Visualization of bubble coalescence in bubble chains ...

International Journal of Multiphase Flow 105 (2018) 159–169

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International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow

Visualization of bubble coalescence in bubble chains rising in a liquid metal O. Keplinger∗, N. Shevchenko, S. Eckert Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany

a r t i c l e

i n f o

Article history: Received 6 December 2017 Revised 26 March 2018 Accepted 3 April 2018 Available online 19 April 2018 Keywords: Liquid metal GaInSn Bubble chain Bubble coalescence X-ray radiography Two-phase flow

a b s t r a c t Bubble coalescence in liquid metals was studied by considering the case of a bubble chain rising in the eutectic alloy GaInSn. The experiments were performed in a flat vessel with a rectangular cross section. High frame-rate X-ray radiography was used for visualizing the interaction between the bubbles. Essential process parameters such as bubble sizes, bubble shapes, velocities and distance of their closest approach are obtained from image processing. Different coalescence schemes occurring inside the bubble chain are discussed and demonstrated. The results are compared to collision cases where the bubbles bounce off each other. The material properties of the liquid metal differ significantly from those of water or other transparent fluids. In particular, the low viscosity, the high density and the high surface tension result in low values of the Mo number, Mo ≈ 2 × 10−13 and high Reynolds numbers of Re ∼ 104 . Nevertheless, the process of bubble approach, collision and coalescence was found to proceed in a qualitatively similar way as reported by previous studies for the case of water or highly viscous fluids. From the analyzed data, it was difficult to define a quantitative criterion that would allow predicting whether a pair of colliding bubbles would coalesce or bounce off. The observations indicate that the turbulent flow in the immediate vicinity of the bubbles has an important influence on whether coalescence occurs or not. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Liquid metal two-phase flows are of particular importance for technological processes in metallurgy and metal casting. For example, the secondary-metallurgical treatment of liquid steel relies on the injection of purge gas for improving the steel cleanliness. Objectives are the enhancement of mixing and homogenization and the separation of undesired inclusions by flotation. The efficiency of the flotation process is determined not only by the properties of the inclusions in the melt, but also by the size and the specific surface area of the dispersed gas phase. Fundamental analysis indicated that using smaller bubbles in flotation is the most effective approach because it increases the probability of collision between bubbles and particles and reduces the probability of particle detachment at the liquid-gas interface (Tao, 2005). Bubble interaction which leads to coalescence or separation plays an important role with regard to the resulting bubble size distribution and the interfacial area within the melt. Coalescence reduces the number of the bubbles and decreases the gas-liquid interfacial area. Bubble interactions are influenced by both the material properties of the fluid, in particular the liquid viscosity and the surface

∗

Corresponding author. E-mail address: [email protected] (O. Keplinger).

https://doi.org/10.1016/j.ijmultiphaseflow.2018.04.001 0301-9322/© 2018 Elsevier Ltd. All rights reserved.

tension, and the flow structure around the bubbles that controls the probability for collision and the contact time. It is well-known that coalescence of two bubbles takes place in three main steps: 1) bubbles come into a close contact to each other (i.e. collision) meanwhile being separated by a thin liquid film; 2) this liquid film then drains; 3) when the liquid film reaches a critical thickness a film rupture occurs resulting in coalescence. The primary force promoting film thinning is the capillary pressure, which is induced by variations in the curvature of the gas-liquid interface. When the film becomes very thin the intermolecular forces (i.e. the Van der Waals forces) become significant and the bubble surfaces will start to attract each other resulting in a higher pressure in the film than in the gas phase. Since this force will strongly increase with decreasing the film thickness it will eventually account for instability and rupture of the film. If the thinning mechanism lasts longer than the bubble contact time the coalescence does not occur (Chesters, 1991; Chesters and Hofman, 1982; Kim and Lee, 1987; Marrucci, 1969; Oolman and Blanch, 1986; Orvalho et al., 2015; Tse et al., 1998). On the other hand, turbulent shear flow and vehement bubble collisions could lead to bubble breakup. All three processes, bubble collision and deformation, film drainage and film rupture are highly complex and very difficult to quantitatively study. Numerical simulations rely on coalescence models, which must be verified experimentally.

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O. Keplinger et al. / International Journal of Multiphase Flow 105 (2018) 159–169

