A New Analytical Solution for Solving the Smoluchowski Equation Due to Nanoparticle Brownian Coagulation for Non-Self-Preserving System

Abstract

The Smoluchowski equation has become a fundamental equation in nanoparticle processes since it was proposed in 1917, whereas the achievement of its analytical solution remains a challenging issue. In this work, a new analytical solution, which is absolutely different from the conventional asymptotic solutions, is first proposed and verified for nonself-preserving nanoparticle systems in the free molecular regime. The Smoluchowski equation is first converted to the form of moment ordinary differential equations by the performance of Taylor expansion method of moments and subsequently resolved by the separate variable technique. In the derivative, a novel variable, g = m 0 m 2 /m 1 2 , where m 0 , m 1 and m 2 are the first three moments, is first revealed which can be treated as constant. Three specific models are proposed, two with a constant g (an Analytical Model with Constant g (AMC), and a Modified Analytical Model with Constant g (MAMC)), and another with varying g (a finite Analytical Model with Varying g (AMV)). The AMC model yields significant errors, while its modified version, i.e., the MAMC model, is able to produce highly reliable results. The AMV is verified to have the capability to solve the Smoluchowski equation with the same precision as the numerical method, but an iterative procedure has to be employed in the calculation.

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

“…To date, the TEMOM has been widely used in research on aerosol dynamics and nanoparticle formation (Yu and Lin, 2010;Lin et al, 2012;Xie and He, 2013). In addition, numerous recent studies have applied the analytical solution for the PBE based on the TEMOM governing equation (Chen et al, 2014;Xie, 2014;Yu et al, 2014b. The TEMOM has no prior requirement regarding the particle size spectrum, which is similar to the quadrature method of moments (McGraw, 1997), and the number of governing moment equations is equal to the order of the Taylor-series expansion.…”

The asymptotic stability of the Taylor-series expansion method of moment model for Brownian coagulation

“…This approach makes no prior assumption on the shape of particle size distribution (PSD), and provides high computational efficiency and acceptable accuracy compared with other numerical method. Moreover, the form of TEMOM model for Brownian coagulation is simple enough to be solved asymptotically and analytically [7][8][9][10][11]. These solutions have also been verified by the quadrature method of moments (QMOM) numerically [12].…”

Steady-state solutions for particles undergoing Brownian coagulation and breakage by the TEMOM model

“…With the relative simple construction, the Taylor-series expansion method of moments (TEMOM) has a great advantage on the analytical and asymptotic analysis of particle size evolution [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In this paper, we will develop a TEMOM model considering Brownian coagulation both in the free molecule and continuum regime in coincidence with an equal size multiple breakage process, and the corresponding steady-state solutions are obtained.…”