colloidal-hydrodynamic theory of flotation

Please use CPS: physchem/0103026 in any reference to this article 1

COLLOIDAL-HYDRODYNAMIC THEORY OF FLOTATION N. N. Rulyov Institute of Biocolloid Chemistry of Ukrainian National Academy of Sciences, Kiev, Ukraine Phone/fax: +380-44-8078, e-mail: [email protected]

Recent advances in the field of colloidal-hydrodynamic flotation theory are reviewed. Factors limiting the flotation of particles of all size classes are analyzed. The role of surface and hydrodynamic forces in the elementary act of flotation is investigated, as is the dependency of efficiency of particle capture by a bubble on the energy of their collision and the rate of its dissipation. Kinetic aspects of the flotation process are examined with consideration of the phenomenon of aggregation of particles and coalescence of bubbles. A brief list is presented on the main problems in flotation theory and certain promising paths of development of flotation technology are suggested. If we are to Judge by an old German engraving, it may be assumed that flotation was known as early as the Middle Ages. Officially, however, its age may be reckoned from 2 July 1877, i.e., from the moment of issue of German Patent No. 42, Class 22 to the Bessel brothers for a method of purification of graphite which was a prototype of modern flotation technology. However, the level of basic knowledge was so low that one could not even speak of any scientific basis for its development. It is sufficient to observe that for over 50 years. It was not clear why under certain conditions mineral particles adhere to the surface of a fluid. The answer to this question was obtained only in the 1930s, when the science of surface forces was developed Cl-33. Since this time, investigations in the field of flotation have had a scientific basis, which was rapidly broadened with development of fundamental knowledge in the field of physical and colloidal chemistry, and also in the field of physicochemical hydrodynamics. Due to the fact that flotation technology is based on the difference in capacity of minerals for adhering to an interface, the majority of Investigations were directed toward use of various flotation reagents to control this property. For this reason, the thermodynamic method acquired a principal position in flotation theory. However, even in the 1930s it was clear that kinetic factors play an important part in the flotation process [3]. The first attempts to develop a kinetic theory of flotation date to the mid-1930s and late 1940s [4-9]. However, the inadequate level of fundamental knowledge in the field of microhydrodynamics of disperse systems delayed these investigations. Some progress was observed in the late 1950s due to study of the role of nonequilibrium electrical surface forces in the elementary act of flotation of small particles [10], but these investigations were not adequately followed up. Only in the early 1970s did there appear an objective scientific basis for development of kinetic flotation theory, the need for which was already quite apparent. The depletion of rich deposits and the resulting necessity of use of lean and difficulty cleanable ores, with comprehensive utilization of mineral raw materials, as well as the problem of energy conservation and protection of the environment, posed a number of complex problems in the field of improvement of flotation technology. It appears that a solution to these problems is possible only by means of a strict quantitative flotation theory, based on simultaneous examination of colloidal-chemical and hydrodynamic aspects of the process. In order to have an idea of the complexity of this problem, it is sufficient to make a short list of the colloidal-hydrodynamic subprocesses which take place in flotation: orthokinetic heterocoagulation, coagulation, flocculation, coalescence, and the opposite processes in laminar and turbulent flows; convective diffusion and formation of a dynamic adsorption layer of SAS; the formation, thinning, and breakage of free and wetting films; sedimentation of concentrated suspensions and emulsions; and drainage and breakup of the froth. Of course, examination of all these phenomena under the conditions of flotation is impossible without using the latest advances in the field of ’surface forces, capillary hydrodynamics, theory of nonequilibrlium surface phenomena, microrheology of disperse systems, and colloid stability theory. Although this colloidal-hydrodynamic theory Is still far from completion, all the results obtained in the last decade and a half are sufficient for presentation in the present survey, and deserve the attenuation of a wide range of specialists in the field of flotation science and technology.

1.

BASIC THEORY OF ADHESION OF A MINERAL PARTICLE TO AN INTERFACE

1.1 Surface forces and the disjoining pressure isotherm. It is known that even after attachment of a particle to a bubble in the flotation process, a polymolecular film of fluid remains between them (particles with a maximally hydrophobic surface constitute an exception). This shows that both during the stage of convergence and in the stage of attachment there is an interaction between the particle and the bubble due to far-acting forces, i.e., forces the radius of effect of which is much greater than the size of the molecules. The appearance of these forces, termed "surface" forces [11], is due to the presence of surface layers at the interface of any adjacent phases. Within the limits of these layers the composition and intensive properties of the fluid differ from the bulk properties. In films, the thickness of which is less than the sum of the thicknesses of the transition layers bounding them, the normal pressure tensor component becomes different from the pressure of the fluid in the bulk, which gives rise to a gage (positive or negative) pressure in the film, termed the disjoining pressure [12]. The disjoining pressure theory is of enormous significance for understanding of the principle? of flotation. In order to demonstrate this it is sufficient to note that such key concepts for flotation as the wetting contact angle and its hysteresis, the attachment induction time, particle hydrophobicity and floatability, force of attachment of the particle to the bubble, effect of

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Please use CPS: physchem/0103026 in any reference to this article 2 particle shape and its surface terrain on floatability, etc., received a quantitative interpretation only on the basis of the theory of disjoining pressure of wetting films [12-14]. According to modern concepts [11,12], the disjoining pressure n acting in a thin film of fluid with thickness h may be presented in the first approximation as the sum of additive contributions from various surface force components

Π (h) = Π m (h) + Π e (h) + Π s (h) + Π a (h) + Π c (h)

where Π m (h ) is the molecular component of the disjoining pressure, determined by dispersion interaction forces of the

substrate and film molecules, Π e (h ) is the ion-electrostatic component of the disjoining pressure, due to overlap of the

diffusion ionic layers of the charged surfaces of the film, Π s (h ) is the structural component of the disjoining pressure, due

to the change in structure of polar fluids, for example, water, near lyophilic surfaces; and Π a (h ) is the adsorption component of the splitting pressure, due to the nonuniform diffusion distribution of nonionic dissolved substances in the film due to differences in the interaction energy of solution and solvent molecules with the surfaces bounding the film. The redistribution of the molecules may be due to the electrical field of the surface charges, dispersion forces, and structural changes of the boundary layers, changing the dissolution capacity of the solvent. Π c (h ) is the steric component of the disjoining pressure, which is related to the interaction of adsorbed layers of SAS and polymers when they overlap. The plot of Π (h ) , measured during isothermal thinning or thickening of the film, was termed the disjoining pressure isotherm (DPI). The form of the disjoining pressure isotherm is greatly dependent on the contribution of the above mentioned surface force components. Various components may make the principal contribution, depending on the physicochemical properties of the substrate and the fluid, and also the substances dissolved in it. The molecular component is the most universal, since dispersion forces act between any molecules. The contribution of the other components may be greater or less depending on the charge of the film surfaces, the polarity of the fluid, the adsorption properties of the dissolved substances, the concentration and Ionic composition of the electrolyte, pH and other factors. The ion-electrostatic forces may have the greatest radius of effect. For example, in the case of a solution of 1-1 electrolyte in water at a concentration of 10 moles/liter, this radius is 100 nm. In the thickness region of h < 50 nm molecular forces become significant; they are exhibited more greatly the greater the difference in optical properties of the substrate material and the fluid. If the fluid has pronounced polar properties in the thickness region of h < 10 nm there begin to appear structural disjoining forces increase sharply the region of h < 5 nm. The radius of effect of the structural forces is greatly dependent on the properties of the substrate, and when it is well wetted by the fluid the radius may reach several tens of nanometers. On hydrophobisation of the substrate the structural forces sharply decrease, resulting in attraction with poor wetting. The adsorption component of the surface forces appears in the region h < 5 nm; it may also overlap the effect of the molecular forces. Finally, the radius of effect of the steric component is determined by the thickness of the adsorption layers of surfactants and polymers. Fig.1. Isotherm of disjoining pressure Π (h ) of plane-parallel wetting film of thickness h (a) and corresponding profile of film between spherical particle of radius R p and bubble with radius R b (b). The vertical and horizontal shading correspond to the region of attraction and repulsion, respectively, of surface forces. The depth of shading is proportional to Π (h ) .

