Rear stagnant cap

The Rear Stagnant Cap and Bubble Buoyant Velocity at Small Re... 

When the rear stagnant cap and the angle T increase, the collision efficiency decreases. Thus, at uniform surface retardation, i.e. under condition (8.71) and during the rear stagnant cap formation, the mechanisms of DAL effect on the transport stage differ qualitatively. In the former case, with increasing surfactant concentration, the normal component of velocity and, respectively, the flow of particles uniformly decrease over the leading surface. In the latter case, the area admissible for sedimentation of particles decreases. 

Using the formulas for the hydrodynamic field of a bubble carrying a rear stagnant cap (cf. Section 8.7), we can calculate the effect of the cap on collision efficiency. It is unlikely that such work is of interest since the theory described in Section 8.7 is restricted to small Reynolds numbers. Thus, we caimot expect agreement between this theory and reality since the leading surface must be either completely or strongly retarded, according to experimental data by Okazaki (1964). 

In transient state the DAL has a slight effect on the transport stage if the rear stagnant cap covers a smaller part of the surface. If the rear stagnant cap is not too small and characterised by the angle 9 (cf Section 8.6) essentially less than 7t/2, the possibility of its effect depends substantially on the mechanism of fixation of particles on a bubble surface (see Appendix lOD). 

Reasonable investigations under these conditions are restricted by the state of the DAL theory which has been developed so far only for conditions of very strong and weak surface retardation (cf Section 8.6). Collision efficiency has been derived only for potential flow conditions (Sutherland, 1948). With increasing surfactant concentration up to c[ (Eqs 8.135 -8.136), a beginning decrease of bubble velocity may be expected. A respective rear stagnant cap results in a decrease of collision efficiency only when attachment of the particle is accomplished not due to the of instability of the water interlayer at some thickness h but under the effect of attraction forces (Appendix lOB). 

The possibility of detachment increases with bubble size and its rising velocity. Surface retardation at Re > 40 due to the presence of surfactant can appear. At low surface activity the whole surface is retarded almost uniformly which prevents particles from detachment. At high surface activity an increase of surfactant concentration only yields a larger rear stagnant cap, while the rest of the surface is not very strongly retarded. 

However, one can assume that detachment of small particles from the stagnant cap is unlikely. The bubble surface of the rear stagnant cap is strongly retarded and the normal component of liquid velocity is much lower than that at the upper not retarded part. This can prevent detachment of sufficiently small particles from the stagnant cap. 

Since in the present section the inertia-ffee transport of particles is considered, we can restrict ourselves to aggregates of a sufficiently small size, approximately tens of microns. At small degrees of packing in an aggregate, its sedimentation rate can be so small that their transport from below to the rear stagnant cap is possible against gravity. 

Eq. (10.63) can be hilfilled at low and high surface activity of surfactant. It is not recommended to use a surfactant of very high smface activity (cf. condition (8.72)) in the absence of a rear stagnant cap since the adsorption layer is carried away to the rear pole of the bubble. Addition of such surfactant does not enhance particle deposition on the leading surface of bubbles and does not prevent their detachment from the rear surface. The surfactant molecules are concentrated in the close neighbourhood of the rear pole of the bubble and detachment of particles can occur from any part of the surface in the vicinity of which the normal velocity components are directed to the liquid. 

The extension of the rear stagnant cap takes place at increasing surfactant concentration, adsorption and retardation of the surface which also results in a decrease of inertia forces. Any variation of surface concentration and its gradient at any point of the surface results in a change of the velocity gradient. This is a local effect. 

Extension of the rear stagnant cap far from the front pole of a bubble exerts an effect on velocity distribution even in the neighbourhood of the front pole. This is a non-local effect. 

As already mentioned in Section 10.2., extension of the rear stagnant cap results in an earlier separation of the hydrodynamic flow, i.e. decreases and the interception angle also decreases. 

It is well known that even an insignificant expansion of the rear stagnant cap leads to a substantial increase of energy dissipation and to a substantial decrease of surface velocity. Nevertheless, as long as the secondary flow embraces a very small part of the bubble surface, its direct effect on the primary flow is possibly insignificant. 

Appendix 8A Small Rear Stagnant Cap Of Bubble At High Reynolds Numbers... 

A different situation arises when h = 0 and fixation of particles happens due to surface attraction forces. These forces can exceed the gravitation force so fixation of particles at the lower pole of the bubble becomes possible. The appearance even of a small rear stagnant cap in the neighbourhood of the lower pole results in a change of the course of the grazing trajectory. 

Thus, if an experiment shows a sensitivity of the collision efficiency to the appearance of a small rear stagnant cap, it demonstrates that fixation is controlled by the mechanism of surface attraction forces. An opposite experimental result would point to particle fixation according to mechanism controlled by the break of the wetting film. A quantitative description of collision efficiency results in qualitatively different formulas depending on the controlling mechanism, which contain h in one case and the Hamaker constant in the other case. 

Attachment of particles on the rear stagnant cap of bubbles of a size between millimeters and centimicrons can combine the advantages of two processes discussed above ... 

There is not much chance of fixed particles becoming detached because of the strong retardation of the bubble surface in the rear stagnant cap zone. 

Only small amounts of cationic surfactant are needed. Since attachment takes place on the rear stagnant cap, the drift of adsorbed surfactant from the leading bubble surface and its... 

A description of the DAL combining the rear stagnant cap and the hydrodynamic vortexes in its neighborhood appears to be necessary, which is a logical conclusion of a large series of investigations described in Chapter 8. It becomes especially vital in coimection with the problem of intensification of microflotation and in particular in two-stage microflotation. 

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