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 No 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 во 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 10~15m3, и = 1, No = 109/4.189 particle/m3, в0 = 1.53x
103 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.