Bubble interaction and bubble coalescence have been considered by many numerical and experimental studies in water and other transparent viscous liquids (Annaland et al., 2005; Chan et al., 2011; Chen et al., 1998; Chen et al., 2005; Chen et al., 2011; Cheng et al., 2010; Chesters, 1991; Crabtree and Bridgwater, 1971; de Nevers and Wu, 1971; Fan and YIin, 2013; Feng et al., 2016; Gupta and Kumar, 2008; Hasan and Zakaria, 2011; Higuera, 2005; Islam et al., 2015; Kamp et al., 2001; Katz and Meneveau, 1996; Kim and Lee, 1987; Mahbubur Razzaque et al., 2003; Oolman and Blanch, 1986; Orvalho et al., 2015; Paulsen et al., 2014; Prince and Blanch, 1990; Sanada et al., 2009; Sanada et al. 2005; Tse et al., 1998; Ueyama et al., 1993; Yang et al., 2007). In particular, the rapid development of numerical methods nowadays allows highlyresolved, three-dimensional simulations of coalescence processes. However, many numerical studies are widely limited to the case of almost laminar flows in very viscous fluids whereas the experiments mainly focus on water flows. It is questionable whether these data can be easily transferred to liquid metal systems. Heavy liquid metals such as steel differ from many transparent fluids, among other things, due to the distinct higher values for density ρ l and surface tension σ , which lead to significant differences in the Morton number Mo = μl 4 ρ g/ρ l 2 σ 3 (μl denotes the dynamic viscosity and ρ stands for the density difference between liquid and gas) by several orders of magnitude. Moreover, the motion of a gas bubble in a heavy liquid metal is characterized by high Reynolds numbers Re = ρ l uT dB /μl . Typical values of 5 mm for the bubble diameter dB and a terminal velocity uT of 0.3 m/s in liquid steel result in a Re number of about 20 0 0. By contrast, the differences are not so dramatic for the Eötvös number Eo = ρ gdB 2 /σ or the Weber number (We = ρ l uT 2 dB /σ ) which are determined by the ratio ρ l /σ . Several theoretical studies applied the potential flow approximation to analyze the interaction of gas bubbles in the limit of low Weber numbers (We ≤ O(1)) (Chesters and Hofman, 1982; Kumaran and Koch, 1993a, b). The bubbles can be assumed as nondeformable spheres under these conditions as long as the Re numbers remain in a range up to 200. Kumaran and Koch (1993a, b) showed that two horizontally adjacent bubbles move towards each other and collide if their horizontal distance falls below a critical value. However, if the initial orientation of the line connecting the bubbles is closer to the vertical axis, the bubbles take much longer to approach each other. For the arrangement of two vertically aligned bubbles it is predicted that the bottom bubble can be repelled away from the wake of the top bubble. There is a rotational movement of the lower bubble around the upper one with the result of a horizontal alignment. The results obtained by such simplified simulations of the bubble dynamics were called into question by Yuan et al. (1994). They performed a numerical study of the interaction of two bubbles moving in the direction of their line of centers in a Re number range up to 200. The authors found that the bubble interaction is strongly affected by viscous effects as well as by the transport and diffusion of vorticity. Moreover, they suggested the existence of an equilibrium distance at which the wake effect and the inertial repulsion balance. However, this finding is in contrast to corresponding experiments conducted by Katz and Meneveau (1996) for tiny bubbles (dB < < 1 mm) in water at Reynolds numbers ranging between 0.2 and 35. A potential reason for the discrepancy is the deformation of the bubbles. In these experiments, the wake of the leading bubble induces a relative motion of the bubbles leading to coalescence. While the relative velocity during the approach of the bubbles at a diameter of 0.35 mm increases with decreasing distance until just before the contact, the relative velocity of smaller bubbles decreases prior to coalescence. Significant progress in numerical simulations of multiple bubble dynamics have been achieved during the last two decades

