Research Projects

Population balance equations (PBEs) are the fundamental mathematical tools that are used to model dispersed phase systems. These systems have entities (e.g., particles, droplets or crystals) distributed according to one or more properties such as size, volume and composition. PBEs are extensively used in diverse range of industrial and natural settings that include various particulate processes such as breakage, aggregation, growth, nucleation etc. It is challenging to solve PBEs analytically or numerically due to their non-local, non-linear and high-dimensional nature. Additionally, the presence of both differential and integral operator makes it even more complex. The method of moments, stochastic methods, section-based methods (finite element, finite volume, fixed pivot etc.), and semi-analytical methods are the traditional solution techniques. They often struggle to deliver high accurate results, preserve physical quantities (number and mass), adapt seamlessly to higher dimensions and be computationally efficient. In contrast, Discontinuous Galerkin (DG) method offers a promising alternative to conventional methods. It is highly accurate, provides geometric flexibility and is well-suited for parallel computation and adaptivity. Despite the mentioned advantages and its potential, DG framework for solving multi-dimensional PBEs with physically relevant kernels remains largely unexplored. In the application of DG method to PBEs, the key challenge is how to handle the non-local terms arising from breakage and aggregation. The challenge multiplies while developing a robust a posteriori error estimates for adaptability and preservation of physical properties in multi-dimensional model. Developing DG framework and addressing these challenges would not only provide accurate solutions to PBEs but also have major practical influence on disciplines including chemical engineering, environmental science and material processing. In this project, we aim to develop DG framework for solving PBEs and construct a posteriori error estimators. We will also build a fully adaptive DG solver which is capable of automatically refining the mesh and polynomial basis while preserving the physical quantities. To check the accuracy of DG method, we will perform benchmarking against exact solutions and section-based methods. In long term, coupling computational fluid dynamics with DG method will enable accurate resolution of both fluid dynamics and property distribution of dispersed phase entities. This approach will be particularly beneficial in crystallization, emulsification, polymerization etc.