1. 2. Bubble-particle adhesion force. Figure la schematically shows a typical DPI of the wetting film Π (h ) as a function of its thickness h. The figure shows that the radius of effect of the surface forces is about h2; in the regions (0, h0) and (h1, h2,) they are repulsion forces, while in the region (h0, h1,) they are attraction forces. It appears that the presence of a deep and broad DPI minimum is a necessary condition of particle attachment to the bubble. We



Please use CPS: physchem/0103026 in any reference to this article 3 assume that the particle has the form of a sphere with radius R p . If, after attachment to the particle, there is a certain detachment force (for example, the force of gravity) F, then the condition of retention of the particle on the bubble takes the form

{

F < Fc = max − ∫ Π(h )dS

}

(1)

where the integration is carried out over the entire surface S of the film separating the particle and bubble (see Fig. 1b). Due to the fact that the film has variable thickness on the surface of the particle, the magnitude and sign of its disjoining pressure in different areas will be different. Consequently, in order to calculate the critical detachment force, or the so-called adhesion force Fc , one must know the profile of the equilibrium wetting film. This profile may be found from the equation expressing the equilibrium of the capillary and surface forces in the wetting film bounding the meniscus [12] Π + σK σ = 2σ / Rb (2) where

σ

is the surface tension, R b is the bubble radius, and K σ is the curvature of the surface of the fluid on the bubble

side. Using the plecewise-linearized form of the DPI on the basis of Eqs. (1) and (2), for Fc we obtain the equation [13,14]

πR p Rb    s1 1 + 2σ Fc = R p + Rb   Π1 R p 

2

   − s 2 1 − 2σ   Π2Rp  

   

2

   

(3)

where Π 1 and Π 2 , are the absolute values of the DPI minimum and maximum (see Pig. la). Since usually

2σ 0 (i.e., there is an attraction between the particle and bubble) only in the case when the condition s > 0 is met. It also follows from (4) that if the particle has a side, the size of which exceeds the bubble radius (i.e. R p >> Rb ), then

Fc = πRb s i.e., the force of adhesion is higher the greater the bubble radius. If Rb >> R p , then Fc = πR p s

(5)

(6)

It is interesting to note that Fc is exactly half the value obtained for the undeformed surface of a bubble. This is explained by the fact that on deformation of the bubble surface, at the site of attachment of the particle part of the effect of the surface forces is consumed to compensate for the capillary pressure inside the bubble. The importance of this factor was first noted in [15] during analysis of the Kabanov-Frumkln-Work equation [l6, 17], which is a special case of Eq. (3). If the minimum on the DPI is not very deep, Π 2 = 0 , Rb >> R p , and the particle radius is very small, so that 2σ / R p Π 1 , then, as follows from (3)

Fc = 4πσ 2 s1 / R p Π 12

(7)

In this case, as follows from (7), the force of adhesion increases with a decrease of the particle radius and with expansion of the DPI minimum. This case may be realized in the flotation of very small particles (on the order of a micron) by their attachment at the second minimum of the DPI [12]. 1. 3. Stability of wetting film and attachment induction time. It follows from the above discussion that stable particle retention at the interface is possible even in the presence of a surface force repulsion barrier corresponding to the DPI maximum. Surmounting of this barrier in the elementary act of flotation may take place due to the energy of impact of the particle against the surface of the fluid, or compressive hydrodynamic and gravitational forces. Attachment of the particle to the surface of the fluid may also take place due to breakage of a thick metastable film, corresponding to the so-called β-leg of the DIP (see Fig. 1a) and the formation of an absolutely stable thin film corresponding to the α-leg of the DPI [12]. The β α transition may take place only as a result of thermal fluctuations of the thickness of the β-film, or under the effect of external factors, for example, an electromagnetic field, ultrasonics, ionizing radiation, or chemical reactions [l8]. The phenomenon of decomposition of β-films was observed even in the first experiments with wetting films [19]. A spontaneous β α transition is completed stochastically and begins with formation of stable a-phase nucleation centers. As a result of rapid growth of the α-phase the initial film is collected in Individual droplets, separated by the B-phase, the fluid from which evaporates or flows through the α-film under the meniscus. The importance of the β α transition for flotation is obvious if we take into consideration the fact that the dynamic thinning of the β-films to their critical thickness h1 is greatly



Please use CPS: physchem/0103026 in any reference to this article 4 hindered by the viscosity of the fluid. As a result of spontaneous breakage of the film, this stage of the elementary act of flotation is excluded. In this case there was intensive study of the dependence of breaking thickness of wetting films on various external conditions, addition of electrolyte, and addition of surfactants. The probability of appearance of α-phase nucleation centers depends on the capillary pressure of the meniscus with which the film is in equilibrium, and the area of the film. It was shown that the probability of breakage of the β-film is higher, the closer the disjoining pressure in the film to the critical value ( Π 2 ) and the greater the area of the β-film; the breakage phenomenon is statistical in nature and may be

characterized by the mean β-film lifetime τ av . It was also shown that the higher the positive disjoining pressure, the smaller

will be the area of stable β-films. Since the form of the DEI and, in particular, the magnitude of Π 2 , will depend on the composition of the substances dissolved in the fluid, it is apparent that they will also determine the critical thickness and mean lifetime of the β-film. This has been shown by numerous experiments (an extensive bibliography on this question is presented in [12]). It was shown that a decrease in the pH of a. solution of KC1 in water leads to rapid breakage of 6-films on a quartz substrate, while the addition of a cationic surfactant sharply decreases their stability. The critical concentration of ammonium alkyltrimethylbromides, leading to breakage of the B-film decreases very rapidly with growth of the alkyl chain. Breakage of B-films of water on a hydrophobized surface is observed even at h 60 nm for films 0.1 mm in diameter; the critical breakage thickness of water inter-layers on the surface of hydrophobized quartz Increases on an increase in dodecylamine concentration. The above discussion makes clear the physical meaning of the common concept in flotation of "attachment induction time" [8]. In particular, the empirical dependence of induction time τ on temperature T [19] Wa

τ = τ 0 e kT

(8)

where τ 0 and Wa , certain constants characterizing a specific flotation system, may be interpreted in the language of wetting film stability theory. The so-called activation energy Wa is a direct analog of the critical disjoining pressure Π 2 while the temperature dependency (8) of the induction time, which is an analog of the mean β-film lifetime τ av , shows that its breakage takes place due to thermal fluctuations of the thickness. 1. 4. Flotation reagents and DPI. It would be incorrect to think that the role of reagents in flotation amounts only to destabilization of B-films. Actually, collector reagents should be chosen so that the DPI has a sufficiently deep and broad minimum in the region of negative values of Π (h) . It is better for this minimum to be in the region of thin films, since in this case wetting hysteresis is more pronounced, which makes the attachment of particles to the interface more stable. Ideal from the point of view of flotation is a DPI in which there is no positive maximum in the thin film "region and there is a very deep negative minimum in the thin film region. In this case, the attachment induction time is zero [19]. It is clear that in each particular case the selection of reagents is made depending on the type of mineral, and thus there cannot be a single reagent for all cases. There are still a few general principles which follow from the properties of surface forces and which may be used in the selection of collector reagents. As was noted above, the most far-acting component of surface forces is the ion-electrostatic component, especially in the case of dilute electrolyte solutions. Thus, it is quite reasonable to assume that the existence of stable β-films in the region of values of h ≥ 50 nm is determined exclusively by electrostatic forces. In the region of h > 10 nm, however, there begins to be a significant contribution from molecular forces, thus, the stability of g-films in the 10-50 nm region is determined by the ratio between ion-electrostatic and molecular components of the disjoining pressure. This conclusion was elegantly proved by a large number of experimental investigations and theoretical calculations [12]. Thus, the selection of collector reagents may be made on the basis of the effect they exert on the magnitude and sign of the charges of the mineral and fluid surfaces, and also on the thickness of the double electrical layer. Polyvalent electrolytes and ionic surfactants may be used most effectively for regulation of the ion-electrostatic forces. For example, by using very small admixtures of polyvalent electrolytes one can not only change the charge of the film surfaces, but also significantly decrease the radius of effect of the electrostatic forces. Still more effective are ionic SAS, since by using them one can not only neutralize the charge of one of the surface films, but also attain a different sign of their charges. In this case, as was shown convincingly using the example of flotation of quartz [20,21] and polystyrene [22], one achieves the maximum extraction. Naturally, in the choice of reagents one should also consider their effect on the rock particles. The situation is more complex with respect to choice of reagents to provide a deep negative minimum of the DPI in the region of thin films. This is due primarily to the absence of a well-developed theory of structural forces, consideration of which is very important, as was shown by comparison of results of DPI measurement for a-films with calculations based on consideration of molecular and ion-electrostatic forces [12]. It is clear only that in order to eliminate the negative effect of structural forces one should use reagents hydrophobizing the surface of the mineral and thereby disrupting the structuring of the water near it. 1. 5. Flotation complex stability. We will now examine mechanisms of stabilization of a flotation complex when a breakup force acts it on. Since wetting hysteresis prevents movement of a meniscus over a solid substrate, before the movement begins the surface of the fluid (for example, a bubble) near the wetting perimeter should be significantly deformed. Here there are two possible variants of particle and bubble disaggregation. If the rate of increase of the breakup force is low, and the time of effect is great, it is more probable that the breakup of the flotation aggregate will take place due to an increase in the wetting angle and movement of the meniscus over the surface of the particle (the hysteresis mechanism



Please use CPS: physchem/0103026 in any reference to this article 5 [2,23]). However, if the rate of increase of the breakup force is great, then near the wetting perimeter there may form a very thin orifice, which can break even at a very short time of effect of the breakup force (the capillary mechanism). On the basis of the approach that suggested in [13,14] it was possible to evaluate the role of various mechanisms of flotation complex breakup: hysteresis and capillary mechanisms. For this it is sufficient to compare the force of surface tension Fσ = 2πσrc and Fc for a film with the critical radius rc . In particular, at Π 2 =0 and 2σ / R p Π 1 Fc . Consequently, it is more probable that breakup of a flotation aggregate will take place by the hysteresis mechanism since movement of the meniscus requires less detachment force than is required for extension and breakage of the bubble orifice. This, however, may take place at the final stage of disaggregation, i.e., there may be a combined mechanism in which there is first movement of the meniscus and a decrease of the wetting perimeter, and then onset of extension and breakage of the bubble orifice. 2.

