Part I: Dynamic evolution of the particle size distribution in particulate processes undergoing combined particle growth and aggregation
Introduction
An important property of many particulate processes is the particle size distribution (PSD) that controls key aspects of the process and affects the end-use properties of the product. Particulate processes are generally characterized by particle size distributions that can strongly vary in time with respect to the mean particle size as well as to the PSD form (i.e., broadness or/and skewness of the distribution, unimodal or/and bimodal character, etc.). For reactive particulate processes, the quantitative calculation of the evolution of the PSD presupposes a good knowledge of the particle nucleation, growth, and aggregation mechanisms. These mechanisms are usually coupled to the reaction kinetics, thermodynamics (e.g., solubility of a reactant in the particulate phase), and other micro-scale phenomena including mass- and heat-transfer between the different phases present in the system.
Particle nucleation often results in the formation of a large number of small particles within a short period of time. Particle growth due to chemical reaction results in an increase of the mean particle size and can affect the form of the PSD, particularly in size-dependent particle growth processes. Finally, particle aggregation and breakage can result in significant changes in the form of the PSD. In the present study, the effects of particle growth and particle aggregation mechanisms on the time evolution of the PSD are thoroughly analyzed. The numerical difficulties arising in the solution of the dynamic population balance equation in the presence of a particle nucleation mechanism are discussed in a follow-up paper.
The time evolution of the PSD is commonly obtained from the solution of the general population balance equation (PBE) governing the dynamic behavior of a particulate process (Hulburt and Katz, 1964; Ramkrishna, 2000). There is a large number of publications dealing with the application of the PBE in various particulate processes, including aerosol dynamics (Seinfeld, 1979, Landgrebe and Pratsinis, 1990, Friedlander, 2000, Scott, 1968), granulation of solids (Adetayo et al., 1995), crystallization (Randolph and Larson, 1988, Hounslow, 1990), liquid-liquid dispersions (Kronberger et al., 1995), microbial cell cultures (Ramkrishna, 1979, Fredrickson et al., 1967), polymerization (Min and Ray, 1974, Sundberg, 1979, Chen and Wu, 1988; Richards et al., 1989, Alvarez et al., 1994, Kiparissides et al., 1994, Yiannoulakis et al., 2001), fluidized bed reactors (Sweet et al., 1987).
In general, the numerical solution of the dynamic PBE for a particulate process, especially for a reactive one, is a notably difficult problem due to both numerical complexities and model uncertainties regarding the particle growth and aggregation mechanisms that are often poorly understood. Usually, the numerical solution of the PBE requires the discretization of the particle volume domain into a number of discrete elements that results in a system of stiff, nonlinear differential or algebraic/differential equations that is subsequently integrated numerically. The general application of the PBE requires the calculation of the aggregation and growth functions. For a reactive particulate systems these functions may depend on bulk and particle concentrations, which change with time. When the reactor kinetics and the particle stability are coupled to the PSD, the PBE and the kinetic equations must be solved simultaneously. The solution of the resulting system of DAEs will be more difficult due to the increased problem size, but also due to a possible increase in the numerical stiffness and index of the system of DAEs.
In the open literature, several numerical methods have been developed for solving the steady-state or dynamic PBE. These include the full discrete method (Hidy, 1965), the method of classes (Marchal et al., 1988, Chatzi and Kiparissides, 1992), the discretized PBE (Batterham et al., 1981, Hounslow et al., 1988), the fixed and moving pivot discretized PBE methods (Kumar and Ramkrishna, 1996a, Kumar and Ramkrishna, 1996b), the high order discretized PBE methods (Bleck, 1970, Gelbard and Seinfeld, 1980, Sastry and Gaschignard, 1981, Landgrebe and Pratsinis, 1990), the orthogonal collocation on finite elements (Gelbard and Seinfeld, 1979), the Galerkin method (Tsang and Rao, 1989, 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 PBE are described in detail. Moreover, in three publications by Kostoglou and Karabelas, 1994, Kostoglou and Karabelas, 1995, and Nicmanis and Hounslow (1996), comparative studies on the different numerical methods are presented. Based on the conclusions of these studies, the discretized PBE method of Litster et al. (1995), the 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.
