In the sectional method implemented in Particle Tracking, the "sections" along the particle-size coordinate represent a range of masses and therefore volumes since the density of the bulk material is assumed to be constant. In this way, the section to which an aggregate belongs is determined by the volume-based diameter of the corresponding completely coalesced sphere. With the two assumptions mentioned on 1 , the sectional method then tracks the number density of aggregates and the number density of primary particles in each section.
The collision frequency kernels for the aggregates are given by Equation 19–50 , Equation 19–51 , and Equation 19–52 in Particle Coagulation . Along with the dependence of the collision kernel on the collision diameter as the collisions diameter evolves through time, the particle dynamics of aggregates is different from that of completely coalesced spherical particles. For pure coalescence, meaning that the aggregation model is not used, the collision diameter is fixed for each representative aggregate from a section; it is the volume-averaged diameter.
The aggregation model is implemented according to the discretized population balance of Kumar and Ramkrishna [154]. Thus, the aggregation rate of the aggregates is given by
 | (19–162) |
 | (19–163) |
In equations Equation 19–162 and Equation 19–163 , m j is the mass of aggregate of class j, T is the gas temperature, μ is the gas viscosity, k B is the Boltzmann constant, and superscripts FM and CN indicate free-molecular and continuum regimes. The collision kernel, which is valid in all three collision regimes (that is, free-molecular, continuum, and transition) is obtained by Fuch’s interpolation function and also depends on the collision diameter of the aggregates. It is written as
 | (19–164) |
In the above equation, G jk is the so-called transition parameter, ϕjk is the speed parameter, and D f is the diffusivity of the aggregate. All of these depend on the collision diameter.
Thus, along with the dependence of the collision kernel on the collision diameter given by Equation 19–162 , Equation 19–163 , and Equation 19–164 , as the collision diameter evolved through time, the particle dynamics of aggregates is different from that of completely coalesced spherical particles. For pure coalescence, meaning that the aggregation model is not used, the collision diameter is fixed for each representative aggregate from a section; it is the volume-averaged diameter.
The aggregation model is implemented according to the discretized population balance of Kumar and Ramkrishna [155]. Thus, the aggregation rate of the aggregates is given by
 | (19–165) |
 | (19–166) |
 | (19–167) |
 | (19–168) |
In Equation 19–162
through Equation 19–172
, 
indicates the pivot size within the size range 
to 
; 
is the Kronecker delta function; and 
and 
are the powers of any two power-law properties of the particle size
distribution (PSD) that are conserved. In this implementation, aggregate number density and
volume (mass) are conserved. The collision frequency kernel 
depends on the collision diameters of representative aggregates from
sections 
and 
. The primary particle aggregation rate is given by
 | (19–169) |
Similar to the aggregates, the distribution factor 
is determined such that the primary particle size is conserved.
Thus,
 | (19–170) |
Using Equation 19–125 and noting that the number of aggregates does not change due to sintering, we can write
 | (19–171) |
 | (19–172) |
The total generation rates of aggregates and primary particles are given by the addition of the aggregation rate and sintering rate.
The parameters for the characteristic fusion time are available for only a few particle materials. Examples where the parameter data are available are typically metals or metal-oxides such as titania (TiO2), silicon (Si), silica (SiO2), etc. For other particulate systems of practical importance, especially carbon soot, such data are not typically available. This difficulty hinders usage of the aggregation model presented above. However, as noted in Particle Aggregation Model , the expression given by Equation 19–126 indicates a power law dependence on primary particle diameter and the power exponent is typically 4. Such strong dependence indicates that aggregates with small primary particles coalesce quickly while those with large primary particles take a long time to fuse. Although the definitions of small and large are relative, it means that the size (diameter) of primary particles in an aggregate can be limiting, depending on the rate of collisions. The primary particles above this limiting diameter are very slow to fuse. Indeed, for soot particles obtained from flame experiments, the limiting size of primary particles is on the order of 20 to 30 nm.
In addition to difficulty in obtaining reliable sintering data, two equations are solved per section when the complete aggregation model is used in contrast to one equation per section when aggregation is not modeled. Moreover, the system of equations is stiffer due to the sintering rate. When combined, this may make the total computational cost substantially higher with the complete aggregation model.
Considering the above issues, one way to include the effect of aggregation without invoking the full aggregation model is to make use of the limiting primary particle size concept. The aggregates with primary particles less than the limiting value can be thought of as completely coalesced spheres while those with larger primary particles are pure aggregates in which primary particles of the limiting size are in point contact with each other. The corresponding two-parameter model is called the simple aggregation model and is available with the sectional method in the Particle Tracking feature. The two user-specified parameters for this model are the limiting diameter for primary particle and the fractal dimension.
In the simple aggregation model for the sectional method, the sections that have a representative aggregate diameter (that is, the volume-averaged diameter) that is less than the user-specified fixed primary-particle diameter will have aggregates that are coalesced spheres while the sections with larger representative diameter will have pure aggregates. Thus, the number of primary particles per aggregate in any section is known a priori. Consequently, the collision diameter and surface area of aggregates are also known a priori.
Although the simple aggregation model is less rigorous, it is computationally efficient and captures the essential physics of the aggregation process. Its usage is recommended especially for soot systems. Due to the computational expense of the complete aggregation model, only the simplified model is available for flame simulations.