FLOTATION AS A SPECIAL CASE OF HETEROCOAGULATION

From the point of view of the modern science of colloids, flotation is a special case of orthokinetic heterocoagulation of coarsely disperse systems. In distinction from perikinetic heterocoagulation, which predominates in the case of very small (below 0.1 micron) particles, orthokinetic heterocoagulation basically depends not on thermal movement of interacting objects, but on their relative motion, determined by sedimentation and nonuniformity (in space and in time) of the hydrodynamic field of the medium in which they are suspended. Three basic mechanisms of particle-bubble collision are known [13,24]: sedimentation, gradient, and inertial mechanisms. The sedimentation mechanism is determined by the relative motion of the particles and bubbles in the field of gravitational force and is universal in nature. The gradient mechanism of heterocoagulation is determined by steric nonuniformity of the hydrodynamic field. Calculations show [24] that its contribution to flotation may be significant (as compared with the sedimentation mechanism) only in the case when the particles and bubbles are of comparable size. This may occur when using preflocculation of particles (in so-called floccular flotation [13,25]). The inertial mechanism is due to the nonstationary nature of the flow in the bubble-particle region. As was shown in [24], the contribution of this mechanism is comparable to that of the sedimentation mechanism in the case where the averaged acceleration field of the fluid is comparable in magnitude to the acceleration of gravity. These conditions may appear in impeller flotation machines. The characteristics of heterocoagulation during flotation are dependent on a number of factors. One of them is that the bubbles are usually much larger than the particles. In addition, the surface of the bubbles may be deformed on collision with a particle, exhibiting elastic properties [26-29]. It should also be taken into consideration that the hydrodynamic field around the bubble is very strongly dependent on its size and the state of the adsorption layer of surfactants on its surface [30,31], which has a great effect on the efficiency of particle capture. The kinetics of the elementary act of flotation is also dependent on the size, shape, and density of the particles being floated [13,26,27,32], which should also be taken into consideration when developing a quantitative theory. Fig. 2. Trajectories of particles of various size and density near a gas bubble: 1) noninertial; 2) Brownian; 3) weakly inertial; 4) strongly inertial; 5) fluid flow line.

As an example, Fig. 2 shows the fluid flow lines (dashed) and the trajectories of particles (solid) of various sizes near a rising bubble in the bubble coordinate system. If a particle is rather small and (or) its density differs little from the density of the fluid, and in this case the following condition is satisfied

St * =

2u b R p2 ∆ρ 9 Rbη

70 µm), where d p is the particle diameter. Depending on which of the indicated classes a particle belongs to, the collision energy will be dissipated to varying degrees due to the following basic processes: 1) the surmounting of long-range hydrodynamic interaction on convergence at great distances; 2) the pressing out of the thin film on short-range hydrodynamic interaction; 3) excitation of capillary waves on the surface of the bubbles; 4) generation of an α-film and movement of the wetting perimeter over the surface of the particle after breakage of the β-film [13]. In the case of small particles, almost all the collision energy can be dissipated at the far approaches, to the bubble due to LHI, which characteristically also determines the coincidence of trajectory of these particles with the corresponding fluid flow lines (Pig. 2, curve 1). Thus, the low level of the initial kinetic energy of collision quantitatively explains the low floatability of small particles, which is completely absorbed due to the LHI. With an increase in the size and (or) density of the particle and transition to the medium size class, much of the energy cannot be dissipated by process 1, as is shown by the deviation of the trajectory of these particles from the flow lines (Pig. 2, curve 3) and the increase in the contribution of process 2 to dissipation. It should also be noted that the contribution of this process to dissipation of the energy of a particle is greatly dependent on its shape and the level or deformation of the bubble surface [32]. Calculations showed [38,26] that for spherical particles the proportion of the energy dissipated in process 2 falls sharply with an increase in the force of impact of the particle against the bubble surface and, consequently the leve1 of bubble deformation. Figure 3 shows the ratio K of



Please use CPS: physchem/0103026 in any reference to this article 7 particle energy after impact W1 and before it W0 as a function of I

−1

.

−1

As may be seen from the figure, at I >100 there is a quasi-elastic impact of the particle against the surface of the bubble. The situation is different for particles of angular form, which can pierce the film with their edges and points [32]. In this case, energy dissipation takes place mainly due to process 4. Finally, for particles of the largest size class the dissipation of energy due to processes 1 and 2 is not practically significant. Calculations showed that the energy ratio K LHI of a large particle after ( W1 ) and before ( W0 ) overcoming the LHI may be expressed by the formula [26] ∞ W1 (− 2 )n = 3∑ n W0 n = 0 (2 n + 3)!! (St )

where St = 8 R p ρ p / 3Rb ρ l f (Re p ) is the Stokes number and

(12)

f (Re p ) is a coefficient in the resistance function of a

sphere versus Reynolds number.

Fig. 3. A plot of coefficient of energy dissipation in a thin −1 film versus reciprocal inertial parameter I .

Fig. 4. A plot of loss of particle kinetic energy to overcome LHI versus reciprocal Stokes number St

−1

.

It may be seen from (12) that the ratio W1 / W0 = K LHI presented in Fig. 4, rapidly approaches 1 on an increase in St , which is explained by the strong deviation of trajectories of these particles from the corresponding flow lines (Pig. 2, curve 4). With respect to process 3 it was shown by calculations [13, 38] based on evaluation of the energy of capillary waves excited on impact of a spherical particle against the surface of the fluid, that the energy losses are about 60-802, which Is in good agreement with experimental results [37]. When there are sharp corners on the particle surface, causing film breakage and formation of a wetting perimeter, the remaining part of the energy may be consumed due to process 4 [13]. In the case of particles of smooth shape, the remaining part of the energy may in many cases exceed the bond energy. Thus, capillary forces can eject a particle from a bubble on the first collision. Calculations showed that for attachment of- a large spherical particle to a bubble, the particle must complete from 1 to 6 jumps over the bubble surface [38], which Is also demonstrated by experimental observations [37]. It should be noted that even when there are corners and edges on a particle it may be ejected from a bubble due to the fact that the rate of expansion of the meniscus is slowed by the viscous resistance of the thin film formed at the moment of Impact [32]. Thus, if the period of collision is much less than the time of expansion of the wetting perimeter, the energy of the Impact, transformed to the surface tension energy of the film, may not be dissipated, and the particle will be ejected from the bubble, which Is- also often observed. Calculations showed [32] that the ejection condition in this case takes the form

 3 3η  θ R 70 µm). The calculation of the CE of large particles is based on a description of the limit trajectory of their motion near the surface of a bubble by numerical methods with consideration of the phenomenon of escape and the energy dissipation due to forcing out of the thin film and excitation of capillary waves. Unfortunately, such computations could be realized only for spherical particles [13,38]. Fig. 6 A plot of capture efficiency E versus particle radius R p at ρ p =7.5 g/cm,

Rb =0.5 (1) and 0.7 mm (2). The numbers on the curves indicates the number of jumps. (The bubble surface is fully retarded) It was shown that the CE of large particles decreases with an increase in the size of the particles and bubbles, and also with an Increase in particle density. Since all these factors determine the energy of particle-bubble collision in the elementary act of flotation, support is again obtained for the assumption that the principal factor limiting the floatability of large particles is not the difficulty of their retention on the bubble after attachment, but the excess collision energy, which cannot rapidly be dissipated. As an example. Pigs. 6 and 7 present calculated plots of E R p and

( )

E (ρ p )for some special cases [13, 38]. The

numbers at each calculation point Indicate the number of jumps completed by a particle before the moment of attachment, in the process of which the kinetic energy of collision is dissipated. It may be seen from the graphs that the number of these Jumps increased with an increase in the particle size and density and also with an increase in bubble size (and hence the bubbles rise velocity). The results show that in the region of large particles the CE changes by the formula

E ∝ R p−3 / 2

(21)

Due to the fact that the presence of sharp corners on the surface of particles significantly decreases the role of NHD, facilitating breakage of the fluid film at the moment of impact, the CE of these particles in the investigated size range is apparently 2-3 times greater than for spheres. Thus, the results obtained in [13,38] should be considered as an evaluation of CE from below. As an illustration of the situation in general Fig. 8 presents a plot of CE versus diameter d p of actually floated spherical particles, calculated on the basis of the theory presented above. Here it was assumed that the bubble surface is fully slowed by the adsorption layer of surfactants, its diameter is 1 mm, and the particle density is chosen as 7.5 g/cm3. The resulting plot rather well reflects the general situation [53] if we consider that the CE of spherical particles is about 2-3 times lower than for particles of angular shape (for example, crystals).