Despite the plethora of published papers on the numerical solution of the PBE, the selection of the most appropriate numerical method for the calculation of the time evolution of the PSD in a particulate process, undergoing simultaneous particle growth and aggregation, is not always easy. In fact, the majority of the published papers refer to a limited range of variation of the respective particle growth and aggregation rates. As a result, the unrestricted application of a numerical method to the solution of a specific PBE problem cannot be guaranteed. The fact that a large number of different numerical methods have been employed for solving the general PBE, underlines the inherent difficulties in obtaining an accurate and stable numerical solution. Common problems related to the numerical solution of the PBE include the inaccurate calculation of the PSD for highly aggregating processes, numerical instabilities for growth-dominated processes, increased stiffness of the system of DAEs for processes involving rapid particle nucleation, and domain errors for high-order aggregation kernels (Dafniotis, 1996; Kumar and Ramkrishna, 1996a). More specifically, the inclusion of both particle growth and aggregation mechanisms in the PBE gives rise to a markedly difficult to solve numerical problem as the growth term imparts the PBE with a hyperbolic nature. For the solution of PBEs characterized by a particle growth dominating term, moving grid methods have been proposed (Tsang and Rao, 1989, Kumar and Ramkrishna, 1997). Although moving and adaptive grid methods are generally attractive for solving particle growth dominated problems, they may not be the optimum choice for aggregation dominated cases. Moreover, for problems involving a fixed-volume source (e.g., particle nucleation or particle inflow) special care is required for the application of the moving and adaptive grid methods.
In what follows, the general population balance equation is first stated and the two numerical methods applied for its solution are described. In the third section of the paper, a systematic comparison of the numerically calculated PSDs to available analytical solutions is carried out for particulate processes characterized by a simple particle aggregation kernel (i.e., constant or sum) and a zero or linear particle volume growth rate model. The accuracy and stability of the numerical solution (i.e., with respect to the resolution of the full PSD or/and its respective moments) is examined over a wide range of variation of the dimensionless aggregation and growth times. Thus, the conditions of applicability of each numerical method (i.e., DPBE and OCFE) are established. Subsequently, the two numerical methods are applied to particulate processes characterized by a constant particle aggregation kernel and a nonlinear particle growth rate model, exhibiting either a 1/3 or a 2/3 dependence on the particle volume. The effect of the volume domain discretization on the performance (i.e., stability, accuracy) of the numerical solution is examined thoroughly. Finally, the use of a moment-weighting method or the addition of an artificial diffusion term to the original PBE is investigated in order to improve the performance of the numerical method (e.g., correction of the calculated PSD, elimination of undesired oscillations in the solution).
Section snippets
Numerical solution of the population balance equation
To follow the dynamic evolution of the PSD in a particulate process, a population balance approach is commonly employed. The distribution of the particulates (e.g., solid particles, liquid droplets, microbial cells, etc.) is considered to be continuous over the volume variable and is commonly described by a number density function, , that represents the number of particles per unit volume in the differential volume size range (V to ). For a dynamic particulate system undergoing
Results and discussion
Detailed numerical simulations were carried out for a number of dynamic particulate processes, undergoing particle aggregation or/and growth. Several particle aggregation rate functions (e.g., including a constant, a first order sum aggregation kernel and a zero-order Brownian aggregation kernel) were considered. The particle growth was assumed to follow the general power-law model,where is a rate constant and the power-law exponent “a” takes values in the range of . Thus, for
Conclusions
The OCFE method was found to be significantly more accurate than the DPBE in resolving the distribution in aggregation-dominated problems. For combined growth and aggregation problems, the DPBE method produced accurate results only for small dimensionless growth times. Moreover, the distribution peak was consistently under-predicted despite the fairly accurate calculation of the distribution moments. The observed deviations in the calculated DPBE moments from the respective analytical solution
Notation
exponent in the particle growth law (Eq. (12)) diameter, m artificial diffusion coefficient, number fraction distribution in element “”. volume fraction distribution in element “”. growth rate constant, /s growth rate function, /s “th” dimensionless moment of the number density function. number density function at node in element , number density function, diameter-based number density function, average number density function in element ,
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