Solution of population balance equations using the direct quadrature method of moments

Marchisio, Daniele; Fox, Rodney O.

doi:10.1016/j.jaerosci.2004.07.009

Cited by 684 publications

(587 citation statements)

References 23 publications

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“…This could be done by solving for a higher number of moments (Marchisio and Fox, 2005) or by using a full discretised size distribution (Rigopoulos, 2007). As well as providing a more accurate prediction of the size distribution the latter may also enable a greater insight into the effect of turbulence on aerosol processes via changes in the size distribution.…”

Section: Discussionmentioning

confidence: 99%

“…If an assumed shape is used to represent the size distribution then it can be characterised by a number of moments (Pratsinis, 1988;Marchisio and Fox, 2005). A greater number of moments will mean that more information about the size distribution is contained in the calculation.…”

Section: Aerosol Size Distributionmentioning

confidence: 99%

“…The problem faced in using a full size distribution in a non-stochastic approach is multiplied by the number of realisations used in Monte-Carlo simulations typical of solution methods for PDF equation. There is, therefore, an advantage in using methods which assume a distribution shape can be characterised by a relatively small number of moments (see Section 2.2 and Pratsinis (1988); Frenklach (2002); Marchisio and Fox (2005) for more detail). If this is done then the object will be to find the joint scalar -size distribution moment PDF in the flow, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Aerosol nucleation and growth in a turbulent jet using the Stochastic Fields method

Garmory

Mastorakos

2008

Chemical Engineering Science

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The Stochastic Fields transported PDF method for turbulent reacting flows has been used to model the nucleation and growth of Dibutyl Phthalate particles in a hot, turbulent jet in a colder background for which experimental data is available.The aerosol population is modelled using an assumed log-normal size distribution.It has been found that neglecting the effect of turbulent fluctuations leads to the peak particle concentration being predicted too close to the jet and the concentration downstream underpredicted. However, this effect was small compared to that of adjusting modelled surface tension. Only by adjusting this was it possible to reproduce correctly the downstream evolution of particle number found in experiment.Particle mass mean diameter was significantly underpredicted at the centre of the jet, which may be due to the inability of log-normal size distribution to capture the distribution in detail. Taking account of turbulent fluctuations leads to increased mean particle size at the edge of the plume. The extent of this increase is strongly dependent on the choice of micromixing timescale.

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Section: Discussionmentioning

confidence: 99%

Section: Aerosol Size Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Aerosol nucleation and growth in a turbulent jet using the Stochastic Fields method

Garmory

Mastorakos

2008

Chemical Engineering Science

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“…While many variants have been proposed, the closure models may be classified into three approaches: the method of moments with interpolative closure (MOMIC) [18], the direct quadrature method of moments (DQMOM) [19] and the hybrid method of moments (HMOM) [20]. MOMIC is the most widely used approach with the equation for the highest moment closed by logarithmic polynomial interpolations and is adopted in the present study [21].…”

Section: Introductionmentioning

confidence: 99%

A computational study of ethylene–air sooting flames: Effects of large polycyclic aromatic hydrocarbons

Selvaraj

Arias

Lee

et al. 2016

Combustion and Flame

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An updated reduced gas-phase kinetic mechanism was developed and integrated with aerosol models to predict soot formation characteristics in ethylene nonpremixed and premixed flames. A primary objective is to investigate the sensitivity of the soot formation to various chemical pathways for large polycyclic aromatic hydrocarbons (PAH). The gas-phase chemical mechanism adopted the KAUST-Aramco PAH Mech 1.0, which utilized the AramcoMech 1.3 for gas-phase reactions validated for up to C2 fuels. In addition, PAH species up to coronene (C 24 H 12 or A7) were included to describe the detailed formation pathways of soot precursors. In this study, the detailed chemical mechanism was reduced from 397 to 99 species using directed relation graph with expert knowledge (DRG-X) and sensitivity analysis. The method of moments with interpolative closure (MOMIC) was employed for the soot aerosol model. Counterflow nonpremixed flames at low strain rate sooting conditions were considered, for which the sensitivity of soot formation characteristics to different nucleation pathways were investigated. Premixed flame experiment data at different equivalence ratios were also used for validation. The findings show that higher PAH concentrations result in a higher soot nucleation rate, and that the total soot volume and average size of the particles are predicted in good agreement with experimental results. Subsequently, the effects of different pathways, with respect to pyrene-or coronene-based nucleation models, on the net soot formation rate were analyzed. It was found that the nucleation processes (i.e., soot inception) are sensitive to the choice of PAH precursors, and consideration of higher PAH species beyond pyrene is critical for accurate prediction of the overall soot formation.

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“…Therefore, various numerical algorithms have been developed for solving PB equations such as method of moment [114][115][116][117], method of characteristics [108,[118][119][120][121], Monte Carlo techniques [122,123], and discretization methods including finite element technique [119,124,125], cell average methods [107], hierarchical solution strategy based on multilevel discretization [126], method of classes [82,95], fixed and moving pivot method [127,128], and finite difference/volume methods [90,119,[129][130][131]. Table 1 summarises these numerical solution methods with the further reviews below.…”

Section: Efficient Solution Of Pb Equationsmentioning

confidence: 99%

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

2016

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Crystal morphology is known to be of great importance to the end-use properties of the crystal product and affect the down-stream processing such as in filtration and drying, but was previously regarded as too challenging to achieve automatic closed-loop control. As a consequence, previous work has focused on control the crystal size distribution (CSD) where the size of a crystal is often defined as the diameter of a sphere that has the same volume of the crystal. This paper reviews very promising new advances made in recent years in morphological population balance models for modelling and simulation of crystal shape distribution (CShD), measurement and estimation of crystal facet growth kinetics, as well as in 2D and 3D imaging for on-line characterisation of crystal morphology and CShD. A framework is presented integrating various components in order to achieve the ultimate objective of model-based closed-loop control of CShD. The knowledge gaps and challenges that require further research are also identified.

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Solution of population balance equations using the direct quadrature method of moments

Cited by 684 publications

References 23 publications

Aerosol nucleation and growth in a turbulent jet using the Stochastic Fields method

Aerosol nucleation and growth in a turbulent jet using the Stochastic Fields method

A computational study of ethylene–air sooting flames: Effects of large polycyclic aromatic hydrocarbons

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

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