fig. 6. The two correlations also show that liquid viscosity has a negative correlation with the trailing bubble rising velocity. Due to their negligence of bubble shape deformation accompanying with bubble interaction, the two correlations all underestimate trailing bubble velocity. For smaller bubble of 1-mm diameter, the interaction of two bubbles becomes complicated. As shown in Figs. 7 and 8, the two bubbles tend to slowly reduce the separation distance and rise together until coalescence finally occurs. For case 1 and case 4, the pairs undergo several time acceleration and deceleration when the bubble separation distance is below 3 radii which may be interpreted by the potential flow field caused by the trailing bubble, and this force then pushes the leading bubble to move upward. As reported by literature [14], an equilibrium distance between the two bubbles rising in-line might therefore be attained. However, on account of its inherent instability in practice
and that the equilibrium distance is also relating to the two bubble initial separation distance, the existence of the equilibrium distance is very difficult to demonstrate [15]. 3.5. Surface tension influence As shown in Fig. 9, with the increasing liquid surface tension, trailing bubble rising velocity and aspect ratio decrease, while surface tension has no influence on leading bubble rising velocity and aspect ratio. At
1.7
case 1:UB/U0 case 4:UB/U0 case 5:UB/U0 case 7:UB/U0 case 9:UB/U0
1.6 1.5 1.4
U/U0
3.4. Viscosity influence
case 1:UA/U0 case 4:UA/U0 case 5:UA/U0 case 7:UA/U0 case 9:UA/U0
1.3 1.2 1.1 1.0 0.9 0
10
20
30
40
50
t /s Fig. 7. Influence of liquid viscosity on U/U0 as a function of t as bubble diameter at 1 mm.
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
12 10 8
L/R
In addition, for small bubbles (d ≤ 2 mm), when surface tension increases four times, the two bubble is hard to coalesce. Fig. 10 shows two bubble distance with time for different surface tension. For case 5 when bubble diameter is at 2 mm, the two bubbles undergo 13.63 s to coalesce. However, for case 6 when bubble diameter also at 2 mm and with four times surface tension, the two bubbles keep a constant distance and rise together until the initial distance is set as 0.4Lmin. The surface tension with high values means that bubble shape mostly keep spherical during bubble interactions, which is consistent with the assumption of previous works, finding the existence of an equilibrium distance [14].
case 1 case 4 case 5 case 7 case 9
6 4
4. Conclusions
2 0
10
20
30
40
50
t /s Fig. 8. Vertical distance between two bubbles as a function of t as bubble diameter at 1 mm.
a low surface tension, the trailing bubbles deform more easier while approaching, leading to an increase of the trailing bubble velocity, which is also reported by Hasan [26,37,38] that the pair of vertically aligned ellipsoidal bubbles showed stronger attraction than the spherical bubble pair. The Fan and Rodríguez correlations present a better matching result when surface tension is larger in which condition bubble deformation is hard and bubble shape is nearly spherical, consistent with correlations assumption.
The Volume-of-Fluid method (VOF) was used to simulate the interaction of two in-line bubbles of equal size rising in viscous liquid. The simulated conditions included a wide range, that was Re (0.007-25), Mo (3.9E43.9×10-3-9.8) and Eo (0.04-50), which ensured under laminar regime. The method was validated by comparison with the present experimental measurements and existing correlations. The effect of bubble diameter, liquid viscosity and surface tension on the interaction of two bubbles were studied systematically. The following conclusions could be obtained: 1) For the interaction of two in-line bubbles: To bubble velocity, the leading bubble has minor change and the trailing increases. To bubble shape, the leading bubble becomes flat and the trailing becomes elongated. 2) For large bubbles, bubble diameter has a positive influence on the trailing bubble velocity, in contrast with surface tension and liquid viscosity, but little influence on the leading bubble.
case 5,d: 6 mm, -UB/U0
2.0
case 6,d: 6 mm, 5 -UB/U0
1.8
case 6,d: 6 mm, 5 -UA/U0 case 5,d: 8 mm, -UA/U0 case 6,d: 8 mm, 5 -UB/U0
1.4
E/E0
U/U0
case 5,d: 8 mm, -UB/U0
1.6
case 5, d: 6 mm, -EA/E0
1.6
case 6, d: 6 mm, 5 -EB/E0
1.4
case 5, d: 8 mm, -EB/E0
case 6, d: 6 mm, 5 -EA/E0 case 5, d: 8 mm, -EA/E0 case 6, d: 8 mm, 5 -EB/E0
1.2
case 6, d: 8 mm, 5 -EA/E0
case 6,d: 8 mm, 5 -UA/U0 Fan-2003 Rodriguez-2013
1.2
case 5, d: 6 mm, -EB/E0
1.8
case 5,d: 6 mm, -UA/U0
1.0 0.8
1.0
0.6
1
1
10
10
L/R
L/R
Fig. 9. Influence of surface tension on U/U0 and E/E0 as a function of L/R.
case 5,d:2mm,
12
case 6, d
7 6
8
5
L/R
L/R
10
6
4
4
3
2
2
-2
0
2
4
6
8
10
12
14
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
t /s
t /s
Fig. 10. Vertical distance between two bubbles varying with time.
