Direct Quadrature Method of Moments (DQMOM)
This is another version of the quadrature-based methods that has been proposed by Marchisio and Fox (Marchisio and Fox 2005) to overcome some disadvantages of QMOM when dealing with (1) multivariate distributions and (2) systems with a strong dependency of the disperse-phase velocity on the internal variables (Strumendo and Arastoopour 2008). DQMOM differs from QMOM because the abscissas and the weights of the quadrature approximation are tracked directly (rather than the moments, as in QMOM). Furthermore, an explicit expression for the particle distribution function is given in terms of a summation of Dirac delta functions. However, this method also suffers from the previously mentioned drawbacks of QMOM and CQMOM due to the use of Dirac delta function approximation.
Finite Size Domain Complete set of Trial Functions Method of Moments (FCMOM)
Strumendo and Arastoopour (Strumendo and Arastoopour 2008, 2010) introduced a new version of method of moments called Finite size domain Complete set of trial functions Method of Moments (FCMOM), which can be regarded as the fourth category of these methods. The method has unique advantages including fast convergence to the exact solution and provision of the solution of PBE in terms of the moments of the distribution and the reconstructed distribution function itself, which makes it distinct from other available approaches. The method has been validated against available analytical solutions or self-similar solutions for the growth, aggregation, dissolution, and simultaneous growth and aggregation cases in both univariate and bivariate homogeneous flows. In addition, FCMOM has been formulated for inhomogeneous systems without breakage or aggregation, and its performance was excellent in all of the cases (Strumendo and Arastoopour 2008, 2010). The applicability of the method in simulations of a complex system (e.g., inhomogeneous with particulate processes) using the multiphase CFD approach was recently studied by Abbasi and Arastoopour (2013). In this chapter, we will discuss the fundamentals of PBE and FCMOM and the coupling of FCMOM with the two-fluid model (TFM).