Solutions of inverse problems in population balances—I. Aggregation kinetics

Wright, H.; Ramkrishna, Doraiswami

doi:10.1016/0098-1354(92)80021-z

Cited by 45 publications

(45 citation statements)

References 23 publications

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“…This simplified the inverse problem to a parameter estimation problem at the expense of a loss of generality. On the other hand, the methods pioneered by Ramkrishna et al [15] do not strongly constrain the functional form of K(m 1 , m 2 ). These authors address the problem of ill-posedness using a procedure known as Tikhonov regularisation.…”

Section: Introductionmentioning

confidence: 99%

“…The non-triviality of the inverse problem comes from the fact that, since we seek to determine a function of two variables from a function of one variable, we would generally expect it to be ill-posed. Notable previous work on this problem includes the work of Onishi and coworkers in the atmospheric science context [12] and the work of Ramkrishna and coworkers in the chemical engineering context [13][14][15]. Onishi et al [12] address the problem of ill-posedness by using significant prior knowledge about droplet coalescence in turbulent conditions to put strong constraints on the functional form of the kernel.…”

Section: Introductionmentioning

confidence: 99%

“…These authors address the problem of ill-posedness using a procedure known as Tikhonov regularisation. In a previous note [16] we explored the ability of the method in [15] to solve the inverse problem for kernels of the form,…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear least-squares method for the inverse droplet coagulation problem

2013

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If the rates, K(x, y), at which particles of size x coalesce with particles of size y is known, then the mean-field evolution of the particle-size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates, K(x, y) from measurements of the particle size distribution. We assume that K(x, y) is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining, K(x, y), a function to two variables, then reduces to a simpler problem of determining a function of a single variable plus two exponents, µ and ν, which characterise the scaling properties of K(x, y). The price of this simplification is that the resulting least squares problem is nonlinear in the exponents µ and ν. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear least-squares method for the inverse droplet coagulation problem

2013

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“…Determination of the collision frequency function is a complex task in most of the models 6 and it is very difficult to determine it from experimental data. However, an alternative way of retrieving the 7 kernels based on experimental data is to solve the inverse problem [51,52]. Braumann and Kraft studied the 8 inverse problem occurring in a multidimensional population balance model describing granulation employing 9 linear response surfaces [39] and second order response surfaces [53].…”

mentioning

confidence: 99%

“…Therefore, experiments have to be carried out in 12 order to identify and measure the unknown model parameters, e.g. aggregation rate constants [51][52][53]73]. …”

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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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