On a multivariate population balance model to describe the structure and composition of silica nanoparticles

Shekar, Shraddha; Menz, William J.; Smith, A. J.; Kraft, Markus; Wagner, Wolfgang

doi:10.1016/j.compchemeng.2012.04.010

Cited by 34 publications

(52 citation statements)

References 31 publications

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“…These equations are summarized in Table 1. For the binary tree model, the derived properties are analogous to those used in previous studies (Sander et al 2009;Shekar et al 2012a;Menz and Kraft 2013). Of key importance is the surface area:…”

Section: Derived Propertiesmentioning

confidence: 93%

“…Only when coagulation becomes important (relative to sintering) does the binary-tree model require more time, and in this case it requires orders of magnitude more. In the partial-sintering zone, particles containing more than one primary experience "merge" events, where two distinct primaries are combined into one due to sintering (Shekar et al 2012a). These events alter the binary tree structure and as such, for large trees (i.e., particles with many primaries), more time is required to regenerate the tree structure after a node is lost.…”

Section: Computational Resourcesmentioning

confidence: 98%

“…The process through which this is done is described in detail by Shekar et al (2012a). The physical interpretation of Equations (12)-(15) is illustrated in Figure 1.…”

Section: Particle Processesmentioning

confidence: 99%

“…The computational particles may interact with each other through a series of stochastic jump processes which define how the particles change with time. These methods have been applied to a variety of systems (Celnik et al 2007;Sander et al 2009Sander et al , 2011Shekar et al 2012a;Chen et al 2013;Menz and Kraft 2013). However, the main disadvantage of stochastic methods is that it is difficult to include spatial inhomogeneity Patterson and Wagner 2012).…”

Section: Introductionmentioning

confidence: 99%

“…This work investigates the conditions under which use of these models is appropriate; and where their use could represent a significant oversimplification. This is accomplished with a generic stochastic population balance solver (Goodson and Kraft 2002;Morgan et al 2006;Patterson et al 2006b;Celnik et al 2007) which has been mathematically described and numerically investigated elsewhere (Sander et al 2009;Shekar et al 2012a;Menz and Kraft 2013).…”

Section: Introductionmentioning

confidence: 99%

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The Suitability of Particle Models in Capturing Aggregate Structure and Polydispersity

Menz

Kraft

2013

Aerosol Science and Technology

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This work presents the mathematical formulation of particle models commonly used in aerosol dynamics of nanoparticles. A detailed numerical study is conducted in order to understand under which conditions these models differ. The silica model of Shekar et al. (2012b) and silicon model of Menz et al. (2013) taken from the literature are analyzed using three different particle models, demonstrating that substantial errors can be incurred when using a particle model inappropriate for a particular modeling application. These errors can potentially be reflected in fitted parameters; should a parameter estimation procedure be employed with such models. It is concluded that the characteristic coagulation and sintering timescales define the conditions under which the use of a particular model is reasonable.

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Section: Derived Propertiesmentioning

confidence: 93%

Section: Computational Resourcesmentioning

confidence: 98%

“…The process through which this is done is described in detail by Shekar et al (2012a). The physical interpretation of Equations (12)-(15) is illustrated in Figure 1.…”

Section: Particle Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Suitability of Particle Models in Capturing Aggregate Structure and Polydispersity

Menz

Kraft

2013

Aerosol Science and Technology

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Error analysis in stochastic solutions of population balance equations

Zhou

Jiang

Chan

2020

Applied Mathematical Modelling

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

Boje

Akroyd

Sutcliffe

et al. 2017

Chemical Engineering Science

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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

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On a multivariate population balance model to describe the structure and composition of silica nanoparticles

Cited by 34 publications

References 31 publications

The Suitability of Particle Models in Capturing Aggregate Structure and Polydispersity

The Suitability of Particle Models in Capturing Aggregate Structure and Polydispersity

Error analysis in stochastic solutions of population balance equations

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

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