Solution of population balance equations using the direct quadrature method of moments
Introduction
The evolution of particulate and other dispersed-phase systems is modeled by using the population balance equation (Randolph & Larson, 1987). The population balance equation (PBE) is a continuity statement written in terms of the internal coordinate (Ramkrishna, 2000) and, in turn, the internal coordinate is usually defined to be a scalar (e.g., particle length or volume) or an array (e.g., particle volume and surface area). The solution of the PBE for particulate systems such as precipitation and crystallization, particle formation in aerosols, and fluidization, usually requires a detailed description of the fluid dynamics and the interactions between mixing and chemical reactions (if present). For this reason computational fluid dynamics (CFD) has become a standard tool for modeling these types of systems. The solution of the PBE inside CFD codes is a very interesting and difficult subject, and in recent years many approaches have been considered.
Among the many available, the moment method introduced by Hulburt and Katz (1964), seems to be a very promising approach for coupling with CFD. In fact, it requires the use of a limited number of scalars (i.e., about four to six lower-order moments), which keeps the problem tractable when applied to complicated multiphase flows. The main issue is the so-called closure problem, related to the difficulties in writing the transport equations of the moments of the particle size distribution (PSD) in terms of lower-order moments (Frenklach & Harris, 1987). A very efficient numerical method for dealing with this closure problem is the quadrature method of moments (QMOM) (McGraw, 1997), which is based on the product-difference (PD) algorithm formulated by Gordon (1968). The QMOM has recently been validated by Marchisio et al. (2003b) and by Marchisio et al. (2003a) and implemented in a CFD code (Marchisio et al., 2003c) showing its great potential. However, two main problems have been detected with the standard implementation: (1) the solution of the transport equations for the moments of the PSD instead of the PBE itself, makes it difficult to treat systems where there is a strong dependency of the dispersed-phase velocity on the internal coordinates (e.g., fluidized bed and bubble column modeling); and (2) in the case of bi-variate PBEs (Rosner et al., 2003) use of QMOM becomes quite complex, as explained by Wright et al. (2001) and by Rosner and Pyykonen (2002) and, especially for CFD applications, existing numerical algorithms are intractable.
In the present work a new approach, namely the direct quadrature method of moments (DQMOM), is formulated, validated and applied. First the method is derived for a PBE with one internal coordinate and then it is extended to two internal coordinates. The extension to multivariate cases and some possible applications to multiphase flows are also discussed.
Section snippets
Closure problem and quadrature approximation
Our principal focus is the solution of the PBE for CFD applications. As already mentioned, one of the most popular approaches for solving the PBE is the moment method, where internal coordinates are integrated out and governing equations are derived in terms of the desired moments. If the moment transform is applied to the PBE, it is possible to derive a continuity equation (i.e., the zero-order moment) and transport equations for mean properties. However, a number of unclosed terms are
DQMOM for Monovariate PBEs
For the monovariate case ( ), QMOM has been shown to predict quite accurately the lower-order moments of the PSD in the case of size-dependent growth by McGraw (1997), aggregation by Barrett and Webb (1998), aggregation and breakage by Marchisio et al. (2003b), by using as few as three nodes (i.e., ) (Marchisio et al., 2003a). We will begin by deriving the DQMOM transport equations for the monovariate case, and then show in Section 4 that DQMOM predicts the same information as QMOM.
The
Equivalence with QMOM
For the monovariate case, DQMOM is mathematically equivalent to QMOM, but has the advantage of being directly soluble without resorting to the PD algorithm. In this section, DQMOM predictions are compared with the QMOM predictions for specific cases in order to demonstrate this equivalence. As already mentioned, QMOM has been tested in our previous work and was found to provide excellent predictions as compared to methods that solve directly for the PSD (Vanni, 2000) even with . In addition,
DQMOM for Bivariate PBEs
For monovariate PBEs we have shown that QMOM and DQMOM yield identical results. Thus, either approach can be used with very similar computational cost. The real utility of DQMOM becomes strikingly apparent for multivariate PBEs for which QMOM is no longer computationally tractable for implementation in a CFD code. Indeed, the DQMOM transport equations for multivariate cases are a straightforward extension of the monovariate case, and the additional computational cost scales like the number of
Bivariate applications of DQMOM
Particulate systems are often studied in terms of two internal coordinates: particle volume ( ) and particle surface area ( ). During aggregation and breakage the particle volume is constant, but the surface area changes. However, during particle growth the volume and surface area both increase. On the other hand, during particle fusion and sintering the volume is constant, but the surface area decreases. The bivariate formulation of this problem is very interesting for several reasons. For
DQMOM for multivariate PBEs
We will close the discussion by noting that the multivariate version of Eq. (50) follows by inspection: where the mixed-moment source term is defined by By choosing an independent set of low-order moments , Eq. (65) generates a linear system that can be used to find the unknown source terms
Conclusions
The direct quadrature method of moments (DQMOM) has been formulated and validated for monovariate and bivariate PBEs. The extension to multivariate PBEs has also been formulated. The DQMOM approach can be used to approximate the solution to a PBE written in terms of properties such as age, length and volume. The method is based on a quadrature approximation that provides the best-possible closure for the moments of the PSD for a given number of nodes N. However, unlike the original quadrature
Acknowledgements
The authors thank the U.S. National Science Foundation (CTS-9985678 and CTS-0112571) and the U.S. Department of Energy (DE-FC07-01ID14087) for support of this work. The authors would also like to thank Dr. Liguang Wang for preparing Fig. 8.
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