A new numerical approach for the simulation of the growth of inorganic nanoparticles

Morgan, Neal; Wells, Clive G.; Goodson, Mike; Kraft, Markus; Wagner, Wolfgang

doi:10.1016/j.jcp.2005.04.027

Cited by 67 publications

(45 citation statements)

References 26 publications

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“…In the past 30 years, numerous scholars have attempted to solve the PBE. Three main numerical methods have been proposed and evaluated: the method of moments (Hulburt and Katz, 1964;Lee et al, 1984;McGraw, 1997;Frenklach, 2002;Marchisio et al, 2003;Park, 2003;Yu et al, 2008a), the sectional method (Landgrebe and Pratsinis, 1990), and the stochastic particle method or Monte Carlo method (Kraft, 2005;Morgan et al, 2006;Kruis et al, 2012;Zhao and Zheng, 2013). Both the advantages and disadvantages of these three methods were compared by Kraft (2005).…”

Section: Introductionmentioning

confidence: 99%

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

Zhang

Liu

et al. 2015

Aerosol Air Qual. Res.

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We verified the effect of the Taylor expansion order on the accuracy of the Taylor-series expansion method of moment (TEMOM) model in both the free molecular and continuum-slip regimes. The ordinary differential equations for moments with fourth-order Taylor expansion were first derived for fractal-like agglomerates, which were further compared to the existing TEMOM model with third-order Taylor expansion. We confirmed that the TEMOM model with a fourth-order Taylor expansion is less accurate than that with a third-order Taylor expansion. Moreover, the scope of application of the TEMOM model with a fourth-order Taylor expansion is limited. The existing TEMOM model with a third-order Taylor expansion was verified as the most reliable model for solving population balance equations for agglomerates undergoing Brownian coagulation.

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Section: Introductionmentioning

confidence: 99%

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

Zhang

Liu

et al. 2015

Aerosol Air Qual. Res.

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“…After Eqs. (16) and ( 17 ) are introduced into Eq. (11) , the errors of the proposed analytical solution for non-self-preserving aerosols are largely reduced, and their precision can be ensured.…”

Section: Non-self-preserving Aerosolsmentioning

confidence: 99%

“…However, the PBE associated with a particle-size-dependent coagulation kernel cannot be precisely solved. Various methods for solving the PBE have been proposed in the last century, including the method of moments [6][7][8][9][10][11] , sectional method [12][13][14][15] , and Monte Carlo method [16][17][18][19] , with almost all methods belonging to the numerical solution. For the numerical solution, time-consuming iterative algorithms, such as the 4th-order Runge-Kutta method, must be performed.…”

Section: Introductionmentioning

confidence: 99%

A new analytical solution for agglomerate growth undergoing Brownian coagulation

Liu

Jin

et al. 2016

Applied Mathematical Modelling

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a b s t r a c tWe proposed a new analytical solution for a population balance equation for fractal-like agglomerates. The new analytical solution applies to agglomerates with any mass fractal dimensions. Two well-known numerical methods, including the Taylor-series expansion method of moments and the quadrature method of moments, were selected as references. The reliability of the analytical solution with three mass fractal dimensions of 1.0, 2.0, and 3.0 in the continuum-slip regime was verified. The accuracy of the new analytical solution is affected by both the Knudsen number and the mass fractal dimension. The new analytical solution can be further improved in accuracy by introducing a correction factor to the originally derived mathematical formula. The new analytical solution was finally confirmed to possess potential for replacing the numerical solution in the continuum-slip regime.

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“…The study was conducted on a single processor, however, and advances in computer processing power since then as well as the availability of moderate-size clusters at a reasonable price means that such simulations are now well within the capacity of modern computing systems. Morgan et al [163] applied a stochastic PBE to simulate the formation of silica, titania and iron oxide particles in a 1-D laminar premixed flame environment. Rosner and Pyykonen [212] coupled a two-dimensional PBE resolved by a QMOM approach with CFD to model the formation of alumina in a counterflow diffusion flame reactor.…”

Section: Soot Formation and Nanoparticle Synthesismentioning

confidence: 99%

Population balance modelling of polydispersed particles in reactive flows

Rigopoulos

2010

Progress in Energy and Combustion Science

136

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Polydispersed particles in reactive flows is a wide subject area encompassing a range of dispersed flows with particles, droplets or bubbles that are created, transported and possibly interact within a reactive flow environment -typical examples include soot formation, aerosols, precipitation and spray combustion. One way to treat such problems is to employ as a starting point the Newtonian equations of motion written in a Lagrangian framework for each individual particle and either solve them directly or derive probabilistic equations for the particle positions (in the case of turbulent flow). Another way is inherently statistical and begins by postulating a distribution of particles over the distributed properties, as well as space and time, the transport equation for this distribution being the core of this approach. This transport equation, usually referred to as population balance equation (PBE) or general dynamic equation (GDE), was initially developed and investigated mainly in the context of spatially homogeneous systems. In the recent years, a growth of research activity has seen this approach being applied to a variety of flow problems such as sooting flames and turbulent precipitation, but significant issues regarding its appropriate coupling with CFD pertain, especially in the case of turbulent flow. The objective of this review is to examine this body of research from a unified perspective, the potential and limits of the PBE approach to flow problems, its links with Lagrangian and multifluid approaches and the numerical methods employed for its solution. Particular emphasis is given to turbulent flows, where the extension of the PBE approach is met with challenging issues. Finally, applications including reactive precipitation, soot formation, nanoparticle synthesis, sprays, bubbles and coal burning are being reviewed from the PBE perspective. It is shown that popualtion balance methods have been applied to these fields in varying degrees of detail, and future prospects are discussed.

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A new numerical approach for the simulation of the growth of inorganic nanoparticles

Cited by 67 publications

References 26 publications

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

Verification of Expansion Orders of the Taylor-Series Expansion Method of Moment Model for Solving Population Balance Equations

A new analytical solution for agglomerate growth undergoing Brownian coagulation

Population balance modelling of polydispersed particles in reactive flows

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