The Fundamental Aspects of TEMOM Model for Particle Coagulation due to Brownian Motion—Part II: In the Continuum Regime

The fundamental aspects of the Taylor-series expansion method of moment (TEMOM) model proposed to model the aerosol population balance equation due to Brownian coagulation in the continuum regime is shown in this study, such as the choice of the expansion point u, the relationship between asymptotic behavior and analytical solution, and the error of the high-order moment equations. All these analyses will contribute to the buildup of the theoretical system of the TEMOM model.


Introduction
The population balance equations (PBE) are used to describe the evolution process of aerosol particles in a wide range of physical, chemical, and environmental subjects, such as nucleation, coagulation, diffusion, convection, and so on. When the Brownian coagulation plays a dominant role in such cases where aerosol particles at a high concentration are concerned or where suspended particles have evolved for a long time [1], the PBE for a monovariants system can be written as [2] ( , ) in which ( , ) is the number density concentrations of the particles with volume from to + at time , ( , 1 ) is the collision frequency function between particles with volumes and 1 . Because of its strong nonlinear partial integral-differential structure, the direct solution is very complicated and only a limited number of analytical solutions exist for simple coagulation kernel [3]. So several methods are proposed to solve this equation numerically, such as the sectional method (SM) [4], the Monte Carlo method (MCM) [5], and the method of moment (MM) [6]. With lower computational cost compared to the SM and MCM, the moment method has been widely used and become a powerful tool for investigating evolution processes of aerosol particles [7,8].
By multiplying both the sides of (1) then integrating over the entire particle size distribution (PSD) [9], the growth rate of the particle moment can be obtained as follows: where the moment is defined as One main difficulty of the moment method is the closure of the moment equations. There exist several methods to overcome this bottleneck, including but not limited to the quadrature method of moment (QMOM) [10], the direct quadrature method of moment (DQMOM) [11], and the Taylor-series expansion method of moment (TEMOM) [3,12]. It should be pointed out that the TEMOM has no prior assumption for the PSD using the Taylor-series expansion to achieve the closure and is considered as a promising approach to approximate the PBE for its relative simplicity of implementation and high accuracy [13]. Based on TEMOM model, the important information about the PSD, namely, the particle number density, particle mass, and geometric standard deviation, can be obtained for Brownian coagulation over the entire size regimes, and its results have a great agreement with other moment methods [3,[12][13][14][15][16]. But in these works, some fundamental problems are not clarified, for example, why the expansion point is set to be 1 / 0 instead of other formulas; why the Taylor-series are truncated just at the first three terms; and what about the errors estimation of the present TEMOM model. In the present study, mainly as a methodological introduction, we would like to demonstrate the theoretical analysis to answer these questions for Brownian coagulation in the continuum regime.

Brief Review of TEMOM Model and Its Solutions in the Continuum Regime
At the initial time, the particle size maybe small in the free molecule regime. As time advances, the particle volume will grow due to coagulation between particles, and the particle size will transform to the near continuum regime via the transition regime and finally will tend to the continuum regime [17]. Therefore, the characteristic of particle evolution in the continuum regime is important to understand the coagulation mechanism. The collision frequency function in the continuum regime is where the constant 2 = 3 /2 , is the Boltzmann's constant, is temperature, and is gas viscosity. Substituting (4) into (2), a set of moment equations including integral and fractional moments can be obtained. Using a Taylor-series expansion at = = 1 / 0 for , the fractional moments can be approximated by the combination of integral moments as follows: then the closure of the moment equations will be achieved automatically without any prior assumption about the particle size spectrum. The minimum number of moments for closing the equations is the first three-order moments 0 , 1 , and 2 , which represents or are proportional to the total particle number concentration, and the total particle mass concentration, the total scattering light, respectively. According to the results derived by Yu et al. [3] and Xie and Wang [18], the equations can be rearranged as follows: where the dimensionless moment is defined as It is clear that 1 remains constant due to the rigorous mass conservation requirement, and its initial conditions can be noted as 00 , 20 , and 0 = 00 20 / 2 1 . Equations in (6) are a set of ordinary differential equations and can be solved directly. The main process is described briefly as follows. Because of the same structures of the first and the third equations in (6), the following relationship can be obtained: This equation can be dissolved directly as follows: where 1 is the integration constant, and the dimensionless moment can be expressed as Then substituting (10) into the first formula in (6), the relationship between 0 and can be obtained: in which 2 is the integration constant. Then the second moment 2 and dimensionless particle moment can be calculated by (9) and (10), respectively. As time advances, 0 tends to zero due to coagulation; (11) can be simplified with some limit operation as which is consistent with the asymptotic analysis shown by Xie et al. [18,19].
Abstract and Applied Analysis 3

The Choice of Expansion Point
In the TEMOM model, the choice of the expansion point at = 1 / 0 is not arbitrary. Some researchers think that the expansion for the characteristic size should take account of dispersion in the size spectrum and that is best done by using the well-known log-normal based expression = The detailed derivation is provided in the appendix. The comparison of numerical results between (6) and (13) is shown in Figure 1. And the initial conditions are selected as lognormal distribution: 00 = 1; 1 = 1; 20 = 4/3. The results show that the relative errors are small, and the two sets of equations are equivalent approximately. However, this selection results in more complicated moment equations. In fact, for any expression, when it is operated using the Taylor expansion technique, the selection of the expansion point is not unique. In mathematics, we only need to make sure in the targeted range than the Taylor-series expansion is convergent, and the final constructed moment equations are simple in the form.

The High-Order Moment Equations
The accuracy of the TEMOM model largely depends on the truncation errors of Taylor-series expansion. One method to determine the truncated errors is comparing the results of different TEMOM models, for example, the first threemoment model, the first four-moment model, the first fivemoment model, and so forth. Similar to the derivation of the first three moment equations, is expanded at = and truncated at the first four terms as then the closed first four-moment equations can be obtained as follows: The comparison of numerical results among (5), (14), and (15) are also shown in Figure 1. The results show that the relative errors are small, and the three sets of equations can be considered equivalent nearly. Since the complexity of moment equations increases rapidly with the increasing number of reserved items but with little differences in the numerical results, we usually prefer using the first threemoment model proposed originally by Yu et al. [3].

Conclusion
Without a prior assumption for the shape of particle size distribution, the TEMOM has been considered as a promising method to model the aerosol population balance equation. In this study, the fundamental problems of the TEMOM model in the continuum regime due to Brownian coagulation are clarified, such as the choice of the expansion point and the error of the high-order moment equations. It benefits the understanding of PBE and TEMOM model.
(A.1) Then using (3) and (5), the fractional moments can be approximated by the combination of the first three integral moments, for example, the 1/3 is