# Homogeneous Aggregation

Convergence and accuracy of FCMOM for different homogeneous aggregations has been studied by Strumendo and Arastoopour (2008). For a homogeneous aggregation/coalescence case starting from a Gaussian-like distribution,

where * N_{o}* and are initial number of droplets and initial average droplet size, respectively, and the aggregation/coalescence process is governed by the sum aggregation kernel

where * в_{о}* is the aggregation constant.

Scott (1968) has provided an analytical solution for Eq. (4.2) giving the dimensionless size distribution function at any time:

The aggregation kernel defines the net rate of aggregation/coalescence and depends on:

- 1. Frequency of collisions between droplets of size ? and droplets of size n
- 2. Efficiency of aggregation (i.e., the probability of a droplet of size ? coalescing with a particle of size n)

In Eq. (4.35), ? is dimensionless droplet size defined as ?/?_{0}, and f is dimensionless size distribution function defined asN??r. т is related to dimensionless time, T, by,

and T is defined

For this case, an approach similar to that described in Sect. 4.5.1 was utilized to implement the governing equations [i.e., Eq. (4.18) without the convective term] for the finite Smoluchowski equation. Initial moments were calculated from Eq. (4.33), while the model parameters were adopted from Scott (1968) with ?_{0} = 4.189 x 10~^{15}m^{3}, * и =* 1,

*10*

**N**_{o}=^{9}/4.189

**particle/m**^{3}, в

_{0}

*1.53x*

**=**10^{3} 1/s, and ?_{max} was set to 13?_{0}.

Figure 4.4 shows that the simulation results obtained using 12 moments at t = * 450 s* (т = 0.5) are in excellent agreement with the analytical solution.