Fluid Dynamics Studies on Bottom Liquid Detachment from a Rising Bubble Crossing a Liquid–Liquid Interface

Abstract

The detachment regimes and corresponding detachment height of lower liquid from a coated bubble during the bubble passage through an immiscible liquid–liquid interface were studied. High-speed imaging techniques were used to visualize the lower liquid detachment from a rising bubble near the interface. Analysis of industrial slag samples by a scanning electron microscope (SEM) was also carried out. The results indicate that the detachment height of lower liquid from a rising bubble showed a distinct correlation to penetration regimes. Bubble size and a fluid’s physical properties exerted a significant influence on the detachment height of the lower liquid. The detachment height for medium bubbles (Weber number: 4~4.5; Bond number: 2.5~7.5) varied significantly with increasing bubble size, which contributes to the lower liquid entrainment in the upper phase due, significantly, to the higher detachment height and large entrainment volume. The maximum detachment height for large bubbles is limited to approximately 100 mm due to the early detachment with the liquid column at the interface though large bubbles transporting a larger volume of lower liquid into the upper phase.

1. Introduction

The penetration of a single bubble through the liquid–liquid interface is a common dynamic process in a wide range of industrial applications, such as the steelmaking and copper smelting processes [1,2,3,4,5], petrochemical process [6], biological engineering, nuclear reactor safety [7,8,9], heat transfer [10,11], etc. For example, in the steelmaking process, it is very common that rising bubbles are formed by gas injection through tuyeres or porous plugs during ladle argon gas treatment, RH vacuum treatment, and a continuous casting mold to increase the thermal and chemical homogenization of the melts as well as the non-metallic inclusions’ removal by bubble adhesion and bubble wake flow in the molten steel [12,13]. Generally, it entails mass transfer in multiphase systems, resulting in the lower liquid entrainment into the upper phase. Due to the immense industrial relevance, a fundamental understanding of the phenomena of bubble penetration through a liquid–liquid interface is indispensable.

Currently, approximately 50% of copper concentrate smelting is carried out by bath smelting furnaces, including top-blown (e.g., Ausmelt), side-blown (e.g., Tenniente), and recently developed bottom-blown technologies. An immiscible liquid–liquid system containing Cu-enriched molten sulfide phase (matte) and the molten slag is formed. Interfacial tension between copper matte and slag is reported between 0.022 N/m and 0.110 N/m with matte grade from 0 to 80%. The average viscosities of bottom-blown smelting furnace (BBF) slags and Teniente Converter (TC) slags are 0.180 Pa$·$s and 0.124 Pa$·$s, respectively. The densities of slags and mattes in both BBF and TC are 3.5 $\times$ 10³ kg/m³ and 4.7 $\times$ 10³ kg/m³, respectively. Mechanical entrainment by rising bubbles was demonstrated to be responsible for the majority of copper matte entrainment in the smelting slags [2,3,4,7,14]. The dynamic film mass transfer by a single bubble in the contact of gases and liquids is a common operation [15,16,17,18,19]. The film and column from a rising gas bubble may disintegrate into many droplets, leading to entrainment, e.g., jet entrainment [18,20]. R. Minto et al. [4] reported fine copper particles’ entrainment in the reverberatory slag (viscosity 10 cP) using argon bubbles in the copper/slag system (1200 °C). Metal droplet entrainments by bubble penetration behaviors through the iron and slag interface were observed by using X-ray techniques [3,15,21]. A. Scheludko et al. [22,23,24] proposed that there exists a critical thickness leading to the film rupture, and the theory of the critical thickness of the rupture of free thin liquid films. The bubble force balance could still be maintained during the film thinning period from a macroscopic view until the film reaches a critical film thickness [25,26]. Y. Mori et al. [27] reported that the lower limit of bubble diameter to pass through the interface between superposed liquid layers fell in the range of 3.4 to 4.4 mm in glycerol–water and Freon 113 systems. J. Mercier et al. [28] found that medium gas bubbles between 3.4 and 5.5 mm could maintain the water layer after the bubble passes through the interface between superposed liquid layers. M. Islam et al. [29] made a numerical investigation to study bubble-assisted matte transportation into the slag phase.

