where v and
Λ are particle volumes and
K is the aggregation kernel. The equation only considers binary aggregation. The first right-hand side term describes the source (or birth) of new particles with a volume
v that result from the aggregation of smaller particles with volumes of
Λ and
v-
Λ. The second term describes the sink (or death) of particles with a volume
v as they together with any other particle of a volume between 0 and infinity form a particle with a volume larger than
v.
The volume of the two aggregating particles, Λ and
v-Λ, have been replaced by the particle sizes
λ and (L
3-
λ3)
1/3. The discretized form of
Equation 3-177 becomes
where Kj,k is the aggregation kernel for two colliding particles belonging to intervals
j and
k and
δij is the Kronecker delta defined as
Furthermore, the set Si contains all index combinations of
j and
k such that two particles of sizes
Lmid,j and
Lmid,k can form a larger particle with a size that lies within size interval
i.
The index λi,j, in
Equation 3-179, represents the interval that ensures that the two colliding particles from intervals i and j are included in the index pair set
Sλi,j.
