Comparison of the accuracy and performance of quadrature-based methods for population balance problems with simultaneous breakage and aggregation

Evaluation of hydrodynamic behavior of the perforated gas distributor of industrial gas phase polymerization reactor using CFD-PBM coupled model

“…Quadrature method of moments proposed by McGraw(1997) and based on the quadrature approximation can be determined using the lower-order moments by resorting to the product-difference (PD) algorithm. QMOM was extensively used to be coupled in CFD framework in gas-solid and gas-liquid FBRs (Silva et al, 2010;Yan et al, 2012). The main drawback of QMOM is that all different particle diameters have the same gas velocity.…”

2D CFD-PBM simulation of hydrodynamic and particle growth in an industrial gas phase fluidized bed polymerization reactor

“…-As drawback, the phase average velocity of the different solid phases must be known to solve the transport equations for the moments. (Claudotte et al, 2010;Silva et al, 2010;Yan et al, 2012a) SQMOM -Using advantage of CM and QMOM and minimize their drawback to the inversion of large sized moment problems as required by QMOM. -The SQMOM can track accuracy of any set of low-order moments with the ability of the shape of the distribution reconstruction.…”

“…A brief comparison on various PBE solving methods is illustrated in Table 1. Table 1 QMOM has been extensively validated for a wide range of problems with different internal coordinates (Buffo et al, 2012;Claudotte et al, 2010;Silva et al, 2010;Yan et al, 2012b). One of the main restrictions of QMOM approach is that the solid phase is represented A c c e p t e d M a n u s c r i p t 6 through the moments of the distributions and the phase average velocity of the different solid phases must be known to solve the transport equations for the moments.…”

“…Several methods have been proposed to solve the PBE, such as discretized population balance (DPB) or class method (CM) (Kumar and Ramkrishna, 1996), standard method of moment (SMM) (Marchisio et al, 2002), the method of moments (MOM) (Diemer and Olson, 2002;Hutton et al, 2012), quadrature method of moments (QMOM) (Buffo et al, 2012;Silva et al, 2010), sectional quadrature method of moments (SQMOM) (Attarakih et al, 2009b), direct quadrature method of moments (DQMOM) (Fan et al, 2004b;Mazzei et al, 2009) and fixed pivot quadrature moments (FPQMOM) (Yao et al, 2015). The details of the mathematical and numerical issues involved in these methods have been discussed by Ramkrishna et al (2000) and Rigopoulos et al (2010).…”

Computational fluid dynamic modeling of fluidized-bed polymerization reactors

“…Silva et al [107] analyzed the different quadrature based methods for solving the PBE. From their work, they found that parallel parent and daughter classes (PPDC) has poor convergence and QMOM, DQMOM had similar accuracy, but the DQMOM was the most efficient.…”