## INTRODUCTION

An important property of suspension polymerization processes is the particle size distribution (PSD), which controls key aspects of the process and affects the end-use properties of the product. Suspension polymerization processes are generally characterized by PSDs that can vary in time with respect to the mean particle size as well as to the PSD form (i.e., broadness and/or skewness of the distribution, unimodal and/or bimodal character, etc.). The quantitative calculation of the evolution of the PSD presupposes a good knowledge of the droplet/particle breakage and coalescence mechanisms. These mechanisms are coupled with the reaction kinetics, thermodynamics and other microscale phenomena, including mass and heat transfer between the different phases present in the system.

The time evolution of the PSD is commonly obtained from the solution of a population balance equation (PBE), governing the dynamic behaviour of the dispersed liquid monomer droplets that are being polymerized to solid polymer particles. The numerical solution of the dynamic PBE is a notably difficult problem due to a number of numerical complexities and model uncertainties regarding the particle breakage and coalescence mechanisms and are often poorly understood. It commonly requires the discretization of the particle volume domain into a number of discrete elements and the subsequent numerical solution of the resulting system of stiff, nonlinear differential or algebraic/differential equations (DAEs). In the open literature, a number of numerical methods have been reported on the steady-state and dynamic solutions of the general PBE. These include the full discrete method (Hidy, 1965), the method of classes (Marchal et al., 1988; Chatzi and Kiparissides, 1992), the discretized PBE (DPBE) methods (Batterham et al., 1981; Hounslow et al., 1988), the fixed and moving pivot DPBE methods (Kumar and Ramkrishna, 1996a,b), the high-order DPBE methods (Bleck, 1970; Gelbard and Seinfeld, 1980; Sastry and Gaschignard, 1981; Landgrebe and Pratsinis, 1990), the orthogonal collocation on finite elements (OCFE) (Gelbard and Seinfeld, 1979), the Galerkin method (Nicmanis and Hounslow, 1998) and the wavelet-Galerkin method (Chen et al., 1996). In the reviews of Ramkrishna (1985), Dafniotis (1996), and Kumar and Ramkrishna (1996a), the various numerical methods available for solving the general PBE are described in detail. Moreover, extensive comparative studies have been presented in the publications of Kostoglou and Karabelas (1994, 1995), Nicmanis and Hounslow (1996) and in a recent series of papers by Kiparissides and coworkers (Alexopoulos et al., 2004; Alexopoulos and Kiparissides, 2005; Roussos et al., 2005; Meimaroglou et al., 2006). On the basis of the conclusions of these studies, the DPBE method of Litster et al. (1995), the fixed pivot method of Kumar and Ramkrishna (1996a), the Galerkin and the orthogonal collocation on finite-element methods were found to be the most accurate and stable numerical techniques for the numerical solution of the PBE.

The dynamic evolution of the PSD in a particulate process can also be obtained via stochastic Monte Carlo (MC) simulations. Spielman and Levenspiel (1965) were the first to employ an MC approach to study the effect of particle coalescence in a two-phase particulate reactive system in well-mixed reactors. Later, Shah et al. (1977) developed a general MC algorithm for time varying particulate processes. In 1981, Ramkrishna (Ramkrishna, 1981) established the precise mathematical connection between population balances and the MC approach. In MC simulations, the dynamic evolution of the PSD is inferred by the properties of a finite number of sampled particles. Based on the method employed for the determination of the sampling time step, MC simulations can be distinguished into time-driven (Domilovskii et al., 1979; Liffman, 1992; Debry et al., 2003) and event-driven ones (Garcia et al., 1987; Smith and Matsoukas, 1998; Tandon and Rosner, 1999; Kruis et al., 2000; Efendiev and Zachariah, 2002). In regard to the total number of simulated particles, MC methods can be further classified into constant number and constant volume MC methods. A more detailed description of the characteristics of the various MC formulations can be found in a number of publications (Maisels et al., 2004; Meimaroglou et al., 2006; Zhao et al., 2007).

In the present study, two numerical approaches, namely, the fixed pivot technique (FPT) and a stochastic Monte Carlo (MC) algorithm are applied for solving the general PBE, governing the PSD developments in a methyl methacrylate (MMA) free-radical batch suspension polymerization reactor, in terms of the process conditions (i.e., monomer to water volume ratio, temperature, type and concentration of stabilizer, energy input into the system, etc.) and polymerization kinetics. To the best of our knowledge, this is the first time that the two numerical methods (i.e., the FPT and MC) are applied for the calculation of the dynamic evolution of the PSD in a batch suspension polymerization reactor, using a comprehensive model taking into account all the physical and chemical phenomena in the polymerization process. The validity of both numerical methods is examined via a direct comparison of model predictions with experimental measurements on the average particle diameter and the droplet/particle size distributions for both non-reactive and reactive systems.