Please use CPS: physchem/0103026 in any reference to this article 11 Fig.7. A plot of capture efficiency E versus particle density ρ p : R b = 0.5 mm, R p = 90 µm (retarded bubble surface).

Fig. 8 A plot of capture efficiency E versus particle diameter d p ( d b = 1 mm, ρ p =7.5 g/cm3):

E I ∝ d p−2 / 3 ,

5.

E IV ∝ d −p3 / 2

E II ∝ d 3p / 2 ,

KINETICS OF THE FLOTATION PROCESS

Under real conditions the intensity of the flotation process is determined not only by the kinetics of the elementary act of flotation, but also by a number of accompanying processes determined by the simultaneous participation of a large number of bubbles in the flotation process. The most important are: processes of coagulation, flocculation, and heterocoagulation of solid phase particles; processes of formation, aggregation, and coalescence of bubbles; processes of formation and breakup of the froth layer, etc. It is clear that in order to describe mass transfer in a flotation machine one should simultaneously consider all the above mentioned processes, the examination of which should be based on a colloidal-hydrodynamic approach. 5.1. The kinetic equation of flotation. Under uncrowded conditions the kinetics of flotation in a batch machine are described by the equation of [5, 36]: nt = n0 exp(− Kt ) , where nt and n0 are the current and initial particle concentration, and K is a constant characterizing the rate of particle removal. If the efficiency of particle capture by a bubble allows expansion by the small parameter R p / Rb , i.e., E =

∞

∑ε n= 0

n

( R p / Rb ) n where ε n is the expansion coefficient, and

the distribution of particles and bubbles by size is the gamma-distribution then, as calculations have shown [13, 36], the following relationship exists between K and the ε n coefficients:

  4 Rbm Γ ( p b + 5) 1  2  3  1  2  1            ≅ + + + ε + + + + ε1 K 1 1 1 1 1 1 0   p b  p b  p b  p b  p b  p p  3qp b5   2  1  1  2  R pm  1  2   1 + + 1 + 1+ ε2 2 + 1+ 1+ 1+ p b  p p  p p  Rbm  p p  p p  


R pm Rbm 3 pp

+

 R 3pm ε 3 + ...  R3 bm 

(22)


Please use CPS: physchem/0103026 in any reference to this article 12 where Γ(x) is the gamma-function, q is the bubbling rate, i.e., the volume of gas passing through a unit of chamber cross section per unit of time, R pm

and Rbm are the mean radii, and p p and p b are the parameters in the particle (p) and

bubble (b) gamma-distributions, respectively. Thus, Eq. (22) permits one to relate the experimentally observed quantities K , q , R pm , and Rbm to the ε n coefficients, which may be calculated only on the basis of the colloidal-hydrodynamic theory of the elementary act of flotation. On the basis of (22), it is easy to show that in the case of monodisperse systems, when p p ,b → ∞ , the above equation takes a simpler form

E=

4R p 3q

K

(23)

Of course, Eqs. (22) and (23) describe the simplest, one might say ideal case, since their derivation does not reflect processes accompanying flotation: coalescence of bubbles, aggregation (coagulation) of particles, precipitation of particles from the froth layer, etc. Usually, these processes greatly confuse the overall picture, making it difficult to distinguish the factors limiting the rate of the flotation process. The attempt to describe the kinetics of the flotation process generally (without a detailed examination of subprocesses) gave rise to a great number of phenomenological models (for a general bibliography on this question, see [56]). Unfortunately, their value is small due to the lack of understanding of the relationship between the phenomenological parameters and the physicochemical characteristics of the flotation system. This practically negates the predictive ability of the phenomenological theories. More predictive in this sense appears a detailed study of the main subprocesses of flotation with their subsequent consideration in description of the flotation process as a whole [13]. 5.2. The hydrodynamic field in the flotation chamber and hydrodynamic suppression of flotation. The great significance of the hydrodynamic situation in the flotation machine for flotation is well known. Even in the absence of moving elements (for example, an impeller), in the chamber of a plunger machine there may appear very strong convective flows. These flows, as has been shown [57], are due to the convective instability of the fluid under conditions of bubbling. It was shown that even with a uniform distribution of bubble sources on the bottom of the chamber there arise systems of eddies, in which the velocity of the fluid u c may be evaluated by the formula

uc ≈ where g is the acceleration due to gravity,

ν

l qg 3 ν

(24)

is the kinematic viscosity of the medium, q is the mean bubbling rate, and

l is the horizontal width of the eddy. On the basis of (24) it is easy to show that the rate of the convective flows may be many times greater then .the bubble sedimentation rate, which under certain conditions should lead to their accumulation in the pulp and to coalescence. It was also shown that the shear gradients arising in these flows have the magnitude

Gη = where

qgl νλ (1 − λ / l )

(25)

λ is the width of the transition layer between the ascending and descending flows. It follows from (25) that even at comparatively low bubbling rates q there may arise very strong shear gradients Gη , which can break up the froth layer and the flotation aggregates [57]. This explains, for example, the phenomenon of hydrodynamic suppression of electro-flotation of solids at a bubbling rate of q ≥ 3⋅10-3 cm/s [58].

Fig.9 Ratio of the specific capacities of an n__

__

stage and single-stage flotation line Q n / Q1 as a function of the required level of solids removal α. 5.3. Efficiency of multistage flotation plants. A known negative feature of agitation of slurry is the decrease in specific capacity of the



Please use CPS: physchem/0103026 in any reference to this article 13 flotation machine, i.e., the ratio of capacity to volume of the flotation chamber. Thus, use of multistage systems has been proposed. Calculations showed [59] that the specific capacity 2L of these systems by volume of slurry processed at the same total volume of the working chambers is

K (1 − α ) 1 / n Qn = n[1 − (1 − α ) 1 / n ] __

(26)

where n is the number of stages and a is the assigned degree of solids removal. As an illustration of the efficiency of __

__

utilization of multistage flotation lines. Fig. 9 shows plots of the ratio Q n / Q1 versus α for various values of n . Figure 9 shows that an effective increase in specific capacity may be achieved even with use of 2-3 stages; further division of the stages is not so efficient and may prove economically unfeasible. 5.4. Role of particle aggregation in flotation. Depending on the goals of flotation, Particle aggregation (coagulation, flocculation) may play either a positive or a negative role. In the flotation treatment of water aggregation is desirable, since the enlargement of the floated objects caused by it, as was shown above, leads to an increase in the particle capture efficiency E and, consequently, to an increase in the extraction rate K [25, 60]. Aggregation is especially important for flotation removal of fine particles, the capture efficiency of which is very low [25]. In the case of flotation enrichment of minerals the aggregation of particles of different type, i.e., heterocoagulation, interferes with the separation of minerals, while aggregation of particles of only one type, on the contrary greatly favors selective extraction. If the size of the particles being floated is less than 0.5 micron, their aggregation takes place by a perikinetic mechanism and is weakly dependent on the hydrodynamic field in the flotation chamber. This usually occurs in ionic flotation. However, if the size of the particles exceeds 0.5 micron, aggregation takes place by the orthokinetic (gradient) mechanism and is very strongly dependent on the level of nonuniformity of the field of fluid velocities in the flotation machine. In [61-63] there was an investigation of the effect of gradient aggregation on the process of extraction of small particles. Calculations [61,633 showed that the contribution of aggregation to flotation may be significant only in the case where the bulk concentration of particles exceeds a certain critical level C c ≅ 0.1E / β , where β is the efficiency of particle aggregation in a simple shear field [64,65]. This conclusion was supported by experiments on the flotation of monodisperse latex particles 3 µm in diameter [62]. It was shown that aggregation significantly changes the flotation kinetics if the concentration of these particles exceeds 106 cm-3. Here it should be noted that in the 1950s, there was a prolonged discussion in the literature on the order of the equation, which should be used to describe the kinetics of the flotation process. The results presented in [61-63] show graphically that there is no single answer to this question if the particles being floated are aggregate-stable, as frequently occurs In practice. For example, if the Initial particle concentration exceeds the critical concentration, gradient aggregation of particles and aggregate flotation first predominates, since bubbles much more effectively capture the aggregates than single particles. With a decrease in particle concentration aggregation rapidly slows and, in the final, it does not play such significant role as compared with flotation, i.e., the process will be determined by flotation of Individual particles. Thus, in the course of particle extraction the process kinetics changes from second to first order. These theoretical conclusions are supported by experimental results on the flotation of small particles of scheelite [66], where it was shown that after addition of a flocculant the order of extraction kinetics changes from 1 to 2. 5.5. Effect of bubble coalescence on flotation kinetics. As was noted above, under conditions of bubbling there practically always appear convective flows, the velocity of which may greatly exceed the bubble sedimentation velocity. It is clear that the nonuniformity of the hydrodynamic field caused by these flows will lead to aggregation and coalescence of gas bubbles by a gradient mechanism. This occurs particularly in the case of very small bubbles, the velocity of which is low. It is easy to show that the rate of disappearance of primary bubbles due to coalescence is expressed by the formula [13,67]

