Aggregation Brownian

For Brownian aggregation, the aggregation kernel can be written as follows (Elimelech et al., 1995 Sandkuhler et al., 2003) ... 

This expression was obtained by setting the shear aggregation rate to the Brownian aggregation rate and solving for the size a to which this corresponds assuming the colloid stability ratio, W is unity. 

Solution The radii of particles when the shear aggregation rate is equal to the Brownian aggregation rate are calculated by the preceding equation with y varied between 1 to 100 sec , T = 298 K, rj = 0.01 poise. The results follow ... 

With 0.05 jLim latex particles in a glycerol-salt solution, large-scale shear aggregates have been shown to have the same fractal dimension as that of Brownian aggregates (i.e.. Dp = 1.8 0.1) when the shear rate is less than 1500/sec [88]. Due to shear allegation, the aggregation rate for these experiments was much faster than that of comparative Brownian aggregation rates. 

FIGURE 6.2.1 The self-similar distribution function for (i) the constant aggregation frequency (dotted line), (ii) the sum frequency (continuous line), and (hi) the Brownian aggregation frequency (dot-and-dash line). (From Wright and Ram-krishna, 1992. Reprinted with permission from Elsevier Science.)... 

The frequency evaluated at one of the particle sizes equated to zero need not always exist For example, the Brownian aggregation frequency is unbounded when one of the particle sizes approaches zero. This is a reflection of the fact that aggregation is virtually certain between two particles when one of them is much smaller than the other and consequently capable of very rapid diffusional motion relative to the other much larger particle. In this case a(0, /y)/ cannot be expected to exist. 

Calculations were made for Brownian aggregation for both a constant volume of mixing and a constant number of particles in the volume of... 

FIGURE 7.4.2 Predictions of the cumulative number fraction in Brownian aggregation for five particles in the volume of mixing from Monte Carlo simulation compared with (i) population balance equation, (ii) product density analysis using closure hypothesis (7.4.13). 

The case of constant kernel is similar to the situation that was analyzed in Sect. 4.1. The fast aggregation problem with the constant kernel was exactly solved by Smoluchowski in 1917 [73]. This model is based on the more complicated kernel for 3D Brownian aggregation ... 

Here k is the Boltzmann constant, which is equal to 1,381, 10 J/K. For small particles (< 0.1 pm), Brownian aggregation is a very rapid process (time constants 1 s). 

It follows from Example 6a that the time constant for Brownian aggregation is inversely proportional to the cube of the particle diameter. That means that the aggregation slows down as the particles grow. When they reach a certain size, the shear aggregation will take over. 

For a given precipitation, one of the few variables that can be used to influence aggregation is the liquid viscosity. When the precipitation would be carried out with the same rate in a solution of higher viscosity, both the surface growth and the Brownian aggregation will be slowed down. Since the final particle size is determined by aggregation, an increased viscosity will result in smaller particles. This has been confirmed experimentally (Eshuis et al., 1994). 

The final particle size will be determined predominantly by the flow conditions, as was shown in the preceeding section. However, the effects of nucleation, surface growth and Brownian aggregation, that lead to the intermediate particles, will still be visible in the final structure of the end product. The size of the primary particles may determine the internal structure of the porous particles that are finally formed (e.g. the internal surface area). The size of the intermediate particles may determine the macroporosity of the final particles, and also their strength. 

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