Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation

Highlights: • New approaches for solving coagulation-fragmentation equation are developed. • Both schemes are easy to code and robust to implement on any kind of grids. • Both schemes are easy to extend for higher dimensional problems. • Analytical solutions are derived for 2D & 3D coagulation-fragmentation equations. • New approaches show better accuracy and efficiency than the existing method. This study focuses on development of two approaches based on finite volume schemes for solving both one-dimensional and multidimensional nonlinear simultaneous coagulation-fragmentation population balance equations (PBEs). Existing finite volume schemes and sectional methods such as fixed pivot technique and cell average technique have many issues related to accuracy and efficiency. To resolve these challenges, two finite volume schemes are developed and compared with the cell average technique along with the exact solutions. The new schemes have features such as simpler mathematical formulations, easy to code and robust to apply on nonuniform grids. The numerical testing shows that both new finite volume schemes compute the number density functions and their corresponding integral moments with higher precision on a coarse grid by consuming lesser CPU time. In addition, both schemes are extended to approximate generalized simultaneous coagulation-fragmentation problems and retains the numerical accuracy and efficiency. For the higher dimensional PBEs (2D and 3D), the investigation and verification of the numerical schemes is done by deriving new exact integral moments for various combinations of coagulation kernels, selection functions and fragmentation kernels.