Experimental studies were carried out to understand the bubble passage through the immiscible liquid–liquid interface [18,22,30,31,32]. A rising bubble at the liquid–liquid interface stemmed from the interface deformation and thinning rates, and different-sized bubble groups showed different flow regimes [22,33]. The residence time of the bubble at the liquid–liquid interface was documented, and the time spent at the interface varies with bubble diameter, interfacial resistance, viscoelastic, and the shearing–thinning behavior of the heavy liquid [18,32]. K. Singh et al. [10,34] reported the effects of physical properties on two fundamental quantities, i.e., retention time and retention height. M. Tanno et al. [17] studied the formation of the liquid column and found interfacial tension shows a slight influence on the column height, while the increasing density of the lower liquid is responsible for the reduced column height. N. Mao et al. [35,36] revealed the formation and detachment of the enclosing water film as a bubble crosses the water–oil interface.

However, there is no mature evaluation system to predict bubble influence on the lower liquid entrainment and clarify the relationship between bubble penetration behavior and lower liquid entrainment possibility. The objectives of the present work are to visualize the bubble behaviors at the interface to overcome the difficulty of observation in high-temperature experiments, and investigate the detachment heights of the lower liquid from a rising gas bubble after crossing a liquid–liquid interface and the correlation with the lower liquid transport regimes to explore copper matte entrainment by bubbles in smelting slags. In the present study, copper slags from bottom-blown smelting and side-blown smelting processes were used to determine matte entrainment by bubbles. This finding can provide clear knowledge of different bubble penetration regimes and the corresponding detachment height, which is vital to the guidance for predicting lower liquid entrainment mechanisms in two immiscible liquids.

2. Experimental Methodology

2.1. Experimental Setup

The cold model experimental setup shown in Figure 1a was used to capture bubble penetration behaviors through a liquid–liquid interface to simulate bubble penetration through a copper matte–slag interface. The system consisted of a 600 mm high transparent glass container (80 mm × 80 mm × 600 mm), in which a single bubble was generated through a glass inlet tube connecting with a syringe pushed by a syringe pump. A high-speed camera (FASTCAM SA1.1, Photron Ltd., Tokyo, Japan) was placed at the level of the interface between immiscible liquids in front of the glass cube to capture the bubble penetration behaviors at 500 fps with 1280 × 720 pixel resolution. A light source (XGY-II, Zhejiang Xinguangyang Lighting Co., Ltd., Haining, China) in the opposite direction of a high-speed camera was used to provide dispersed light through a light shading board to illuminate liquids in the glass cube. Air was fed from a syringe pushed by an infusion pump (Kd Scientific 780100, KD Scientific Inc., Holliston, MA, USA) to control the airflow rate precisely. The bubble size could be controlled through airflow rate and custom-made replaceable glass nozzles. The bubble size and lower liquid detachment height in the upper phase were quantified by Image analysis. Detachment height H measured by the high-speed camera is defined as the maximum displacement of the lower liquid from the interface at the quiescent state in the upper liquid after detaching from the rising bubbles, as shown in Figure 1b.