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3) For small bubbles, the interaction of two bubbles is complicated. Two bubbles tend to move together. Especially for high viscous or high surface tension liquid, the bubble pairs undergo several times acceleration and deceleration when bubble distance is lower than 3 radii. 4) By comparing with the correlations, it can be concluded that bubble deformation plays an important role at bubble interaction, which causes the correlations underestimate the trailing bubble velocity. References [1] H.A.N. Wang, Z.Y. Zhang, Y.M. Yang, H.S. Zhang, Numerical investigation of the interaction mechanism of two bubbles, Int. J. Mod. Phys. C 21 (01) (2010) 33–49. [2] A.A. Kulkarni, J.B. Joshi, Bubble formation and bubble rise velocity in gas-liquid systems: a review, Ind. Eng. Chem. Res. 44 (16) (2005) 5873–5931. [3] J.F. Harper, The motion of bubbles and drops through liquids, Adv. Appl. Mech, Elsevier 12 (1972) 59–129. [4] B. Bunner, G. Tryggvason, An examination of the flow induced by the motion of many buoyant bubbles, J. Visual 2 (2) (1999) 153–158. [5] X. Chai, X. Liu, J. Xiong, X. Cheng, Numerical investigation of bubble wake properties in the moving liquid with LES model, J. Nucl. Sci. Technol. 53 (11) (2016) 1870–1880. [6] J.B. Joshi, K. Nandakumar, G.M. Evans, V.K. Pareek, M.M. Gumulya, M.J. Sathe, M.A. Khanwale, Bubble generated turbulence and direct numerical simulations, Chem. Eng. Sci. 157 (2017) 26–75. [7] H.A. Jakobsen, H. Lindborg, C.A. Dorao, Modeling of bubble column reactors: progress and limitations, Ind. Eng. Chem. Res. 44 (14) (2005) 5107–5151. [8] J. Crabtree, J. Bridgwater, Bubble coalescence in viscous liquids, Chem. Eng. Sci. 26 (6) (1971) 839–851. [9] D. Bhaga, M. Weber, In-line interaction of a pair of bubbles in a viscous liquid, Chem. Eng. Sci. 35 (12) (1980) 2467–2674. [10] I. Komasawa, T. Otake, M. Kamojima, Wake behavior and its effect on interaction between spherical-cap bubbles, J. Chem. Eng. Jpn 13 (2) (1980) 103–109. [11] S. Narayanan, L.H.J. Goossens, N.W.F. Kossen, Coalescence of two bubbles rising in line at low Reynolds numbers, Chem. Eng. Sci. 29 (10) (1974) 2071–2082. [12] J. Katz, C. Meneveau, Wake-induced relative motion of bubbles rising in line, Int. J. Multiphase Flow 22 (2) (1996) 239–258. [13] J.F. Harper, On bubbles rising in line at large Reynolds numbers, J. Fluid Mech. 41 (04) (1970) 751–758. [14] H. Yuan, A. Prosperetti, On the in-line motion of two spherical bubbles in a viscous fluid, J. Fluid Mech. 278 (1994) 325–349. [15] M. Watanabe, T. Sanada, In-line motion of a pair of bubbles in a viscous liquid, Jsme. Int. J. B-Fluid. T 49 (2) (2006) 410–418. [16] J. Zhang, L.S. Fan, On the rise velocity of an interactive bubble in liquids, Chem. Eng. J. 92 (1–3) (2003) 169–176. [17] J.R. Muñoz, A. Soria, E. Salinas-Rodríguez, On the relative motion of two spherical bubbles rising in line and interacting by a laminar wake, AIChE Annual Meeting, Conference Proceedings 2005, pp. 689–693.
9
[18] J. Ramírez-Muñoz, A. Soria, E. Salinas-Rodríguez, Hydrodynamic force on interactive spherical particles due to the wake effect, Int. J. Multiphase Flow 33 (7) (2007) 802–807. [19] J. Ramírez-Muñoz, A. Gama-Goicochea, E. Salinas-Rodríguez, Drag force on interacting spherical bubbles rising in-line at large Reynolds number, Int. J. Multiphase Flow 37 (8) (2011) 983–986. [20] J. Ramírez-Muñoz, E. Salinas-Rodríguez, A. Soria, A. Gama-Goicochea, Hydrodynamic interaction on large-Reynolds-number aligned bubbles: drag effects, Nucl. Eng. Des. 241 (7) (2011) 2371–2377. [21] S.A. Baz-Rodríguez, J. Ramírez-Muñoz, A. Soria, In-line interaction between two spherical particles due to a laminar wake effect, Int. J. Multiphase Flow 39 (2012) 240–244. [22] J. Ramírez-Muñoz, S. Baz-Rodríguez, E. Salinas-Rodríguez, E. Castellanos-Sahagún, H. Puebla, Forces on aligned rising spherical bubbles at low-to-moderate Reynolds number, Phys. Fluids 25 (9) (2013), 093303. [23] S.A. Baz-Rodríguez, J. Ramírez-Muñoz, A. Soria, J.C. Sacramento-Rivero, Hydrodynamic interaction of two spherical bubbles rising in-line: a semi-analytical approach, Chem. Eng. Commun. 201 (5) (2014) 674–687. [24] M. Gumulya, R.P. Utikar, G.M. Evans, J.B. Joshi, V. Pareek, Interaction of bubbles rising inline in quiescent liquid, Chem. Eng. Sci. 166 (2017) 1–10. [25] J. Feng, X. Li, Y. Bao, Z. Cai, Z. Gao, Coalescence and conjunction of two in-line bubbles at low Reynolds numbers, Chem. Eng. Sci. 141 (2016) 261–270. [26] N. Hasan, Z.b. Zakaria, Computational approach for a pair of bubble coalescence process, Int. J. Heat Fluid Flow 32 (3) (2011) 755–761. [27] M. Cheng, J. Hua, J. Lou, Simulation of bubble-bubble interaction using a lattice Boltzmann method, Comput. Fluids 39 (2) (2010) 260–270. [28] E. Delnoij, J.A.M. Kuipers, W.P.M. van Swaaij, Computational fluid dynamics applied to gas-liquid contactors, Chem. Eng. Sci. 52 (21) (1997) 3623–3638. [29] C.W. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39 (1) (1981) 201–225. [30] D. Ma, M. Liu, Y. Zu, C. Tang, Two-dimensional volume of fluid simulation studies on single bubble formation and dynamics in bubble columns, Chem. Eng. Sci. 72 (2012) 61–77. [31] J. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (2) (1992) 335–354. [32] D.L. Youngs, Time-dependent multi-material flow with large fluid distortion, Numerical Methods for Fluid Dynamics, 1982. [33] R. Moissis, P. Griffith, Entrance effects in a two-phase slug flow, J. Heat Transf. 84 (1) (1962) 29–38. [34] A.M.F.R. Pinto, M.N. Coelho Pinheiro, J.B.L.M. Campos, Coalescence of two gas slugs rising in a co-current flowing liquid in vertical tubes, Chem. Eng. Sci. 53 (16) (1998) 2973–2983. [35] D. Bhaga, M.E. Weber, Bubbles in viscous liquids: shapes, wakes and velocities, J. Fluid Mech. 105 (1981) 61–85. [36] R. Turton, O. Levenspiel, A short note on the drag correlation for spheres, Powder Technol. 47 (1) (1986) 83–86. [37] Z. Yu, H. Yang, L.-S. Fan, Numerical simulation of bubble interactions using an adaptive lattice Boltzmann method, Chem. Eng. Sci. 66 (14) (2011) 3441–3451. [38] M. Manga, H.A. Stone, Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions, J. Fluid Mech. 300 (1995) 231–263.
