TY - JOUR
T1 - A rational function approximation to the effectiveness factor for multi-particle interactions in dense-phase dry milling
AU - Capece, M.
AU - Dave, R.
AU - Bilgili, E.
N1 - Funding Information:
We gratefully acknowledge partial financial support from the National Science Foundation Engineering Research Center for Structured Organic Particulate Systems (NSF ERC for SOPS) through the Grant EEC-0540855 . The corresponding author (E.B.) thanks Dr. Pavol Rajniak of Merck for seeding the germ of this work via insightful comments on our presentation at the AIChE Annual Meeting (E. Bilgili, M. Capece, Paper No: 392f, Salt Lake City, UT, Nov. 2010).
PY - 2012/11
Y1 - 2012/11
N2 - Non-linear population balance models (PBMs) have recently been shown to be superior to their linear counterparts for quantifying the breakage kinetics in dense-phase dry milling processes, where mechanical multi-particle interactions are significant. At the particle ensemble scale, particles of different sizes interact causing a population dependence of the specific breakage rate function. Non-linear PBMs explicitly account for these multi-particle interactions through a population-dependent functional called the effectiveness factor. Due to the inherent difficulty and complexity of proposing and selecting an appropriate form of the effectiveness factor, a general and systematic approach is desired to assess breakage kinetics in an unbiased manner. In this study, toward addressing the need for a general form of the effectiveness factor, we propose a rational function approximation. First, computer generated effectiveness factor was approximated by a rational function. Then, evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill was fitted with various forms of the effectiveness factor in the context of linear and non-linear PBMs. Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate various forms of the effectiveness factor. The statistical analysis suggests that a rational function may replace specific forms of the effectiveness factor for a less restrictive analysis of the multi-particle interactions. Additionally, fitting a non-linear PBM using the rational function approximation allowed us to describe the dense-phase dry milling data accurately. Accordingly, this study enables experimenters to quantify the impact of the multi-particle interactions in a robust, systematic, and unbiased manner via the rational function approximation to the effectiveness factor within the context of the non-linear PBM.
AB - Non-linear population balance models (PBMs) have recently been shown to be superior to their linear counterparts for quantifying the breakage kinetics in dense-phase dry milling processes, where mechanical multi-particle interactions are significant. At the particle ensemble scale, particles of different sizes interact causing a population dependence of the specific breakage rate function. Non-linear PBMs explicitly account for these multi-particle interactions through a population-dependent functional called the effectiveness factor. Due to the inherent difficulty and complexity of proposing and selecting an appropriate form of the effectiveness factor, a general and systematic approach is desired to assess breakage kinetics in an unbiased manner. In this study, toward addressing the need for a general form of the effectiveness factor, we propose a rational function approximation. First, computer generated effectiveness factor was approximated by a rational function. Then, evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill was fitted with various forms of the effectiveness factor in the context of linear and non-linear PBMs. Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate various forms of the effectiveness factor. The statistical analysis suggests that a rational function may replace specific forms of the effectiveness factor for a less restrictive analysis of the multi-particle interactions. Additionally, fitting a non-linear PBM using the rational function approximation allowed us to describe the dense-phase dry milling data accurately. Accordingly, this study enables experimenters to quantify the impact of the multi-particle interactions in a robust, systematic, and unbiased manner via the rational function approximation to the effectiveness factor within the context of the non-linear PBM.
KW - Dry milling
KW - Effectiveness factor
KW - Multi-particle interactions
KW - Particle breakage
KW - Population balance
KW - Rational function
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U2 - 10.1016/j.powtec.2012.06.054
DO - 10.1016/j.powtec.2012.06.054
M3 - Article
AN - SCOPUS:84865198700
SN - 0032-5910
VL - 230
SP - 67
EP - 76
JO - Powder Technology
JF - Powder Technology
ER -