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

Abstract

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

Modelling soot formation in a benchmark ethylene stagnation flame with a new detailed population balance model

“…The primary particle model is the most detailed particle model discussed in the literature, which describes a nanoparticle as a sequence of primary particles [23][24][25]. As for the numerical methods to solve the PBEs, the three most common methods in the literature include the method of moments [26][27][28][29], discrete sectional methods [30][31][32][33] and stochastic methods [20,25,[34][35][36][37][38][39]. The first two methods are generally restricted to one or two internal particle dimensions.…”

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

“…The performance of the hybrid approach is compared with a single particle type space model in which the discrete ensemble describes the full type space, and primary particles are represented by stochastic entities in the ensemble alongside aggregate particles. The latter has been the standard approach for detailed population balance models to date and is well documented in the existing literature [19,55,17]. Because the detailed particle model describes primary particles as spheres, the two approaches are expected to be equivalent for the same particle processes.…”

“…shared surface area and centre-to-centre distance between particles [11]. Detailed particle models have been used to study synthesis of soot [12,13,14], SiO 2 [15,16], silicon [17] and TiO 2 [18,19,11]. Detailed particle models have been shown to provide important additional information when the particle system is polydisperse or the coagulation and sintering timescales are similar [20].…”

A two-step simulation methodology for modelling stagnation flame synthesised aggregate nanoparticles

“…Stochastic methods allow for the extension of the particle model to include a very detailed description of each particle and allow key physical details to be included, providing a powerful tool to investigate the mechanisms that control particle growth and morphology. Detailed particle models have been used to simulate soot [19], silicon [20], silica [21] and titania [22,23].…”