2.2. Materials

The silicone oils of different viscosities (10 cP, 50 cP, 200 cP, 500 cP) were used as the upper liquid layer (thickness: 400 mm) to simulate different copper smelting slag viscosities, and water solutions (thickness: 200 mm) were used to simulate copper matte. SDS (sodium dodecyl sulphate) powder was added to the solution to adjust surface tension with negligible density and viscosity variation, and a drop volume tensiometer was used to measure liquid surface tension and control the SDS addition amount. Glycerol solutions of different concentrations (80%, 70%, 60%, 40%) were used to reduce the density and surface tension interference. CaCl₂ was used to change the lower liquid density due to its high solubility in water. The interfacial tension between the lower liquid and upper liquid was changed by changing the lower liquid surface tension by SDS surfactant (sodium dodecyl sulfate). Water, glycerol, CaCl₂, and silicone oils were used to create a series of liquid−liquid interface systems. The surface tensions, viscosities, and densities of the liquids were acquired from the manufacturer and verified by a tensiometer, viscosimeter, and densimeter in the laboratory, respectively. The liquid combinations were chosen to cover a broad range of interfacial tensions, viscosities, and density ratios, as shown in

Table 1. Physical properties of upper and lower liquids used in the cold model experiments.

Lower Phase				Upper Phase
Material	Viscosity/cP	Density/ g·cm−3	Surface Tension/ mN·m−1	Material	Density/ g·cm−3	Surface Tension/ mN·m−1
Glycerol 80%	60.1	1.21	48.06	10 cP silicone	0.930	20.1
Glycerol 70%	22.5	1.19	48.26	50 cP silicone	0.959	20.7
Glycerol 60%	10.8	1.16	48.11	200 cP silicone	0.970	21.1
Glycerol 40%	3.72	1.10	48.66	500 cP silicone	0.970	21.1
CaCl2 40%	8.48	1.42	58.03
CaCl2 30%	3.33	1.30	58.28
CaCl2 20%	1.81	1.19	57.00
Water (SDS)	1.00	1.00	42.89
	1.00	1.00	49.20
	1.00	1.00	55.48
	1.00	1.00	62.36
	1.00	1.00	72.14

.

In the cold model, water viscosity (1.00 cP) is similar to copper matte viscosity, and silicone oil of 200 cP is close to slag viscosity. Furthermore, interfacial tension between water and silicone oil in the water model is between 22.19 and 51.44 mN/m, which is in the range of 0.022 N/m to 0.11 N/m as reported in the literature. Dimensionless numbers based on the fluid properties and bubble size were proposed to evaluate the bubble penetration behaviors at the interface.

2.3. Image Processing

The diameters of bubbles and liquid droplets were measured by Image J. Four bubble shapes were observed in cold model experiments: spherical, oblate ellipsoidal, oblate ellipsoidal cap, and spherical cap from small scales to large scales. Due to the deformation of rising bubbles, the equivalent bubble diameter d_eq was defined as

$$d_{eq}={\left(d_h^2{d}_v\right)}^{1/3}$$

(1)

where d_h and d_v are horizontal and vertical diameters, respectively [22].

3. Results

3.1. Bubble Passages at the Interface between Immiscible Liquids

The studies of bubble passages at the interface and their corresponding detachment regimes between the bubble and the lower liquid phase are presented in this section. A single bubble injected into the bottom layer will rise to the interface due to the buoyancy force and reach the stable terminal velocity in several milliseconds. Different bubble penetration behaviors and detachment regimes between the bubbles may come up depending on the bubble and physical properties of the two immiscible liquid layers. In the present experiments, three detachment regimes are classified based on the bubble size: Figure 2a–c show the thin liquid film detachment from small bubbles, Figure 2d–f present the lower liquid shell detachment from medium bubbles, and Figure 2g–i indicate lower liquid column detachment from large bubbles.

3.1.1. Thin Liquid Film Detachment from Small Bubbles

Figure 2a–c show the penetrating process of a small bubble sized 2.5 mm in diameter from water into the silicone oil. It can be seen that the interfacial force tended to prevent the bubble from crossing the interface and made it dwell at the interface for several milliseconds. Trapped bubbles undergo low interface deformation and low thinning rates. The thin bubble cap film would drain gradually from the bubble top surface due to the pressure gradient. When the bubble cap film reached critical film thickness, the gas bubble broke up the interface and ascended to the top (Figure 2a) [24,25,37]. The short, denser liquid bridge connecting the bubble and interface as shown in Figure 2b broke eventually and a small proportion of the liquid film at the micron scale remained stable on the bubble surface. The stability of thin liquid film on small gas bubbles is controlled by film surface tensions, and gravity and other forces can be ignored. The surface tension of thin denser liquid film makes it difficult to peel off from the bubble’s bottom surface to form a free denser liquid globule as shown in Figure 2c. The thin imperceptible liquid film is believed to be responsible for the lower liquid entrainment by microbubbles in metal–slag systems due to the stable attachment determined by the surface tension [1,4,5].

