Znd. Eng. Chem. Res. 1990,29,955-967 Annual Meeting, Phoenix, AZ, Jan 25-28,1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 267. Dell, C. C.; Jenkins, B. W.The Leeds Flotation Column. Seventh International Coal Preparation Congress, Sydney, 1976. Dobby, G. S.; Finch, J. A. Mixing Characteristics of Industrial Flotation Columns. Chem. Eng. Sci. 1985,40,1061. Geankoplis, C. J. Transport Processes and Unit Operations; Allyn and Bacon: Boston, 1978. Hucko, R. E.; Gala, H. B. Promising Advanced Coal Preparation Technologies for Reducing SO2 Emissions. Prepr. Pap.-Am. Chem. Soc., Diu. Environ. Chem. 1988,28 (No. l), 211-215. Idlas, S. A.; FitzPatrick, J. A.; Slattery, J. C. Conceptual Design of Packed Flotation Columns. Znd. Erg. Chem. Res. 1990,preceding paper in this issue. Jameson, G. J.; Nam, S.; Young, M. M. Physical Factors Affecting Recovery Rates in Flotation. Miner. Sci. Eng. 1977,9,103. Kawatara, S.K.; Eisele, T. C. Column Flotation of Fine Coal. In Fine Coal Processing; Mishra, S . K., Klimpel, R. R., Eds.; Noyes Publications: Park Ridge, NJ, 1987; p 414. Killmeyer, R. P.; Hucko, R. E. Interlaboratory Comparison of Advanced Froth Flotation Processes. Society of Mining Engineers
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Annual Meeting, Las Vegas, NV, Feb 27-March 2,1989; Preprint 89-137. Laplante, A. R.; Yianatos, J.; Finch, J. A. On the Mixing Characteristics of the Collection Zone in Flotation Columns. In Column Flotation 88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 69. Mori, S.; Okamato, H.; Hara, T.; k o , K. Kinetics Studies of Fluorite Flotation. h o c . 15th Znt. Min. Process. Cong., Cannes 1986,3, 155-162. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Szatkowski, M.; Freyberger, W. L. Model Describing Mechanism of the Flotation Process. Trans. Znst. Min. Metall. 1985,94,C61C70. Treybal, R. E. Mass-Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980.
Received for review September 18, 1989 Revised manuscript received February 23, 1990 Accepted March 6,1990
Rate of Collection of Particles by Flotation Dongming Li,+Joseph A. FitzPatrick,*and John C. Slattery*pt Department of Chemical Engineering and Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208-3120
The capture of a single spherical particle by a bubble in froth flotation is analyzed to estimate both the induction time and the rate of flotation. Both London-van der Waals forces and electrostatic double-layer forces are recognized. The induction time is the time required for the thinning and rupture of the liquid film between the spherical particle and the bubble. With the assumption that the effects of the electrostatic double layer can be neglected, the predicted result for the induction time describes the trends seen in prior experimental studies. When the effects of the electrostatic double layer cannot be neglected, better selectivity between particles having different surface potentials is predicted at intermediate values of the electrolyte concentration. When the effects of the electrostatic double layer can be neglected, an expression for the rate of flotation can be derived up to a proportionality factor. This allows us to draw several qualitative conclusion regarding the rate constant that are supported by experimental observations. Froth flotation is a process in which particles are captured selectively from a suspension by air bubbles. The capture of a particle by a bubble can be thought of as occurring in a series of stages: bubble-particle approach, thinning and rupture of the liquid film between them, and formation of a stable particle-bubble aggregate. The attachment of mineral particles to air bubbles is the most fundamental requirement for successful flotation. When as a result of mixing a solid particle is brought into near contact with an air bubble for a sufficiently long period of time, a thin liquid film forms between them and begins to drain (Schulze, 1984). The thin film is not bounded by parallel planes. As a drop or bubble approaches an interface, it develops a dimple: the film is thicker at ita center than a t its rim (Derjaguin and Kussakov, 1939; Allan et al., 1961;Platikanov, 1964;Hartland, 1967, 1969; Hodgson and Woods, 1969;Hartland and Woods, 1973;Burrill and Woods, 1973). As the thickness of the draining film becomes sufficiently small (about 10oO A), the effects of the disjoining pressure attributable to the London-van der Waals forces and to any electrostatic double layer become significant. The liquid film drains *Current address: Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122. Department of Chemical Engineering. t Department of Civil Engineering. 0888-5885/90/2629-0955$02.50/0
until coalescence occurs, and the particle is attached to the bubble. We usually refer to the time required for the thinning and rupture of the liquid film between particle and bubble as the i n d u c t i o n time. The induction time depends not only upon the disjoining pressure but also upon many other variables including the particle size, the bubble size, the surface tension, and the viscosity of the continuous phase. Particles having a shorter induction time will be more easily captured, and they will be said to have a higher selectivity. Experimentally, there have been attempts to characterize the induction time by measuring the particles of different species captured by an air bubble on the tip of a capillary tube that is forced against a bed of particles for a short period of time (Laskowski and Iskra, 1970; Schulze, 1984;Yordan and Yoon, 1985;Yoon and Luttrell, 1985;Ye and Miller, 1988). From the viewpoint of theory, Ralston (1983)and Schulze (1984)have pointed out that an analysis of the induction time which assumes a parallel-plane film misses the strong influence of film dimpling upon film drainage. In what follows, we develop a more complete hydrodynamic theory for the thinning of a dimpled liquid film between a bubble and a solid particle under the influence of disjoining pressure attributable both to London-van der Waals forces and to an electrostatic double layer. Our primary objective is to determine the induction time. 0 1990 American Chemical Society
956 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990
t
L*
r*
Figure 1. Thin liquid film formed as a small bubble approaches a small solid particle.
Statement of the Problem Figure 1shows a draining liquid film formed as a small bubble approaches a small solid particle. Our frame of reference is stationary with respect to the solid particle. A number of assumptions will be required. (i) The solid particle is assumed to be a sphere. Outside the neighborhood of close approach to the particle, the bubble is also assumed to be spherical. This is equivalent to assuming that the Bond number
where Ap* e p(c)' - P (B)'
(2)
is the magnitude of the density difference between the continuous phase and the bubble phase, p(C)* is the density of the continuous phase, p(B)*is the density of the bubble phase, g* is the magnitude of the acceleration of gravity, R is the radius of the bubble, and y is the equilibrium surface tension. (The superscript * denotes a dimensional variable.) (ii) Viewed in the cylindrical coordinate system of Figure 1, the liquid-gas interface is axisymmetric: z* = h;(r*,t*)
(3)
(iii) Outside the neighborhood of close approach between the bubble and the particle, the dependence of hi upon r* is sufficiently weak that
