Quadrature method of moments QMOM

The quadrature method of moments (QMOM) is a presumed PDF approach that determines the unknown parameters by forcing the lower-order moments of the presumed PDF to agree with the moment transport equations (McGraw 1997 Barrett and Webb 1998 Marchisio et al. 2003a Marchisio et al. 2003b). As with the multi-environment presumed PDF method discussed in Section 5.10, the form of the presumed PDF is... 

This part of the chapter is devoted to a few of the popular numerical discretization schemes used to solve the population balance equation for the (fluid) particle size distribution. In this section we discuss the method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the discrete method, the chzss method, the multi-group method, and the least squares method. 

In the quadrature method of moments (QMOM) developed by McGraw [131], for the description of sulfuric acid-water aerosol dynamics (growth), a certain type of quadrature function approximations are introduced to approximate the evolution of the integrals determining the moments. Marchisio et al [122, 123] extended the QMOM for the application to aggregation-breakage processes. For the solution of crystallization and precipitation kernels the size distribution function is expressed using an expansion in delta functions [122, 123] ... 

The quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM) were introduced in Chapter 3 as equivalent methods for solving a homogeneous GPBE. In fact, the DQMOM was derived by Marchisio Fox (2005) primarily for the purpose of solving spatially inhomogeneous multivariate moment-transport equations. Unlike for the univariate case, where the moment-inversion algorithm is uniquely defined for a given set of moments, the QMOM in the multivariate case is much... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

In the quadrature method of moment (QMOM) a few moments of the number distribution function/ are tracked in time directly, just as for the standard MOM, but in this approach the requirement of exact closure is replaced by an approximate closure condition that allows the method to be applied under a much broader range of conditions. This method was first proposed by McGraw [151] for modeling aerosol dynamics and has later been extended to aggregation and breakage processes in crystallization by Marchisio et al. [141, 142]. 

The sectional quadrature method of moments (SQMOM) represents a hybrid between the sectional method and the quadrature method of moments (QMOM) and was proposed by Attarakih et al. [10] in order to solve the PBE for poly-dispersed systems. The novel idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as breakage and coalescence. 

Another method representing an extension of the QMOM method has obtained increasing attention for particulate systems during the last years. According to Fan et al [46], one of the main limitations of the QMOM is that the solid phase is represented through the moments of the distribution, thus the phase-average velocity of the different solid phases must be used to solve the transport equations for the moments. Thus, in order to use this method in the context of multiphase flows, it is necessary to extend QMOM to handle cases where each particle size is convected by its own velocity. In order to address these issues, a direct quadrature method of moments (DQMOM) has... 

The value of cr is determined by fixing one additional moment (a total of 21V + 1 moments, i.e. an odd number of moments). In order to distinguish between moment methods using Eq. (3.81) and those using Eq. (3.82), we will refer to the former as the quadrature moment of moments (QMOM) and the latter as the extended quadrature method of moments (EQMOM) (Yuan et al, 2012). The principal advantage of using the EQMOM instead of the QMOM is that with one additional moment it is possible to reconstruct a smooth, nonnegative NDF that exactly reproduces the first 21V + 1 moments. However, there are several... 

Lage, P. L. C. 2011 On the representation of QMOM as a weighted-residual method -the dual-quadrature method of generalized moments. Computers Chemical Engineering 35(11), 2186-2203. 

Below a Matlab script implementing the tensor-product QMOM for a simple bivariate case described in this section is reported. The required inputs are the number of nodes for the first (Nl) and for the second (N2) internal coordinates. Since in the formulation described above the moments used for the calculation of the quadrature approximation are defined by the method itself, no exponent matrix is needed. The moments used are passed though a matrix variable m, whose elements are defined by two indices. The first one indicates the order of the moments with respect to the first internal coordinates (index 1 for moment 0, index 2 for moment order 1, etc.), whereas the second one is for the order of the moments with respect to the second internal coordinate. The final matrix is very similar to that reported in Table 3.8. The script returns the quadrature approximation in the usual form the weights are stored in the weight vector w of size N = Mi M2, whereas the nodes are stored in a matrix with two rows (corresponding to the first and second internal coordinate) and M = M1M2 columns (corresponding to the different nodes). 

At this point, one teehnieai question remains how wiii the muitivariate quadratures be defined If a brute-foree QMOM were used, there would be only one ehoiee sinee ail of the transported moments are reproduced with such methods. However, sinee brute-foree QMOM are numerieaiiy iii-eonditioned, a more likely choice will be to use the CQMOM (see Section 3.2.3 for details). As described in Yuan Fox (2011), when the CQMOM is used for spatial transport eare must be taken when choosing the permutation. For example, with V (i.e. a 3D phase spaee) there are F = 6 permutations of the CQMOM, but only two of them give the eorreet spatial flux in the x direction (i.e. permutations 123 and 132). In order to accommodate the two CQMOM permutations, the update proeedure deseribed above is modified as follows ... 

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