An extended quadrature method of moments for population balance equations

Abstract

a b s t r a c tPopulation balance equations (PBE) for a number density function (NDF) arise in many applications of aerosol technology. Thus, there has been considerable interest in the development of numerical methods to find solutions to PBE, especially in the context of spatially inhomogeneous systems where moment realizability becomes a significant issue. Quadrature-based moment methods (QBMM) are an important class of methods for which the accuracy of the solution can be improved in a controlled manner by increasing the number of quadrature nodes. However, when a large number of nodes is required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Moreover, oftentimes pointwise values of the NDF are required, but are unavailable with existing QBMM. In this work, a new generation of QBMM is introduced that provides an explicit form for the NDF. This extended quadrature method of moments (EQMOM) approximates the NDF by a sum of non-negative weight functions, which allows unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. Here, we use EQMOM to solve a spatially homogeneous PBE with aggregation, breakage, condensation, and evaporation terms, and compare the results with analytical solutions whenever possible. However, by employing realizable finite-volume methods, the extension of EQMOM to spatially inhomogeneous systems is straightforward.

Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications

“…The EQMOM approach approximates the unknown NDF with a weighted sum of smooth, non-negative kernel density functions δ σ (L, L α ) [36,57]:…”

“…The 2N α quantities W αγ , called secondary weights and abscissae, respectively, are computed using the standard Gaussian quadrature formulae for known orthogonal polynomials to the kernel NDF [36,57]. β α 1 γ 1 α 2 γ 2 is the aggregation kernel for the bubbles of size L α 1 γ 1 and L α 2 γ 2 ; a αγ is the breakage kernel for the bubbles size of L αγ ; and b αγ represents the daughter distribution function.…”

“…Compared with CM, the QMOM can consider a wide range of bubble sizes with a reduced number of equations for the moments of the NDF. However, in some evaporation and combustion problems (Fox et al [35]; Yuan et al [36]), the value of the NDF for null internal coordinates needs to be known, which is not the case if the QMOM method is used. DQMOM solves the equations for weights and abscissae directly.…”

“…DQMOM solves the equations for weights and abscissae directly. Shortcomings related to the conservation of moments affect the DQMOM approach since weights and abscissas are not conserved quantities (Yuan et al [36]). In order to overcome these limitations, Yuan et al [36] introduced the Extended Quadrature Method of Moments (EQMOM), which enables the shape of NDF to be reconstructed from a moment set using continuous kernel density functions instead of Dirac delta functions.…”

Modeling and simulation of mixing in water-in-oil emulsion flow through a valve-like element using a population balance model

“…A great query around polydisperse multiphase flow simulation is to choose the method for solving the population balance equation (PBE). There are different numerical approaches, for example, Monte Carlo stochastic methods (Meimaroglou and Kiparissides, 2007;Irizarry, 2008), weighted residuals methods (WRM) (Subramanianand and Ramkrishna, 1971;Hulburt and Akiyama, 1969;Dorao and Jakobsen, 2006;, methods of classes (MoC) (Lister et al, 1995;Kumar and Ramkrishna, 1996) and methods of moments (MoM) (McGraw, 1997;Marchisio and Fox, 2005;Lage, 2011;Favero and Lage, 2012;Yuan et al, 2012;Petitti et al, 2013). However, there is no method that is accurate, efficient and robust at the same time for the general solution of the PBE.…”