3.1.2. Lower Liquid Shell Detachment from Medium Bubbles

Figure 2d–f illustrate the process of a medium gas bubble (3.49 mm diameter) transporting the denser liquid into the upper phase. It was found that the thick lower liquid layer enveloping the bubble surface could only be caused when the bubble could penetrate the interface immediately. Medium gas bubbles are seen with direct interface penetration and a perceptible thick lower liquid layer at the bubble’s bottom after penetration. A large volume of the lower liquid could be transported higher in the upper liquid due to the stable attachment, making a large volume of lower liquid entrainment possible. Generally, the medium gas bubble’s and water droplets’ detachment will go through four stages: the bubble penetrating the interface (Figure 2d), water column formation (Figure 2e), water column breakup, and the water droplets’ detachment from a rising bubble (Figure 2f). The water column formed when the gas bubble attempted to penetrate the water–silicone oil interface, and a visible stable and thick water layer covered the surface of the gas bubbles (Figure 2f). After rising to a certain height, the water layer drained and was suspended underneath the gas bubble in the form of a droplet due to the higher specific gravity. Note that the medium gas bubble penetration regimes do not necessarily happen in all superposed liquid systems, depending on the liquids’ physical properties.

3.1.3. Lower Liquid Column Detachment from Large Bubbles

Large gas bubbles are associated with rapid penetration through an interface with a long liquid column due to the large momentum. However, bubbles tend to detach from the lower liquid column near the interface when the column ruptures, and the scattered droplets would settle down quickly. Figure 2g–i show the large bubble penetration process through the interface. Because of the much larger bubble buoyancy force than interfacial tension, the bubble could pass through the interface directly with a long liquid column behind as the bridge between the bubble and interface (Figure 2h). Generally, the lower liquid layer on the bubble surface tends to rupture before the long liquid column is scattered into several droplets (Figure 2i). The rupture scenario is seen with high interface deformation and high drainage rates. This bubble penetration regime is always happening in different liquid systems when the bubble reaches a critical bubble size [38]. The large gas bubbles have a greater capability to entrain a larger volume of lower liquid, however, with a limited detachment height below 100 mm.

3.2. Detachment Height of Lower Liquid from a Rising Bubble

3.2.1. Influence of Interfacial Tension

Figure 3a shows the influence of the interfacial tension on the detachment height of lower liquid from a rising bubble in the cold model experiments. The combination of silicone oil 50 cP and water with sodium dodecyl sulfate (SDS) was adopted to change the interfacial tension determined as the surface tension difference between two immiscible liquids. Bubbles ranging from 3.2 mm to 10 mm in diameter and interfacial tension (22.19~51.44 mN/m) were tested. The density and viscosity remained constant with varying interfacial tensions. Tendencies between bubble size d and detachment height H were drawn to reveal their relationships. As shown in Figure 3a, for bubbles in the range of 3.0 mm and 4.5 mm, the lower liquid was more likely to be transported up to 400 mm in silicone oil by rising bubbles in the form of a stable enveloping layer on the bubble surface. The lower liquid droplet detachment height decreased significantly with increasing bubble size. For bubbles larger than 4.5 mm, the detachment height at different interfacial tensions was restricted to no more than 100 mm, which is equivalent to the column height of the lower liquid column height. This is because large bubbles induced a lower liquid column when penetrating the interface due to the large momentum, but did not transport the liquid layer enveloping the bubble surface stably. The large droplets from the scattered column were carried upward for a short distance in the bubble wake, which is in agreement with M. Tanno’s study [17].