dN ∝ − q 5 / 2 / Rb2 dt

(27)

where N is the numerical concentration of the bubbles and q is the bubbling rate. It may be seen from formula (27) that at a constant bubbling rate even a slight decrease in bubble radius leads to a strong Increase in the rate of bubble coalescence. Consequently, if the depth of the bubbled layer of water and the bubbling rate are sufficiently great, small part of the bubbles of original radius reach the surface. It is known that even at a low bubbling rate, a group of Interacting nonstationary hydrodynamic eddies is formed in a slurry; if the layer of water Is sufficiently deep, these eddies are situated in several layers by layer depth. The velocity of the fluid In these eddies may be an order of magnitude or more great than the velocity of the primary bubbles. Consequently, almost all the primary bubbles will accumulate In the lowest eddy level, nearest the bubble source, where the eddies have the highest velocity. Due to coalescence the bubbles are enlarged and, reaching the size at which their sedimentation rate exceeds the velocity of the convective flows, they enter the upper layers and then the froth. Thus, the finely disperse, most flotation-active bubbles treat only that part of the volume of fluid which is most closely adjacent to the bubble source. As a result, the efficiency of flotation treatment will depend on the proportion of this volume. This conclusion is supported by experiments on electro-flotation of fine particles [67], in which it was shown that in the upper part of the flotation chamber 60% comprises bubbles completing one, and 40% two coalescence. It is clear that in order to decrease the negative effect of bubble coalescence on microflotation the sources of




microbubbles should be distributed in some manner by flotation chamber depth and varied so that the microbubbles are more or less uniformly distributed over the entire volume. Here it should be taken into account that an extreme increase in the bubbling rate or a decrease in the bubble size will lead to rapid coalescence and enlargement of the bubbles, and this will not give the desired results. It is also clear, and this is supported by experimental observations [67,68], that with an increase in the bubbling rate q the region of Increased flotation activity, in which the bubbles have the minimum size, will be drawn increasingly closer around the source of the bubbles. This is due to the fact that, as follows from [27], the rate of bubble coalescence is proportional to q 26. 6. KINETIC FEATURES OF FLOTATION OF SMALL PARTICLES A basic feature of flotation of fine particles is the low efficiency of particle capture by a bubble which, as was shown above, is due to the sharp weakening of the Influence of Inertia and an Increase in the negative effect of the LHI on a decrease in particle size. It follows from 4.2 that an increase in CE may be achieved in two ways: a decrease in the size of the bubbles or (and) an increase in particle size due to aggregation [25, 69]. However, this apparently simple solution gives rise to new problems. For example, on a decrease in bubble size there is a significant decrease in the velocity of their rise which, first, limits the bubbling rate q [36] and, second, sharply decreases the velocity of separation of mineralized bubbles from the fluid due to their sedimentation. As a result there is first a decrease in the particle removal rate K ∝ q and, second, a decrease in the flow capacity of the flotation machine for liquid phase which is very undesirable, for example, for flotation treatment of water. In the case of particle aggregation, however, other problems arise, related to the stability of the aggregates and the force of cohesion of the aggregates and bubbles [26], as well as the problem of aggregation of different particles, preventing selective flotation [70]. Thus arises the necessity of a rigorous theoretical examination of the problem of optimization of the colloidal-hydro-dynamic parameters of flotation systems.

__

Fig. 10. A plot of the optimal bubbling rate q m (a), bubble radius Rbm (b), and the maximum specific capacity Qm (c) versus particle radius R p at α = 0.95 6.1. Optimization of the hydrodynamic regime. In [71], there was a detailed examination of the hydrodynamic conditions of flotation of small particles in a system of the countercurrent type in order to determine the optimal bubbling rate q m and bubble radius Rbm . The results of the calculations are presented in Fig. 10a and 10b, where the shading designates the regions of values of q m and Rbm in which the capacity of the flotation machine differs by not more than 10% from its __

maximum Qm , shown in Fig. 10c. Here it was assumed that the depth of the flotation layer is 50 cm, and the level of particle recovery is α = 95%. It follows from Fig. 10 that with a decrease in the particle radius all these quantities decrease, while the limitations on q m and Rbm become more stringent. 6.2. Floccular flotation. As was shown in the preceding section, with a decrease in Particle size there is a decrease not only in the capacity of the flotation plant, but also in the optimum bubble size. For example, at R p < 4 µm, the bubble radius should be less than 20 microns. Unfortunately, generation of such small bubbles in sufficient quantity is a rather difficult problem. In addition, they rapidly dissolve, and at high gas concentration, as shown in paragraph 5.5, they very rapidly coalesce and enlarge. Thus, in the case of very small particles it is more desirable to use not extremely small bubbles, but to provide for enlargement of the particles due to their aggregation or flocculation. Here it should be taken into account, however, that use of flocculation in flotation also has its limitation [25]. The fact is that on an Increase in the particle aggregate size there is an increase not only in the efficiency of capture, but also the hydrodynamic detachment forces acting This article is available from: http://preprint.chemweb.com/physchem/0103026



on it from the flow around the bubbles. As was shown in [25], in floccular flotation the bubble size should not exceed a maximum level Rb max determined by the formula

Rb max = K A ( R A / Rb ) Fc1 / 3

(28)

where Fc is the force at which the flotation complex of the aggregate with the bubble breaks up; R A is the radius of the aggregate; and K A ( R A / Rb ) is a function reflecting the hydrodynamic breakup forces acting on the flotation complex, presented in Fig. 11, from which it can be seen that at R A ≥ Rb the maximum size of the bubbles is determined only by the force of adhesion with the aggregate. Apparently, in order to carry out floccular flotation of particles it is necessary to use regimes providing stable aggregates and good adhesion of/aggregates to the bubbles. Otherwise, the bubbles, if they are too large, will simply break up the particle aggregates, and not float them. Thus, it follows that in floccular flotation it is also desirable to use comparatively small bubbles. These conclusions are very well supported by experience in flotation water treatment [46]. Fig. 11. A plot of K A ( R A / Rb ) 6.3. Selective flotation of small particles. It is well known that with a decrease in particle size there is a significant decrease not only in the intensity of the flotation process, but also in the selectivity of particle removal. It is clear that to carry out selective flotation of a mixture of different particles the following conditions should be met: 1) the efficiency of capture of particles of one type by a bubble should be at least higher than the efficiency of capture of particles of other types; 2) aggregates consisting of particles of different types should be absent or unstable. Assume an stable to aggregation mixture of geometrically identical particles of two types, the efficiency of capture of which by a bubble is E1, and E2, respectively, and initial numerical concentrations in the water are n01 and n02. If the flotation is not carried out under crowded conditions, the particle concentration in the solution will change in accord with a first-order reaction equation (see paragraph 5.1). Determining the flotation selectivity coefficient as K s = (n1 / n2 ) /(n01 / n02 ) , where n1 and n2 are the numerical particle concentrations in the concentrate, there was obtained in [13,70]:

(

)(

K s = 1 − e − K1t / 1 − e − K 2 t

where, according to (22), K 1, 2

),

(29) is the particle removal intensity constant, proportional to E1, 2 . Using this equation, it is easy

to show that at K 1 > K 2 >0 the selectivity monotonically decreases with time, while the process itself may be carried out in the time ∆t ≤ 1 / K 1 . Here the degree of recovery of the desired component ε 1 = ( n01 − n0 t ) / n01 reaches only 63%. At

∆t > 1 / K 1 the degree of recovery of the desired component ε 1 increases, but there is a significant drop in selectivity. Taking the above into consideration and keeping in mind present economic requirements for utilization of natural resources, condition 1) should be replaced by a more strict condition, i.e.: 3) only the efficiency of capture of particles of the desired component should be nonzero. It is clear that in order to satisfy condition 3) it is necessary for the bubbles to attach and retain only particles of a single type in the rising process. Thus, there is no necessity for calculating the capture efficiency E1 and E 2 ; it is sufficient only to evaluate the balance of forces favoring and hindering attachment of particles of one type to a bubble [70,72]. As was shown in paragraph 1.2, the forces of attachment of particles of the first and second types to the bubble FA1, 2 may be evaluated as a1 R p and a 2 R p , respectively (assuming a1 , > a2 ,). On the other hand, particles already fixed in the trailing section of the bubble are acted on by detachment forces due to the effect of the force of gravity Fg and viscous forces of the flow around the bubble FH . These, as was shown in [72], change in accordance with the formula ( Fg + FH ) ~ R p . It is clear that to carry out 3

selective flotation it is necessary to satisfy the condition:

