Analysis of nonlinear batch grinding in stirred media mills using self-similarity solution

Rao, B. Venkoba; Datta, Amlan

doi:10.1016/j.powtec.2006.07.020

Cited by 7 publications

(4 citation statements)

References 15 publications

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“…Bilgili has shown that if the environment factor is taken in the form of eq where every particle is affected differently by the environment, the shape of the particle size distribution continues to evolve even at long grinding times. On the other hand, it is known that if the environment factor is considered in a lumped form, self-similar behavior can be observed at a later period of grinding. , Similarity behavior also exists for an aggregative breakage equation as shown by Kostoglou and Karabelas . In this section we shall show that for the present model (eq ) self-similar behavior exists.…”

Section: Similarity Analysis Of Nonlinear Breakage Equationsupporting

confidence: 64%

“…On the other hand, it is known that if the environment factor is considered in a lumped form, self-similar behavior can be observed at a later period of grinding. 21,23 Similarity behavior also exists for an aggregative breakage equation as shown by Kostoglou and Karabelas. 24 In this section we shall show that for the present model (eq 11) self-similar behavior exists.…”

Section: Similarity Analysis Of Nonlinear Breakage Equationsupporting

confidence: 62%

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Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

Chakraborty

Ramkrishna

2011

Ind. Eng. Chem. Res.

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In this work, we introduce mechanistic considerations in nonlinear milling of particulate material. We show that the mobility of the powder can have a significant effect on the breakage frequency. During grinding, the change in particle size leads to changes in the mobility of the powder which in turn alters the breakage rate. The mechanistic modeling, being able to accommodate different responses of powder mobility for wet and dry grinding, is able to distinguish naturally between dry and wet grinding, an attribute not possessed by other nonlinear models. It is shown that the rich variety of phenomena observed in grinding mills emanates from the interplay between opposing factors that influence the breakage environment in a mill. Such factors include viscosity, density, and compaction of the powder. For dry grinding, the change in powder porosity and viscosity leads to a variety of nonlinear effects. For wet grinding the viscosity and density of the slurry impart a nonmonotonic variation in the breakage rate with particle size. A population balance equation (PBE) for environment dependent breakage has been formulated by considering the foregoing effects. Similarity analysis of the model equation shows the existence of self-similar behavior even for the environment dependent breakage. Experimental investigation of such self-similarity provides an effective diagnostic process for validating model framework. Furthermore, when combined with the methodology of solving inverse problems (e.g., Sathyagal, A. N.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1377, an effective approach becomes available for model identification.

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Section: Similarity Analysis Of Nonlinear Breakage Equationsupporting

confidence: 64%

Section: Similarity Analysis Of Nonlinear Breakage Equationsupporting

confidence: 62%

Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

Chakraborty

Ramkrishna

2011

Ind. Eng. Chem. Res.

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“…Equation ( 1) is supported by two different mechanistic modeling approaches. First, an analytical solution of the PBM for batch milling, which is identical to that for continuous plug-flow milling, can be used to predict the variation in the mean particle size x m = C/t 1/N , for sufficiently long milling times [54,75]. Here, the constant C depends on the parameters of the specific breakage rate function S and self-similar breakage distribution function, and N is the power-law exponent of the size-dependent S. This relation can be rearranged to give a linear relation in t:…”

Section: Discussionmentioning

confidence: 99%

Development of a Semi-Mechanistic Modeling Framework for Wet Bead Milling of Pharmaceutical Nanosuspensions

Clancy,

Guner,

Chattoraj

et al. 2024

Pharmaceutics

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This study aimed to develop a practical semi-mechanistic modeling framework to predict particle size evolution during wet bead milling of pharmaceutical nanosuspensions over a wide range of process conditions and milling scales. The model incorporates process parameters, formulation parameters, and equipment-specific parameters such as rotor speed, bead type, bead size, bead loading, active pharmaceutical ingredient (API) mass, temperature, API loading, maximum bead volume, blade diameter, distance between blade and wall, and an efficiency parameter. The characteristic particle size quantiles, i.e., x10, x50, and x90, were transformed to obtain a linear relationship with time, while the general functional form of the apparent breakage rate constant of this relationship was derived based on three models with different complexity levels. Model A, the most complex and general model, was derived directly from microhydrodynamics. Model B is a simpler model based on a power-law function of process parameters. Model C is the simplest model, which is the pre-calibrated version of Model B based on data collected from different mills across scales, formulations, and drug products. Being simple and computationally convenient, Model C is expected to reduce the amount of experimentation needed to develop and optimize the wet bead milling process and streamline scale-up and/or scale-out.

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“…In this case, Equation (2) cannot be solved iteratively by using the matrix equations. Alternatively, some scaling (selfsimilar) functions [7][8][9][10][11][12] can be used as the selection function to solve kinetic PBM, as long as the selected function describes the non-linear breakage kinetics [5,13,14]. All the above discussion indicates that the solution of the kinetic PBM gets complicated because of the uncertainties in the breakage rate.…”

Section: Introductionmentioning

confidence: 99%

A Computational Algorithm to Understand the Evolution of Size Distribution with Successive Breakage Events at Grinding

Camalan

2021

The 2nd International Electronic Conference on Mineral Science

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The kinetic population balance model (PBM) is widely used to predict the particle size distributions of grinding products. However, the model may not be solved if the rate of particle accelerates or decelerates in the mill hold-up, i.e., non-first-order breakage. This study presents a computational algorithm coupled with a pseudo-matrix model to simulate the product size distributions (PSDs) of successive breakage events at grinding. The algorithm’s applicability and accuracy were validated against PSDs taken from different grinding equipment. The advantages of the algorithm are as follows—(i) time can be implicitly or explicitly added to the algorithm. (ii) The parameters required to run the algorithm are quite few. (iii) The proposed algorithm can predict PSDs in the normal or abnormal breakage region. Even a short-time grinding test will be sufficient to estimate the parameters if abnormal breakage effects are reduced or eliminated. (iv) The algorithm can work with arbitrary sets of parameters that are irrelevant to the mill feed and mill type. Also, the algorithm’s framework shows that grinding is not a chaotic process; yet it may be due to the surface/gravitational attraction forces between particles and grinding media.

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Analysis of nonlinear batch grinding in stirred media mills using self-similarity solution

Cited by 7 publications

References 15 publications

Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

Development of a Semi-Mechanistic Modeling Framework for Wet Bead Milling of Pharmaceutical Nanosuspensions

A Computational Algorithm to Understand the Evolution of Size Distribution with Successive Breakage Events at Grinding

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