**Author details**

geometric standard deviation must be less than a certain value. In addition, the accuracy of numerical calculation for capturing fractional moments at an initial stage can be largely increased. In the generalized TEMOM, the closure function shown in Eq. (8) is changed, in which the moment sequence is composed not only by integer moments but also by fractional moments. Thus, higher‐order Taylor series expansion can be achieved for the closure function,

Unlike the TEMOM, the closure of moment governing equations in the log MM is achieved by assuming the PSD to log‐normal size distributions. This method was first investigated by Cohen and Vaughan whose work covers both Brownian and gravitational coagulation [73]. This work forms the basis of the computer code HAARM and aerosol dynamics model MAD [74], and the latter finally became key part in some atmospheric forecast models, such as WRF/ chem. Thanks to works from scientists, including Lee [44, 75], and Pratsinis [14], the log MM becomes one of several main methods of moments today, and especially, it has been applied

Similar to the TEMOM, the log MM also requires to first obtain the moment equations as shown in Eq. (7). Then, the task is to use its own closure function to achieve the closure of moment equations. In this method, the closure function is obtained on basis of the log‐normal size

<sup>9</sup> exp ln <sup>2</sup>

2 0 2

s

are involved, it can be further numerically solved together with Eq. (10).

**3.4. Quadrature method of moments**

<sup>1</sup> ln ln . 9

In theory, once the PBE was converted to forms such as Eq. (7), in which unresolved moments

The QMOM and its variants such as DQMOM are regarded as the mostly used method of moments in the implementation of the PBM [47, 48]. This method achieves great success in that it has no any physical assumptions and has no requirement for the form of dynamics kernels. Thus, this method can deal with all relevant aerosol dynamics problems. Unlike the

æ ö <sup>=</sup> ç ÷ ç ÷ è ø

æ ö <sup>=</sup> ç ÷

2 2

è ø

2 1 g 3/2 1/2 0 2 <sup>=</sup> *<sup>m</sup> <sup>v</sup> m m*

> 2 1

*m m m*

s

*<sup>k</sup> m mv k <sup>k</sup>* (10)

making it much more accurate function.

in computational fluid dynamics software, Fluent.

distribution assumption, which has the following expression, where

0 g

**3.3. Log-normal method of moments**

40 Aerosols - Science and Case Studies

Mingzhou Yu1,2\* and Liu Yueyan1

\*Address all correspondence to: yumingzhou1738@yahoo.com

1 China Jiliang University, Hangzhou, China

2 Key Laboratory of Aerosol Chemistry and Physics, Chinese Academy of Science, Xi'an, China