(see for instance Liao et al., 2014; Prosperetti and Tryggvason, 2009; Schwarz and Fröhlich, 2014). In particular, numerical simulations using a lattice Boltzmann method were reported by Gupta et al. (2008) and Cheng et al. (2010). Gupta and Kumar (2008) considered the interaction of two or three bubbles for Eo numbers of O(1) and Mo numbers of O(10−4 ). The authors emphasize that the bubble dynamics is strongly governed by vortex pattern generated in the wake of the leading bubble. Calculations of the co-axial and oblique bubble coalescence of two bubbles were conducted by Cheng et al. (2010) for an Eo number of 116, a Re number of 4 and cover a Mo number range between 5 and 260. The simulations demonstrate an immersion of the trailing bubble in the wake of the leading one, accompanied by bubble deformation in form of stretching the bubble shape in vertical direction. Simultaneously with the elongation of the trailing bubble, the leading bubble takes the shape of an oblate ellipsoidal cap. While the bubble shapes remain symmetric in case of co-axial coalescence, a remarkable asymmetric deformation of the trailing bubble becomes apparent in oblique coalescence. Simulations of bubble clouds within the same study showed a tendency of bubbles to form clusters. Hasan and Zakaria (2011) used the volume of fluid (VOF) approach to study the coalescence of bubble pairs under laminar flow conditions for a Re number of 10. The authors focused on the role of the surface tension by varying the Eo number between 5 and 50 and the Mo number between O(10−2 ) and O(10). It was observed that bubbles will merge later in case of higher surface tension, which was explained by lower bubble deformation. A recent experimental and numerical study by Feng et al. (2016) considered bubble coalescence in various aqueous solutions of glycerin at large Morton numbers (Mo = 1 … 266) and small Reynolds numbers (Re = 0.1 … 2). The authors observed an acceleration of the trailing bubble in the wake of the leading one until the moment of collision where the trailing bubble slows down distinctly. Moreover, it was found that coalescence is retarded by lowering of liquid viscosity and surface tension. It is unclear to what extent the above-mentioned studies are relevant for liquid metals, since metallic fluids are characterized by very low Mo numbers in the order of O(10−13 ) and, in particular, the motion of bubbles in heavy liquid metals is associated with very large Re numbers in the order of O(103 ). Therefore, systematic investigations of two-phase flows and bubble behavior in liquid metals are important and desirable. Wang et al. (2017) used VOF method with a continuum surface force (CSF) model to study coaxial bubble coalescence in Argon-steel systems. This approach was verified by comparing numerical simulations and experiments in water. The calculations carried out in the parameter range covering the situation in the argon/steel system show a qualitatively similar behavior of the bubbles as already reported for bubbly flows in water. However, corresponding data from liquid metal experiments to validate these results are not available. Since liquid metals are generally more difficult to handle than water experiments and the opacity of the metals does not allow the use of powerful optical methods for bubble detection, so far there are only very few experimental studies on the dynamics of bubble motion in liquid metals. Local conductivity probes (Eckert et al., 20 0 0a, b; Iguchi et al., 1997; Manera et al., 2009; Oryall and Brimacombe, 1976; Xie et al., 1992; Xie and Oeters, 1994), ultrasound techniques (Timmel et al., 2010; Z. H. Wang et al., 2017b; Zhang et al., 2005) or inductive methods (Gundrum et al., 2016; Lyu and Karcher, 2016) can be used to detect gas bubbles in liquid metals. However, all these methods are subject to certain restrictions. For example, conductivity probes are intrusive and provide only local information. Ultrasonic methods encounter problems when the number of bubbles in the measuring volume increases with increasing gas content. Multiple echoes at the bubble interfaces can create signal artifacts causing serious difficulties in


video-camera optics

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scintillation screen vessel with GaInSn slit X-ray source

mirror collimators Fig. 1. Schematic drawing of the experimental setup.

data processing. Inductive methods provide a contactless method for bubble detection in liquid metal flows, but, the reconstruction of the bubble shape is still challenging and the interpretation of the signals becomes exceedingly difficult in case of interaction of multiple bubbles. X-ray radiography has proven to be an efficient method for bubble detection in liquid metals which is based on the absorption contrast between the liquid and gas phase (Davis et al., 1978; Fröhlich et al., 2013; Iguchi et al., 1995; Shevchenko et al., 2013; Timmel et al., 2015; Vogt et al., 2015; Wang et al., 1999). The main limitation for this technique is the thickness of the fluid volume that can be investigated due to high X-ray absorption coefficients for liquid metals. The accuracy of the X-ray radiography technique for determining bubble parameters is discussed in detail in (Keplinger et al., 2017). Another irradiation method is the neutron radiography (Mishima et al., 1999; Saito et al., 2005) which allows the investigation of thicker measuring volumes but at the expense of image contrast. The topic of bubble coalescence in liquid metal experiments was mentioned in (Fröhlich et al., 2013) but no comprehensive studies were presented. In the present work we study the occurrence of bubble coalescence in a bubble chain rising in a liquid metal. Experiments were carried out in GaInSn, a ternary alloy that is liquid at room temperature and whose material properties are very similar to those of liquid steel. The dynamics of the bubble interactions were visualized by means of X-ray radiography using high-speed video imaging. Many events of bubble collision were analyzed, either resulting in coalescence or not. 2. Experimental setup Bubble visualization experiments were carried out at the X-ray laboratory at HZDR. The capability of the X-ray radioscopy for visualizing liquid metal two-phase flows has already been demonstrated in previous work (Timmel et al., 2015). The scheme of the experimental setup is shown in Fig. 1. A continuous divergent polychromatic X-ray beam was produced by an industrial Xray tube (GE ISOVOLT 450KV/25–55 from GE Sensing & Inspecting Technologies GmbH) operating at a voltage of 320 kV and a current of 14 mA. The X-ray beam penetrates the liquid metal along the narrow extension of the container as shown in Fig. 1. The nonabsorbed part of the beam that can be estimated by the BeerLambert law I = I0 e−μx , where I0 is the primary beam intensity, μ is the X-ray attenuation coefficient and x is the thickness of the liquid metal, impinges on a scintillation screen (SecureX HB from Applied Scintillation Technologies) that is attached to the outer wall of the container. The X-ray intensity is converted into visible light in the scintillator, which is deflected over a mirror and projected by a lens system (custom-made design by Thalheim-Spezial-Optik) onto the sCMOS sensor plane of the high-speed video camera (pco.edge 5.5 of PCO). The field of view of 30 × 110 mm2 allows for achieving

