Agglomeration rate

Assuming that sedimentation is slow compared to the first two coUision mechanisms, the overaU agglomeration rate, —dN/dt, is... 

Thus in a mixed system, as e.g. in a stirred tank, the rate of agglomeration additionally depends on the shear field and therefore on the energy dissipation e in the vessel. Furthermore, in precipitation systems solution supersaturation plays an important role, as the higher the supersaturation, the stickier the particles and the easier they agglomerate (Mullin, 2001). This leads to a general formulation of the agglomeration rate... 

According to Smoluchowski s theory (equation 6.53), the agglomeration rate increases proportional with the fluid shear rate 7... 

Agglomeration rates also depend on the level of supersaturation in the reaetor and on the power input. Wojeik and Jones (1997) found a linear inerease of the agglomeration kernel with the growth rate. Therefore, the level of supersaturation was aeeounted for by Zauner and Jones (2000a) using the relation... 

In all such laboratory studies, plant conditions and compositions should be employed as far as possible. Agglomeration rates tend to increase with the level of supersaturation, suspension density and particle size (each of which will, of course, be related but the effects may exhibit maxima). Thus, agglomeration may often be reduced by operation at low levels of supersaturation e.g. by controlled operation of a batch crystallization or precipitation, and the prudent use of seeding. Agglomeration is generally more predominant in precipitation in which supersaturation levels are often very high rather than in crystallization in which the supersaturation levels are comparatively low. 

This additional Eq. (18) was discretized at the same resolution as the flow equations, typical grids comprising 1203 and 1803 nodes. At every time step, the local particle concentration is transported within the resolved flow field. Furthermore, the local flow conditions yield an effective 3-D shear rate that can be used for estimating the local agglomeration rate constant /10. Fig. 10 (from Hollander et al., 2003) presents both instantaneous and time-averaged spatial distributions of /i0 in vessels agitated by two different impellers color versions of these plots can be found in Hollander (2002) and in Hollander et al. (2003). 

Fig. 11 presents the results of some 30 simulations for various conditions and two impeller types in terms of the mean agglomeration rate constant observed in the various simulations vs. the vessel-averaged shear rate found in the simulations. The simulations all started from the dotted curve for relating local agglomeration rate constant to local shear rate. A clear decrease in the maximum of /i0 as well as a shift toward higher average shear rates was found which are caused by the local nature of the nonlinear flow interactions only. These... 

Fig. 10. Results of LES-based simulations of an agglomeration process in two vessels one agitated by a Rushton turbine (left) and one agitated by a Pitched Blade Turbine (right). The two plots show the agglomeration rate constant fl0 normalized by the maximum value, in a vertical cross-sectional plane midway between two baffles and through the center of the vessel. Each of the two plots consists of two parts the right-hand parts present instantaneous snapshots the left-hand parts present spatial distributions of time-averaged values after 50 impeller revolutions. Reproduced with permission from Hollander et al. (2003).

Fig. 11. The discrepancy between the original kinetic relation due to Mumtaz et al. (1997) and the observed relation between mean agglomeration rate constant fl0 and volume-averaged shear rate. Symbols refer to individual numerical simulations (LES). RT stands for Rushton Turbine, PBT for Pitched Blade Turbine. Reproduced with permission from Hollander et al. (2001b).

An example Hollander et al. (2001a) nicely demonstrated how the strong inhomogeneities in stirred-tank flow result in unpredictable scale-up behaviour and that the impact of the detailed hydrodynamics and of the non-uniform spatial particle distribution on agglomeration rate is larger and more complex than usually assumed their study once more illustrated the risks of scale-up on the basis of keeping a single non-dimensional number. Sophisticated CFD, especially on the basis of LES, offers an attractive alternative indeed. 

These rate constants are compared for two cases in Fig. 7.3. It follows that heterogeneity in particle size can significantly increase agglomeration rates. 

Example 7.1 Effects of Particle Size on Agglomeration Rate... 

Compare the agglomeration rate of an aqueous suspension containing 104 virus particles per cubic centimeter (d = 0.01 pm) with that of a suspension containing, in addition to the virus particles, 10 mg liter1 bentonite (number cone. = 7.35 106 cm3 d = 1 pm). The mixuture is stirred, G = 10 sec1, and the temperature is 25° C. Complete destabilization, a = 1, may be assumed. (This example is from O Melia, 1978.)... 

Thielmann, F., Naderi, M., Ansari, M. A., Stepanek, F. The effect of primary particle surface energy on agglomeration rate in fluidised bed granulation. Powder Technol., 181, 2008, 160-168.M. J. Valazza, G. G. Wada. Creating a successful partnership with a contract manufacturer. Pharmaceutical Technology Europe, 13(5), 2001, 26-34. 

It must be kept in mind that the efficiency of the coagulation process in practice is not solely determined by the agglomeration rate the attainment of certain desirable floe properties must be included in deliberations directed toward the optimization of the process. 

For nonspherical particles, Muller (1928) postulated that since the diffusion equation applicable to aerosol problems is the same (except for definition of terms) as the general equation for electric fields (Laplace s equation), there should be analogs among the electrostatic terms for various properties of coagulation. For example, the potential should be analogous to particle number concentration, and field strength to particle agglomeration rate. Zebel (1966) pointed out that... 

The second method for aerosol coagulation in turbulent flows arises because of inertial differences between particles of different sizes. The particles accelerate to different velocities by the turbulence depending on their size, and they may then collide with each other. This mechanism is unimportant for a monodisperse aerosol. For a polydisperse aerosol of unspecified size distribution, Levich (1962) has shown that the agglomeration rate is proportional to the basic velocity of the turbulent flow raised to the 9/4 power, indicating that the agglomeration rate increases very rapidly with the turbulent velocity. Since very small particles are rapidly accelerated, this mechanism also decreases in importance as the particle size becomes very small, being most important for particles whose sizes exceed 10-6 to 10"4 cm in diameter. In all cases brownian diffusion predominates when particles are less than 10-6 cm in diameter. 

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