Kernel aggregation

For monodisperse partieles, [V + L"][L + L" ] = 4, the aggregation kernel beeomes independent of partiele volume and the aggregation rate is given by... 

Smoluehowski also presented a simple theory of aggregation kineties assuming eollisions of perfeet eolleetion effieieney to prediet spherieal partiele size distributions in a uniform liquid shear field of eonstant veloeity gradient. The aggregation kernel is then expressed as... 

The significance of this novel attempt lies in the inclusion of both the additional particle co-ordinate and in a mechanism of particle disruption by primary particle attrition in the population balance. This formulation permits prediction of secondary particle characteristics, e.g. specific surface area expressed as surface area per unit volume or mass of crystal solid (i.e. m /m or m /kg). It can also account for the formation of bimodal particle size distributions, as are observed in many precipitation processes, for which special forms of size-dependent aggregation kernels have been proposed previously. 

A semi-empirical form of orthokinetic aggregation kernel due to Thompson (1968) is adopted for illustrative purposes only, viz. 

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

In order to take particle-particle interactions into account, a stability ratio W is included which relates the collision kernel /So to the aggregation kernel /3agg. The stability ratio W depends on the interaction potential aggregation rate without to the rate with interactions additional to the omnipresent van der Waals forces. For Brownian motion as dominant reason for collisions, the stability ratio W can be calculated according to Eq. (6) taken from Fuchs [ 10]. In case of shear as aggregation mechanism, the force dip/dr relative to the friction force should rather be considered instead of the ratio of interaction energy relative to thermal energy. 

Three types of interactions contribute to the lattice energies of the (small, aperiodic) crystals depicted in Fig. 10 kernel — electron-pair attractions (the only interactions that favor ion-aggregation) kernel-kernel repulsions and electron-pair — electron-pair repulsions. 

The mechanistic implications of the GPLE have been only partially discussed. Standard models cannot be used to justify the use of a steady-state distribution because they were developed using only aggregation kernels. However, there is no fundamental reason why steady-state configurations do not exist as shown by Puentes and Gamas [6] based on an analysis of the surface free energy corresponding to a crystallite distribution. 

The results just presented show that steady state particle size distributions are possible for a wide variety of coagulation-fragmentation conditions. In the case of homogeneous kernels the shape of these distributions is a slightly distorted Gaussian, and it approaches experimental results. In spite of the fact that the analytical solutions derived in this work are necessarily restricted, it can be expected that solutions with shapes farther from normal distributions are possible once the fragmentation and aggregation kernels are modified. 

Eor higher values of Kn, we move into the free-molecular (or Epstein) regime where the following aggregation kernel is used ... 

This aggregation kernel and, in general, the phenomenon of particles aggregating because of Brownian motion, is labeled as perikinetic. 

This is often referred to as the orthokinetic aggregation kernel. 

In the case of a constant aggregation kernel it is possible to obtain... 

Aggregation (kernel based) SVM R library (el071) High... 

Here p (in m /s) is the rate corrstant of aggregation or the aggregation kernel which depends on the particle volrrmina u and (y-u). 

The perikinetic aggregation is controlled by diffusiort, and the aggregation kernel of collisions in a monodisperse suspertsion of particles with the size L is... 

The fact that the origin and mechanisms of particle aggregation are principally understood and can even be quantitatively predicted by means of aggregation kernels and stability ratio (cf. Sect. 5.2.3) does not imply that such a prediction is easy to accomplish or that the interfacial properties can be easily derived from stability measurements. Indeed, the stability theory ignores the origin of interparticle forces and is indifferent towards their temporal evolutions. If, however. 

Next, we illustrate the solution of the aggregation problem for the case of the dimensionless aggregation kernel given by x + x. ... 

A quick demonstration may be made of the constant aggregation kernel, which allows exact calculation of the self-similar solution. We let a x, x ) = a, and recognize it to be homogeneous as described by (5.2.14) with m = 0 so that the scaling size h t) is proportional to t. Equation (5.2.16) may be solved by Laplace transform, which is left as an exercise to the reader. However, the reader may more readily verify that for this case by... 

Self-similar behavior from numerical calculations have been shown for the case of Brownian motion by Friedlander and Wang (1966) for which the aggregation kernel is given by... 

---