Chemical Engineering Science 63 (2008) 3988--3997 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles Mahendra N. Nandanwar, Sanjeev Kumar ∗ Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India ARTICLE INFO ABSTRACT Article history: Received 27 December 2007 Received in revised form 5 April 2008 Accepted 28 April 2008 Available online 6 May 2008 Keywords: Population balance modelling Discretization methods Multi-dimensional population balance equations Modelling and simulation In this work, we show that straight forward extensions of the existing techniques to solve 2-d population balance equations (PBEs) for particle breakup result in very high numerical dispersion, particularly in directions perpendicular to the direction of evolution of population. These extensions also fail to predict formation of particles of uniform composition at steady state for simultaneous breakup and aggregation of particles, starting with particles of both uniform and non-uniform compositions. The straight forward extensions of 1-d techniques preserve 2 n properties of non-pivot particles, which are taken to be number, two masses, and product of masses for the solution of 2-d PBEs. Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112--4125] have recently proposed a new framework of minimal internal consistency of discretiza- tion which requires preservation of only (n + 1) properties. In this work, we combine a new radial grid [proposed in 2008. part I, Chemical Engineering Science 63, 2198] with the above framework to solve 2-d PBEs consisting of terms representing breakup of particles. Numerical dispersion with the use of straight forward extensions arises on account of formation of daughter particles of compositions different from that of the parent particles. The proposed technique eliminates numerical dispersion completely with a radial distribution of grid points and preservation of only three properties: number and two masses. The same features also enable it to correctly capture mixing brought about by aggregation of particles. The proposed technique thus emerges as a powerful and flexible technique, naturally suited to simulate PBE based models incorporating simultaneous breakup and aggregation of particles. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Population balance equations (PBEs), first proposed by Hulburt and Katz (1964), Randolph (1964), and Fredrickson et al. (1967), find applications in both physical and biological processes. Particulate systems in which particles continuously change their identity can be described by PBEs with internal state of particle as continuous variable. Crystallizers, liquid--liquid and gas--liquid contactors, fer- menters, fluidized beds, polymer reactors, etc. (Ramkrishna, 2000) are few of the process equipment which have been simulated us- ing PBEs. In a number of industrial processes (granulation for exam- ple), particles need to be identified with more then one independent (internal) variable. In general, if n internal variables are required to identify a particle uniquely, the population balance equations re- quired to characterize such processes become n-dimensional in na- ture. In many such processes, breakup and aggregation of particles ∗ Corresponding author. Tel.: +91 80 2293 3110; fax: +91 80 2360 8121. E-mail address: sanjeev@chemeng.iisc.ernet.in (S. Kumar). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.04.054 occur simultaneously. A general PBE in n-d space for simultaneously and independently occurring processes of breakup and aggregation of particles is given by n(v, t) t = 1 2  Q(v ′ , v ′′ )n(v ′ , t)n(v ′′ , t) × P(v ′ + v ′′ |v) dv ′ dv ′′ -  ∞ 0 Q(v ′ , v)n(v, t)n(v ′ , t) dv ′ +  (v, v ′ )(v ′ )n(v ′ , t) dv ′ - (v)n(v, t) (1) where v is a vector of n internal attributes of particles, n(v, t) dv is number of particles in range v to v + dv, Q(v, v ′ ) is aggregation frequency, (v) is breakage frequency, (v, v ′ ) dv is average number of particles formed in range v to v + dv when a particle of state v ′ breaks, and P(v ′ , v ′′ |v) is the probability of formation of a particle with attribute v when two particles with attributes v ′ and v ′′ ag- gregate. When a particle can be adequately identified for its role in a process system by just one internal variable, for example its size, the above general equation leads to 1-d PBE, and vector v reduces to