Process 1 Collision Efficiency

For a batchwise flotation process, the flotation recovery (the mass of particles recovered in a given time t) R is given by:

{3GhEcEAEs\ , , .. R = 1 - exp - t (-2Jy— ) = 1 " exp( - tk)

where G is the volumetric gas flow rate of a swarm of bubbles of diameter db passing through a particle suspension of volume V and depth h, and:

3GEcEAEsh 2dbV

where EC is the collision efficiency, EA is the attachment efficiency and ES is the stability efficiency of the bubble-particle aggregate. This dissection of capture efficiency into three parts was originally proposed by Derjaguin and Dukhin (1960-61) and focuses attention on the three zones of bubble-particle capture where, in order, hydrodynamic interactions, surface forces and forces controlling bubble-particle aggregate stability are dominant.

The flotation rate constant k is directly analogous to that obtained in chemical reaction kinetics. Its value will be partly determined by the substep(s) in bubble-particle collision, attachment and detachment processes, as well as by physical variables such as G. (For a constant G and constant bubble size distribution, db will be an appropriate average.)

Equation [2] has been shown to apply, for example, to a system of monodisperse polystyrene latex particles floating under batchwise conditions. A plot of ln (1 — R) versus t yields the rate constant k. For systems that are polydisperse in particle size and/or in which particles of different hydropho-bicities are present, the recovery then becomes the sum of a series of exponential terms and the plot of

Table 1 Key papers in understanding fundamental flotation substeps (details of references are given in Further Reading)

Date Area ofresearch

1948 A fundamental paper by Sutherland on the kinetics of the flotation process appeared in Australia. This paper invoked induction time, described particle size effects in flotation, and catalysed other similar approaches. While it was preceded by other efforts, this paper was the first comprehensive effort to describe recovery, size and time data in a fundamental manner.

1960-61 In Moscow, Derjaguin and Dukhin produced a key paper on the theory of flotation of small and medium-sized particles. Hydrodynamics, surface forces and diffusiophoresis were all used in this theory. This seminal work resulted in an acceleration of fundamental flotation research worldwide.

1972 Blake and Kitchener, working together in London, published some very careful measurements of the thickness of aqueous films on hydrophobic quartz surfaces. Film thicknesses, measured as a function of salt concentration, were shown to depend on the electrical double layer force. Film instability occurred on hydrophobic surfaces at film thicknesses less than about 60 nm. This value, which was smaller than the range of the electrical double layer force, represented the combined effects of hydrophobic force, surface heterogeneities and external disturbances. Blake and Kitchener's film thickness studies hinted at the length dependence of hydrophobic forces, information which was subsequently obtained by surface force experiments after 1982.

1976 Scheludko and colleagues in Bulgaria considered how particles might become attached to a liquid surface and developed the capillary theory of flotation.

1977 Anfruns and Kitchener published the first measurements of the absolute rate of capture of small particles in flotation. This was the first critical test of collision theory under conditions where the bubble and particle surface chemistry was characterized and controlled.

1983 Schulze published a key textbook on the physicochemical substeps that are important in flotation, drawing on a wide range of hydrodynamic, surface chemical and engineering information. Originally published in German, once translated into English the book captured an international audience.

c. 1980-present There has been a strong interest in developing reliable collision models (Dai etal., 1998). The surface force apparatus and, recently, the atomic force microscope colloid probe technique, have provided very useful insight into electrical double layer, van der Waals and hydrophobic forces (Israelachvili, 1985; Fielden etal., 1996). Thin film drainage has been investigated between a rigid and a deformable interface (Miklavcic et al., 1995). Attachment efficiencies have been measured (Hewitt et al., 1995). Reliable methods for measuring contact angles on particles have been developed (Diggins et al., 1990). Major theoretical and experimental advances in describing dynamic contact angles on well-defined surfaces have been made (Blake, 1993).

ln (1 — R) versus t will show curvature, reflecting the different contributions to the recovery from the various particles present in the mixture.

In the metallurgical literature, R versus t data are frequently analysed by assuming that the pulp consists of 'fast' and 'slow' floating components, allowing the respective rate constants (kf and ks) and fractions (ff and fs) to be determined. Although this is a gross simplification of the real multicomponent situation, much valuable information may be gleaned from such an analysis. In fact the latter is frequently used to examine the flotation behaviour of particles of a specific size range in flotation circuits, where the behaviour of an individual flotation cell or bank of cells may be approximated to a batchwise process.

Derjaguin and Dukhin were the first to distinguish three zones of approach of a bubble and a particle on the basis of the different kinds of force in each zone (Figure 1). This model is a very useful one and helps to identify the various contributions to capture efficiency. However, it should not be taken to mean that there are well-defined boundaries between each zone; rather they grade into one another, the importance of the various contributing effects in each zone being more accurately identified as further information becomes available.

Zone 1 is a region far from the bubble surface where hydrodynamic forces are dominant, controlling EC in eqn [1]. Hydrodynamic drag forces act to sweep the particle around the bubble, viscous forces tend to retard this relative motion between the two, while particle inertial and gravitational forces move the particle towards the bubble.

Figure 1 Hydrodynamic (1), diffusiophoretic (2) and surface force (3) zones of interaction between a bubble and a particle. (Reproduced with permission from Derjaguin BV and Dukhin SS (1960-61). Theory of flotation of small and medium-size particles. Transactions of the Institute of Mining and Metallurgy 70: 221-246, Figure 1).

Figure 1 Hydrodynamic (1), diffusiophoretic (2) and surface force (3) zones of interaction between a bubble and a particle. (Reproduced with permission from Derjaguin BV and Dukhin SS (1960-61). Theory of flotation of small and medium-size particles. Transactions of the Institute of Mining and Metallurgy 70: 221-246, Figure 1).

Broadly speaking, all models of collision efficiency predict that EC decreases with particle size at constant bubble size down to a particle diameter of about 0.5 |im. Then, Brownian diffusion probably takes over as the predominant capture mechanism (although this has not been proven), the collision efficiency increasing with decreasing size as the tiny particles (virtually 'solute molecules') move towards the bubble surface. In 1948 Sutherland made the first significant contribution to the treatment of collision efficiency. His hydrodynamic treatment of the process of particle and bubble approach in zone 1 was carried out without any consideration of particle inertia, bubble deformation or film thinning, deficiencies that were in part recognized by Sutherland and Wark in 1955.

The Sutherland theory, based on potential theory or streamline flow, shows that the concentration, C, of mineral floated at a time t is related to its initial concentration, C0, by the recovery, R as:


where Rb and Rp are the bubble and particle radii, Vt is the bubble-particle relative velocity, X is the induction time, NB is the number of bubbles per unit volume, and $ is the fraction of particles retained in the froth following bubble-particle attachment. The reader should note the relationship between eqns [2], [3] and [4], which are the basis for a firstorder model, largely based on pulp microprocesses. Despite the deficiencies of the Sutherland model, his 'first approximation theory' yields results that are in fair agreement with experimental determinations of particle trajectories, touching angles and collision efficiencies, obtained from model experiments performed in a vertical flow tube with individual particles and a single bubble. For more detailed treatments of the hydrodynamic aspects of bubble-particle collision the reader is referred to the extensive literature available.

The inability of collision theories to describe adequately the collection process between bubbles and smooth and angular particles was vividly demonstrated by Anfruns and Kitchener in 1977. Their experiments, the first measurements of absolute rate of capture, gave results in only fair agreement with collision theory, assuming every collision resulted in capture of their very hydrophobic particles.

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