Collision and coalescence

Figure 9 shows schematically a collision between two particles which may result in rebound or coalescence. In coalescence, the particles stick together because a bonding strength sufficient to resist the separating forces has been established. If the particles are smaller than a few micrometres, van der Waals forces of attraction suffice to overcome the competitive effects of gravity and kinetic energy of the particles. This means that aggregates are formed by agitation of the dry powder (Ho and Hersey, 1979). Normally, there must be free surface liquid present to agglomerate by liquid bonding. It is likely that the primary particles of the powder exhibit elastic deformation by impact so that the established pendular bondings must supply the bonding strength required to absorb the relative kinetic energy of the particles. In contrast, moist agglomerates may show plastic deformation by the impact through which kinetic energy is dissipated partly or fully by the deformation. This means a lower bonding strength is required to achieve coalescence of plastically deformable bodies compared with coalescence of elastic-brittle bodies of the same size.

The pendular bonding strength described by equation 2 shows that the strength is dependent on agglomerate porosity and surface tension. The equation applies to static conditions while the collision outlined in Fig. 9 is dynamic. Experience from production practice shows that wet granulation of a 'difficult' formulation may be facilitated by increasing the concentration of the binder liquid or by changing to a binder with a higher molecular weight, i.e. by applying a more viscous binder liquid.

It is well established that the viscosity influences the adhesion forces exerted by a liquid in dynamic conditions (Bowden and Tabor, 1964). This can be illustrated by reference to a model with two flat, parallel and circular plates separated by a film of liquid. Suppose the liquid is an 8% m/m solution of PVP K90 which has a surface tension 68mNm~' and viscosity of lOOmPas (Ritala etal., 1986). If the distance between the plates is 0.1 ¡j.m and the area of contact is 1 cm2, the adhesive force between the two plates due to the surface tension is 272N (or about 28 kg), c.f. equation 3.

The force F required to separate the plates within a certain time, t, is given by the following equation (Bowden and Tabor, 1964):

where h is the initial distance between the plates and k is the final distance. In case of complete separation, i.e. l/k2 = 0, equation 8 predicts that the force required to separate the plates (R = 0.564 cm) within 1 s is about 23 800 N. The apparent adhesive force is, thus, increased by a factor of 87 compared with the adhesive force in the static state. The internal strength of the liquid probably cannot resist the required tensile force so it will rupture at lower forces.

The example illustrates that the strength of liquid bondings under dynamic conditions may be significantly higher than the strength at static condi-





Fig. 9 Collision between two particles with relative velocity 2u0. The collision results in rebound with relative velocity 2u (u < ua) or coalescence due to bonding forces created by free surface liquid.


Fig. 9 Collision between two particles with relative velocity 2u0. The collision results in rebound with relative velocity 2u (u < ua) or coalescence due to bonding forces created by free surface liquid.

tions. The factor by which the cohesive force is increased by the change depends on the viscosity of the liquid and the relative velocity of the solid particles.

The described effect of the liquid viscosity is likely to affect the growth rate in agglomeration processes. There is, however, only limited evidence for the effect. Ritala etal. (1986) have, in a comparison between different binders, shown that the binder concentration has a slight effect upon the granule growth in wet granulation in a high shear mixer, but the effect was much smaller than that of the liquid surface tension.

Ennis etal. (1991) have recently provided an interesting discussion of the coalescence between colliding particles. They showed that the cohesive force of the moving pendular liquid bridge comprises a capillary force component and a viscous force component, the latter being dominant. The prerequisite to a successful coalescence is that the dimensionless Stokes' number Stv, defined as:

does not exceed a critical value, which is dependent on the strength of the pendular bond created by the collision. In equation 9, m is the mass of the particle, r is the particle diameter, u0 is the initial particle velocity and r¡ is the viscosity of the liquid. Stv equals the ratio between the relative kinetic energy of the particles (lAm[2u0]2) and the work done by the pendular bond at rebound (Fvis2h where Fvis is the viscous bonding force and h is the thickness of the surface film on the particle). Intuitively, one can see that if the pendular bond can supply energy that exceeds the kinetic energy of the particles, the bond strength suffices to keep the particles together. Granulation regimes that apply to different granulation methods can be established on the basis of the Stokes' number.

The collision between particles outlined in Fig. 9 implies that the particle surfaces recover elastically, and may separate without residual deformation, as is the case when the particles are crystalline materials. With particle agglomerates, it is more likely that the collision is not truly elastic so that the impact results in a plastic deformation of the surfaces. The dissipation of the kinetic energy of the agglomerates results in heating of the mass which is the basic reason for the close correlation between power consumption and granule growth in high shear mixers (Holm etal., 1985a).

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