On the solution of population balance equations by discretization—II. A moving pivot technique

Kumar, Sanjeev; Ramkrishna, Doraiswami

doi:10.1016/0009-2509(95)00355-x

Cited by 418 publications

(213 citation statements)

References 11 publications

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“…In the absence of a growth mechanism, Kumar and Ramkrishna [22] showed that a similar system can also be constructed for pure aggregation and breakage kinetics. Here, the pivots are allowed to move within the bin which they characterize, giving an indication of how the number density distribution varies across the bin as a result of aggregation and breakage events.…”

Section: Review Of Moving and Adaptive Grid Discretization Schemes Fomentioning

confidence: 99%

An explicit adaptive grid approach for the numerical solution of the population balance equation

Sewerin

Rigopoulos

2017

Chemical Engineering Science

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Section: Review Of Moving and Adaptive Grid Discretization Schemes Fomentioning

confidence: 99%

An explicit adaptive grid approach for the numerical solution of the population balance equation

Sewerin

Rigopoulos

2017

Chemical Engineering Science

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“…All these methods are, however, constrained with respect to the choice of grid, as they are based on exploiting the properties of the geometric progression. More recent works by Litster et al [138] and Kumar and Ramkrishna ([124], [125]) have succeeded in extending the concept of DPBs to finer grids; the latter also offers the advantage of conserving any two chosen moments. In [126] a method particularly suited to problems with growth involving discontinuities was shown, based on the method of characteristics.…”

Section: Discretisation Methodsmentioning

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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“…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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On the solution of population balance equations by discretization—II. A moving pivot technique

Cited by 418 publications

References 11 publications

An explicit adaptive grid approach for the numerical solution of the population balance equation

An explicit adaptive grid approach for the numerical solution of the population balance equation

Population balance modelling of polydispersed particles in reactive flows

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

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