FA2 < −( Fg + FH ) < FA1 This article is available from: http://preprint.chemweb.com/physchem/0103026

(30)



As an example, Fig. 12a schematically shows a plot of the forces of attachment of particles of the first and second types FA1 , and FA2 , as well as the detachment forces ( Fg + FH ) versus particle radius. It may be seen from the figure that the above-indicated condition is satisfied only in a certain rather narrow range of values R1 1, it is easy to show that the rate of particle sedimentation may be evaluated by the formula

u p = 16 R p ∆ρg / 2 ρ l

(32)

Assuming, in particular, R p = 50 µm, ∆ρ = 3⋅103 kg/m3, we have u p = 9 cm/s. Thus, the velocity of the convective flows should be 20-30 cm/sec as a minimum, which is comparable to or even somewhat exceeds the velocity of the large bubbles used In flotation ( R b = 0.5-1 mm). Since the velocity of rise of mineralized bubbles is significantly lower than the velocity of free bubbles, it is clear that by maintaining free particles In the suspended state we hinder removal of even particles fixed on bubbles Into the froth and increase the probability of their detachment on collision with other particles. It is easy to show that at 50% filling of the bubble surface by particles its buoyancy will be positive only when the following condition is satisfied

Rb >

3∆ρ Rp 2ρ l

(33)

However, if we take into consideration the fact that the velocity of a mineralized bubble should greatly exceed the velocity of convective flows, the inequality in (33) should be intensified by a factor of two or even three. In this case for R p = 50 microns, ∆ρ = 3 g/cm3 we obtain R b >>225 µm, i.e., for flocculation of these particles we should use bubbles with radius on the order of 0.5-1 mm. Unfortunately, as was shown in paragraph 4.4, an increase in bubble size has a very negative effect on the efficiency of particle capture due to the great collision energy and the phenomenon of rebound of the particle from the bubble. Thus, satisfying the first two specifications (maintaining particles in the suspended state and removing the mineralized bubbles from the slurry), we eliminate the efficiency of capture of the particles by the bubbles. To this we should also add that with an increase in the bubble size there is a significant increase in the detachment force acting on a fixed particle from the flow of fluid around the rising bubble. It is easy to show [72] that for large bubbles ( R b ≥l mm) this force becomes comparable to or even greater than the force of gravity, which naturally hinders flotation even more. Thus, a vicious circle arises, which can be broken only by cardinal restructuring of the traditional flotation scheme. 7.1. Froth separation. The principal specifications which should be satisfied by flotation machines for large particles are very clearly formulated in [74]. Unfortunately these specifications are far from being satisfied by all flotation machines. The most successful solution to the problem of flotation of large and superlarge (on the order of 1 mm or more) particles is the principle of froth separation [75], which was used as a basis for development of flotation machines in which the feed is not into the slurry, but directly into the froth layer [76,78]. This minimizes the energy of collision of the particles and bubbles, and hence excludes the rebound phenomenon. In addition it is no longer necessary to create powerful convective flows or to use very large bubbles. Similar results may also be obtained in a flotation column if one uses comparatively large bubbles, but high bubbling rates. Since the velocity of bubble rise decreases sharply on an increase in the gas saturation of the slurry, which is approximately proportional to the bubbling rate [79], the bubble-particle collision energy decreases and the rebound phenomenon will be absent. This achieves high levels of capture efficiency and minimizes the probability of detachment of fixed particles. These conclusions are supported by the results of [80], in which it was shown that with a certain bubbling rate, the rate of removal of mercury droplets in a flotation column begins to increase rapidly, reaching a maximum. The discovery of methods of froth separation is a clear example of solution of a complex flotation problem, which would require violation of old, established principles.

η and velocity V p of particle movement in it versus resulting shear stress P . Fig. 13. A schematic plot of froth viscosity

7.2. Motion of a mineral particle in a froth layer. In spite of the significant advances made in the theory of froth separation [80], one should recognize that it is far from perfect. It appears that the main difficulty is related to correct description of the motion of a mineral particle in a froth layer, which in the first approximation may be considered as a non-Newtonian fluid characterized by certain limit shear stress and viscosity, depending on the isotherms of the disjoining pressure in the films separating the particle from the bubbles and in the films separating Individual bubbles. It is known that the viscosity η of a solid-like structure such as a froth layer is a function of the shear stress P , which is schematically shown in Fig. 13. As may be seen from the graph of This article is available from: http://preprint.chemweb.com/physchem/0103026

η (P) , in the region P < Ps ,


Please use CPS: physchem/0103026 in any reference to this article 18 where Ps is the limit stress, the froth layer behaves as a solid (η = ∞ ). In the region Ps < P < Pd where P. is the limit

η max which is determined by the rate of formation of contacts and the force of cohesion of individual bubbles in the froth. Finally, in the region P > Pd , the froth viscosity decreases sharply to a certain η min , which corresponds to breakage of individual contacts between bubbles. This may apparently take place when the water content of ’the froth is high. Naturally, Ps and Pd , will also depend on this factor. Figure 13 schematically shows (dashed curve) the particle velocity V p in the froth layer as a function of the shear dynamic (or Bingham) shear stress, the froth viscosity will have a certain finite, but high value

stress

Pp created by it, which may approximately be evaluated as the ratio of particle mass to the area of its surface, i.e., as Pp = R p ρ p g / 3

(34)

It follows from (34) and Fig. 13 that the velocity of a particle in the froth changes abruptly at the points R p = 3Ps / ρ p g and R p = 3Pd / ρ p g . It is clear that if the velocity of particles in the froth is less than the velocity of the froth itself, they will be carried from the flotation column. Here we will examine how the particle velocity in the froth depends on the surfaceproperties of the froth. We will assume that the force of cohesion of particles of one type with adjacent bubbles is much less than the force of cohesion of the bubbles with one another. Then during movement this particle as it were stretches the bubbles, and the shear stress created by it will be determined by Eq. (34). Then we will assume that the force of cohesion of a particle of another type with the bubbles is greater than the force of cohesion of the bubbles to one another. It is clear that these particles can move in the froth only with the bubbles attached to them. Here the shear stress created by a particle clad in a "coat" of bubbles (Fig. 14) may be evaluated as

P( p + b ) = R p ρ p g /(1 + Rb / R p ) 2

(34)

Fig.14. Particle (shaded) in a froth layer. It follows from (35) that in the case where Rb ≥ R p the inequality Pp >> Pp + b is satisfied. Thus, by selection of the reagents and particle size one can satisfy the Inequality

Pp > Pd > Pp +b

(36)

In this case (see Fig. 13) the particles of the first type will move in the froth at a velocity much greater than the velocity of the particles of the second type. In addition, for the particles of the second type one can satisfy the Inequality Ps > Pp + b when they will not move through the froth at all. Thus, the selective flotation of large particles by the method of froth separation depends not only on the rheological properties of the froth layer and the size of the particles, but also on the ratio of the forces of cohesion of the particles with the bubbles and the bubbles with one another, which are determined by the isotherms of the disjoining pressure of the corresponding (wetting and free) films of fluid. 8. FLOTATION TECHNOLOGY DEVELOPMENT PROSPECTS 8.1. Unsolved problems. It follows from the above discussion that simultaneous examination of the colloidalchemical and hydrodynamic aspects of the flotation process permits one to better understand the nature of the factors limiting it, and also to determine the main problems, solution of which will determine the further development of flotation technology. These problems include the following: a. Submicron particles. Creation of coagulants and flocculants with an acutely selective effect, providing high floccule stability and density. Study of the kinetics of coagulation and flocculation in laminar and turbulent flows with high shear velocities. Study of the coalescence of gas bubbles and the rate of separation of gas-liquid emulsions in nonuniform hydrodynamic fields as a function of the composition and concentration of the flotation reagents. b. Small particles. Development of an experimental technique of measurement of the forces of cohesion of the particles with a bubble and with one another. Study of the stability of particle aggregates in a nonuniform hydrodynamic field as a function of their size and the composition of the reagents. c. Medium particles. Study of the stability of wetting films of fluid as a function of various factors (reagent composition, temperature, and electromagnetic and sonic fields). Study of the kinetics of thinning and breakage of wetting films forming in the process of collision of a particle with a bubble. d. Large particles. Study of the kinetics of the process of dissipation of kinetic energy in the elementary act of flotation. Study of the stability and rheological properties of flotation froths. Especially acute at the present time is the problem of flotation (selective and non-selective) of small particles, which This article is available from: http://preprint.chemweb.com/physchem/0103026



