Original Research PaperScale-up procedure of parameter estimation in selection and breakage functions for impact pin milling
Graphical abstract
Introduction
Milling is an intended size reduction process as an integral unit operation in the particulate solids processing industries such as mineral, chemical, food and pharmaceutical engineering. However, milling is well known to be an energy-intensive process, requiring a massive amount of energy consumption in the milling process for the desired size reduction. For example of large ball mills, less than 1% of the total energy is spent on the fracture process whilst 85% is dissipated as heat, making the milling process notoriously inefficient in particle breakage [3]. As a result, early studies on the milling process is mainly focused on the relationship between specific energy and size reduction such as Rittinger law, Kick law and Bond law [4]. These theories are based on a specific size modulus of the particle but do not take into account the distribution modulus [5]. It is pointed out that Bond model and Rittinger model may be applied to the same set of grinding data whilst Kick model is applicable for coarse grinding.
With the advancement of computational modelling methods, considerable efforts have been made to promote the understanding of energy consumption and size reduction over a milling process. Amongst these computational methods, population balance model (PBM) has been widely used to predict the product size distribution as a Model Driven Design (MDD) approach. For example, Austin and Bagga [6] studied the dry milling in ball mills based on the first-order kinetics PBM. It was demonstrated that the breakage rates are lower for smaller particle sizes and slows down due to the accumulation of fines in ball milling. Vogel and Peukert [7] presented an approach to quantify the grinding behaviour of different materials by constructing a master curve for breakage probability. Two material parameters, and were introduced to represent the grinding property.where is breakage probability; is mass-specific kinetic energy; and are particle size and impact number. The two material parameters were determined by single-particle impact tests. Vogel and Peukert [1] presented an upscaling study from single particle impact behaviour to impact mill modelling. The two material parameters were applied to the systematic and multiscale modelling of two grinding mill, i.e. sieve hammer mill and air classifier mill. Luciani et al. [8] described a model-based approach to scale-up high-shear wet milling processes using the population balance equation. The Austin model [6] was adopted to describe the specific breakage rate and the breakage distribution function for its milling process. Very good correlations between the breakage rate and the shear number were found. However, more work is needed to determine the scale-up factor from the mechanistic perspective.
This paper presents a scale-up procedure of parameter estimation of selection function and breakage function from single particle impact breakage to inform the predictions at the process scale of an impact pin mill. The selection and breakage functions for particle breakage in the literature are briefly reviewed, which can be well suited in the context of population balance model. Single particle breakage tests are conducted to provide the database for the parameter estimation in Vogel and Peukert model [1]. The estimated parameters in the particle level are upscaled in an impact pin mill using the population balance model, which is implemented in the software gPROMS (Process Systems Enterprise, UK) [2]. The scale-up procedure by specifying the parameters from single particle level to the process level of PBM demonstrates an approach to help predict the size reduction process subject to the prevailing mechanism in an impact pin mill and other milling processes alike.
Section snippets
Population balance model for particle breakage
Population balance modelling has been widely used to track particle attribute evolution in the particulate processes. The dimension of the population balance model is dependent on the number of phases (solid, liquid, and gas) to be considered. Most population balance models in the literature are one-dimensional, usually considering only the variation of particle size or volume. A key feature of the population balance model is the inclusion of rate process kernels such as consolidation,
Experimental protocol
Prior studies on single particle breakage test focus on understanding the effects of impact velocity and impact angles on the selection function and breakage function [22], [23]. The experimental protocol of single particle impact breakage is designed with the aim of providing avenues to determine the selection and breakage functions from a small number of tests. The experimental protocol and granule breakage probability and daughter size distribution measurements have been previously reported
Parameter estimation in single particle impact
The input parameters in the chosen selection function and breakage function can be categorized into four types: material-dependent parameters (e.g. ), a process-dependent parameter (e.g. ), size-dependent parameters (e.g. ), and fitting parameters (, , and ). Note that the product of particle size and the specific threshold energy , i.e. is assumed to be constant [1]. As a result, the size-dependent parameter is inversely proportional to the particle
Scale-up procedure
The scale-up procedure of milling operations is a critical step in the comminution industry. The scale-up procedure for a grinding mill using population balance models is first introduced by Herbst and Fuerstenau [39]. The grinding process is sub-divided as breakage, transport and classification and the application of Bond Model in a ball milling is discussed. The scale-up procedure could be divided into three steps according to Datta and Rajamani [40]. First, lab scale milling experiments in a
Conclusions
This paper has presented a scale-up procedure of parameter estimation in the selection function and breakage function, which is able to help predict the particle size reduction using the population balance model. The representative selection functions and breakage functions in the literature are briefly reviewed, which can be well suited in the context of population balance model (PBM). It was found that the majority of the selection functions and breakage functions are empirical. As a prelude
Declaration of Competing Interest
The authors declared that there is no conflict of interest.
Acknowledgement
The authors gratefully acknowledge the financial support from the International Fine Particle Research Institute, National Natural Science Foundation of China (Grant No. 51909146 and 51909147) and Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Grant No. Z018005. We would like to thank Prof. Jin Y. Ooi and Dr. Alvaro Janda from the University of Edinburgh for many helpful discussions. The
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