Author's personal copy Direct quadrature spanning tree method for solution of the population balance equations A. Vikhansky CD-adapco, United Kingdom article info Article history: Received 23 June 2012 Received in revised form 20 August 2012 Accepted 22 August 2012 Available online 31 August 2012 Keywords: Population balance Method of moments Coagulation Method of particles Smoluchowski equations Coalescence abstract A new method for solution of the multivariate population balance equations (PBEs) is presented in this work. The method uses M quadrature abscissas and weights to close the PBE. Unlike other similar methods, e.g., the direct quadrature method of moments (DQMoM) and the quadrature method of moments (QMoM), the proposed method neither inverts a badly scaled matrix nor solve a time-consuming eigenvalue problem. The method does not use any a priori discretization of the phase space and as the particles’ size distribution (PSD) widens and shifts toward large characteristic sizes due to coagulation, the abscissas adaptively follow the PSD. The particles are linked by a spanning tree; the tree works as a pipeline redistributing the mass across the system and ensuring that each computational particle accounts for a prescribed fraction of the total mass and therefore the method is given the name: direct quadrature spanning tree method (DQST). & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Equations of population balance provide a general mathematical framework for modelling of particulate systems (Ramkrishna, 2000). Starting from the famous Boltzmann equation (Bird, 1976) equations of population balance span over the wide range of physical, technological and environmental applications such as mixing (Pope, 1985; Spielman & Levenspiel, 1965), liquid/liquid dispersion (Hounslow & Ni, 2004; Tsouris & Tavlarides, 1994; Vikhansky & Kraft, 2004a; Vikhansky, Kraft, Simon, Schmidt, & Bart, 2006), soot formation (Balthasar & Kraft, 2003), combustion (Rigopoulos, 2010), dynamics of atmospheric aerosols (Piskunov & Petrov, 2002; Vikhansky & Kraft, 2004b), breakage and agglomeration of powders (Lin, Lee, & Matsoukas, 2002; Mort, 2005; Smith & Matsoukas, 1998), growth of microbial cell population (Henson, 2003), polymerization (Immanuel & Doyle, 2003) and crystallization (Bermingham, Verheijen, & Kramer, 2003). In the framework of multivariate population balance modelling, each particle is characterized by an N-dimensional vector of internal coordinates x (Ramkrishna, 2000), which might include size, concentrations of admixtures, and porosity. The general form of the equation of population balance reads as @nðt, xÞ @t ¼ Bðnðt, xÞÞÀDðnðt, xÞÞ  Lðnðt, xÞÞ, ð1Þ where nðt, xÞ is the number density of the particles, Bðnðt, xÞÞ and Dðnðt, xÞÞ are the birth and death rates of the particles due to coagulation and breakage, respectively. For a monovariate Smoluchowski equation the formulae for Bðnðt, xÞÞ and Dðnðt, xÞÞ read Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jaerosci Journal of Aerosol Science 0021-8502/$-see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jaerosci.2012.08.004 E-mail address: alexander.vichansky@cd-adapco.com Journal of Aerosol Science 55 (2013) 78–88