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Article
Interaction of two in-line bubbles of equal size rising in viscous liquid☆ Zhen Tian, Xi Li ⁎, Youwei Cheng, Lijun Wang College of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 24 April 2019 Received in revised form 23 May 2019 Accepted 4 June 2019 Available online xxxx Keywords: Bubble column reactor Computational fluid dynamics Bubble interaction Bubble rise velocity Volume-of-fluid method Bubble aspect ratio
a b s t r a c t The interaction of bubbles is the key to understand gas–liquid bubbling flow. Two-dimensional axis-symmetry computational fluid dynamics simulations on the interactive bubbles were performed with VOF method, which was validated by experimental work. It is testified that several different bubble interactive behaviors could be acquired under different conditions. Firstly, for large bubbles (d: 4, 6, 8, 10 mm), the trailing bubble rising velocity and aspect ratio have negative correlations with liquid viscosity and surface tension. The influences of viscosity and surface tension on leading bubble are negligible. Secondly, for smaller bubbles (d: 1, 2 mm), the results are complicated. The two bubbles tend to move together due to the attractive force by the wake and the potential repulsive force. Especially for high viscous or high surface tension liquid, the bubble pairs undergo several times acceleration and deceleration. In addition, bubble deformation plays an important role during bubble interaction which cannot be neglected. © 2019 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.
1. Introduction Bubble dynamics in gas–liquid flow plays a significant role in many industrial and natural processes, such as coal and petroleum processing, chemical engineering and cleaning [1]. A great amount of the early work concentrates on single bubble rising velocity [2,3]. However, in practical applications bubble swarm mostly emerges [4], and hydrodynamics of each bubble would be evidently affected by the wake of the neighboring bubbles, which becomes more apparent in the near wake region [5,6]. It is considerable importance to understand two bubble interactions for developing reliable models suitable for the momentum exchange between two phases when gas volume fractions are high and bubble interaction cannot be ignored [7]. Thus, it is crucial to understand two bubble interactions as the separation distance among them gets smaller. Experimentally, Crabtree [8], Bhaga [9] and Komasawa [10] investigated the spherical cap bubble rising in line and provided one method of calculating the coalescence velocity of the bubbles by the asymptotic wake theory, which neglected the bubble shape and radial position, and then deduced the empirical equation of the average apparent wake velocity contribution. Narayanan [11] studied bubble pairs rising velocity in aqueous glycerin solutions at Reynolds number range from 0.5 up to 80 and indicated that the coalescence of bubbles was followed the analysis of weightless solid spheres when Re b 7. When Re N 7, the coalescence of bubbles was under the influence of two additional velocities produced owing to the wake structure. Katz [12] conducted ☆ Supported by the National Natural Science Foundation of China (91334105). ⁎ Corresponding author. E-mail address: [email protected] (X. Li).
experiments about air bubbles rising in line in distilled water of small size (d ≤ 475 μm, Re ≤ 35, Eo ≤ 0.027) and pointed out that the relative velocity of two bubbles was dependent on bubble diameter. For large bubbles (349 μm ≤ d ≤ 475 μm), the relative velocity increased as the distance decreased. For small bubbles (d ≤ 349 μm), the relative velocity decreased as the bubble distance decreased which could be owing to a possible repulsive force caused by pressure gradient. Theoretically, Harper [13], using the irrotational first approximation in larger Reynolds number and inviscid liquid, found that there existed a repulsive force between two bubbles rising in line which was inversely the fourth power of the separation distance between them. Yuan [14] conducted numerical research on two spherical bubbles motion of equal size moving along their line of centers and drew the conclusion that an equilibrium distance was reached when two bubbles rose in line, ignoring shape deformation, at which the force induced by the wake interaction effect equaled to that by the pressure gradients. Watanabe [15] also analytically and numerically studied bubble pairs rising in line for Reynolds number from 5 up to 150 and predicted that a stable equilibrium distance was presented between bubble pairs, which was also owing to the equilibration between viscous attractive forces and the potential repulsive forces. However, their numerical results were not consistent with the experimental results, which was owing to the different initial bubble distance. Fan [16] considered that the viscous drag force played an dominant role and inertial force was negligible, and three mathematical models were developed for calculating the trailing bubble velocity under the action of leading bubble for intermediate Reynolds number range [Re∼ O(100)]. In addition, several similar correlations [17–23] have been put forward using an analytical approximation method, in which a new theoretically reasonable
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
artificial origin has been introduced. For low to moderate Reynolds numbers (Re ≤ 50), Ramírez-Muñoz [22] pointed out that the following three forces (inertial forces, buoyancy and quasi-steady drag) played an important role in pushing the trailing bubble motion in the leading bubble wake region. However, both the history and added-mass body acceleration forces were negligible. For moderated to large Reynolds numbers (50 ≤ Re ≤ 300), Baz-Rodríguez [23], by introducing a theoretically justified artificial origin and adapting the type of analytical solution suitable for far wake region, proposed an equation applying to calculate the axial velocity distribution. In addition, due to an upward repulsion force was generated, induced by the potential flow field ahead of the trailing bubble, the leading bubble was pushed and moved upward together. As a result of the analysis, an equilibrium distance among bubble pairs was existed, which was in accordance with the conclusions of Yuan [14] and Watanabe [15]. In view of the preceding arguments, it is obvious that some uncertainties still exist concerning the interaction of two in-line bubbles. Experiments mainly focus on the spherical or spherical cap bubble, which neglect ellipsoid bubble. Theoretical methods are often based on some assumption and simplification on account of the strong nonlinearity caused by bubble deformations. The correlations put forward till now on the bubble drag force are still restricted to spherical bubbles [6]. While detailed analytical formulas based on some assumptions provide a better understanding of underlying principles, they are not generally suitable for real industrial situations in which liquid viscosities or surface tension might be significant. In this article, therefore, the VOF method was exploited to simulate bubble behavior. To validate the model, numerical results were compared with those of the experiments for trailing bubble. The effects of bubble diameter, liquid viscosity, surface tension on the interactive bubble were systematically researched within the laminar flow regime, in order to improve comprehension of bubble pair interaction rising in line from a numerical standpoint. 2. Numerical Simulation 2.1. Continuity and momentum equations As indicated by previous experimental and simulated work [1, 24–27], the inline configuration is found to be stable and bubble shape is axisymmetric for interaction of two in-line bubbles of equal size rising in viscous liquid, which is also validated by experimental work. A 2D-
axisymmetry model compared with three-dimensional simulation is enough to describe bubble rising trajectory and shape during bubble interactions. Thus in this study, a simplified physical model of simulation process is adopted for minimizing the computational costs. The simulations are conducted in transient and axisymmetric models by using Ansys Fluent, and the axis of symmetry is along the center of bubble column. Both gas and liquid phases are assumed as incompressible fluids, and the flow is Newtonian and laminar. Continuity and momentum equations are as follows [28]: ∇u¼0