the highest possible frame rate of 180 frames per second. An exposure time of 3 ms resulted in a good signal-to-noise ratio without significant blurring of the bubble images due to their high rising velocities. The eutectic composition of the ternary alloy GaInSn which is liquid at room temperature was used as liquid metal and filled up to a height of 144 mm into a vessel made of acrylic glass with a rectangular cross section of 144 × 12 mm2 . Data on the thermophysical properties of the eutectic GaInSn alloy can be found in (Plevachuk et al., 2014). Essential material properties at room temperature are as follows: density 6.33 g/cm3 , kinematic viscosity 3.7 × 10−7 m2 /s and surface tension 0.585 N/m. GaInSn proves to be a very good choice for the modeling of metallurgical processes, since the material properties are very similar to those of liquid steel and the low melting point allows easy operation without the installation of heating systems. The inert Argon gas was injected through a single long bevel orifice with an inner diameter of 0.785 mm (Sterican®). This nozzle was made of stainless steel and positioned in the middle of the cross section just above the bottom of the vessel. A gas flow control system (MKS Instruments) is used to regulate the Ar flow in a range from 10 to 1200 cm³/min. The system is normed to Nitrogen at standard conditions (1 bar, 0 °C). Thus, the desired gas flow is adjusted by applying a correction factor. The experiments are conducted at room temperature and normal atmospheric pressure. The image analysis was performed using Matlab scripts for extracting diverse parameters as bubble sizes, bubble trajectories and rising velocities. The image processing was conducted as follows: in a first step a shading correction was performed by subtracting a mean reference image measured at zero gas flow rate. Separation of individual bubbles and bubble clusters from the background was obtained by using a threshold value. All pixels whose brightness is above the threshold value were assigned to the bubble area. The corresponding threshold value was derived from a calibration measurement considering a configuration of 5 and 10 mm diameter glass balls surrounded by GaInSn at a zero gas flow rate. As a next step the images were analyzed using the Matlab function ‘regionprops’ which allows to determine bubble properties like perimeter, area, center of mass, etc. The bubble area was converted from pixel to metric values using the scaling of the image. With the assumption of spherical bubble shapes the mean bubble diameter was calculated according to dB = 2 AB /π , where AB is the projection area of a bubble in the X-ray images. The rising velocity was deduced from the bubble trajectories. Interfaces of bubbles which move in the immediate vicinity or even overlap, were determined manually. A detailed description of the image processing can be found in (Keplinger et al., 2017).

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a)

b)

c)

d)

Fig. 2. Sequence of single frames illustrating coalescence between two bubbles rising in a bubble chain for different Ar gas flow rates: a) 200 cm3 /min, b) 500 cm3 /min, c) 800 cm3 /min, d) 1200 cm3 /min (examples for coalescence events are marked by arrows).

3. Results and discussion A variety of experiments were performed for different gas flow rates (Qg ) in a range up to 1200 cm3 /min. Fig. 2 presents selected image sequences showing examples for bubble coalescence and characteristic changes that the structure of the two-phase flow undergoes for growing gas flow rate. As expected, both the

bubble size and the release frequency increase as the Ar volumetric flow increases. The rising bubbles form a bubble chain at flow rates below 800 cm3 /min (see Fig. 2a–c). In this range single bubbles detach from the injection nozzle while in the case of 1200 cm3 /min shown in Figure. 2d the bubbles are ejected as clusters which makes the subsequent analysis rather difficult. As can be seen in Fig. 2, the bubble chain is not straight but rather


Number of events per second

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6 Bubble clustering

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2

0 0

200

400

600

800

1000 1200

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Gas flow rate (cm /min) Fig. 3. Number of coalescence events per second as a function of the Argon gas flow rate.

curved. This pattern does not remain stable for high gas flow rates (Qg ≥ 800 cm3 /min), but is subject to periodic oscillations, with the bubble chain moving back and forth in the observation plane. The resulting S-shape of the bubble chain is typical for this aspect ratio of the fluid container and the chosen range of the gas flow rates. As long as the gas volume flow is small (Qg ≤ 200 cm3 /min), the release frequency is low and as a result the distances between the bubbles are large enough so that bubble wake does not affect the behavior of the subsequent bubble and interaction and collision between the bubbles hardly occur. Under these conditions the probability of coalescence is very low, but it grows continuously with an increase in gas throughput. The number of coalescence events as a function of the gas flow rate in our experimental geometry is shown in Fig. 3. An ascertainable number of coalescence events can be observed for Qg = 200 cm3 /min (see Fig. 2a) corresponding to ∼0.06 events per second for 28 ± 4 bubbles ejected per second. This value grows up to ∼5.19 events per second for 40 ± 3 bubbles ejected per second at Qg = 800 cm3 /min (see Fig. 2c). This is also the highest value for Qg , in which one could still assume that isolated bubbles rise in the form of a chain. The phenomenon of bubble clustering occurring at Qg > 800 cm3 /min does not allow to accurately determine the injection conditions and the resulting number of coalescence events. Please note that the bubble dynamics is very sensitive with respect to the conditions at the gas injection. Therefore, it has been decided to focus the analysis of bubble coalescence within this study to the situation corresponding to the highest coalescence rate occurring in a chain of single bubbles at Qg = 800 cm3 /min. For further analysis and discussion within this paper we have chosen several exemplary coalescence events from the high-speed videos. An important criterion for the selection of these events was that the process of approach and coalescence took place almost completely within the observation field of the camera. Moreover, the analysis is restricted to the situation of a fully developed flow regime which is achieved about one minute after the initiation of the Ar gas flow. In this stage, the originally straightforwardly rising bubble chain becomes bended leading to the above-mentioned S-shape of the bubble trajectories (Fig. 2). Fig. 4 illustrates an example for approach and consequent coalescence occurring between two approximately equal-sized bubbles rising in a bubble chain at a gas flow rate of 800 cm3 /min. The figure shows a sequence of enlarged image sections having a size of 17.5 × 19.2 mm2 . The position of the viewing windows was adapted by tracking the center-of-mass of the two bubbles. The entire interaction process can be viewed in the video Video1.avi (sup-