The interfacial tension also has an obvious influence on detachment height. Increasing the upper liquid surface tension and decreasing the lower liquid surface tension would be beneficial to the liquid layer’s stability on the bubble surface [2]. For the same-sized medium bubbles (3~4.5 mm), the dense liquid detachment height reduces with increasing interfacial tension from 22.19 mN/m to 34.78 mN/m. When the interfacial tension is larger than 41.66 mN/m, the detachment height of all bubbles will be restricted under 100 mm, whereas that of lower interfacial tension tends to exhibit a larger value.

3.2.2. Influence of Upper Liquid Viscosity

Figure 3b shows the lower liquid detachment height of bubbles in different viscosity silicone oils. The water as a lower liquid could keep density at 1000 kg/m³ and surface tension at 50 mN/m by adding different amounts of SDS. Bubbles smaller than approximately 3.5 mm in diameter were not observed to transport a perceptible amount of lower liquid into silicone oils. The detachment height of the lower liquid descended notably with increasing bubble size from 3.5 to 4.3 mm and leveled off with increasing bubble size. Moreover, the detachment height decreased with increasing upper liquid viscosity from 10 cP to 500 cP for the medium-sized bubbles (3.5~4.3 mm). However, the increasing upper liquid viscosity made the rising bubble and attached liquid layer separate more easily due to the large viscous force in the rising process. In 500 cp silicone oil, each of the bubbles could only cause a liquid column without a stable enveloping liquid layer being carried upward.

3.2.3. Influence of Lower Liquid Viscosity

The larger dense liquid viscosity decreased the bubble impact momentum at the interface and also retarded the liquid column rupture in the upper liquid, leading to the smaller detachment height. The influence of bubble size and the lower liquid viscosity effects are shown in Figure 3c. Taking advantage of the high solubility of glycerol in water, glycerol solutions with a viscosity varying from 3.72 cP to 60.1 cP were used as the lower liquid and 100 cP silicone oil as an upper liquid. The larger lower liquid viscosity would make the lower liquid layer attachment to the bubble surface difficult, and reduce the high detachment height possibility. Thus, for a bubble in the medium size range, the detachment height would increase with decreasing lower liquid viscosity. When the liquid viscosity reached up to 60.1 cP, the bubbles in the medium and large size range could only cause a glycerol solution column near the interface with scattered droplets near the interface. The detachment height was slightly influenced by the large bubble size due to the large column length caused by them, which, however, was not comparable with that of medium bubbles.

3.2.4. Influence of Lower Liquid Density

The density of the lower liquid could not only influence the settlement velocities of the lower liquid droplets in the upper liquid but also the attachment stability to a rising bubble surface. Due to the higher solubility of CaCl₂ in water, CaCl₂ solution density was changed from 1000 kg/m³ to 1416 kg/m³ as the lower liquid phase, and 100 cP silicone oil was used as the upper liquid. As shown in Figure 3d, the bubbles in the medium size range (3.2~4.6 mm) could carry lower liquid with densities at 1000 kg/m³, 1189 kg/m³, and 1298 kg/m³ to a higher position in the silicone oil. However, the detachment height increased with decreasing lower liquid density, despite the unobvious detachment height difference between 1189 kg/m³ and 1298 kg/m³. For the system of density 1416 kg/m³, when the bubble was smaller than 4.0 mm, the lower liquid could also be transported high by rising bubbles; however, not as high as in the systems of lower density. Generally, when the bubbles are larger than 4.6 mm, there is no significant difference among the detachment heights of lower liquids with varying densities.

