Assuming spherical particles, equation Equation 19–50 can be written as
 | (19–54) |
where
 | (19–55) |
and 
is the Boltzmann constant.
Following Method II proposed by Frenklach and Harris[135], the coagulation effect on the r -th size moment is defined as
 | (19–56) |
where
 | (19–57) |
Or, by lumping all the terms inside the summation, the coagulation terms for the r -th moment can be rewritten as
 | (19–58) |
where
 | (19–59) |
Note that by definition the function 
is symmetric, that is, 
. The summations in 
can be resolved in terms of the particle size moments, whole and
fractional, positive and negative as
 | (19–60) |
where
 | (19–61) |
The whole-order positive moments are obtained by solving their own transport equations, that is, the equations of size moments. The fractional-order positive moments are determined by logarithmic interpolation between the whole positive moments, that is,
.
The fractional-order negative moments, on the other hand, are computed by logarithmic extrapolation from the whole positive moments
Derivation and evaluation of 
and 
are described in detail in references[134],
[135]. The final forms of the free-molecular coagulation source terms for the
equations of moments are
 | (19–62) |
 | (19–63) |
and
 | (19–64) |
With an assumption of a spherical particle, the collision kernel for the continuum regime as given by Equation 19–51 can be written in terms of particle class i and j. Additionally, while retaining the Cunningham slip correction factor [144], the following expression is written.
 | (19–65) |
The slip correction factor is written as 
= 
= 1 += Kn. The source terms for the moments equations
for collisions in the continuum regime then become
 | (19–66) |
Details of the derivation of coagulation source terms in the continuum regime can be found in the paper by Kazakov and Frenklach[144] .
As mentioned earlier, the collision frequency in the transition regime is approximated by the harmonic mean of the two limiting values[145].
 | (19–67) |
Note that G 1 is always zero. That is, the total number of bulk species molecules in particles (or total particle mass) is not affected by coagulation although the total number of particles decreases.