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plementary material). The process presented here is fairly close to the ideal co-axial type of coalescence where the bubbles rise along the same vertical line and the rupture of the liquid film occurs between the bottom surface of the leading bubble and the top surface of the trailing bubble. In the first frames (1–9), it becomes clear that the trailing bubble obviously moves faster in the wake of the leading one and finally catches it. The shape of both bubbles is significantly deformed and far from being an ellipsoid. The acceleration of the trailing bubble causes an elongation of the bubble in the vertical direction while the leading bubble takes more and more the form of a spherical cap. A similar behavior of the bubbles was reported by numerical studies considering the case of co-axial bubble coalescence in highly viscous fluids. The simulation results revealed the formation of a liquid jet behind the leading bubble that not only deforms the lower surface of the leading bubble, but also induces a change of the trailing bubble resulting in a pear-like elongated shape (Islam et al., 2015; Legendre et al., 2012; Zahedi et al., 2014). In the further stage of the process the bubbles approach each other and collide (frames 10–15) resulting in coalescence (frame 16). Here, due to limitations of both spatial and temporal resolution, it is difficult to identify the exact moment when the coalescence (i.e. film rupture) happens, but, likely frame number 16 illustrates the absorption of the trailing bubble by the leading one. Up to the time of frame 15 one can still see the trace of a liquid film between the bubbles. The image sequence suggests that the size of the trailing bubble becomes smaller while the size of the leading bubble increases. In frame number 13 the borders of the leading and the trailing bubble profiles are marked by dashed and dotted lines, respectively. A characteristic feature of the bubble collision is a flattening and increasing curvature of the leading bubble while the trailing bubble pushes in from below. After the two bubbles have merged the lower surface returns to a convex shape and a more spherical shape is obtained (frame 17). This effect is governed by the surface tension which acts as a force reducing the surface energy. The frames 17–24 show the subsequent motion of the coalesced bubble. Please note, that the mean diameter of the coalesced bubbles reaches values of about 11–14 mm. Thus, the shape and the movement of the bubble are strongly influenced by the sidewalls being separated by a gap of 12 mm. In general, the number of bubble coalescence events observed in the experiments was much smaller than the number of bubble collisions. If two bubbles collide there is a possibility that the bubble could either coalesce or bounce off each other. An example of such a collision without subsequent coalescence is shown in Fig. 5. The entire interaction process can be viewed in the video Video2.avi (supplementary material). Just as in the situation of the coalescing bubbles, the rising motion starts with an acceleration of the trailing bubble in the wake of the leading bubble (frames 10– 12), followed by subsequent collision (frames 13–19). In this case as well, the collision causes a distinct flattening of the leading bubble. But, instead of merging, the leading bubble makes a sideways movement sliding along the trailing bubble, the liquid film in between does not rupture. Finally, both bubbles separate and move apart (frames 19–32). It can be assumed that with growing distance between the bubbles the influence of the bubble wake on the motion of the trailing bubble becomes weaker. Fig. 6a and b compares the trajectories of 3 different pairs of colliding bubbles, in which coalescence occurs or does not occur, correspondingly (more examples are presented in supplementary material). The solid and dashed lines in the graphs represent leading and trailing bubbles and the dotted lines the coalesced ones. First of all, it is noticeable that all bubble trajectories differ strongly from a straight line, which in turn is due to the S-shape of the bubble chain. The trajectories of the trailing bubbles and the leading bubbles coincide during the initial state of the bubble rise, but then they deviate significantly from each other. This phenomenon

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Fig. 4. Sequence of single frames (enlarged image section) illustrating the interaction of two bubbles which results in coalescence. Time step between two frames is 1/180 s. Image size is 17.5 × 19.2 mm2 . Ar gas flow rate is 800 cm3 /min.