Note that the stability of an indiscernible thin liquid film or micro-droplets attached to the small or large gas bubbles is mainly determined by surface tension and interfacial tension between liquids and gas bubbles, which is not discussed here [2]. To conclude, bubbles in the range of 3.0 mm and 4.5 mm in diameter more likely formed a stable lower liquid layer on the surface, as shown in Figure 2d–f, and caused a larger detachment height. A fluid’s physical properties also have a significant influence on the detachment height.

3.3. Bubble Terminal Rising Velocities

The terminal velocity of a single bubble is of fundamental importance for studying a series of bubble behaviors when penetrating the interface between two immiscible liquids, as shown in Equation (2).

$${\displaystyle \frac{\pi }{6}}{d}_{eq}^3\rho g={\displaystyle \frac{\pi }{8}}\rho V^2{d}_{eq}^2{C}_D$$

(2)

where C_D is the dragging coefficient; ρ, d_eq, and V denote the liquid density, bubble equivalent diameter, and terminal rising velocity. The velocity of an air bubble has been studied by numerous investigators in different experimental conditions. Many empirical or semiempirical equations of C_D have been proposed as a function of Re [39]. For small bubbles maintaining an approximately spherical or nearly spherical shape at low Reynold number, Stokes solution with C_D = 4(ρ_l − ρ_g)gd_eq/3ρV² can provide a reasonable prediction. For pure systems, the terminal velocity was described by Hadamard and Rybcznski. Travis made a detailed analysis of bubble terminal velocity in experiments by considering C_D determined by Loth’s theory for clean and contaminated bubbles. The results showed that bubbles rise with mobile surface conditions [22].

A series of experiments were carried out to determine the suitable bubble terminal velocity equations in the experiments, and the bubble velocities in 50 cP silicone oil are plotted in Figure 4. Furthermore, the velocity of a single bubble coated by lower liquid after passing through the interface in 50 cP silicone oil and water solution systems with different interfacial tensions was also investigated and compared with bubbles after detachment from the lower liquid shell.

The results showed that the bubble terminal velocity for mobile surface conditions is larger than that for immobile surface conditions. Bubble rising velocity measured by the high-speed camera in 50 cP silicone oil were predicted well by Travis’ theory that the bubbles rise with mobile surface conditions. Stokes and H-R equations were only applicable for bubbles with a low Reynolds number (i.e., small bubble size). It is detectable that as the bubble passes through the interface, the terminal rising velocities of bubbles coated by lower liquid decrease 10~20 mm/s compared with uncoated bubbles due to the gravity force of the lower liquid coating on the rising bubble surface.

3.4. Correlation of the Detachment Height and Non-Dimensional Numbers

The characteristics of a bubble can be further understood through dimensionless numbers which involve various bubble parameters. The detachment height of the lower liquid from a rising bubble is closely associated with the corresponding detachment regimes, mainly involving two significant competing processes of the collision: (i) lower liquid film drainage, and (ii) interface deformation [33]. Essentially, the principal forces dominating the phenomenon are viscosity force, inertia, and interfacial tension. Thus, four typical dimensionless numbers are investigated to characterize the lower liquid detachment height from the rising bubble.

The Reynolds number (Re) based on the terminal bubble rising velocity obtained in the dense or light phase represents the ratio of inertial forces to viscous forces and is closely related to the bubble shape or passage of time (i.e., time required to cross the interface) and also investigated here (Equation (3)) [9]. The Weber number (We) as the ratio of kinetic to surface energy using the bottom liquid density and interfacial tension, denoting the necessary kinetic energy required to induce the interface deformation, was applied by Travis, as shown in Equation (4) [22]. The Bond number (Bo) denoting buoyancy vs surface tension and Archimedes (Ar) denoting buoyancy vs viscous force to classify the bubble shapes and flow regimes at the interface were also investigated in the previous literature, as shown in Equations (5) and (6). Equations of dimensionless numbers are defined as follows:

$$Re={\displaystyle \frac{{\rho }_{B}V{d}_{eq}}{{\mu }_B}}$$

(3)

$$We={\displaystyle \frac{{\rho }_L{V}^2{d}_{eq}}{{\sigma }_I}}$$

(4)

$$Eo={\displaystyle \frac{{\rho }_{B}g{d}_{eq}^2}{{\sigma }_I}}$$

(5)

$$Ar={\displaystyle \frac{{{\rho }_B\left(g{d}_{eq}^3\right)}^{1/2}}{{\mu }_B}}$$