have two aspects: resource conservation and ecology. The imperfection of present flotation technology Involves not only enormous losses of mineral raw materials, lost in the sludge, but also Incurs a great damage’ to the environment by its discharges. As in the case of large particles, progress may be made only by a radical change in the technology used, developing over many decades with an orientation toward particles of the medium size class. 8.2. Turbulent microflotation. A promising trend in this field is the technology termed "turbulent microflotation" [82,83,13]. The essence of this technology is that flotation basically takes place in a long narrow channel, through which the slurry, saturated with microbubbles of gas, is passed. Here the rate of flow of the mixture is- maintained so as to ensure a turbulent flow regime, providing for three processes: aggregation of particles, heterocoagulation of particles and bubbles, and aggregation- and coalescence .of mineralized bubbles. Correct selection of the concentration of microbubbles, the type of reagents, and the point of their addition provides highly efficient adhesion of the particles to the microbubbles and aggregation of the microbubbles. As a result, at the outlet from the channel there are formed large aggregates consisting of particles and microbubbles with very high buoyancy, and thus easily separated from the fluid in small-scale froth separators of tube design. Theoretical and experimental investigations of "turbulent microflotation" [13] give reason to believe that the specific capacity of flotation machines operating on this principle will permit us to obtain more than a 100-fold gain as compared with existing systems with regard to small particles. Obviously, this method may also be used successfully for selective microflotation. As was noted above, the principal factor limiting selective flotation is heterocoagulation of different particles, which is proportional- to the product of their concentrations, i.e., ~ αn1 n 2 , where α is the efficiency of heterocoagulation. The flotation rate, for example, of particles of the first type is ~ ENn1 , where N is the bubble concentration and E is the efficiency of capture, which on a decrease in bubble size approaches α . Obviously, on dilution of the slurry, for example, by a factor of m , the heterocoagulation rate 2

decreases by a factor of m , while the flotation rate only decreases by a factor of m . Thus, at sufficiently high dilution one can always achieve a situation where the heterocoagulation of different particles becomes insignificant as compared with flotation. One should, however, take into consideration the fact that here there is an increase by a factor of m in the volume of slurry treated, which makes this method uneconomical when using the usual equipment. The situation is different with regard to the method proposed above, the main advantage of which is high water capacity, allowing more than 100-fold dilution, which decreases the heterocoagulation of different particles by a factor of 104 and thus opens new possibilities in the field of mineral flotation treatment. In spite of the fact that the theory presented here is far from complete, we can already use it as a basis to obtain a quantitative, or at least semi-quantitative explanation of all the basic principles of the flotation process, i.e.: the dependency of floatability of particles on their size, density, and shape; the dependency of flotation activity of bubbles on their radius and the level of slowing of the surface by an adsorbed layer of surfactants; the dependency of intensity of the flotation process on the bubbling rate, particle’ and bubble size, and aggregate stability and coalescent capacity of the bubbles; and- the dependency of the level of flotation selectivity on the size and surface properties of the particles, etc. The theory presented above also permits one to determine the factors limiting the capacity of flotation machines depending on the size of the particles and bubbles, and also on the hydrodynamic regimes used in them. Finally, this theory permits us to find the most promising ways of developing flotation technology (floccular flotation and turbulent microflotation). Thus, there is every reason to believe that further development of the colloidal-hydrodynamic theory of flotation will provide still deeper understanding of this, at first glance simple, but basically very complex technological process, and will aid in its improvement. REFERENCES . V. Derjaguin, Elastic properties of thin layers of water, Zhurn. Fiz. Khimii, vol. 3, no. 13, pp. 29-41, 1932. P. A. Rebinder, M. E. Linets, M. M. Rimskaya, et al., Physicochemistry of Flotation Processes [in Russian], ONTIZ, Moscow, 1933. 3. A. N. Frumkin, Physicochemistry of flotation theory, Uspekhi Khimii, vol. 11, no. 1, pp. 1-15, 1933. 4. 0. S. Bogdanov, Mechanics of the process of mineralization of air bubbles in a flotation slurry, Dis. Dokt. Tekhn. Nauk, Mashinopis, Moscow, 1946. 5. G. S. Beloglazov, Principles of the Flotation Process [in Russian], Gos. Izd-vo Nauch. Tekhn. Lit. po Cher. 1 Tsvet. Metallurgll, Moscow, 1947. 6. Z. V. Volkova, On the problem of the flotation mechanism, Zhurn. Fiz. Khimii, vol. 8, no. 2, pp. 197-207, 1936. . L. Sutherland, Kinetics of the flotation process, J. of Phys. Coil. Chem., vol. 52, no. 3, PP. 394-424, 1948. 7. 8. M. A. Eigeles, Kinetics of adhesion of mineral particles to an air bubble in flotation suspensions, DAN SSSR, vol.- 24, no. 4, pp. 342-346, 1939. 9. I. Sven-Nll'son, Significance of time of contact between a mineral and an air • bubble in flotation, in: New Investigations in the Field of Flotation Theory [in Russian], ONTI, Moscow, 1937. 10. . V. Derjaguin and S. S. Dukhin, Theory of flotation of small and medium size particles, Trans. Inst. Min. Mefcall. vol. 70, part 5, pp. 221-246, I960. 11. . V. Deryaguin, N. V. Churaev, and V. M. Muller, Surface Forces [in Russian], Nauka, Moscow, 1985. 12. . V. Derjaguin and N. V. Churaev, Wetting Films [in Russian], Nauka, Moscow, 1984.

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13. N. N. Rulyov, Colloidal-hydrodynamic theory of flotation, Dis. Dokt.. Khim. Nauk, Mashinopis, Kiev, 1987. 14. N. N. Rulyov and S. S. Dukhin, Fixation of particles on the surface of a bubble in flotation and the disjoining pressure of wetting films, Kollold. Zhurn. vol. 45, no. 6, pp. 1146-1153, 1983. 15. V. I. Melik-Gaikazyan, Interfacial interactions, in: Physicochemistry of Flotation [in Russian], Nauka, Moscow, 1983. 16. . N. Kabanov and A. M. Prumkin, Size of gas bubbles given off in electrolysis, Zhurn. Fiz. Khimii., vol. 4, no. 5, PP. 538-548, 1933. 17. I. V. Uork, Principles of Flotation [in Russian], Metallurgizdat, Moscow, 1943. 18. S. S. Dukhin, N. N. Rulyov, and D. S. Dimitrov, Coagulation and Dynamics of Thin Films [in Russian], Nauk. Dumka, Kiev, 1986. 19. M. A. Eigeles and M. L. Volova, Kinetic investigation of effect of contact time, temperature, and surface condition on the adhesion of bubbles to mineral surfaces, Mineral Processing. Int. Congress (London, April I960), Preprint No. l4, group III, pp. 1-14. 20. A. Bleier, E. D. Coddard, and L. D. Kulkarni, Adsorption and critical flotation conditions, J. Colloid and Interface Sci., vol. 59, no. 3, pp. 490-504, 1977. 21. . . Dibbs, L. L. Sirois, and R. Bredin, Some electrical properties of bubbles and their role in the flotation of quartz, Can. Metall. Quarterly, vol. 13, no. 2, pp. 395-408, 1974. 22. G. L. Collins and G. L. Jameson, Double-layer effects in the flotation of fine particles, Chem. ’Eng. Sci., vol. 32, no. 3, PP. 239-246, 1977. 23. . S. Bogdanov, On the role of wetting hysteresis in flotation, Obogashchenie Rud, no. 3, p. 27, 1979. 24. N. N. Rulyov and A. A. Baichenko, Mechanisms of capture of particles by a bubble in a turbulent flow, Kollold. Zhurn., vol. 48, no. 2, pp. 302-310, 1986. 25. N. N. Rulyov, S. S. Dukhin, and V. P. Semenov,. On the effect of aggregation of particles on the elementary act of nonlnertial flotation, Kollold. Zhurn., vol. 41, no. 2,pp. 263-271, 1979. 26. N. N. Rulyov and S. S. Dukhin, Dynamics of thinning of fluid film on inertial impact of a spherical particle against the surface of a bubble in the elementary act of flotation, Kolloid. Zhurn., vol. 48, no. 2, pp. 302-310, 1986. 27. N. N. Rulyov, S. S. Dukhin, and A. G. Chaplygin, Efficiency of capture of floated particles during multiple inertial reflections, Kolloid. Zhurn., vol. 49, no. 5, PP. 939-948, 1987. 28. P. E. Whelan and D. J. Brown, Particle-bubble attachment In froth flotation, Bull. Inst. Min. Met., no. 59, PP. 181-192, 1956. 29. H. J. Shultse and G. Gottschalk, Experimental investigations of the hydrodynamic interaction of particles with a gas bubble, Kollold. Zhurn., vol. 43, no. 5, PP. 934-943, 1981. 30. N. N. Rulyov Hydrodynamics of a rising bubble, Kollold. Zhurn., vol. 42, no. 2,pp. 252-263, 1980. 31. N. N. Rulyov and E. S. Leshchov, On the effect of surface-active substances on the hydrodynamic field of a bubble, Kollold. Zhurn., vol. 42, no. 3, PP. 521-527, 1980. 32. N. N. Rulyov, Effect of particle surface terrain on the efficiency of particle capture by a bubble in the elementary act of flotation, Kollold. Zhurn., vol. 50, no. 6, pp. 1151-1157, 1988. 33. N. N. Rulyov, Efficiency of capture of particles by a bubble in noninertlal flotation, Kolloid. Zhurn., vol. 40, no. 5, PP. 898-908, 1978. 34. N. N. Rulyov and E. S. Leshchov, Efficiency of flotation capture of small non-inertial particles by a bubble of gas rising at moderate Reynolds numbers’, Kolloid. Zhurn., vol. 42, no. 6, pp. 1123-1127, 1980. 35. N. N. Rulyov, Theory of certain experimentally established principles of flotation of small particles, Kollold. Zhurn., vol. 40, no. 6, pp. 1202-1204, 1978. 36. N. N. Rulyov, . V. Derjaguin, and S. S. Dukhin,. Kinetics of flotation of small particles by a collective of bubbles,. Kollold. Zhurn., vol. 39, no. 2, pp. 314-323, 1977. 37. H. Stechemesser, H. J. Shulze, and B. Radoev, The present state of experimental and theoretical studies on kinetics of bubble-particle Interaction, in Surface Forces [in Russian], Proc. Seventh International Conference (Moscow, December 1985), Nauka, Moscow, 1985. 38. N. N. Rulyov and A. G. Chaplygin, Role of dissipation of kinetic energy In the , elementary act of flotation of large bubbles, Kollold. Zhurn., vol. 50, no. 6, pp. 1144-1150, 1988. 39. N. N. Rulyov, Efficiency of capture of Brownian particles by a gas bubble during flotation, Kolloid. Zhurn., vol. 41, no. 4, pp. 742-749, 1979. 40. N. N. Rulyov, A. A. Vinnlchenko, and T. Z. Sotskova, Effect of surface forces on the efficiency of capture of Brownian particles of polystyrene by an air bubble during flotation, Kolloid. Zhurn., vol. 43, no. 4, pp. 678-684, 1981. 41. N. N. Rulyov, E. S. Leshchov, and V. D. Nazarov, Role of ion-electrostatic forces in the elementary act of flotation, Khimlya 1 Tekhnologiya Vody, vol. 2, no. 5, PP. 395-402, 1980. 42. S. S. Dukhin, N.. N. Rulyov, E. S. Leshchov, and Yu. Ya. Eremova, Negative effect of inertial forces on the kinetics of flocculation of particles- and on flotation water treatment, Khimiya 1 Tekhnologiya Vody, vol. 3, no. 5, PP. 387-395, 1981. 43. G. Reay and G. Ratcliff, Effects of bubble size and particle size of collection efficiency, Can. J. Chem. Eng., vol. 53, no. 4, pp. 481-485, 1975. 44. G. Reay and G. Ratcliff, Removal of fine particles from water by dispersed air flotation, Can. J. Chem. Eng., vol. 51, no. 2, pp. 178-185, 1973.