Fig. 5. Sequence of single frames (enlarged image section) illustrating interaction between two bubbles without coalescence. Time step between two frames is 1/180 s. Image size is 17.5 × 21.5 mm2 . Ar gas flow rate is 800 cm3 /min.

results from the interaction of the trailing bubbles with the complex wake structures generated by the leading bubbles. A plot of the height of the bubble centers as a function of time (Fig. 6c and d) shows that the vertical position of the leading bubble can be approximated by a straight line while the height position of the trailing bubble has a pronounced curved shape before the collision takes place. This is a clear indication of the trailing bubble in the wake. In contrast, it can be concluded that before collision the leading bubble moves almost uniformly upwards without significant influence by the trailing bubble. Such a behavior was also found in the numerical calculations by Wang et al. (2017a). Our measurements showed terminal velocities of 0.42 ± 0.06 m/s for the leading bubble. During the collision,

the velocity of the trailing bubble is decelerated down to the velocity values of the leading bubbles. After coalescence, the height of the merged bubble follows an almost straight line (Fig. 6c, dotted), but with a slightly different inclination angle corresponding to a higher velocity of the larger bubble after the coalescence due to the increased buoyancy force. The bubble diameter increases from values of 9 ± 1 mm (this value is in a good agreement with the value calculated directly from the bubble detachment frequency) to 12 ± 2 mm after coalescence (Fig. 6e and f). There exist a number of theoretical models for predicting the bubble size after gas injection through a single nozzle (see for instance Davidson and Harrison, 1963; Davidson and Schuler, 1960; Gaddis and Vogelpohl, 1986; Kumar and Kuloor, 1970). Despite of the fact, that these models do


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Fig. 6. a) Bubble trajectories before (solid and dashed) and after (dotted) coalescence, b) bubble trajectories of the leading and trailing bubbles without coalescence (solid and dashed), c) the height positions of the bubble centers before (solid and dashed) and after (dotted) coalescence, d) the height positions of the leading and trailing bubbles without coalescence (solid and dashed), e) bubble diameters before (solid and dashed) and after (dotted) coalescence, f) bubble diameters of the leading and trailing bubbles without coalescence (solid and dashed).

not cover the parameter ranges for liquid metals we found a very good match between our experimental bubble size data and their predictions. In situations without coalescence the bubble trajectories intersect, but then separate from one another and, self-evidently, the distance between their centers starts to increase (Fig. 6b). In this case the height position as a function of time of the leading bubbles can be approximated by a line during the whole observation period (Fig. 6d) while the pronounced curved shape for the trailing bubbles indicates a remarkable effect of acceleration by the wake of the leading bubble here, too. After collision, the slope of all curves becomes almost linear. The distance of the closest approach of the bubble centers during collision corresponds to values of 4.0 ± 1 mm for both cases with and without coalescence. 4. Coalescence of two bubbles inside a rising cluster of three bubbles Even more complex schemes of bubble coalescence are conceivable inside the bubble chain. In particular, many events were observed during the experiments where coalescence occurred within a cluster of three bubbles. In our experiments bubble clustering starts at gas flow rate of 800 cm3 /min, while at 1200 cm3 /min the occurrence of single bubbles is completely displaced by clusters. The observed coalescence schemes are displayed in Fig. 7. The clus-

ter itself is formed from separately ejected single bubbles which due to flow structures gather together and rise in close vicinity to each other. Schemes a and b show the coalescence between two directly adjacent (i.e. leading and middle or middle and trailing, respectively) bubbles and could be described in good approximation by the coalescence examples presented in the previous section. Scheme c illustrates a special case when the coalescence occurs between the leading and the trailing bubble while the middle bubble rises close beside it. The number of such coalescence events where the leading and trailing bubbles merge is not very large. Two examples will be presented below. Fig. 8 displays two examples according to scheme c which are highlighted by solid and dashed lines, respectively. The entire interaction process can be viewed in the video Video3.avi (supplementary material). In the corresponding images the injection of individual bubbles can be clearly detected (see the areas highlighted in frames 1 and 12). The motion of both the middle and the trailing bubble is accelerated in the wake of the precursory bubbles (i.e. in the wake of the leading and middle bubbles, respectively). This acceleration phase leads to a distortion of the shape of the middle and the trailing bubbles. The bubbles are stretched along the direction of their movement (frames 1–7 and 12–18). The trailing bubble undergoes a stronger elongation since its motion is influenced by both wakes of the middle and the leading bubble. The rapid approach of the trailing bubble from below influences the pres-

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Fig. 7. Coalescence schemes occurring in a cluster of three rising bubbles: a) coalescence between leading and middle bubble, b) coalescence between middle and trailing bubble, c) coalescence between the leading and the trailing bubble.

