Stochastic interacting particle systems and nonlinear kinetic equations

Chemical Engineering Science

Sutcliffe

et al. 2017

This paper uses a network of ideal flow reactors and a detailed population balance model to study the evolution of the size and shape distributions of pigmentary titanium dioxide, formed under industrial synthesis conditions. The industrial reactor has multiple reactant injections, a tubular working zone in which the exothermic reaction is completed, and a cooling zone. A network of continuously stirred tank reactors is used to model variation in composition around the feeds and plug flow reactors with prescribed temperature gradients are used to describe the working and cooling zones. The quality of the industrial product depends on its morphology, and this is influenced by factors including temperature and throughput. In this paper, a multivariate particle model is accommodated using a stochastic method and the particle morphology is characterized in terms of the distributions of primary and aggregate particle diameters, number of primary particles per particle and neck radii of connected primary particles. Increasing temperature or residence time is shown to produce larger particles. Qualitative similarities are highlighted between such findings and previous studies. The throughput studies are also in qualitative agreement with empirical industrial experience. There is scope for extending and improv

“…Theoretical convergence of the population balance equation as N (max) → ∞ has been studied by Eibeck and Wagner (2003); Patterson (2013Patterson ( , 2016 and Wells (2006).…”

Section: Physical Systemmentioning

confidence: 99%

Detailed population balance modelling of TiO 2 synthesis in an industrial reactor

Boje

Chemical Engineering Science

Sutcliffe

et al. 2017

“…doubling [46] and mass flow algorithms [43] and weighted particle methods [47,48,49], and improve efficiency e.g. fictitious jumps and majorant kernels [50], linear process deferment algorithm [51]. A split solution method has been proposed for studying gelation processes, to reduce the chance of stochastic effects forming metastable states [52]: the ODEs for particles smaller than size N 1 are treated deterministically, those for particles of sizes between N 1 and N are treated stochastically, and larger particles are removed (the gelled mass).…”

Section: Introductionmentioning

confidence: 99%

A hybrid particle-number and particle model for efficient solution of population balance equations

Boje

Journal of Computational Physics

Kraft

2019

This work presents a hybrid particle-number and particle model to improve efficiency in solving population balance equations for type spaces spanning spherical and aggregate particles. The particle-number model tracks simpler, spherical particles cheaply by storing only the number of particles with a given one-dimensional internal coordinate, while the particle model allows resolution of the detailed aggregate structure that occurs due to collision and coagulation between particles by storing distinct computational entries for each particle. This approach is exact if primary particles are defined by their monomer count and the particle-number model increments in single monomers. A stochastic method is used to solve the population balance equations for the combined type space. The hybrid method works well for large ensembles (> 2 12 particles) with a detailed particle model, where per

“…[1, 9, 22-29]) and stochastic method (see, e.g., Refs. [11,[30][31][32][33][34][35]). These methods often encompass a trade-off between physical detail and computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

“…In the stochastic method the particle population is represented by an en-semble of stochastic particles and the particle processes are treated probabilistically [36]. The stochastic solution has been proven to converge to the deterministic solution of the PBE [33]. The method easily allows a highly detailed particle description; however, under certain conditions, the computational time [34] and memory requirement [35] can be intractable.…”

Section: Introductionmentioning

confidence: 99%

Extension of moment projection method to the fragmentation process

Yapp

Journal of Computational Physics

et al. 2017

The method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics. Recently, the moment projection method (MPM) was proposed and validated for particle inception, coagulation, growth and, more importantly, shrinkage; here the method is extended to include the fragmentation process. The performance of MPM is tested for 13 different test cases for different fragmentation kernels, fragment distribution functions and initial conditions. Comparisons are made with the quadrature method of moments (QMOM), hybrid method of moments (HMOM) and a high-precision stochastic solution calculated using the established direct simulation algorithm (DSA) and advantages of MPM are drawn.