Simultaneous Quadrature Method of Moments for the Solution of Population Balance Equations, Using a Differential Algebraic Equation Framework

The quadrature method of moments (QMOM) is a recent technique of solving population balance equations for particle dynamics simulation. In this paper, an alternative solution for the QMOM is described and thoroughly tested, which is based on the formulation and simultaneous Solution of a semi-explicit differential algebraic equation (DAE) system. The DAE system consists of the ordinary differential equations resulting from the application of the method of moments, as well as a system of nonlinear algebraic equations derived by applying the quadrature theory for the approximation of the moments. It is shown that the proposed approach provides an efficient procedure for evolving the quadrature abscissas and weights from the QMOM. The Jacobian matrix of the DAE system is provided analytically to make the solution more robust. The DAE-QMOM. method is compared to the well-established method for solving QMOM based on the product difference (PD) algorithm. The numerical results are compared to the analytical solutions in the case of breakage, aggregation, growth, and nucleation mechanisms. Excellent agreements are found on the moment evolution predicted by both methods. However, the DAE-QMOM method is found to be more accurate and robust than the PD-QMOM in some cases. Additionally, the DAE-QMOM is also capable of providing the solution significantly faster than the PD-QMOM method.