The application of the method of moments to soot-particle formation was first reported by Frenklach and coworkers [[24] , [7] , [25] ]. This method describes the average properties of a particle population. The method of moments tracks the evolution of an aerosol system by moments of its particle-size distribution function. The use of moments rather than the actual form of the particle-size distribution function overlooks variations among individual particles in the aerosol system. Also, the actual particle-size distribution function cannot be derived from its moments unless an assumption is made regarding the form of the distribution function. Since in many practical applications only the average properties of the aerosol system are sought, the history and properties of individual particles may not be important such that this approximation can still be quite useful. The loss in details of the particle-size distribution function due to the use of method of moments is compensated by computational speed with which the moments are calculated, which reduces the demand for computing resources. Detailed descriptions of the method of moments are reported elsewhere by Frenklach et al. [[24] , [7] , [25] ].

Without making any assumptions about the form of the particle-size distribution function, the method of moments can provide overall properties of a particle system such as number density, total particle volume fraction, total particle surface area density, and average particle size. To express these overall properties in terms of particle-size moments, we first define the particle-size moments.

Given a particle-size distribution function , where represents a measure of particle size, for example, particle mass or particle diameter, the r -th moment of this particle-size distribution function is defined in Equation 8–1 .

(8–1)

The Particle Tracking module uses particle class, which is defined as the number of bulk species molecules in a particle core, as the measure of particle size. Both particle mass, and particle volume, are proportional to particle class. Because particle classes are discrete numbers, the number of class particles can be represented by a discrete function and Equation 8–1 is equivalent to Equation 8–2 .

(8–2)

In the following sections, the summation notation of Equation 8–2 will be used in formulation expressions and derivations.