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# Classical Method of Moments (MOM)

In this approach, the functional form of the property (i.e., size) distribution function is assumed, while the unknown parameters in the distribution function are independent of the internal variables and can be computed as a function of the moments (Barrett and Webb 1998). Although in some cases these methods can provide good results and at the same time are not computationally intensive, the obvious drawbacks are that they are not general and they require that the functional form of the distribution function does not change during the process (Strumendo and Arastoopour 2008).

# Quadrature Method of Moments (QMOM)

In this approach, no explicit assumption is made regarding the form of the size distribution function, and the integrals appearing in the moment equations are computed numerically by means of quadrature formulas. This technique was first presented by McGraw (1997) and later applied to different processes by others (Barrett and Webb 1998; Marchisio et al. 2003a, b). Using this approach, McGraw computed the evolution of the moment equations correctly and efficiently (from a computational point of view). Different from the methods of the first category, this method can be considered general because no explicit assumption is made regarding the functional form of the distribution function. On the other hand, in QMOM, the solution is given in terms of the moments, while the size distribution function disappears from the governing equations. The reconstruction of the distribution function from the moments (Diemer and Olson 2002a, b), if possible, can be rather complex (Strumendo and Arastoopour 2008). In QMOM, reconstruction of the distribution function is achieved using the moment-inversion algorithm through approximation of the distribution function by Dirac delta functions. However, the positivity of the number density function cannot be guaranteed by QMOM (Yuan 2013). To overcome this problem, the conditional quadrature method of moments (CQMOM) has been proposed by Yuan and Fox (2011). In CQMOM, the moment- inversion algorithm is based on one-dimensional (1D) adaptive quadrature of conditional velocity moments and is shown to always yield realizable distribution functions (i.e., nonnegative quadrature weights). CQMOM can be used to compute exact N-point quadratures for multi-valued solutions (also known as the multivariate truncated moment problem) and provides optimal approximations of continuous distributions. In order to control numerical errors arising in volume averaging and spatial transport, an adaptive 1D quadrature algorithm is formulated for use with CQMOM (Yuan and Fox 2011). However, one drawback still exists with CQMOM, which is the inability of the method to provide explicit values for the density function.

To overcome this problem, Yuan et al. (2012) proposed a method called extended quadrature method of moments (EQMOM) by generalizing the quadrature formula with kernel density functions with finite or infinite support parameters. The parameter value is determined by fixing one additional moment. The advantage of this method over the QMOM is that, with one additional moment, it is possible to reconstruct a smooth and nonnegative distribution function that closely reproduces the moment set (Marchisio and Fox 2013). Compared to CQMOM, it uses explicit values for the distribution function. For details of this method, see Yuan (2013).

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