# New Analytical and Numerical Solutions of the Particle Breakup Process

Abdelmalek Hasseine1, *, Mark W. Hlawitschka2, Waid Omar3, Hans-Jörg Bart2
1 Laboratory of Civil Engineering, Hydraulic, Sustainable Development and Environmental, University of Biskra, Biskra, Algeria
2 Chair of Separation Science and Technology, TU Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, Germany
3 Al-Balqa Applied University, Faculty of Engineering Technology, Department of Chemical Engineering P.O. Box 15008, Marka 11134, Jordan

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Laboratory of Civil Engineering, Hydraulic, Sustainable Development and Environmental, University of Biskra, Biskra, Algeria; E-mail: hasseine@yahoo.fr

## Abstract

### Objective:

In this work, we obtained the analytical and approximate solutions of the population balance equations (PBEs) involving the breakup process in batch and continuous flow by applying the Adomian decomposition method and piecewise continuous basis functions, respectively.

### Methods:

The key to the advanced numerical method is to represent the number distribution function of the dispersed phase through the orthogonal Chebyshev basis polynomials. It is a straightforward and effective method that has the advantage of simultaneously giving the distribution and the different required moments. Therefore, it does not require the construction of the distribution from moments computations obtained by the transformation of the initial problem and the lost information.

### Results:

The performance of this numerical approach is evaluated by solving breakup equation and comparison against analytical solutions obtained from the Adomian decomposition method, which generally allows the analysis of this approach.

### Conclusion:

The numerical results obtained by the present numerical method were compared with the new analytical solutions of the PBE. It was found that both piecewise continuous basis functions and analytical solutions have comparable results.

Keywords: Population balance, Decomposition method, Piecewise continuous solution, Bioreactors, Monte Carlo methods, Nanoparticle physics.