Analytical solutions of the particle breakage equation by the Adomian decomposition and the variational iteration methods

• The breakage process for batch and continuous flow systems is studied.
• The Adomian decomposition method (ADM) and Variational iteration method (VIM) are used to solve the PBE.
• Approximation of particle breakage mechanisms with assumed functional forms for breakage frequencies.
• The ADM and VIM propose analytical solutions of the PBE for the breakage.

The breakage in batch and continuous systems has attained high interest in chemical engineering and granulation from a process and from a product quality perspective. The wet granule breakage process in a high shear mixer will influence and may control the final granule size distribution. In this work, we developed analytical solutions of the particle breakage using the population balance equation (PBEs) in batch and continuous flow systems. To allow explicit solutions, we approximate particle breakage mechanisms with assumed functional forms for breakage frequencies. This new framework for solving (PBEs) for batch and continuous flow systems proposed in this work uses the Adomian decomposition method (ADM) and the variational iteration method (VIM). These semi-analytical methods overcome the crucial difficulties of numerical discretization and stability that often characterize previous solutions in of the PBEs. The results obtained in all cases show that the predicted particle size distributions converge exactly in a continuous form to that of the analytical solutions using the two methods.