Extended method of moments

The most important full CLD methods are the extended methods of moments, such as the probabihty generating function method and coarse-graining-based techniques the fixed pivot method and the discrete weighted Galerkin formulation. In what follows, the most important aspects of these methods are highlighted. 

In the extended method of moments, a CLD is reconstructed based on calculated moments, as introduced above. The simplest way is a predefined mass CLD, such as the two-parameter Wesslau distribution (Pladis and Kiparissides, 1988) ... 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

Detonation, Free Volume Theory of Multi-component Fluid Mixtures. The free volume theory of the liq state is extended to multi-component fluid mixts by using the method of moments in the treatment of the order-disorder problem. The results of this extension are given in the article by Z.W. 

Another attempt (which in fact was the first attempt) to extend the method of moments to the thermal properties of harmonic oscillators for large values of u was the y-method (4). This method is based on the first-order approximation of the G( m)-expansion, which can be written in the form... 

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... 

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). 

ECQMOM extended conditional quadramre method of moments... 

Brenner (1980) has explored the subject of solute dispersion in spatially periodic porous media in considerable detail. Brenner s analysis makes use of the method of moments developed by Aris (1956) and later extended by Horn (1971). Carbonell and Whitaker (1983) and Koch et al. (1989) have addressed the same problem using the method of volume averaging, whereby mesoscopic transport coefficients are derived by averaging the basic conservation equations over a single unit cell. Numerical simulations of solute dispersion, based on lattice scale calculations of the Navier-Stokes velocity fields in spatially periodic structures, have also been performed (Eidsath et al., 1983 Edwards et al., 1991 Salles et al., 1993). These simulations are discussed in detail in the Emerging Areas section. 

Analogous equations can be used with any other instantaneous distribution. This relatively easy integration extends the use of instantaneous distributions to transient reactor operation and considerably broadens the use of this powerful technique. When compared with the method of moments, instantaneous distribution allows for the complete prediction of CLD and CCD instead of only averages when contrasted to the full solution of the population balances, the method of instantaneous distributions provides the same information at a much shorter time using a more elegant solution, allows the modeler to analyze the problem with a simple glance at the equation, and can even be implemented on simple commercial spreadsheets for easy calculation. 

We will start with a very simple homopolymerization model that includes only initiation, propagation, transfer to hydrogen, -hydride elimination and imimolecular catalyst deactivation, as depicted in Table 2.4. From our previous discussion of the standard model for polymerization with coordination catalysts, it is known that several steps are not included in Table 2.4. It will be shown, however, that general expressions for population balances and the methods of moments starting with this simplified mechanism can be developed and later they can be extended, rather easily, to include more polymerization steps. 

Because of the variety of coalescence kernels, it is impossible to develop a generalized structure for reduced order models using the method of moments. A special kernel model is assumed in this work. The methodology can be extended to the development of moment models with different kernel structures. The sample kernel model is assumed as ... 

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]. 

---