ð1Þ
h i ∂ðρuÞ þ ∇ðρuuÞ ¼ −∇p þ ∇ μ ∇u þ ∇T u þ F s þ ρg ∂t
ð2Þ
where p represents the pressure. Fs stands for the source term that represents surface tension contribution. The density and viscosity in Eq. (2) are estimated supposing linear contributions of two phases when a computational cell is occupied by two phases. ρ ¼ ρl þ ρg ð1−F Þ
ð3Þ
μ ¼ μ l þ μ g ð1−F Þ
ð4Þ
2.2. Volume fraction equation The Volume Fraction Equation (VOF) [29] is applied to simulate interaction of the two in-line bubbles of equal size rising in Newtonian viscous liquids. In the VOF model, a single momentum equation is established to express the two fluids, and each phase volume fraction in all computational cells is marked by the volume ratio function F which is utilized to distinguish each phase. When the indicator function F equals to 0 or 1, it means that the cell is occupied by pure gas or pure liquid, respectively, and when F is within 0-1, it means that the cell is occupied by both gas and liquid [30]. The transport equation of fluid volume function is as follows: DF ∂F ¼ þ ðu ∇ÞF ¼ 0 Dt ∂t
ð5Þ
Fig. 1. Illustration of the hydrodynamic features of bubble depicted in the present study.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
8 <0 F ¼ 0bFb1 : 1
in bubbles in interface in liquid fluid
ð6Þ
2.3. Continuum surface tension equation The continuum surface force model [31] is applied to compute the surface tension force for cells existing gas–liquid interface. The piecewise-linear approach put forward by Youngs [32] is applied as the geometric reconstruction scheme to represent the phases interface. Fs ¼ σ
ρκ∇F l 0:5 ρg þ ρl
3
Table 1 Parameters selected for the current study Case
T /K
ρl /kg·m−3
μ /mPa·s
σ /mN·m−1
d /mm
Mo
1 2 3 4 5 6 7 8 9
293 – – 323 373 – 423 – 473
970 970 970 951 921 921 859 859 829
300.3 99.4 39.8 19.2 99.4 99.4 59.6 59.6 39.8
20.3 20.3 20.3 19.2 16.5 82.5 14.4 72.0 12.6
1, 2, 4, 6, 8, 10 1, 2, 6 1, 2, 6 1, 2, 4, 6, 8, 10 1, 2, 4, 6, 8, 10 1, 2, 6, 8 1, 2, 4, 6, 8 2, 6 1, 2, 4, 6
9.8 0.12 2.9×10-3 2.0 0.23 1.9×10-3 4.8×10-2 3.9×10-4 1.4×10-2
ð7Þ
where κ = ∇ ⋅ n, n = (n/|n|), n = ∇ Fq. 2.4. Model geometry and solution method The rectangle computational domain is with a width of 12 radii and a height 180 radii, which is enough to eliminate the boundary effect [25]. In addition, the geometry structured grids are used with typical grid size of R/70 equably, which is enough to rule out the mesh effect and ensure high computing accuracy, especially for bubble shape during interaction. The pressure-based Navier–Stokes solution algorithm is used as the solution method, where a finite volume methodology is applied to discretize the governing equations. The Pressure-Implicit with splitting of Operators (PISO) pressure–velocity coupling scheme is adopted in the momentum equation, which is remarkable for transient calculations. The discretization scheme applied for pressure equation is pressure staggering option (PRESTO!). The QUICK scheme is used as discretization scheme for momentum equation. The Euler explicit formation is adopted to solve the volume fraction equation where the maximum Courant number is specified as 0.25. Implicit body force is taken into account to avoid poorly segregated algorithms convergence, which may be caused by large body forces existing in multiphase flow, such as surface tension forces and buoyancy force induced by high density differences. Variable time stepping is applied to automatically adjust the time step by specifying the global courant number as 0.25. The value of absolute criteria is 10−4 for velocity and continuity residual as a convergence criterion. As reported by Komasawa [10], the bubble behavior, such as rising velocity, bubble geometry and liquid velocity profile around a bubble, for free rising bubble had no significant differences with that of bubble held stationary. Considering that fixed frames needed large computational domains necessarily, which then consumed much time, the leading bubble position was set as moving reference frames. This method implied that a velocity-inlet boundary condition was adopted at the top of the region with a uniform velocity. The magnitude of velocity was equivalent to the velocity of the isolated bubble, and the inlet was consisted of liquid phase. In addition, a downward moving wall velocity of the same value as the isolated bubble and a no slip wall condition were applied in the wall boundary condition. Symmetry boundary condition was adopted at the axis. A pressure outlet boundary condition was specified at the bottom of the computational domain. Firstly, the single leading bubble was simulated to acquire flow distribution, such as axial velocity, radial velocity and volume fraction of two phases, inside and around the bubble by using above assumption, which indicated appropriate results on account of the position of the leading bubble undergoing only very small transformation. Then the flow distribution around the leading bubble was copied and pasted into a new position under the leading bubble with a separation distance equivalent to 75% of a minimum distance between bubbles (Lmin), above which there was no interaction between bubbles [33,34]. The stabilization length of the liquid flow field under the leading bubble (Lmin) was defined to highlight the influence of the bubble in its surroundings.
The standard used to assess the values of Lmin was based on determining the position where the absolute value of the normalized difference between the inlet velocity Uin and the column centerline velocity Uz was equivalent to 0.01 (|Uz-Uin |/Uin = 0.01). By this means, the initial flow field for the simulation of two in-line bubbles, with identical bubble shape, was created, divided by a requested distance, where the two bubble interaction was roughly ignored. For ease of understanding, the features depicted are schematically illustrated in Fig. 1. Fig. 1 (a) gives a global view of the following hydrodynamic features: the bubble velocity, minimum interaction distances below (Lmin) the leading bubble. Fig. 1(b) gives a global view of two interaction bubble at initial condition. 2.5. Experiment In addition, some experimental work has been carried out to validate the numerical method. The experimental setup contained two parts: bubble column and high speed camera system. The bubble column consisted of a cylindrical stainless-steel column with dimensions 50mm diameter and 600-mm height. On both the front and rear sides of the column three pairs of high strength compression quartz windows, with a viewing area of 20 mm × 100 mm, were uniformly installed along the test section. A thermostat was installed outside of the column, heated by temperature controlling systems. The experiment was carried out at temperature from 293 K up to 473 K. Nitrogen from the highpressure cylinders was introduced into the bottom of column through various diameter orifice nozzle (do: 0.5 mm, 1.12 mm, 2.5 mm). A pair of bubbles was released in succession from a single nozzle through quick-closing valve, and shared the same axis of symmetry during interaction. Silicone oil was used as the liquid phase. The initial liquid level exceeded the nozzle about 500 mm. The transient bubble rising velocity and bubble shape were obtained by a high-speed camera (Mikrotron CUBE7). The orifice was used to calibrate the image pixel resolution. Image resolution was 0.03 mm·pixel−1, and errors caused by distinguishing bubble edges was under 3 pixel. The relative error was
Fig. 2. Simulated drag coefficient of single bubble compared with experimental data, Bhaga and Turton model.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
under ± 4.5% due to the minimum diameter of the bubbles that was about 2 mm. In addition, with increase of bubble diameter, the relative error decreased. Liquid density was computed by measuring mass and volume separately. Liquid viscosity and surface tension was measured by rotational rheometer (HAAKE RS6000) and
video-based, contact angle device (OCA 20) separately. Uncertainties estimated at 95% confidence were 2.0% and 1.0% for viscosity and surface tension, respectively. Table 1 shows the physical properties of the silicone oil at different temperature. The simulation conditions are also presented in Table 1.
