Collision of a small rising bubble with a large falling particle
Graphical abstract
Introduction
The gas–liquid–solid three-phase systems are found in many industrial applications. An important one is the separation of solid materials by flotation. This process is based on the ability of some solids to remain attached to the gas–liquid interface. Particles of such a solid then agglomerate with bubbles and are floated to the liquid surface, from which they can be easily separated. The flotation was originally used for the separation of coal or mineral particles from the mined ore deposits. Owing to its simplicity and high efficiency, the flotation is nowadays also used for separation of oil sands, print inks in paper-recycling, waste water treatment and also for the separation of various plastic materials in their recycling process. In the standard flotation process, the particles are usually much smaller than the bubbles (we refer to these size proportions briefly as mineral flotation thereinafter). Oppositely in the case of plastics flotation, the particles are of comparable or even bigger size than bubbles. The inverted size proportion changes the mechanics of bubble–particle interaction (Alter, 2005). To develop a suitable model for predicting the efficiency of the plastics flotation, it is required to get the detailed description of the bubble–particle interaction, i.e. of their approach, attachment and eventual detachment (Nguyen and Schulze, 2004). The theoretical models for predicting the flotation efficiency (e.g. Dai et al., 2000, Nguyen and Schulze, 2004) were developed for the mineral flotation. Due to modified mechanics, these models cannot be used for the plastics flotation with the inverted size proportions.
The first theoretical description of collision of a small particle with a large bubble was provided by Sutherland (1948). His model is based on the fact that the liquid streamlines are denser near the equator of the rising bubble. Particles move along the streamlines and some of them approach the bubble to a distance, which is smaller than the particle radius, and thus collide with the bubble surface. Sutherland assumed potential flow around a spherical bubble, and particles, which perfectly follow the liquid motion, and obtained an expression for the collision efficiency. Many conceptually similar models were developed later. Their reviews are available (e.g., Dai et al., 2000). For example, Flint and Howarth (1971) considered motion of the particle relative to the moving liquid due to particle settling. The model of Yoon and Luttrell (1989) covers all the flow regimes (Stokes, intermediate and potential) around the bubble. Other models consider a relative motion of the particle and liquid due to inertial and gravity forces (e.g., Schulze, 1989). The GSE model (Dai et al., 1998, Dai et al., 1999, Ralston et al., 1999) considers the particle inertia as well as (im)mobility of the bubble surface, and solves the particle motion via Basset–Boussinesq–Oseen equation. Recently, Huang et al. (2012) studied the particle motion using the rigorous equation for particle motion (including lift and Basset forces acting on the particle) and a fully resolved flow around the bubble.
The above mentioned models were developed for interactions of small particles with large bubbles and they are not suitable for the case of plastics flotation with inverted size proportions. Some studies of the small bubble–large particle interactions were published (e.g. Singh, 1998, Drelich et al., 1999, Shen et al., 2002, Pascoe and O'Connell, 2003, Basařová et al., 2010); however, these works provide mostly experimental results on the flotation yield as a function of various parameters, but not a fundamental study of the interaction mechanics.
The aim of this work is to broaden the knowledge of hydrodynamic interactions that occur between bubbles and solids of a comparable size or between larger particle and smaller bubble. Focusing on collision processes between a single rising bubble (Db < 1 mm) and a larger falling solid particle in stagnant liquid, a model describing the bubble trajectory is given here. The model assumes potential flow around the particle and balances forces acting on the bubble, leading to a differential equation for its motion. The bubble is considered spherical and can be either clean (with mobile interface) or contaminated (immobile interface). The particle is also spherical and moves vertically downward with a steady velocity. The model is validated experimentally by comparing observed trajectories with the computed one. The collision efficiencies are evaluated. At the end, it is found that the resulting efficiencies are governed mostly by the buoyancy and interception mechanisms and the inertial effects are only minor. For this situation, a simple expression for the collision efficiency is derived.
Section snippets
System of reference
The coordinate system is introduced as shown in Fig. 1. The frame of reference, either Cartesian (x, z) or spherical (r, θ), moves together with the particle, and its origin is fixed in the particle's centre. The particle radius is denoted Rp and the velocity of particle fall is Up. The liquid has density ρ and viscosity η. The density of gas inside the bubble is ρg.
In the frame of reference, which moves together with the particle, the liquid is observed to move upward with the velocity Up at
Materials and apparatus
Pure water (distilled, de-ionised and de-mineralised water) was used at 25 °C for the measurements. The pH value was 6.13 and conductivity 1.6 μS/cm. In addition, an aqueous solution of the surface-active agent terpineol (Fluka Company) was also used at a concentration of 187 mg/l (this solution is referred as contaminated water thereinafter). The surface tension (measured by du Nouy's ring method) was 71.6 mN/m and 63 mN/m for the pure and contaminated water, respectively. The particle was a smooth
Bubble trajectory
As a typical example, Fig. 4 shows data obtained within one experimental set. These data are provided for pure water, bubble diameter of 0.5962 mm and particle settling velocity of 50 mm/s. Several trajectories differing in the initial horizontal position of the bubble (x0, see legend) are shown. The grey lines illustrate the trajectory computed using the model with corresponding initial conditions. The positions of bubble centre in consecutive movie frames are denoted by symbols. The uppermost
Conclusions
The collision of a small rising bubble with a larger spherical particle falling through a stagnant liquid is studied. A theoretical model describing the bubble trajectory and velocity evolution is given and its results are successfully validated by comparing with experimental measurements. For both mobile and immobile bubble surfaces, the horizontal deflection of the bubble trajectory prior to the collision increases linearly with the initial bubble distance from the axis. The proportionality
Acknowledgements
This work was supported by the Czech Science Foundation (project 101/11/0806) and by Specific University Research (MSMT no. 21/2011).
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