(6)

where ρ_B, μ_B, and σ_I denote the density and viscosity of the lower liquid, and interfacial tension; d_eq serves as the bubble equivalent diameter. The relationships between dimensionless numbers and the lower liquid detachment height were studied as shown in Figure 5.

In Figure 5a, the detachment height descends with increasing Re for a specific liquid–liquid combination, while no obvious overall relationship between detachment height and Reynolds is observed. Obviously, all bubble penetrations occurred in the laminar flow regime with the maximum Re up to 2291 to simulate bubble penetration through the copper matte interface in most cases in the quiescent settlement zone. This is probably because detachment height is closely related to bubble shape, which is not solely dependent on the Reynolds number but also the Bond number and Moton number [39]. However, no matter whether there are small bubbles with a spherical shape or large bubbles with a non-spherical shape, a larger Re will lead to the unsteady wake at the rear surface of a rising bubble, which contributes to the smaller detachment height of the lower liquid from the bubble.

As we can see in Figure 5b, the Weber number as indicative of the kinetic energy required to cause interface deformation shows a good correlation with detachment height between the bubble and lower liquid. The results indicate that the critical bottom interfacial Weber number for a bubble to induce a high detachment height (>100 mm) in different liquid–liquid combinations is approximately in the range of 4~4.5, which is remarkably consistent with Travis’ previous study. Bubbles with a Weber number <4 tend to be spherical or nearly spherical bubbles trapped at the interface at the first impact on the interface. In this condition, trapped bubbles penetrate the interface with the drainage of the liquid film between the bubble and the interface with an indiscernible lower liquid film surrounding it [4]. With the increasing Weber number, detachment height decreases dramatically due to the shell rupture prior to the rupture of the long column induced by large bubbles as shown in Figure 5b. This dimensionless number can be regarded as an important criterion to predict the possibility of a gas bubble causing a large lower liquid entrainment volume and higher detachment height, which provide fundamental guidance for industrial application.

Bo and Ar numbers plotted in Figure 5c,d are two important dimensionless parameters to determine the bubble shape and their corresponding penetration behaviors. Evidently, the values of Bo in the range of 2.5~7.5 exert a more significant influence on the detachment height between the lower liquid and a rising bubble. Detachment height decreases dramatically with increasing Bo, and levels out below 100 mm when Bo is larger than 7.5. However, the relationship between detachment height and the Ar number does not appear to have a uniform correlation in different liquid–liquid combinations, while the detachment height more likely undergoes an obvious decline with an increasing Ar number when Ar < 1000. This is mainly because a high Archimedes number represents a toroidal shape with a large bubble size, which makes it difficult for the lower liquid shell to remain stable at the bubble surface, leading to the detachment prior to the rupture of a long column dragged by a large rising bubble. A Grace diagram involving Reynolds numbers (Re) and Eotvos numbers (Eo = Bo) is illustrated in Figure 6 to characterize the penetration detachment height by dimensionless number, with the results in cold model experiments.

4. Industrial Applications

In the copper-making process, bubbles were generated through the oxidation of concentrate to form bubbles or the injection of O₂-enriched gases, as shown in Equation (7).

$${\mathrm{CuFeS}}_2\left(\mathrm{s}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)+{\mathrm{SiO}}_2\left(\mathrm{s}\right)\to \left(\mathrm{Cu}\text{-}\mathrm{Fe}\text{-}\mathrm{S}\right)\left(\mathrm{l}\right)+\mathrm{FeO}\text{-}{\mathrm{SiO}}_2\left(\mathrm{l}\right)+{\mathrm{SO}}_2\left(\mathrm{g}\right)$$

(7)

To identify the sources of matte entrainments, as-received slag samples from the bottom-blown smelting furnace (BBF) slags, Teniente Converter (TC) slags, and laboratory slags were analyzed by Scanning Electron Microscope (Sigma Cryo-SEM, Zeiss, Germany), as shown in Figure 7. In Figure 7a,b, matte droplets are observed to be attached to bubbles in the industrial slag samples, which is also demonstrated by laboratory slag samples in Figure 7c,d.