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. V. Derjaguin, S. S. Dukhin, and N. N. Rulyov, On the role of hydrodynamic interaction in the flotation of small particles, Kolloid. Zhurn., vol. 38, no. 2, pp. 251-257, 1976. . V. Derjaguin, S. S. Dukhin, and N. N. Rulyov, Microflotation: Water Treatment and Concentration [in Russian], Khimiya, Moscow, 1986. . V. Derjaguin, S. S. Dukhin, and N. N. Rulyov, Kinetic theory of flotation of ^U particles, Surface and Colloid Science, Plenum Press, New York-London, pp. 71-113, J. F. Anfruns and J. A. Kitchener, Rate of capture of small particles in flotation, Trans. Inst. Min. Metall., London, vol. 86, pp. 9-15, 1977. . V. Derjaguin, S. S. Dukhin, and N. N. Rulyov, A method of capillary hydrodynamics of thin films in flotation theory, Kolloid. Zhurn., vol. 39, no. 6, pp.. 1051-1059, 1977. S. S. Dukhin and N. N. Rulyov, Hydrodynamic interaction of a solid spherical particle with a bubble in the elementary act of flotation, Kolloid. Zhurn., vol. 39, no. 2,pp. 270-275, 1977. N. N. Rulyov, V. . Ososkov, A. N. Purich, and L. D. Skrylev, Kolloid. Zhurn., vol. 40, no. 6, pp. 1132-1138, 1978. N. N. Rulyov, V. . Ososkov, and L. D. Skrylev, Efficiency of capture of droplets of petroleum emulsion by an air bubble during flotation, Kolloid. Zhurn., vol. 39, no. 3, PP. 590-594, 1977. W. J. Trahar and L. J. Warren, The floatability of very fine particles. A review, Int. J. Mineral Processing, vol. 3, no. 1, pp. 103-131, 1976. . . Kremer and R. P. Nagaev, On interaction of a particle with a spherical bubble in the case of low Stokes numbers, Prikl. Matemafcika i Mekhanika, vol. 43, no. 3,pp. 657-663, 1979. H. S. Tomlinson and M. 0. Fleming, Flotation rate studies, Mineral Processing. Sixth Internat. Congress (Cannes, June 19 3), Pergamon Press, London, pp. 562-573, 1965. Yu. . Rubinshtein and Yu. A. Filipov, Kinetics of Flotation [in Russian], Nedra, Moscow, 1980. N. N. Rulyov and V. M. Rogov, A two-dimensional model of convective flows appearing during micro-flotation, Khimlya i Tekhnologiya Vody, vol. 5, no. 3, pp. 195-199, 1983. A. M. Golrnan, Separation and concentration of dissolved substances by flotation methods, Itogi Nauki i Tekhniki (Ser. Obogashchenie Poleznykh Iskopaemykh), VINITI, Moscow, 1980. N. N. Rulyov and V. P. Gorshkov, Role of slurry agitation in the operation of a batch flotator and ways of increasing its capacity, Khimlya 1 Tekhnologiya Vody, vol. 2, no. 3, pp. 214-217, 1980. N. N. Rulyov, Effect of sublate particle size on kinetics of ionic flotation, Khimlya i Tekhnologiya Vody, vol. 1, no. 2, pp. 3-8, 1979. N. N. Rulyov, Role of gradient coagulation in the flotation of small particles, Kolloid. Zhurn., vol. 45, no. 1, pp. 99-107, 1983. N. N. Rulyov, . N. Lazarenko, and V. M. Rogov, Gradient coagulation under conditions of microflotation, Kolloid. Zhurn., vol. 47, no. 5, PP. 1132-1137, 1985. . S. Leshchov, N. N. Rulyov, and V. M. 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Golrnan Effect of sublate particle size on the kinetics of ionic flotation, in: Intensification of Processes of Concentration of Mineral Feedstock [in Russian], Nauka, Moscow, 1981. N. N. Rulyov and S. S. Dukhin, Effect of particle size on selectivity of flotation, Kolloid. Zhurn., vol. 46, no. 4, pp. 775778, 1984. N. N. Rulyov and S. S. Dukhin, Optimization of the hydrodynamic parameters of plants treating water by flotation, Khimiya i Tekhnologiya Vody, vol. 2, no. 3, pp. 217-220, 1980. . V. Derjaguin, S. S. Dukhin, and N. N. Rulyov, Effect of particle size on heter-ocoagulation in the elementary act of flotation, Kolloid. Zhurn., vol. 39, no. 4, pp. 680-691, 1977. P. A. Arp and S. G. Mason, The kinetics of flowing dispersion doublets of rigid spheres (theoretical), Colloid and Interface Sci., vol. 1, no. 1, pp. 21-43, 1977. 0. S. Bogdanov, The floatability of large particles, in: Physlcochemlstry of Flotation Theory [in Russian], Nauka, Moscow, 1983. V. A. 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78. Yu.. V. Ryadov and N. F. Meshcheryakov, Current trends in the field of flotation of large particles, in: Present Status and Prospects for Development of Flotation Theory [in Russian], Nauka, Moscow, 1979. 79. N. N. Rulyov, Collective velocity of rise of bubbles, Kolloid. Zhurn., vol. 39, no. 1, pp. 80-85, 1977. 80. G. Para, M. Zembala, and A. Pomlanowski, Physicochemical aspects of the model mercury flotation. Part 3. An attempt at explaining the flotation activity of some floaters, Polish J. Chem., vol. 54, no. 1, pp. 77-86, 1980. 81. V. I. Mellk-Gaikazyan, Froth separation, in: Physicochemistry of Flotation Theory [in Russian], Nauka, Moscow, 1983. 82. N. N. Rulyov, Kinetics of emulsion flotation In a turbulent flow, Khimiya i Tekhnologlya Vody, vol. 1, no. 1, pp. 9-13, 1979. 83. N. N. Rulyov, S. S. Dukhin, and V. P. Gorshkov, USSR Inventor’s Certificate 939393, ICN CO 2 P 1/24, A method of extracting finely disperse particles suspended in a fluid, Byul. no. 24, publ. 30 June 1982.



colloidal-hydrodynamic theory of flotation

colloidal-hydrodynamic theory of flotation

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