## Primer On Particle Distribution And Dynamics

13.2.1 Particle Properties

A case in which the source particles are of one size allows the description of the mass of a particle in terms of the number of monomers present in it (e.g., using the index i), as well as the number concentration (Ci) of such particles. For more typical situations, the distribution in particle size is usually given in terms of the cumulative particle size spectrum N (s), the number of particles smaller than size s, or the differential size spectrum n(s)

(Note that symbols are also defined in Table 13.1.) Aggregates are not solid spheres that conserve volume when they combine. Theoretical studies31,32 and observations15,33-35 have shown that the density declines as aggregate size increases. This increase is usually described using fractal scaling between mass and length:

where m is the particle mass, r is the particle radius, often identified with the radius of gyration, and Df is the fractal dimension. If volume were conserved, Df would equal 3. Observations on aggregated systems yield values of Df ranging from 1.3 to 2.3.15,33-36

### 13.2.2 Particle Collision Rates

The description of collision rates between particles is the foundation of physical coagulation theory. The rate of collision between two different size particles present at number concentrations of Ci and Cj is

where ¡ij is the particle size-dependent rate parameter known as the coagulation kernel. The three different mechanisms used to describe particle collision rates and their rate constants are Brownian motion, ¡j,Br; shear, ¡y,h and differential sedimentation, ¡y,ds. The total ¡ij is usually assumed to be the sum of these three.20,37 The rectilinear formulations are the simplest expressions for these terms and are calculated

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