Spectral solution of population balance equations using barycentric Lagrange polynomials: Application to stirred tank reactor hydrodynamics

Abstract The population balance equation (PBE) is a fundamental tool for modelling the evolution of particle size distributions in dispersed multiphase systems, such as those involving droplets, bubbles, or solid particles. An accurate and efficient numerical solution of the PBE is essential for understanding and optimizing a wide range of processes in chemical and process engineering. In this study, a spectral method based on barycentric Lagrange polynomial interpolation is developed for solving the PBE. The method is applied to several representative cases, including pure growth, pure breakage, pure coalescence, and combined breakage–coalescence. In each case, numerical results are compared against analytical solutions and against the quadrature method of moments (QMOM) by evaluating the first four moments, demonstrating excellent agreement. The method is further validated using experimental data from a liquid–liquid extraction batch reactor, where simultaneous droplet breakage and coalescence occur. A computational cost study shows that while both the barycentric and standard formulations yield the same high level of accuracy, the barycentric formulation significantly reduces computational time, especially as the number of collocation points increases. These results underscore the effectiveness of the barycentric spectral method as a robust, accurate, and computationally efficient framework for modelling particulate processes in chemical and process engineering applications.