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

Mingzhou Yu1,2, Martin Seipenbusch3, Jinghua Yang1, Hanhui Jin 4

• 1 China Jiliang University, Hangzhou 310028, China
• 2 Institute of Earth Environment, Chinese academy of sciences, (null), China
• 3 Institute for Mechanical processes Engineering and Mechanics, Karlsruhe Institute of Technology, (null), Germany
• 4 Institute of Fluid Engineering, Zhejiang University, (null), China

Revised: June 9, 2014
Accepted: July 22, 2014

Yu, M., Seipenbusch, M., Yang, J. and Jin, H. (2014). A New Analytical Solution for Solving the Smoluchowski Equation Due to Nanoparticle Brownian Coagulation for Non-Self-Preserving System. Aerosol Air Qual. Res. 14: 1726-1737. https://doi.org/10.4209/aaqr.2014.03.0059

## HIGHLIGHTS

• A new analytical solution for solving the Smoluchowski equation is first proposed and verified.
• The new solution extends beyond the conventional asymptotic solutions.
• The new solution is suitable for non-self-preserving aerosols.

## 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 non-self-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 = m0m2/m12, where m0, m1 and m2 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.

Keywords: Analytical solution; Taylor-expansion method of moments; Smoluchowski equation; Nanoparticle Brownian coagulation; Free molecular regime 