An adaptive direct quadrature method of moment for population balance equations

Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) for population balance equations (PBE) have been shown to be accurate and computationally efficient for Isotropic systems or when used with computational fluid dynamics (CFD) codes. However, numerical difficulties can arise for cases where there is a large variation of moments or where two abscissas have similar values. Previous study has demonstrated that introducing an appropriate adjustable factor to the QMOM, the numerical difficulty can be alleviated in some cases with an additional benefit of improving numerical accuracy or significantly reducing computational time. However, no reliable method is available to determine the optimal adjustable factor that allows the highest possible accuracy to be obtained while maintaining computational efficiency. In this work, an adjustable factor is introduced to the DQMOM and a novel procedure is proposed that enables the optimal adjustable factor to be found for a given problem. A number of test cases including pure aggregation, pure breakage, pure growth, aggregation and breakage, aggregation and growth have been carried out. Our results show that the proposed method is capable of either improving numerical accuracy or reducing the computational time for a variety of problems. The novelty of this method is that the optimal adjustable factor is determined based on the actual particle size distribution at a given time, thereby reducing error accumulation. It also allows other factor-searching procedures to be incorporated in a straightforward manner without influencing the adaptive DQMOM (ADQMOM)itself.