Bubble-Induced Entrainment Between Stratified Liquid Layers

Three-phase flows occur in numerous industrial applications, in free and porous media (nuclear safety, petroleum engineering, and so on). Many biomedical and chemical processes involve mixtures of three or more fluids, for example double emulsions, which play critical roles in applications such as prolonged drug delivery systems, entrapment of vitamins and flavor encapsulation for food additives. Considering a configuration of two immiscible liquids in stratified layers, it is known that a gas bubble of sufficient size rising through the stratified layers can entrain some of the heavier liquid from the lower layer into its wake and transport it to the upper layer of lighter liquid. This entrainment phenomenon has significant effects on both heat and mass transfer, and is encountered in a number of industrial applications. For example, in chemical processing, gas bubbles are sometimes released into pools of stratified liquids to induce mixing and phase transport. In nuclear-reactor safety applications, bubbles of non-condensable gas rise through stratified liquid pools of molten fuel and materials. In metallurgical processing, bubbles of oxygen and other gases may bubble through pools of molten metal with overlying pools of molten silica slag (Ref. 1).

However, computational models for direct simulation of three-phase flow are rather rare in comparison with the large body of research on two-phase flow. The current model, based on the predefined multiphysics coupling between the Laminar flow and Ternary phase field interfaces, simulates three-phase flow (gas and two liquids) with large density and viscosity differences.

This benchmark model simulates a single gas bubble which rises through the interface between two stratified liquid layers thereby inducing entertainment, and validates the results against literature (Ref. 2).

Model Setup

When a gas bubble rises in a two-stratified-liquid-layers configuration, the bubble can either become captured at the interface, or penetrate into the light phase, possibly leading to entrainment of the heavy phase. The criterion is based on a macroscopic balance between buoyancy and surface tension forces, and states that the bubble will penetrate the liquid-liquid interface, if its volume, V, satisfies (Ref. 2),

where g denotes the gravity constant, σ denotes the surface tension, and 1, 2, 3 are the suffixes for gas, heavy liquid, and light liquid, respectively. This criterion has been validated both experimentally and numerically (Ref. 1, Ref. 2).

This model is one of the case considered in Ref. 2. The physical properties of the fluids used in the simulation are listed in Table 1 and Table 2.

Table 1: Surface tension between three phases.
Parameters	Surface tension (N/m)
, (between the gas and the two liquids)	0.07
(between the two liquids)	0.05

Table 2: Physical properties of three phases.
	Density (kg/m3)	Viscosity (Pa·s)
Bubble	1	1e-4
Heavy liquid	1200	0.15
Light liquid	1000	0.1

Inserting the physical parameters above in Equation gives Vc = 8.8726·10−8 kg/m3. Because the volume of a sphere is given by V = 4πr3/3, the critical bubble radius can be estimated to be rc = 2.778 mm.

The radius of the bubble in the simulation is r = 8 mm. Since r > rc, the gas bubble will penetrate the liquid-liquid interface. Due to its relatively large volume, it will furthermore entrain some of the heavy liquid into its wake and transport it into the light liquid forming small droplets of heavy liquid in the upper layer. The mobility M0, which is taken as a function of the phase field variables, is defined in such a way that it becomes zero in each pure phase,

, where Mconst = 2·10−5 m3/s.

The model is axisymmetric; Figure 1 shows its geometry. The mesh contains around 36150 elements with linear shape functions. The simulation time interval is between 0 and 0.65 s, which is around the time the bubble needs to rise through the given column.

Figure 1: Geometry of the computation domain.

Results and Discussion

The simulation results are shown in Figure 2. Initially, a gas bubble is placed underneath the surface between a heavy liquid and a light liquid, see Figure 2(a). The gas bubble then passes into the upper pool and a volume of heavy liquid is clearly seen to be entrained in its wake, having been pulled through the interfaces between the two liquid layers, see Figure 2(b). If the bubble were of insufficient size, the levitated column would fall back to the lower pool and the gas bubble would continue to rise through the upper pool without having caused any entrainment. This is not the case in current model. Here, the size of the bubble is sufficiently large that the levitated liquid column rises to a height such that it eventually pinches off from the lower layer, see Figure 2(c). As the column elongates, it becomes hydro-dynamically unstable and pinches off into a series of small droplets. A small amount of the heavy liquid adheres to the bubble, see Figure 2(d), and can be considered to have been successfully entrained into the upper fluid layer.