1

11

12

22

23

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Fig. 8. Sequence of single frames illustrating coalescence between two bubbles inside a cluster of three rising bubbles. Time step between two frames is 1/180 s. Ar gas flow rate is 800 cm3 /min.

sure field around the middle with the result that the elongation of the middle bubble is significant lower than for the trailing bubble. At the same time, the middle bubble pushes the leading one from below causing its flattening. When the bubbles come close to each other and collide, either due to particular flow pattern or because of the collision asymmetry the leading and trailing bubbles approach one another, while at the same time the middle bubble

is pushed aside (frames 10–16 and 21–26). Then, the leading and the trailing bubbles merge (frames 17 and 27). Frames 27–33 show the subsequent motion of the coalesced and middle bubbles of the second example, while the merged and middle bubbles resulting from the first coalescence event have already left the field of view. Fig. 9 presents measured data with respect to bubble positions and bubble size. The trajectory plot demonstrates that all bubbles


Fig. 9. a) Bubble trajectories of two selected events: before (solid and dash-dotted) and after (dotted) coalescence, (dashed) of the middle bubble; b) corresponding height positions of the bubble centers: before (solid and dash-dotted) and after (dotted) coalescence, (dashed) of the middle bubble; c) Bubble sizes: before (solid and dash-dotted) and after coalescence (dotted), (dashed) of the middle bubble.

move along almost the same path up to a height of about 25 mm, whereas significant deviations become apparent above this position (Fig. 9a). The temporal evolution of the height reveals that the leading bubble is caught up by both the middle and the trailing bubble after some time. Again, the slope of the lines is a measure for the bubble velocity showing that the trailing bubble is exposed to the highest acceleration (Fig. 9b). Moreover, the trailing bubble velocity remains larger than those of the other two bubbles for a longer time allowing the trailing bubble to overtake the middle bubble and to get closer to the leading one. These effects are a direct result of the interaction of the bubbles with the wakes. When the leading and trailing bubbles collide, their velocities equalize and the coalescence takes place. The increasing bubble diameters from 9.8 ± 2.2 mm to 11.4 ± 0.6 mm before and after coalescence, respectively, lead to increasing velocities of the merged bubble due to the buoyancy force (see Fig. 9c). 5. Discussion and conclusions For the first time, an experimental investigation of bubble coalescence in liquid metals has been performed on the basis of a visualization of the dynamics of bubble interaction by means of high-speed video imaging. Bubble interaction in opaque GaInSn has been studied using X-ray radiography. The analysis of the X-ray images allows for determining significant quantities such as bubble size and shape or bubble velocity. The parameter range considered here differs clearly from bubbly flows in water and highly viscous fluids which have been extensively studied in previous papers. Due to the high density and surface tension, the Mo number reaches a very small value of Mo ≈ 2 × 10−13 . The onset of coalescence is observed at 200 cm3 /min when a transition from the single bubble regime to the bubble pairing regime takes place. The pairing regime is characterized by the fact that the initially equidistant bubble chain starts to decay and two consecutive bubbles form a closer group (Badam et al., 2008; Kyriakides et al., 1997). The corresponding closer distance between the bubbles leads to an interaction of the bubble with the wake of the

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preceding bubble which promotes bubble collision. Accordingly, we observed a growing number of coalescence events with increasing gas flow rate. Moreover, the coalescence rate is supposed to depend on the liquid metal height. A higher liquid level in the fluid container increases the bubble residence time inside the liquid metal which subsequently will promote bubble interaction and bubble collision probability. Since the coalescence rate should be higher, the larger the collision frequency is, an increasing coalescence rate is expected with increasing liquid metal height. In the experiments carried out at a gas flow rate of 800 cm3 /min the bubbles show a mean equivalent diameter of about 9 mm and a terminal velocity of about 0.42 m/s. These data result in the following values for the non-dimensional parameters: Re ≈ 110 0 0, Eo = 9.4 and We ≈ 19. At such high values of the Re and the We number the bubble shape becomes non-spherical and undergoes significant deformations. The X-ray images show the occurrence of oblate ellipsoids. The high X-ray attenuation coefficient of the liquid metal restricts the dimension of the fluid vessel D along the X-ray beam. The distance of 12 mm between the side walls is in the same order of magnitude as the size of the bubbles after coalescence. Therefore, the influence of the side walls on bubble characteristics (trajectory, shape and velocity) is probably not negligible especially at high gas flow rates. In general, constraining walls can cause an elongation of bubbles in the vertical direction, suppress secondary motion and alter the wake structure by reducing the wake volume and the rate of fluid circulation within the wake (Clift et al., 1978). Nevertheless, the actual quantitative impact of the wall effects on the coalescence rate is hard to estimate. Krishna et al. (1999) found that the maximum effect of the wall can reduce the rise velocity in water by a factor of about 0.5. Furthermore, the authors report that the bubble rise velocity becomes independent on dB for dB /D > 0.6. However, in our experiments we observed a measurable increase of the rise velocity of the merged bubble after coalescence (see Figs. 6c and. 9b). The case of ideal co-axial coalescence along the vertical line, which is often considered by the numerical simulations, does not occur in this experiment. The bubble chain does not rise along a straight line. Lateral deflections of the bubble path lead to an Sshape, the position and direction of which also oscillates over time. Thus, the situation considered here corresponds more to the case of oblique coalescence. Regardless of the large deviations with respect to the nondimensional parameters, the specific features of the experiment and the turbulent character of the flow, which becomes apparent in the dynamics of the bubble chain, the behavior of two bubbles during coalescence is very similar to that described in the literature for water and viscous fluids in the qualitative sense: •

