Direct quadrature method of moments DQMOM

For fluid particles that continuously coalesce and breakup and where the bubble size distributions have local variations, there is still no generally accepted model available and the existing models are contradictory [20]. A population density model is required to describe the changing bubble and drop size. Usually, it is sufficient to simulate a handful of sizes or use some quadrature model, for example, direct quadrature method of moments (DQMOM) to decrease the number of variables. 

In theory, this model can be used to fix up to three moments of the mixture fraction (e.g., (c), ( 2), and (c3)). In practice, we want to choose the CFD transport equations such that the moments computed from Eqs. (34) and (35) are exactly the same as those found by solving Eqs. (28) and (29). An elegant mathematical procedure for forcing the moments to agree is the direct quadrature method of moments (DQMOM), and is described in detail in Fox (2003). For the two-environment model, the transport equations are... 

To overcome the difficulty of inverting the moment equations, Marchisio and Fox (2005) introduced the direct quadrature method of moments (DQMOM). With this approach, transport equations are derived for the weights and abscissas directly, thereby avoiding the need to invert the moment equations during the course of the CFD simulation. As shown in Marchisio and Fox (2005), the NDF for one variable with moment equations given by Eq. (121) yields two microscopic transport equations of the form... 

The direct quadrature method of moments (DQMOM) begins with a closed1 joint composition PDF transport equation (see Section 6.3). For simplicity, we will consider the high-Reynolds-number form of (6.30) on p. 251 with the IEM mixing model ... 

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. 

This group of methods contains a large variety of solution strategies, but only a few popular techniques like the finite volume (FVM) methods and the direct quadrature method of moments (DQMOM) will be discussed. 

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 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). 

The inhomogeneity of the spray distribution resulting from the atomization process has a strong influence on the spray drying process. The interaction of spray particles may strongly affect the evaporation process. The direct quadrature method of moments (DQMOM) to solve Williams spray equation has been developed by Marchisio and Fox [8]. This method has been applied to sprays by Fox et al. [9] for the first time, and it is suitable to model a poly-disperse droplet/particle flow and the related physical processes. DQMOM has been coupled... 

The direct quadrature method of moments (DQMOM) is used to solve Eq. (9.16), where the joint droplet size, velocity, and temperature distribution,/, are approximated as the sum of the product of weighted Dirac-delta functions of radii, velocities [8, 42], and temperatures. 

The major goal of The direct quadrature method of moments (DQMOM) was to derive transport equations for the weights w and abscissas that can be solved directly and which yield the same moments nk without resorting to the ill-conditioned PD algorithm. Another novel concept imposed is that each phase can be characterized by a weight w and a property vector )i, thus the DQMOM can be employed solving the multi-fluid model describing multi-phase systems. Moreover, since each phase has its own momentum balance in the multi-fluid model, the nodes of the DQMOM quadrature approximation are convected with their own velocities. The DQMOM was proposed by Marchisio and Fox [143] and Fan et al. [53] in order to handle poly-dispersed multi-variate systems. 

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