Fig. 3. Numerical data of the bubbles interface, velocity field and streamlines in the liquid phase, as a function of L/R compared with experimental image at different temperature.
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3. Result and Discussion 3.1. Single bubble rising The first case considered is that an isolated bubble rising in liquid. The accuracy and validity of the numerical simulation are checked firstly, by carrying out some comparisons of the isolated bubble rising simulation results with experimental results and correlations or theoretical equations available in the literature. Single bubble rising in a viscous Newtonian fluid has been researched both numerically and experimentally [1,6,26]. Bhaga [35] and Turton [36] studied the bubble rising velocity in viscous liquid and proposed the CD-Re relationship. As shown in Fig. 2, the simulated gas bubble drag coefficient is in accordance with experimental data and the calculated results by Eqs. (8), (9). In general, the CFD-VOF model used in this paper is practicable. h i1=0:9 C D ¼ ð2:67Þ0:9 þ ð16= ReÞ0:9 CD ¼
ð8Þ
24 0:413 1 þ 0:173 Re0:657 þ Re 1 þ 16300 Re−1:09
ð9Þ
3.2. Sequential images of two in-line bubbles When two bubbles rise in-line, the trailing bubble hydrodynamic behavior are evidently influenced by the leading bubble by changing the
1.8
U/U0
1.6
Experiment d: 2mm-UB/U0 d: 4mm-UB/U0 d: 8mm-UB/U0 Fan-2003 Rodriguez-2013
velocity field and pressure field, which becomes more apparent in the leading bubble wake neighboring region. In this connection, it is specifically significant to obtain bubble pairs interaction behavior along with the two bubble separation distance becoming shorter. In the following sections, in order to access detailed knowledge of the bubble interaction behavior during the approaching process, the interactions of two in-line bubbles rising under buoyancy in a viscous fluid at various conditions are simulated. For convenience, label the leading bubble A and the trailing bubble B. Firstly, it is significant to present a qualitative description on the bubble shape and velocity change process of the two bubbles as the trailing bubble approaches the leading one. Fig. 3 shows simulated sequential images about the description of bubbles shape, vector velocity and axial velocity streamlines as the dimensionless separation distance (L/R) among them becoming shorter, compared with experimental images. The conditions of Fig. 3 are at temperature 293, 423 and 473 K with the bubble diameter of 2 mm and 6 mm. It is demonstrated that the inline configuration is stable, and bubble shape is axisymmetric during bubble interaction from experimental observation. From the observation of Fig. 3, a very close qualitative accordance is observed among the calculated images and photographs. The leading bubble shape remains mostly unchanged, and just the bottom is progressively flattened as the trailing bubble approaches. In addition, elongation of the trailing bubble can be found in both the simulation and experiment image. The reason for this phenomenon could be that the leading bubble wake generates high velocity area at the bottom, which pushes the trailing bubble becomes narrow. Furthermore, as temperature increases, the initial Simulation Experiment d: 2 mm-EB/E0 d: 2 mm-EB/E0 d: 2 mm-EA/E0 d: 4 mm-EB/E0 d: 4 mm-EB/E0 d: 8 mm-EB/E0 d: 4 mm-EA/E0 d: 8 mm-EB/E0 d: 8 mm-EA/E0
1.8 1.6 1.4
E/E0
Simulation d: 2mm-UB/U0 d: 2mm-UA/U0 d: 4mm-UB/U0 d: 4mm-UA/U0 d: 8mm-UB/U0 d: 8mm-UA/U0
1.2
1.4
5
1.0 1.2
0.8 0.6
1.0 1
10
1
10
L/R
L/R
(a) case 1, T :293 K
U/U0
1.8 1.6 1.4
1.8
Simulation Experiment d: 2 mm-EB/E0 d: 2 mm-EB/E0
1.6
d: 2 mm-EA/E0
d: 4 mm-EB/E0
d: 4 mm-EB/E0
1.4
d: 4 mm-EA/E0 d: 6 mm-EB/E0
E/E0
Simulation Experiment d: 2 mm-UB/U0 d:2 mm-UB/U0 d: 2 mm-UA/U0 d:4 mm-UB/U0 d: 4 mm-UB/U0 Fan-2003 d: 4 mm-UA/U0 Rodriguez-2013 d: 6 mm-UB/U0 d: 6 mm-UA/U0
2.0
1.2
d: 6 mm-EA/E0
1.0
1.2
0.8
1.0
0.6
1
10
100
1
10
100
L/R
L/R
(b) case 9, T :473 K Fig. 4. Influence of bubble diameter on U/U0 and E/E0 as a function of L/R.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
Table 2 Correlations for trailing bubble rising velocity Investigator
Correlations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !#) ( " Z dU B U B C D0 Re0 d 12 ν t dU B =dτ pffiffiffiffiffiffiffiffiffiffi ddτ − ¼ 2g 1− þ 1− exp − 16ðL þ dÞ d π t0 dt U0 2 t−τ ! !2 ReA Re W 1 1 ReA 1 ð1−U B Þ dU B ð1−WÞ 1− A −1 ¼ 0 þ 1þ βðU B −WÞ þ x þ x0 16 16 W 0 x þ x0 48 x þ x0 W 0 dx " !# C Re 1 x = L/d x0 = 0.32621 + 0.01689ReA UB∗ = UB/U0 W ¼ DA 1− exp − A 2 16 x þ x0
Fan [16] Rodríguez [22]
−1
C DA ReA 1 1 þ ½8= ReB þ 1=2ð1 þ 3:315= ReB 0:5 Þ β¼ −1 2 16 x þ x0 1 þ ½8= ReA þ 1=2ð1 þ 3:315= ReA 0:5 Þ " !# ! 2 ∂W 2 ∂U C Re 1 1‐2:21 Re‐0:5 B U B B −1 ¼ 0 W ¼ 1− DA 1− exp − A β¼ þ U B β− W βþ C D0 ∂x 3C DA 2 16 x þ x0 ∂x 1‐2:21 Re‐0:5 A
W0 ¼ Rodríguez [23]