•

•

•

The motion of the trailing bubble is strongly affected by the wake structure of the leading bubble: the rising velocity is much larger than the velocity of the leading bubble. The motion of the leading bubble is only governed by the buoyancy and drag forces and by the global flow pattern inside the vessel. The shape of the trailing bubble becomes elongated in the direction of the motion while the leading bubble undergoes strong horizontal deformation towards a spherical cap as a result of the collision. After coalescence the lower surface of the merged bubble returns to a convex shape and the bubble develops into an ellipsoid.

A detailed comparison of events of bubble collision with and without coalescence does not reveal significant differences in the determining parameters which allow for identifying a quantitative criterion for coalescence. The bubble velocities and the distance of

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the closest approach of bubble centers are very similar for both cases and do not seem to be effective as governing parameters for coalescence. Duineveld (1997) suggested a criterion for coalescence of small bubbles (in the order of 1 mm) in pure water relying on a critical Weber number of Wecr = 0.18 based on the relative velocity of approach. Coalescence occurs if We falls below Wecr whereas bubbles bounce and separate for higher We numbers. By contrast, the analysis of our data gave no indication of the existence of such a threshold value for the liquid metal system investigated here. At this point, it should be pointed out that the investigations by Duineveld (1997) were carried out in the range of laminar bubble ascent. The threshold for the occurrence of coalescence is distinctly below the onset of the path instability (We = 3.3). Our experiments were conducted in the turbulent regime, the path instability manifests itself in the S-shape of the bubble chain. Therefore, we assume that the processes of bubble collision, coalescence and bouncing are mainly governed by the structure of the turbulent flow in the vicinity of the bubbles. Despite qualitative similarities in underlying bubble interaction phenomena it is worth to mention that due to considerable differences in surface tension bubbles in water and in GaInSn vary with respect to shape deformation. Keplinger et al. (2017) demonstrated that for the same injection conditions (experimental geometry, nozzle diameter and gas flow rate) the bubble size is ∼3.25 mm and ∼6.35 mm and the maximum deformation is ∼2.17 and ∼1.77 for water and GaInSn, correspondingly. If the bubble size in water is then increased to the same bubble size as in the liquid metal (from 3.25 mm to 6.35 mm), this will enhance surface wobbling. Since bubble-wake dynamics depend not only on fluid viscosity but also on bubble shape deformation, the bubble interaction within a rising bubble chain could change significantly (Fan and Tsuchiya, 1990). During the experiments presented here, we have observed not only bubble coalescence but also bubble break-up. Subsequent analysis of the bubble break-up is a part of an ongoing work. Acknowledgment The authors are grateful to the German Helmholtz Association for the financial support in form of the Helmholtz-Alliance “LIMTECH”. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijmultiphaseflow.2018. 04.001. References Annaland, M., van, S., Deen, N.G., Kuipers, J.A.M., 2005. Numerical simulation of gas bubbles behaviour using a three-dimensional volume of fluid method. Chem. Eng. Sci. 60, 2999–3011. https://doi.org/10.1016/j.ces.2005.01.031. Badam, V.K., Buwa, V., Durst, F., 2008. Experimental investigations of regimes of bubble formation on submerged orifices under constant flow condition. Can. J. Chem. Eng. 85, 257–267. https://doi.org/10.1002/cjce.5450850301. Chan, D.Y.C., Klaseboer, E., Manica, R., 2011. Film drainage and coalescence between deformable drops and bubbles. Soft Matter 7, 2235–2264. https://doi.org/10. 1039/C0SM00812E. Chen, L., Li, Y., Manasseh, R., 1998. The coalescence of bubbles -a numerical study. Third International Conference Multiphase Flow, ICMF’98. ´ M.P., Sanyal, J., 2005. Three-dimensional simulation of bubble Chen, P., Dudukovic, column flows with bubble coalescence and breakup. AIChE J. 51, 696–712. https: //doi.org/10.1002/aic.10381. Chen, R.H., Tian, W.X., Su, G.H., Qiu, S.Z., Ishiwatari, Y., Oka, Y., 2011. Numerical investigation on coalescence of bubble pairs rising in a stagnant liquid. Chem. Eng. Sci. 66, 5055–5063. https://doi.org/10.1016/j.ces.2011.06.058. Cheng, M., Hua, J., Lou, J., 2010. Simulation of bubble-bubble interaction using a lattice Boltzmann method. Comput. Fluids 39, 260–270. https://doi.org/10.1016/ j.compfluid.20 09.09.0 03. Chesters, A.K., 1991. The modelling of coalescence processes in fluid-liquid dispersions : a review of current understanding. Chem. Eng. Res. Des. 69, 259–270.

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