Solution of population balance equations by MWR

Singh, P.; Ramkrishna, Doraiswami

doi:10.1016/0098-1354(77)80004-5

Cited by 61 publications

(23 citation statements)

References 8 publications

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“…7 Specific weighting functions have been used already in the earliest attempts to solve the PBE. See for example the work by Singh and Ramkrishna 26 or Chiu and Christofides 27,28 and references, therein, for problem tailored weighting functions. Parameter dependent weighting functions are treated in depth by Deuflhard and Wulkow 29 in the context of MWR.…”

Section: à1 ð1þmentioning

confidence: 98%

Generalization and numerical investigation of QMOM

et al. 2006

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in Wiley InterScience (www.interscience.wiley.com).A generalized framework is developed for the quadrature method of moments (QMOM), which is a solution method for population balance models. It further evaluates the applicability of this method to industrial suspension crystallization processes. The framework is based on the concepts of generalized moments and coordinate transformations, which have been used already in earlier solution approaches. It is shown how existing approaches to QMOM are derived from the suggested unified framework. Thus, similarities and differences between the various QMOM methods are uncovered. Further, potential error sources involved in the different approaches to QMOM are discussed and assessed by means of a series of test cases. The test cases are selected to be challenging. The error in the QMOM solution is evaluated by comparison to an adaptive, error controlled solution of the population balance. The behavior of a range of different QMOM formulations is analyzed by means of numerical quadrature, dynamic simulation, as well as numerical continuation and bifurcation analysis. As a result of this detailed analysis, some general limitations of the method are detected and guidelines for its application are developed. This article is limited to lumped population balance models with one internal coordinate.

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Section: à1 ð1þmentioning

confidence: 98%

Generalization and numerical investigation of QMOM

et al. 2006

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“…While the method is highly efficient when the physics are simple, the approach does not generalize to complex physics. The method of weighted residuals approximates the distribution by a linear combination of basis functions (Singh and Ramkrishna, 1977). As in the method of moments, this results in a system of ODEs, which can be solved using off-the-shelf solvers.…”

Section: ѩF͑l T͒mentioning

confidence: 99%

High resolution algorithms for multidimensional population balance equations

2004

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“…It was noted by Hulbert and Akiyama (1969) that a larger number of terms of the Laguerre series may be required to approximate +( V, t ) for distributions that are not unimodal. In such a case, it may be advantageous to use the method of weighted residuals with problem specific polynomials, as illustrated by Singh and Ramkrishna (1977).…”

Section: Dt (46)mentioning

confidence: 99%

A population balance approach to the modeling of solid phase reactions

Bhatia

Perlmutter

1979

AIChE Journal

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A population balance approach is used to model solid phase reactions in terms of nuclei of the product phase dispersed in the reactant matrix. A conversion-time relationship is obtained by the solution of a set of moment equations derived from the population balance. Some previously known solutions for both homogeneous and heterogeneous nucleation are developed by this approach to demonstrate the relative ease of so!ution when compared with previously uJed integral techniques. A model for nucleus impingement at low conversions is formulated and applied to published data on the oxidation of cuprous iodide to illustrate the practical use of this technique. The classical Avrami model (Avrami, 1939(Avrami, , 1940(Avrami, , 1941) is modified to include the initial volume of a nucleus, and it is shown that insensitivities in the original model are removed by this improvement. The need for direct measurements of nucleation and growth rates is emphasized. 1975) or lengrhy power series procedures (Von Goler andThe procedure developed here is generally easier than those ot the corresponding integral equations and has the added advantage that the size dependent interaction of nuclei can be readily incorporated through the use of birth and death rate functions. A procedure for the calculation of these functions is presented which is valid for the early stages of the reaction. The fit of the modified Avrami model to a specific set oi' experimental data is compared with that of a model presented in this paper to emphasize the importance of direct measurements of nucleation and growth rates in the determination of a correct impingement model. CONCLUSIONS AND SIGNIFICANCEThe modeling of phase-transformation kinetics is approached through a balance over the number density of nuclei, which in-orporates birth and death functions that arise from nucleus impingement. This approach for obtaining the conversion-time relationship results in a set of first-order differential equations. The technique offers an advantage over the previously used methods becau;e of the relative ease of solution of these equations. It can also be used to calculate the nucleus size density for correIation with experimental data.For comparison purposes, the well-known Avrami model is modified to include a term for the initial volume ok a ical Engineers, 1979. growth nucleus in order to increase the sensitivity of the least-squares fit to the choice of parameter values. The two models are equally satisfactory in fitting a specific set of experimental data in the low to moderate conversion range, showing that conversion-time data alone are insufficient to distinguish between the two models for nucleuj impingement. Direct measurements of nucleation and growth rates are needed to identify the more accurate model.The simplified theory presented does not predict the sigrno:dal shape of the conversion-time curve since it does not account for the higher-order impingement among nuclei at high conversion. The consiraining effect of the outer surface of a reacting ...

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Solution of population balance equations by MWR

Cited by 61 publications

References 8 publications

Generalization and numerical investigation of QMOM

Generalization and numerical investigation of QMOM

High resolution algorithms for multidimensional population balance equations

A population balance approach to the modeling of solid phase reactions

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