bubble shape of two bubbles varies from ellipsoidal to spherical cap and a recirculation zone generally appears under the leading bubbles bottom, and a conspicuous bubble interaction appears at larger separation distances, which reflecting that the minimum stabilization length (Lmin) increases with temperature increase. In addition, when bubble diameter is 2 mm, the leading bubble is evidently pushed by the trailing one for the entire range of distances between bubbles. However, for bubble diameter equals to 6 mm, the location of the leading bubble remains roughly unaffected. 3.3. Bubble diameter influence As shown in Fig. 4, the transmutation of the experimental and numerical results about U/U0 (U0: the isolated bubble velocity. UA: leading bubble velocity. UB: trailing bubble velocity) and E/E0 (E0: isolated bubble aspect ratio. EA: leading bubble aspect ratio. EB: trailing bubble aspect ratio) with the dimensionless separation distance (L/R), are presented respectively with various diameters. The relative error between numerical and experimental results is below 8%. The difference may originate from the difference of the initial bubble distance between simulation and experiment. Several trends could be obtained from the results. For separations exceeds 2 radii, both the leading and trailing bubble rising velocities increase with decreasing distance between bubbles. The leading bubble aspect ratio is gradually decreasing with the decreasing distance, whereas the trailing bubble aspect ratio is gradually increasing. For all diameter bubbles, the bubble diameter has a positive effect on the trailing bubble velocity, consistent with the trailing bubble aspect ratio. However, different trends related to bubble size are observed for the behavior of the leading bubble. With regard to larger bubbles (d ≥ 4 mm), the leading bubble was little affected by the existence of the trailing bubble when the two bubble separation distance exceeds 10 radii. When separation distance under 10 radii, there is slightly increase in bubble velocity and bubble aspect ratio gets smaller. With case 1:UB/U0 case 1:UA/U0 case 4:UB/U0 case 4:UA/U0 case 5:UB/U0 case 5:UA/U0 case 7:UB/U0 case 7:UA/U0 case 9:UB/U0 case 9:UA/U0
1.8
U/U0
1.6
U B ¼ U B =U 0
x ¼ L=d
regard to the smaller bubbles (d ≤ 2 mm), the leading bubble is pushed by the trailing bubble and bubble velocity increases significantly with the decreasing separation distance. The two bubbles tend to rise together, and the separation distance between bubble pairs is slowly reduced, until bubble coalescence takes place. This phenomenon is also experimentally observed by Katz [12] and numerically by Gumulya [24] that for smaller bubbles the leading bubble rising velocity increased considerately (up to 30% of the single bubble velocity), and it was explained by the pressure gradients contribution, which became more dominant for small bubbles. However, for larger bubbles, the wakeinduced relative motion was dominant over the pressure gradients. When separations under 2 radii, for large bubbles, there exists a certain deceleration effect, which may be owing to that an accessional force is generated to move away intervening liquid between the two bubbles, and the bubble aspect ratio of the trailing bubble is also becoming small. Some comparisons about the increase of the trailing bubble velocity are also made between simulated results and the wake correlations proposed by Fan [16] and Rodríguez [22,23] which are presented in Table 2. As shown in Fig. 4, the two correlations also shows that bubble diameter has a positive correlation with the increase of the trailing bubble rising velocity. However, the predicted value underestimates the results. Both the Fan and Rodríguez correlations are based on the assumption that bubble shape is spherical. However, the trailing bubble becomes narrow during actual bubble interaction, which causes the drag force of trailing bubble to become small. Thus the two correlations underestimate the trailing bubble velocity. When the two bubbles distance at 2.5 radii, the bubble rising velocity and aspect ratio, ratio to isolated bubble value, are presented in Fig. 5 respectively with the various bubble diameters. As presented in Fig. 5, the trailing bubble rising velocity has a positive correlation with the diameter, consistent with bubble aspect ratio. However, the leading bubble rising velocity decreases firstly with the diameter and then tends to be roughly constant (UA/U0 = 1.07), and the bubble aspect ratio presents same tread. case 1:EB/E0 case 1:EA/E0 case 4:EB/E0 case 4:EA/E0 case 5:EB/E0 case 5:EA/E0 case 7:EB/E0 case 7:EA/E0 case 9:EB/E0 case 9:EA/E0
1.8
1.6
1.4
E/E0
2.0
x0 ¼ 0:1859 Re0:5 A
1.2
1.4 1.0
1.2 0.8
1.0 0
2
4
6
d /mm
8
10
0
2
4
6
8
10
d /mm
Fig. 5. Numerical result of U/U0 and E/E0 when bubble distance at 2.5 radii.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003
Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
U/U0
1.3 1.2
1.10
Mo: 9.8-EB/E0 Mo: 9.8-EA/E0 Mo: 0.12-EB/E0 Mo: 0.12-EA/E0 Mo: 2.9×10-3-EB/E0 Mo: 2.9×10-3-EA/E0
1.08 1.06 1.04
E/E0
Mo: 9.8-UB/U0 Mo: 9.8-UA/U0 Mo: 0.12-UB/U0 Mo: 0.12-UA/U0 Mo: 2.9×10-3 UB/U0 Mo : 2.9×10-3 UA/U0 Fan-2003 Rodriguez-2013
1.4
7
1.02 1.00 0.98
1.1
0.96 0.94
1.0
0.92
10
10
L/R
L/R
(a) d: 2 mm Mo: 9.8-UB/U0 Mo: 9.8-UA/U0 Mo: 0.12-UB/U0 Mo: 0.12-UA/U0 Mo: 2.9×10-3-UB/U0 Mo: 2.9×10-3-UA/U0 Fan-2003 Rodriguez-2013
1.8
U/U0
1.6 1.4
Mo: 9.8-EB/E0 Mo: 9.8-EA/E0 Mo: 0.12-EB/E0 Mo: 0.12-EA/E0 Mo: 2.9×10-3-EB/E0 Mo: 2.9×10-3-EA/E0
1.8 1.6 1.4
E/E0
2.0
1.2 1.0
1.2
0.8 1.0
0.6 1
10
1
100
10
100
L/R
L/R
(b) d: 6 mm Fig. 6. Influence of liquid viscosity on U/U0 and E/E0 as a function of L/R.
