**1. Introduction**

Aerosol particles usually refer to fine particles in air whose size is smaller than micrometer [1]. This type of particles can be found in a wide range of industrial and natural phenomena such as nanoparticle synthesis [2, 3], aerosol sciences and air pollution [4–7], contamination control in the microelectronics and pharmaceuticals industries [8], and diesel particulate formation [9]. The dynamics characteristics of size of these particles spans from free molecular size regime much less than Kolmogorov length scale to inertial range. Due to Brownian motion, aerosol varies greatly in the degree of stability, even though the aerosol flow convection transport is not involved. Although this type of multiphase system widely emerges in industries and our surroundings, some key issues including the conversion from gas to particle and the subsequent particle growth affected by the surroundings remain unresolved [10]. Unlike some common

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

techniques used in the multiphase flow community, the task of the study on aerosol dynamics is to grasp the interaction between the dispersed particles and the carrier phase and also to obtain the fundamentals of internal processes such as nucleation, condensation, coagulation, and breakage [11].

If the particle formation and subsequent growth are studied theoretically, the theory should cover from simple kinetics theory to continuum theory. The Stokes' law needs to be modified as applied in this field because aerosol particles are usually smaller than molecular mean free path [1]. Some common methodologies in multiphase flow community, such as Euler‐Lagrange and Euler‐Euler, are unsuitable to be used in this field. In fact, in the first decades of last century, studies on dynamics of micro‐ and nanoscale particles are always the focus of physical science [12]. With further requirements of modern industrial nanoparticle synthesis and atmospheric observation, the study on interaction between fine particles and the surrounding becomes more and more important, and thus, it needs to combine the methodology in modern multiphase flow theory with aerosol dynamics to resolve complicated micro‐ and nanoscale particle multiphase problems [13]. The object of these studies is to capture the property, behavior, and physical principle of aerosol particles in air and further apply this knowledge to their meas‐ urement and control.

Besides convection and diffusion transport, the evolution of aerosol particle dynamics arises mainly from internal mechanisms, including homogeneous or heterogeneous nucleation, condensation, coagulation, and breakage. Among these internal mechanisms, coagulation occurs most commonly, but it is yet the most difficult to be treated from the viewpoint of mathematics, because the correlation among all the particles must be concerned separately [14]. Since the pioneering work of Smoluchowski in 1917, the mean‐field theory has been introduced in aerosol collision problems and has been basis for numerous theoretical appli‐ cations. These applications include the derivation of coagulation kernel under different mechanisms, the solution of the governing equations within the Smoluchowski mean‐field theory, and the application of Smoluchowski mean‐field theoretical model to predict the behavior of aerosol multiphase system. In fact, within the Smoluchowski mean‐field theory, some important phenomena such as self‐preserving distribution [15–17], and gelation or asymptotic behavior [18–21] have been thoroughly studied using state‐of‐the‐art technologies such as the method of moments (MOM) [17, 19–23], sectional method (SM) [24–30], stochastic particle method [31–35], and Monte Carlo method [36–42]. The method of moments is more widely used than other methods due to that it requires the least requirement for computational cost as well as the relative simplicity of implementation [36].

Although the method of moments has become a powerful tool for investigating aerosol chemistry physical processes since it was first used in aerosol community by Hulburt and Katz [22], the closure of governing equations in terms of kth moment is not easy to achieve. Up to now, there have been five main techniques proposed to achieve the closure of moment equations, namely the Taylor series expansion MOM (TEMOM) [43], the predefined size distributed method such as log‐normal MOM (log MM) [44, 45] and Gamma MOM [46], Gaussian quadrature MOM (QMOM) and its variants [47, 48], pth‐order polynomial MOM [49], and MOM with interpolative closure (MOMIC) [23]. In recent years, the PBM scheme, which couples the PBE with the Computational Fluid Dynamics (CFD), has been increasingly received attention, and accordingly, it is possible to simultaneously capture the details of the fluid flow and transport, the evolution of the particle size distribution and complex chemical kinetics. For any techniques, consuming computational cost has to be concerned as the solution of Navier‐Stokes equations is involved. Thus, the efficiency of solving the PBE in the imple‐ mentation of the PBM is another important issue besides the accuracy of numerical calculation. Although the QMOM and its variants are the most used scheme for solving the PBE today, it shows disadvantage in efficiency as compared to the TEMOM and log MM. It needs to note here in the log MM, and the log‐normal size distribution has to be employed in the construction of the model, which inevitably weakens its reliability and capability for solving the PBE. It is necessary to construct a new approach with respect to moment equation, which is easy to implement with low computational cost like the log MM and has not the prior requirement for particle size spectrum like the QMOM, to adapt to the requirement of modern complicated particulate industries. In particular, there needs a suitable technique capable of providing explicit moment governing equations for further asymptotic analysis for the PBE [50]. In order to accomplish it, a new promising method of moments based on Taylor series expansion technique has been proposed and successfully applied to resolve some aerosol engineering problems [43]. Relative to the QMOM and log MM, the TEMOM has advantage to give explicit moment governing equations, making it suitable as basis equation for further analytical solution or asymptotic solution of the PBE.

This chapter is outlined as follows: In Section 2, the review of the PBE as well as its solution is presented, in which three main techniques applied for solving the PBE, namely the meth‐ od of moments, sectional method, and Monte Carlo method, are briefly presented; the meth‐ od of moments is highlighted in Section 3, where three predominated methods of moments, including the QMOM, log MM, and TEMOM, are presented separately.
