Sectional models divide the particle-size domain into a finite number of sections.The particle size-distribution, that is, number density of particles, within a section can then approximated by some function, such as delta, piecewise polynomial, etc. In principle, particles of various sizes (can) reside in a given section. The complexity and the overall cost of a sectional method are influenced by the choice of approximation for in-section size-distribution. The resulting formulation is referred to as a discretized population balance. The sectional models thus avoid excessive computational requirements of the discrete model and also avoid oversimplification imposed by the moment models.
The classical population balance equation for the discrete spectrum is called the Smoluchowski equation[136]. Using this equation, the net rate of generation of particles of size k (N k ) is written as:
 | (19–11) |
The first term on the right-hand side is the rate of formation of particles of size k by the collision of particles of size i and j. A factor of ½ is introduced since each collision is counted twice. The second term is the rate of loss of particles of size k by collisions with all other particles. The collision frequency function ( β ) depends on the physical model employed for coagulation.
Aggregation and coagulation are the internal processes that modify the particle-size distribution of a particulate system. The driving force behind these processes is collision among the particles. While the term aggregation is used for non-coalescing collisions, in the case of coagulation, it is assumed that the colliding particles coalesce immediately after the collision and a new (spherical) particle is formed. Coagulation can be thought of as starting from an aggregate-particle that eventually forms a fully coalesced particle. Coagulation and aggregation decrease the total number of particles and the total surface area, while increasing the average particle size. For a chemically reacting flow system, the collision frequency function is a function of the sizes of the colliding particles, the flow field, and gas properties such as temperature and pressure, and viscosity.
Except for a very few idealized cases, finding an analytical solution for the Smoluchowski equation is impossible and numerical techniques must be employed. A number of approximate solution techniques have been proposed. These techniques may be broadly classified as either moment models or section models.
Discretized population balance approaches have been developed by quite a few researchers, for example, Gelbard and Seinfeld[137], Marchal et al.[138], and Landgrebe and Pratsinis[139]. These models vary in their choice of discretization (linear, geometric, etc.), their assumption about the shape of the size distribution in each interval, and their choice of average value of properties for each interval. Hounslow [140] showed that some of these methods gave significant errors in their prediction of either the total volume or the total number of particles. Hounslow et al.[140] and Litster et al. [141], developed a discretized-population-balance (DPB) that ensures correct prediction of the total particle number and volume. They use geometric, adjustable discretization as follows:
 | (19–12) |
In the above equation, q is a positive integer and thus the discretization is adjustable, V is the volume/size, and subscript i is the section index. The higher the value of q is, the finer the resolution. Wynn [142] corrected the formulation of Litster et al.[141] and the population balance for aggregation is written as
 | (19–13) |
The above equation guarantees number density and volume (mass) conservation. To capture nucleation, the corresponding rate term is simply added to the appropriate section. Hounslow also proposed a discretization for the growth term that correctly predicts the zeroth, first, and second moment.
Kumar and Ramkrishna[143] proposed a general discretization technique to preserve any two properties (for example, mass and number density) of particle-size distribution. In addition, they more rigorously tackled numerical issues related to growth rate in addition to nucleation and aggregation. They also formulated techniques such as moving pivots and selective refinement. The discretization (without growth) is written as:
 | (19–14) |
 | (19–15) |
 | (19–16) |
 | (19–17) |
In equations Equation 19–15 through Equation 19–17 , X i indicates the pivot size within the size range V i to V i +1, δ jk is the Kronecker delta function, S(v) is the function representing nucleation, and Θ and ? are the powers of any two power-law properties of the PSD that are conserved. (For example, 0 and 1 for number density and volume (mass)). For the cases when the size distribution within a section is far from being uniform, it is possible to consider the pivot X i = X i (t). The rate of movement of the pivot is proportional to the growth rate.
In Ansys Chemkin, the fixed pivot technique by Kumar and Ramkrishna[143] is used considering its computational efficiency. It can be noted that, using this technique, only the particles of certain "representative" size are thought to exist in the system.