As presented in Fig. 6, Morton number replaces liquid viscosity to describe its influence on bubble interactions. For small bubble of diameter 2 mm, trailing bubble rising velocity increases slightly with the decreasing viscosity under the same bubble distance, while leading bubble velocity has minor decrease. In addition, the influence of viscosity on bubble aspect ratio is small both for the leading and trailing bubbles, for large bubble of 6-mm diameter, trailing bubble velocity increases significantly with the decreasing viscosity, while almost no difference was made on leading bubble velocity. Meanwhile, bubble aspect ratio has the corresponding change with bubble rising velocity. Trailing bubble aspect ratio seems to be very sensitive to the two bubble separation distance, with this sensitivity increase as the viscosity decreases. The results of Fan and Rodríguez correlations are also presented on Fig. 6. The two correlations also show that liquid viscosity has a negative correlation with the trailing bubble rising velocity. Due to their negligence of bubble shape deformation accompanying with bubble interaction, the two correlations all underestimate trailing bubble velocity. For smaller bubble of 1-mm diameter, the interaction of two bubbles becomes complicated. As shown in Figs. 7 and 8, the two bubbles tend to slowly reduce the separation distance and rise together until coalescence finally occurs. For case 1 and case 4, the pairs undergo several time acceleration and deceleration when the bubble separation distance is below 3 radii which may be interpreted by the potential flow field caused by the trailing bubble, and this force then pushes the leading bubble to move upward. As reported by literature [14], an equilibrium distance between the two bubbles rising in-line might therefore be attained. However, on account of its inherent instability in practice
and that the equilibrium distance is also relating to the two bubble initial separation distance, the existence of the equilibrium distance is very difficult to demonstrate [15]. 3.5. Surface tension influence As shown in Fig. 9, with the increasing liquid surface tension, trailing bubble rising velocity and aspect ratio decrease, while surface tension has no influence on leading bubble rising velocity and aspect ratio. At
1.7
case 1:UB/U0 case 4:UB/U0 case 5:UB/U0 case 7:UB/U0 case 9:UB/U0
1.6 1.5 1.4
U/U0
3.4. Viscosity influence
case 1:UA/U0 case 4:UA/U0 case 5:UA/U0 case 7:UA/U0 case 9:UA/U0
1.3 1.2 1.1 1.0 0.9 0
10
20
30
40
50
t /s Fig. 7. Influence of liquid viscosity on U/U0 as a function of t as bubble diameter at 1 mm.
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
12 10 8
L/R
In addition, for small bubbles (d ≤ 2 mm), when surface tension increases four times, the two bubble is hard to coalesce. Fig. 10 shows two bubble distance with time for different surface tension. For case 5 when bubble diameter is at 2 mm, the two bubbles undergo 13.63 s to coalesce. However, for case 6 when bubble diameter also at 2 mm and with four times surface tension, the two bubbles keep a constant distance and rise together until the initial distance is set as 0.4Lmin. The surface tension with high values means that bubble shape mostly keep spherical during bubble interactions, which is consistent with the assumption of previous works, finding the existence of an equilibrium distance [14].
case 1 case 4 case 5 case 7 case 9
6 4
4. Conclusions
2 0
10
20
30
40
50
t /s Fig. 8. Vertical distance between two bubbles as a function of t as bubble diameter at 1 mm.
a low surface tension, the trailing bubbles deform more easier while approaching, leading to an increase of the trailing bubble velocity, which is also reported by Hasan [26,37,38] that the pair of vertically aligned ellipsoidal bubbles showed stronger attraction than the spherical bubble pair. The Fan and Rodríguez correlations present a better matching result when surface tension is larger in which condition bubble deformation is hard and bubble shape is nearly spherical, consistent with correlations assumption.
The Volume-of-Fluid method (VOF) was used to simulate the interaction of two in-line bubbles of equal size rising in viscous liquid. The simulated conditions included a wide range, that was Re (0.007-25), Mo (3.9E43.9×10-3-9.8) and Eo (0.04-50), which ensured under laminar regime. The method was validated by comparison with the present experimental measurements and existing correlations. The effect of bubble diameter, liquid viscosity and surface tension on the interaction of two bubbles were studied systematically. The following conclusions could be obtained: 1) For the interaction of two in-line bubbles: To bubble velocity, the leading bubble has minor change and the trailing increases. To bubble shape, the leading bubble becomes flat and the trailing becomes elongated. 2) For large bubbles, bubble diameter has a positive influence on the trailing bubble velocity, in contrast with surface tension and liquid viscosity, but little influence on the leading bubble.
case 5,d: 6 mm, -UB/U0
2.0
case 6,d: 6 mm, 5 -UB/U0
1.8
case 6,d: 6 mm, 5 -UA/U0 case 5,d: 8 mm, -UA/U0 case 6,d: 8 mm, 5 -UB/U0
1.4
E/E0
U/U0
case 5,d: 8 mm, -UB/U0
1.6
case 5, d: 6 mm, -EA/E0
1.6
case 6, d: 6 mm, 5 -EB/E0
1.4
case 5, d: 8 mm, -EB/E0
case 6, d: 6 mm, 5 -EA/E0 case 5, d: 8 mm, -EA/E0 case 6, d: 8 mm, 5 -EB/E0
1.2
case 6, d: 8 mm, 5 -EA/E0
case 6,d: 8 mm, 5 -UA/U0 Fan-2003 Rodriguez-2013
1.2
case 5, d: 6 mm, -EB/E0
1.8
case 5,d: 6 mm, -UA/U0
1.0 0.8
1.0
0.6
1
1
10
10
L/R
L/R
Fig. 9. Influence of surface tension on U/U0 and E/E0 as a function of L/R.
case 5,d:2mm,
12
case 6, d
7 6
8
5
L/R
L/R
10
6
4
4
3
2
2
-2
0
2
4
6
8
10
12
14
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
t /s
t /s
Fig. 10. Vertical distance between two bubbles varying with time.
