Population Balance Equations for Coupled Particle Growth and Nucleation Processes: Homotopy Analysis

Population balance equations (PBEs) are crucial for characterizing particle property distributions in various industrial and scientific processes, but their inherent nonlinearity when modeling combined particle growth and nucleation presents significant solving challenges, underscoring the importance of developing effective analytical techniques. This study presents an analytical solution to the population balance equations (PBEs) characterizing simultaneous particle growth and nucleation processes. The solution is obtained through the mathematical transformation of the coupled nonlinear PBE system into a series of linear sub-problems, enabling systematic analysis of particle property evolution. The analytical framework provides exact solutions for specific parameter ranges and demonstrates convergent behavior for broader applications. The research methodology involves transforming the complex PBE into a series of simpler linear sub-problems using HAM, employing an exponential function as the initial solution guess, and then calculating these sub-solutions to obtain an overall approximation. The results demonstrate a strong relationship with analytical solutions, showing negligible differences primarily at decimal particle sizes, and reveal that tuning convergence parameters can systematically improve accuracy, reducing deviations by up to 95% compared to initial approximations. This study concludes that the homotopy analysis method’s capacity for controlled convergence enhancement across wide parameter ranges makes it particularly suitable for addressing broader, previously intractable particulate process modeling problems, suggesting its potential application to additional mechanisms such as aggregation and crystal breakage in future PBE research.