1 y R2P 1 RP 2Rhx RP 3 V Rr2 r Rr x Rr Rr

acting on the water film at the particles surface is then computed to be:

assuming that the van der Waals and the electrostatic disjoining pressures are attractive. To calculate the induction time it is necessary to obtain values for the velocity components of the water in the flowing film vr(r, z, t) and vz(r, z, t) in terms of the pressure gradient. For an incompressible Newtonian fluid with constant viscosity, the momentum balance equation for creeping flow for radial symmetry ve = 0 and for negligible pressure variation in the z-direction reduces to:

A solution for vr(z, r) at time t is obtained for both a mobile and an immobile bubble-water interface:

where n"1 for a mobile interface and n"2 for an immobile interface. This, when substituted into the equation of continuity gives vz, which together with vr, gives a solution for the configuration of the bubble-water interface z = h1(r, t)t. From this the change in film thickness at radial position r, with time follows:

A radial position Rh is estimated, as the smallest value of R where the local bubble curvature is the same as that of the undeformed bubble. The force

The induction time is that when F(tind) is deemed just to have exceeded the adhesion energy of the solid-water interface. For the case where the effects of the electrostatic double layer can be neglected, the numerical integration shows an inverse power dependence of the induction time on the van der Waals parameter.

These conclusions have been tested against the experimental data reported by Ye and Miller whose observations were based on forcing a bubble through a solution on to a bed of particles for a predetermined time and recording as the induction time the time at which 50% of the events resulted in attachment.

The theoretical treatment predicts that when the particle radius Rp is very much smaller than the bubble radius Rb, log tind will be a linear function of log Rp. This is consistent with the experimental data of Ye and Miller, although the predicted slopes are slightly lower than those reported.

When the electrostatic effects are allowed for the predictions have to be graphical as a general equation cannot be derived; there are also no experimental data available for checking. From Figure 11 and Table 3 it is, however, clear that these electrostatic effects are very significant.

Li, Fitzpatrick and Slattery have reported values for A and k reproduced here as Table 3 which they used in conjunction with their derived relationships.

The data in the table are given for univalent ions and the surface potentials are given as equal on each surface, but both cases can be easily adapted. When the sign of the surface charges are opposite and the surfaces are consequently mutually attractive, A is negative.

The authors report induction times for a system corresponding to that of Ye and Miller although no direct comparisons with experimental data could be made. Figure 11 is typical of the predictions made.

For electrostatic interactive forces on a 100 | m particle in a 10" 3 molar solution the induction times vary from 10 ms for nei= — 80 mV to more than a second for nei >+ 20 mV.

As the solution molarity falls to 10" 4 mV the induction time for nei =— 80 mV remains at 10 ms but falls to 100 ms for nd =+ 20 mV.

The effect of particle size is also interesting; for a 10"3 molar solution and nel = — 80 mV the induction time is less than 1 ms for a 10 | m particle rising

Figure 11 Induction times as a function of the charge density on particle and bubble surfaces, which are assumed equal but of similar or opposite signs, and the particle radius. (A) n = 10 "3 and (B) n = 10"4 mol L"1. Reproduced with permission from Li, Fitzpatrick and Slattery (1990) Rate of collection of particles by flotation. Industrial Engineering Chemistry Research 29, 955-957.

Figure 11 Induction times as a function of the charge density on particle and bubble surfaces, which are assumed equal but of similar or opposite signs, and the particle radius. (A) n = 10 "3 and (B) n = 10"4 mol L"1. Reproduced with permission from Li, Fitzpatrick and Slattery (1990) Rate of collection of particles by flotation. Industrial Engineering Chemistry Research 29, 955-957.

to 100 ms for a 1000 |im particle. The effect of a reduction in solution molarity to 10"4 does not, however, change the induction times significantly as it does in the previous case.

Although the model refers only to single bubble-single particle contact and the experimental data are similarly idealized, it is nevertheless valuable

Table 3 Showing values of A and k as functions of surface potentials at various solution molarities: 10 ~6,10~4,10~2for univalent ions z"1

±A at M

(mV)

(mV)

(mols/litre)

10 -6

10-

10-2

(molar)

(molar)

(molar)

20

20

58.6

5.86 x103

5.86 x105

40

40

217.9

2.179 x104

2.179 x106

50

50

323.3

3.233 x 104

3.233 x106

60

60

437.9

4.379 x 104

4.379 x106

80

80

673.8

6.738 x 104

6.738 x106

100

100

892.2

8.922 x 104

8.922 x104

k (cm-1)

3.257 x 104

3.257 x 105

3.257 x106

Reproduced with permission from Li D, Fitzpatrick JA, Slattery JC (1990) Rate of collection of particles by flotation. Industrial Engineering Chemistry Research 29, 962, Table II.

Reproduced with permission from Li D, Fitzpatrick JA, Slattery JC (1990) Rate of collection of particles by flotation. Industrial Engineering Chemistry Research 29, 962, Table II.

as it predicts induction times in fundamental terms which surely provides a sound basis for the experimental use of them as a characterizing parameter, for the development of reagent addition strategies for industrial separations.

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