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Z. Tian et al. / Chinese Journal of Chemical Engineering xxx (xxxx) xxx
3) For small bubbles, the interaction of two bubbles is complicated. Two bubbles tend to move together. Especially for high viscous or high surface tension liquid, the bubble pairs undergo several times acceleration and deceleration when bubble distance is lower than 3 radii. 4) By comparing with the correlations, it can be concluded that bubble deformation plays an important role at bubble interaction, which causes the correlations underestimate the trailing bubble velocity. References [1] H.A.N. Wang, Z.Y. Zhang, Y.M. Yang, H.S. Zhang, Numerical investigation of the interaction mechanism of two bubbles, Int. J. Mod. Phys. C 21 (01) (2010) 33–49. [2] A.A. Kulkarni, J.B. Joshi, Bubble formation and bubble rise velocity in gas-liquid systems: a review, Ind. Eng. Chem. Res. 44 (16) (2005) 5873–5931. [3] J.F. Harper, The motion of bubbles and drops through liquids, Adv. Appl. Mech, Elsevier 12 (1972) 59–129. [4] B. Bunner, G. Tryggvason, An examination of the flow induced by the motion of many buoyant bubbles, J. Visual 2 (2) (1999) 153–158. [5] X. Chai, X. Liu, J. Xiong, X. Cheng, Numerical investigation of bubble wake properties in the moving liquid with LES model, J. Nucl. Sci. Technol. 53 (11) (2016) 1870–1880. [6] J.B. Joshi, K. Nandakumar, G.M. Evans, V.K. Pareek, M.M. Gumulya, M.J. Sathe, M.A. Khanwale, Bubble generated turbulence and direct numerical simulations, Chem. Eng. Sci. 157 (2017) 26–75. [7] H.A. Jakobsen, H. Lindborg, C.A. Dorao, Modeling of bubble column reactors: progress and limitations, Ind. Eng. Chem. Res. 44 (14) (2005) 5107–5151. [8] J. Crabtree, J. Bridgwater, Bubble coalescence in viscous liquids, Chem. Eng. Sci. 26 (6) (1971) 839–851. [9] D. Bhaga, M. Weber, In-line interaction of a pair of bubbles in a viscous liquid, Chem. Eng. Sci. 35 (12) (1980) 2467–2674. [10] I. Komasawa, T. Otake, M. Kamojima, Wake behavior and its effect on interaction between spherical-cap bubbles, J. Chem. Eng. Jpn 13 (2) (1980) 103–109. [11] S. Narayanan, L.H.J. Goossens, N.W.F. Kossen, Coalescence of two bubbles rising in line at low Reynolds numbers, Chem. Eng. Sci. 29 (10) (1974) 2071–2082. [12] J. Katz, C. Meneveau, Wake-induced relative motion of bubbles rising in line, Int. J. Multiphase Flow 22 (2) (1996) 239–258. [13] J.F. Harper, On bubbles rising in line at large Reynolds numbers, J. Fluid Mech. 41 (04) (1970) 751–758. [14] H. Yuan, A. Prosperetti, On the in-line motion of two spherical bubbles in a viscous fluid, J. Fluid Mech. 278 (1994) 325–349. [15] M. Watanabe, T. Sanada, In-line motion of a pair of bubbles in a viscous liquid, Jsme. Int. J. B-Fluid. T 49 (2) (2006) 410–418. [16] J. Zhang, L.S. Fan, On the rise velocity of an interactive bubble in liquids, Chem. Eng. J. 92 (1–3) (2003) 169–176. [17] J.R. Muñoz, A. Soria, E. Salinas-Rodríguez, On the relative motion of two spherical bubbles rising in line and interacting by a laminar wake, AIChE Annual Meeting, Conference Proceedings 2005, pp. 689–693.
9
[18] J. Ramírez-Muñoz, A. Soria, E. Salinas-Rodríguez, Hydrodynamic force on interactive spherical particles due to the wake effect, Int. J. Multiphase Flow 33 (7) (2007) 802–807. [19] J. Ramírez-Muñoz, A. Gama-Goicochea, E. Salinas-Rodríguez, Drag force on interacting spherical bubbles rising in-line at large Reynolds number, Int. J. Multiphase Flow 37 (8) (2011) 983–986. [20] J. Ramírez-Muñoz, E. Salinas-Rodríguez, A. Soria, A. Gama-Goicochea, Hydrodynamic interaction on large-Reynolds-number aligned bubbles: drag effects, Nucl. Eng. Des. 241 (7) (2011) 2371–2377. [21] S.A. Baz-Rodríguez, J. Ramírez-Muñoz, A. Soria, In-line interaction between two spherical particles due to a laminar wake effect, Int. J. Multiphase Flow 39 (2012) 240–244. [22] J. Ramírez-Muñoz, S. Baz-Rodríguez, E. Salinas-Rodríguez, E. Castellanos-Sahagún, H. Puebla, Forces on aligned rising spherical bubbles at low-to-moderate Reynolds number, Phys. Fluids 25 (9) (2013), 093303. [23] S.A. Baz-Rodríguez, J. Ramírez-Muñoz, A. Soria, J.C. Sacramento-Rivero, Hydrodynamic interaction of two spherical bubbles rising in-line: a semi-analytical approach, Chem. Eng. Commun. 201 (5) (2014) 674–687. [24] M. Gumulya, R.P. Utikar, G.M. Evans, J.B. Joshi, V. Pareek, Interaction of bubbles rising inline in quiescent liquid, Chem. Eng. Sci. 166 (2017) 1–10. [25] J. Feng, X. Li, Y. Bao, Z. Cai, Z. Gao, Coalescence and conjunction of two in-line bubbles at low Reynolds numbers, Chem. Eng. Sci. 141 (2016) 261–270. [26] N. Hasan, Z.b. Zakaria, Computational approach for a pair of bubble coalescence process, Int. J. Heat Fluid Flow 32 (3) (2011) 755–761. [27] M. Cheng, J. Hua, J. Lou, Simulation of bubble-bubble interaction using a lattice Boltzmann method, Comput. Fluids 39 (2) (2010) 260–270. [28] E. Delnoij, J.A.M. Kuipers, W.P.M. van Swaaij, Computational fluid dynamics applied to gas-liquid contactors, Chem. Eng. Sci. 52 (21) (1997) 3623–3638. [29] C.W. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39 (1) (1981) 201–225. [30] D. Ma, M. Liu, Y. Zu, C. Tang, Two-dimensional volume of fluid simulation studies on single bubble formation and dynamics in bubble columns, Chem. Eng. Sci. 72 (2012) 61–77. [31] J. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (2) (1992) 335–354. [32] D.L. Youngs, Time-dependent multi-material flow with large fluid distortion, Numerical Methods for Fluid Dynamics, 1982. [33] R. Moissis, P. Griffith, Entrance effects in a two-phase slug flow, J. Heat Transf. 84 (1) (1962) 29–38. [34] A.M.F.R. Pinto, M.N. Coelho Pinheiro, J.B.L.M. Campos, Coalescence of two gas slugs rising in a co-current flowing liquid in vertical tubes, Chem. Eng. Sci. 53 (16) (1998) 2973–2983. [35] D. Bhaga, M.E. Weber, Bubbles in viscous liquids: shapes, wakes and velocities, J. Fluid Mech. 105 (1981) 61–85. [36] R. Turton, O. Levenspiel, A short note on the drag correlation for spheres, Powder Technol. 47 (1) (1986) 83–86. [37] Z. Yu, H. Yang, L.-S. Fan, Numerical simulation of bubble interactions using an adaptive lattice Boltzmann method, Chem. Eng. Sci. 66 (14) (2011) 3441–3451. [38] M. Manga, H.A. Stone, Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions, J. Fluid Mech. 300 (1995) 231–263.
Please cite this article as: Z. Tian, X. Li, Y. Cheng, et al., Interaction of two in-line bubbles of equal size rising in viscous liquid, Chinese Journal of Chemical Engineering, https://doi.org/10.1016/j.cjche.2019.06.003