**2. Methods for solving the PBE**

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

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‐

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

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,

cost as well as the relative simplicity of implementation [36].

breakage [11].

32 Aerosols - Science and Case Studies

urement and control.

The study of PBE dates back to 1917 when famous polish scientist, Smoluchowski, first established the discrete governing equation for colloid coagulation, that is, Smoluchowski equation [12]; then, Smoluchowski equation was further developed by Müller in its integral‐ differential form, which finally becomes the basis equation of the PBM [51]. Today, the Smoluchowski equation has developed from its original version only accounting for coagula‐ tion to the present version accounting to almost all aerosol dynamics, including external processes such as particle convection and diffusion transport in air, and internal processes such as nucleation, coagulation, condensation, and breakage [52, 53]. Without loss of generality, the general form of a PBE, accounting for both external and internal processes, can be expressed as: where *n*(*v,xi* ,*t*) is the particle number density for particle volume v, location x, and time t; *uj* is the particle velocity; uth is the thermophoresis velocity; DB is the Brownian diffusion coefficient; Gr is the particle surface growth rate; J is the nucleation rate for the critical monomer volume *v*\*; β is coagulation kernel between two particles; and *a* and *b* are parameters accounting for the breakage rate associated with the turbulence shear force. Equation (1) encompasses almost all physicochemical processes of aerosol with a size smaller than approximately 1 μm and therefore is reliable for studying aerosol dynamics. In particular, an inherent advantage is that it can be coupled with the Navier‐Stokes equation.

$$\begin{aligned} \frac{\partial n(\mathbf{v}, \mathbf{x}\_i, t)}{\partial t} + \frac{\partial \left( u\_j n(\mathbf{v}, \mathbf{x}\_j, t) \right)}{\partial \mathbf{x}\_j} + \frac{\partial \left( (u\_{0i})\_j n \left( \mathbf{v}, \mathbf{x}\_j, t \right) \right)}{\partial \mathbf{x}\_j} \\ = \frac{\partial}{\partial \mathbf{x}\_j} \left( D\_B \frac{\partial n(\mathbf{v}, \mathbf{x}\_j, t)}{\partial \mathbf{x}\_j} \right) + \frac{\partial \left( G\_r n(\mathbf{v}, \mathbf{x}\_i, t) \right)}{\partial \mathbf{v}} + J \left( \mathbf{v}^\*, \mathbf{x}\_i, t \right) \delta \left( \mathbf{v} - \mathbf{v}^\* \right) \end{aligned} \tag{1}$$
 
$$\begin{aligned} + \frac{1}{2} \sum\_{\mathbf{v}'}^{\mathbf{v}} \beta \left( \mathbf{v} - \mathbf{v}', \mathbf{v}' \right) n \left( \mathbf{v} - \mathbf{v}', t \right) n \left( \mathbf{v}', \mathbf{x}\_i, t \right) \mathrm{d}\mathbf{v}' \\ - n \left( \mathbf{v}, t \right) \int\_{\Gamma}^{\infty} \beta \left( \mathbf{v}, \mathbf{v}' \right) n \left( \mathbf{v}', \mathbf{x}\_i, t \right) \mathrm{d}\mathbf{v}' + \int\_{\mathbf{v}}^{\infty} \mathbf{a} \left( \mathbf{v}' \right) b \left( \mathbf{v}' \right) n \left( \mathbf{v}', t \right) \mathrm{d}\mathbf{v}' \\ - a \left( \mathbf{v} \right) n \left( \mathbf{v}, t \right) + \dots, \end{aligned} \tag{1}$$

It needs to note here in Eq. (1), the coordinate (i.e., particle volume *v*) of particle number concentration function might be other quantity, such as particle surface area or charge number, and coordinate number might be more, depending on the specific requirement of study. Even only the particle volume is selected as coordinate of particle number concentration function, the direct numerical solution of Eq. (1) is intractable for most applications due to the extreme large number of independent variables, and it should be further modified using suitable mathematical techniques. To solve the Eq. (1) numerically or analytically, several schemes by different researchers, including the method of moments, sectional method, and stochastic particle method, have been proposed and evaluated. Both advantages and disadvantages of these three methods have been compared in many review articles [36, 53]. In case the coagu‐ lation kernel is simplified with homogeneous assumption, the analytical solution of PBE can be achieved [54–57]. The analytical method has been used to study nanoparticle dynamics in an experimental chamber [58].

Because of relative simplicity of implementation and low computational cost, the method of moments has been extensively used to solve the PBE. In the application of this method, the fractal moment variables inevitably appear in the transfer from the PBE to moment governing equation, which needs to be further treated with different techniques. Due to the low require‐ ment for computational cost, in the last decade, the combination of method of moments and Computational Fluid Dynamics (CFD) technique has been an emerging research field; the task is to investigate the temporal and spatial evolution of nanoparticles under turbulent condi‐ tions.

The information of particle size distribution is lost due to the integral in the transfer from the PBE to moment governing equations in the method of moments, and thus this method is unable to trace the evolution of particle size distribution (PSD) if the reconstruct technique of PSD is not implemented [59]. The sectional method, which divides the PBE into a set of size classes, overcomes the limit of method of moments in tracing the PSD. This method was usually used as an exact solution to validate the method of moments [60] and also widely applied in studies on the evolution of particle size distribution at engineering conditions due to the different dynamical processes including coagulation, condensation, gas‐particle conversion, etc.

An alternative to sectional method and method of moments for solving the PBE is stochastic particle method (or Monte Carlo method) [31–35]. The application of this method, however, is limited because of low efficiency. This method has advantage to capture the evolution of particle size distribution physically and also can be used to obtain some key kernels for aerosol dynamics such as coagulation. Up to now, there have been lots of versions of Monte Carlo methods for solving the PBE, but the coupling between the Computational Fluid Dynamics (CFD) and Monte Carlo method is still limited.
