A new discretization of space for the solution of multi-dimensional population balance equations

Nandanwar, Mahendra Naktuji; Kumar, Sanjeev

doi:10.1016/j.ces.2008.01.015

Cited by 32 publications

(14 citation statements)

References 16 publications

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“…To compare particle population in different cells, the flat representation introduced by Chakraborty and Kumar [21] and Nandanwar and Kumar [23] is used. In the flat representation, cells are first indexed from 1 to I 1 × I 2 and then the normalized particle population in kth cell, e.g., N k = f k 1 ,k 2 x k 1 y k 2 , is plotted against its index k, providing a qualitative comparison of the profiles.…”

Section: Resultsmentioning

confidence: 99%

“…In the flat representation, cells are first indexed from 1 to I 1 × I 2 and then the normalized particle population in kth cell, e.g., N k = f k 1 ,k 2 x k 1 y k 2 , is plotted against its index k, providing a qualitative comparison of the profiles. To enhance the comparison, the following weighted error is used to assess the accuracy of the number distribution quantitatively [23]:…”

Section: Resultsmentioning

confidence: 99%

“…Following, Kumar et al [22] extended CAT to triangular grids and concluded the similar observation. Other applications of sectional methods can be found in [23][24][25][26][27]. Although sectional methods provide accurate results for number density as well as different order moments, their formulations are quite complex; rendering these techniques computationally expensive.…”

Section: Introductionmentioning

confidence: 98%

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An improved and efficient finite volume scheme for bivariate aggregation population balance equation

Singh

Kumar

Bück

et al. 2016

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this work, a finite volume scheme for the numerical solution of bivariate pure aggregation population balance equations on non-uniform meshes is derived. The new method has a simple mathematical structure and it provides high accuracy with respect to the number density distribution as well as different moments. The method relies on weights to conserve the total mass of the system. The new method is compared to a recently developed finite volume scheme by Forestier-Coste and Mancini (2012) for some selected benchmark problems. It is shown that the proposed method is not only computationally more efficient but also more accurate than the method by Forestier-Coste and Mancini (2012).

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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An improved and efficient finite volume scheme for bivariate aggregation population balance equation

Singh

Kumar

Bück

et al. 2016

Journal of Computational and Applied Mathematics

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“…Most important representatives of this group are Finite Volume Methods, [5][6][7][8][9] Finite Difference Methods, [10,11] Spectral Methods [12][13][14] and Finite Element Methods. [15] Despite recent advances using e.g., parallel simulation techniques [16] or alternative discretization shemes, [17] application of discretizationbased solution techniques to high dimensional PBEs is limited due to large computational costs. However, particularly in process engineering problems it is sufficient to look at specific integral quantities of the distribution with respect to the inter-nal coordinates, the so called moments.…”

Section: Introductionmentioning

confidence: 99%

An efficient method for calculating the moments of multidimensional growth processes in population balance systems

Dürr

Kienle

2014

Can J Chem Eng

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Multidimensional growth processes play an important role in many fields of applications, such as crystallization processes and cell culture engineering. The numerical solution of the corresponding multivariate population balance equations is quite challenging as standard discretization-based methods are not efficient for high dimensional problems. For this reason, the distribution dynamics is often characterized by its moments. However, the moment dynamics generally cannot be calculated in closed form. Existing approximation techniques cannot be implemented efficiently in the multivariate framework, in particular for high dimensional problems. This contribution presents an alternative methodology for the efficient approximate computation of moments for multidimensional growth processes using Monomial Cubatures and the Method of Characteristics (MOC). The procedure will be shown to reproduce the moments accurately for one-and two-dimensional examples which feature nonlinear growth rates and coupling to a continuous phase, respectively. Furthermore numerical effort and accuracy will be analyzed for a five-dimensional benchmark problem.

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“…Due to the inherent nature of discretised methods to preserve the properties of the distribution, extensive 3 work has been done particularly on the FP method and the CAT, which have been extended later to improve 4 the applicability with increase in number of dimensions [30,37,74,75]. To compare these developments by 5 solving two-dimensional aggregation PBEs, Kumar et al [76] found that the CAT is quite a stable scheme 6 as compared to the FP method and improves the results both for the number density and for the higher 7 moments.…”

mentioning

confidence: 99%

Model-based analysis of high shear wet granulation from batch to continuous processes in pharmaceutical production – A critical review

Kumar

Gernaey

Beer

et al. 2013

European Journal of Pharmaceutics and Biopharmaceutics

109

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The manufacturing of pharmaceutical dosage forms, which has traditionally been a batch-wise process, is now also transformed into a series of continuous operations. Some operations such as tabletting and milling are already performed in continuous mode, while the adaptation towards a complete continuous production line is still hampered by complex steps such as granulation and drying which are considered to be too inflexible to handle potential product change-overs. Granulation is necessary in order to achieve good flowability properties and better control of drug content uniformity. This paper reviews modelling and supporting measurement tools for the high shear wet granulation (HSWG) process, which is an important granulation technique due to the inherent benefits and the suitability of this unit operation for the desired switch to continuous mode. For gaining improved insight of the complete system, particle-level mechanisms are required to be better understood, and linked with an appropriate meso-or macro-scale model. A brief review has been provided to understand the mechanisms of the granulation process at micro or particle-level such as those involving wetting and nucleation, aggregation, breakage and consolidation. Further, population balance modelling (PBM) and the discrete element method (DEM), which are the current state-of-the-art methods for granulation modelling at micro-to meso-scale, are discussed. The DEM approach has a major role to play in future research as it bridges the gap between micro-and meso-scales. Furthermore, interesting developments in the measurement technologies are discussed with a focus towards inline measurements of the granulation process to obtain experimental data which are required for developing good models. Based on the current state of the developments, the review focuses on the twin screw granulator as a device for * Email address: ingmar.nopens@ugent.be, Tel.: +32 (0)9 264 61 96; fax: +32 (0) continuous HSWG and attempts to critically evaluate the current process. As a result, a set of open research questions are identified. These questions need to be answered in the future in order to fill the knowledge gap that currently exists both at micro-and macro-scale, and which is currently limiting the further development of the process to its full potential in pharmaceutical applications.

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A new discretization of space for the solution of multi-dimensional population balance equations

Cited by 32 publications

References 16 publications

An improved and efficient finite volume scheme for bivariate aggregation population balance equation

An improved and efficient finite volume scheme for bivariate aggregation population balance equation

An efficient method for calculating the moments of multidimensional growth processes in population balance systems

Model-based analysis of high shear wet granulation from batch to continuous processes in pharmaceutical production – A critical review

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