Extended quadrature method of moments

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... 

Yuan, C., Laurent, F. Fox, R. O. 2012 An extended quadrature method of moments for population balance equations. Journal of Aerosol Science 51, 1-23. 

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 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] ... 

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... 

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]. 

If the realizability condition in Eq. (8.52) is satisfied, found from Eq. (8.53) is guaranteed to be realizable. Using the method described above for the PBE (Eq. (8.35)), it is straightforward to extend Eq. (8.53) to second-order time-stepping. The extension to multiple velocity components v in multiple spatial dimensions is a bit more complicated. As described in Yuan Fox (2011) and in Section B.3 of Appendix B, when the CQMOM is used to constmct the multivariate quadratures all permutations must be used in a consistent manner in order to get the correct kinetic energy fiuxes. Nevertheless, for each permutation of the CQMOM, the basic time-stepping formula in Eq. (8.53) is used to update the transported moment set by modifying the definition of K , to include the multivariate moments in the optimal-moment set. Readers interested in more details on multidimensional free transport should consult Yuan Eox (2011) and Section B.3 of Appendix B. 

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