In molten copper matte and slag systems, the flow regimes during the passage of rising bubbles through the matte–slag interface can be predicted by dimensionless numbers We or a Grace diagram. Firstly, the size range of medium bubbles with We between 4~4.5 in the matte–slag system are 3.8~4.0 mm in BBF slags and 3.5~3.7 mm in TC slags, respectively. Bubbles in this size range can transport a considerable amount of matte into the slag phase with a larger detachment height from entrained matte in the slag phase, as shown in Figure 2d–f. However, the detachment heights of medium bubbles are more sensitive to the bubble size and fluid physical properties, as shown in Figure 3a–d. Therefore, how to control bubble size below 3.8~4.0 mm in BBF slags and 3.5~3.7 mm in TC slags by adjusting a fluid’s properties to reduce the matte entrainment by medium bubbles should be further studied. Secondly, larger bubbles tend to cross the matte–slag interface directly with no stable matte layer attachment to the rear of a rising bubble, while a bunch of lower liquid droplets from a scattered liquid column will probably be formed as shown in Figure 2g–i. Note that more than 90% of bubbles that remain in the slags are smaller than 400 μm, while 90% of matte droplets are smaller than 270 μm in both BBF and TC slags. Despite no visible water being directly observed to be attached to the small bubble surface when penetrating the interface as shown in Figure 2a–c, micro bubbles still exert a significant influence on matte entrainment in the slag phase due to their stable attachment to matte droplets by bubble–matte interfacial tension in the slag phase and slow matte settlement velocity returning to the matte phase, which indicates micro bubbles would inhibit the matte droplets settling down to the matte phase. Thus, reducing micro bubbles by complete oxidization of copper matte in the reaction zone will directly facilitate the matte settlement process in the slag in the quiescent settlement zone of a smelting furnace.

5. Conclusions

Bubble penetrations through an immiscible liquid–liquid interface that occurred in the laminar flow regime were investigated by real-time observations in cold model experiments and industrial copper slag sample analysis. Three representative bubble flow regimes through liquid–liquid interfaces were identified based on the bubble size. A Grace diagram and dimensionless numbers for the bubble Reynolds number up to 2291 and the Eotvos number up to 33 were proposed to predict the bubble behaviors in specific liquid–liquid systems.

Small bubbles (We < 4, Eo < 2.5) with a low Re and Eo occurring in the industrial slag samples will dwell at the interface for a while after the first impact on the interface until the thin film between the bubble and the interface reaches a critical film thickness and ruptures. No visible lower liquid was found to be transported into the upper phase directly by small bubbles.

Medium bubbles (4 < We < 4.5, 2.5 < Eo < 7.5) are the leading factor to result in lower liquid entrainment in the upper phase. The detachment heights of the lower liquid from rising medium bubbles are much more sensitive to the bubble size and fluid properties’ variation, which ascends with decreasing interfacial tension, fluid viscosity, and light liquid density. Bubbles with larger Reynolds numbers (Re > 900) tend to appear as ellipsoidal in shape and transport the lower liquid to a limited height despite having a larger lower liquid volume.

The dimensionless numbers would assist in plotting a flow regime map and distinguishing bubble penetration regimes based on bubble size and fluid properties. An evaluation of lower liquid entrainment by rising bubbles was also made by quantifying the detachment height, which could be greatly beneficial for optimizing copper smelting operation for industries.