Process 3 Stability Efficiency and Detachment

Flotation limits for coarse particles The essential problem in understanding bubble-particle aggregate stability is to determine whether or not the adhesive force, acting on the tplc, is large enough to prevent the destruction of the aggregate under the dynamic conditions that exist in flotation. It is important to understand the physics of the problem before moving on to a mathematical description. Let us consider a smooth spherical particle located at the fluid interface. Once the equilibrium wetting perimeter has been established following spreading of the tplc, the static buoyancy of this volume of the particle will act against the gravitational force (Figure 3). The hydrostatic pressure of the liquid column of height Z0 acts against the capillary pressure. The 'other detaching forces' require further discussion. Since they arise from the particle motion relative to the bubble, velocity-dependent drag forces will oppose the detachment of the particle from the bubble. An analysis of these forces is extremely complex and has not been reported to date. Therefore any force balance will necessarily be quasistatic and approximate.

The net adhesive force, Fad, is equal to the sum of the attachment forces, Fa, minus the detachment forces, Fd, i.e.:




P, 1

/ h

flp/ A

\ll I

\ PP


Figure 3 Location of a smooth spherical particle at a fluid interface. (From Schulze (1983) Physicochemical Elementary Processes In Flotation. Amsterdam: Elsevier.)

Figure 3 Location of a smooth spherical particle at a fluid interface. (From Schulze (1983) Physicochemical Elementary Processes In Flotation. Amsterdam: Elsevier.)

The particle will not remain attached to the bubble if Fad is negative but will be present in the liquid phase.

The mathematical description of the various forces that dictate the equilibrium position of particles at liquid-vapour or liquid-liquid interfaces has followed an evolutionary trail. Analogous processes of interest, for example, include pigment 'flushing', where a solid particle is induced to transfer from one liquid phase to another by appropriate surface modification with surfactants, and the stabilization of emulsion droplets by solid particles.

The actual problem of the balance of forces operating on a particle at a liquid-air interface has been considered by Sutherland and Wark, who considered the case of a gas bubble attached to a plane solid surface of infinite extent and used this as a model for bubble-particle adhesion in flotation. Since this work there have been other very notable contributions. It was Princen who proposed the first extensive and generalized treatment of the forces acting on a particle at fluid interfaces. This theory was developed further by Schulze in 1977 and expanded in 1983.

Consider the case of a spherical particle at a liquid-air interface. We assume that the system is in a quasistatic state and that the contact angle corresponds to that for a static system. The dynamic contact angle can depart significantly from the static value, depending in part on the velocity of the tplc. If the particle oscillates around its equilibrium position, the tplc would be expected to move to some extent. Hence a full analysis would need to account for the velocity-dependent drag forces mentioned above and link these to contact angle dynamics. Since this is an intractable problem at present, a simpler approach is necessary.

Let us suppose that a spherical particle of radius Rp is attached to a bubble of radius Rb where Rb is much greater than Rp, as shown in Figure 3. By understanding the forces that operate on the particle, it is possible to calculate the energy of detachment. The forces acting upon the particle are as follows:

• Capillary force, Fc, acting in the vertical direction along the tplc:

Fc = 2nr0y sin <0 = — 2reRpy sin ro[sin(ro + 0)]

where y is the liquid-vapour surface tension.

• Static buoyancy of the fractional volume of the immersed particle, Fb:

Hydrostatic pressure, Fh, of the liquid column of height Z0 on the contact area:

Capillary pressure, Fp, in the gas bubble which acts on the contact area rer0:

Fp = Pynr0

which for a spherical bubble is given approximately as:

Gravitational force, Fg:

where pp is the particle density. Extra detaching forces, Fd, which are denoted approximately and generally as the particle mass multiplied by a generalized acceleration bm in the flotation cell:

It is worth remarking that it is bubble-particle aggregates that are actually accelerated in the flotation device, thus pp is in fact an approximation (Ap = pt — p i).

At equilibrium, the sum of these forces, £ F, must equal zero.

The energy of detachment, Edet, corresponds to the work done in forcing a particle to move from its equilibrium position, heq(ro) at the liquid-vapour interface to some critical point, hcrit(ro), where detachment occurs and the particle moves into the liquid phase. The sum of the various forces, £F, is related to

Edet by:


Equation [13] may be solved by introducing the various forces and carrying out a numerical integration. The detachment process takes place when the kinetic energy of the particle equals Edet. The kinetic energy of the particle is given by freR^pV2, where Vt is the relative (turbulent) velocity of the particle, acquired due to stresses on the bubble-particle aggregate in the turbulent field of the flotation cell, as the aggregate collides with other bubbles or aggregates or due to other modes of excitation. Vt is determined experimentally as the velocity of gas bubbles in the flotation cell and pp is the density of the particle.

The maximum floatable particle diameter based on the kinetic theory, Dmax,K, is given as:

K^pPl g


3 k2

where F is the Helmholtz free energy, L is the contact line, T is the temperature, V is the volume and W is the thermodynamic work. The Young-Dupre equation becomes:

Equation [14] may be solved by numerical integration or by plotting each of the kinetic and detachment energies as a function of Rp at constant y and pp and specified Vt. pj refers to the density of the fluid and y is the surface tension at the liquid-vapour interface. This equation has been shown to describe adequately both the detachment of a sphere from a liquid-vapour interface and the behaviour of hydrophobic angular quartz particles between approximately 30 and 120 |im in diameter under flotation conditions.

Flotation limits for fine particles The only theoretical study to date dealing with the limit of floatability of fine particles was published by Scheludko and co-workers in 1976. The limit is the critical work of expansion required to initiate a primary hole or three-phase contact line during bubble-particle approach - a requirement that is met by the kinetic energy of the particles. The matching of these two quantities enables a minimum particle diameter, Dmin K, for flotation to be obtained:

where k is the line tension, opposing expansion of the tplc. Molecules that are present in a line have a free energy that is different from those at a surface - in fact there is an excess linear free energy and a linear tension in an analogous fashion to that of excess surface free energy and surface tension.

In fact,


The line tension is important for small contact radii and can oppose or reinforce yL/v cos 6. It counteracts the formation of the tplc in Scheludko's theory which neglects thin film drainage and other hydrodynamic effects. Experimental data for hydrophobic angular quartz particles between about 10 and 35 |im in average diameter follow a general trend that is predicted by eqn [15], although quantitative agreement is poor. If a pseudo-line tension embracing surface heterogeneities, replaces k in eqn [15], then this in turn enables Dmin in eqn [15] to be re-expressed in terms of a critical bubble radius below which attachment does not occur. Reconciliation between theory and experiment is then achieved although the concept of pseudo-line tension needs to be placed on a firmer experimental foundation.

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