Investigation of Error Distribution in the Back-Calculation of Breakage Function Model Parameters via Nonlinear Programming

Despite its effectiveness in determining breakage function parameters (BFPs) for quantifying breakage characteristics in mineral grinding processes, the back-calculation method has limitations owing to the uncertainty regarding the distribution of the error function. In this work, using Korean uranium and molybdenum ores, we show that the limitation can be overcome by searching over a wide range of initial values based on the conjugate gradient method. We also visualized the distribution of the sum of squares of the error in the two-dimensional parameter space. The results showed that the error function was strictly convex, and the main problem in the back-calculation of the breakage functions was the flat surface of the objective function rather than the occurrence of local minima. Based on our results, we inferred that the flat surface problem could be significantly mitigated by searching over a wide range of initial values. Back-calculation using a wide range of initial values yields BFPs similar to those obtained from single-sized-feed breakage tests (SSFBTs) up to four-dimensional parameter spaces. Therefore, by searching over a wide range of initial values, the feasibility of the back-calculation approach can be significantly improved with a minimum number of SSFBTs.

[1]  V. Deniz.  The effects of ball filling and ball diameter on kinetic breakage parameters of barite powder , 2012 .

[2]  Leonard G. Austin,et al.  The back-calculation of specific rates of breakage from continuous mill data , 1984 .

[3]  Fengnian Shi,et al.  A specific energy-based size reduction model for batch grinding ball mill , 2015 .

[5]  L. G. Austin,et al.  Comparison of methods for sizing ball mills using open-circuit wet grinding of phosphate ore as a test example , 1987 .

[6]  F. C. Bond.  The Third Theory of Comminution , 1952 .

[7]  Jihoe Kwon,et al.  Investigation of breakage characteristics of low rank coals in a laboratory swing hammer mill , 2014 .

[8]  Timothy J. Napier-Munn,et al.  Is progress in energy-efficient comminution doomed? , 2015 .

[9]  Douglas W. Fuerstenau,et al.  Scale-up procedure for continuous grinding mill design using population balance models , 1980 .

[10]  Leonard G. Austin,et al.  The estimation of non-normalized breakage distribution parameters from batch grinding tests , 1972 .

[11]  Leonard G. Austin,et al.  Modeling of a grinding circuit with a swing-hammer mill and a twin-cone classifier , 1985 .

[12]  Heechan Cho,et al.  Simulation of particle-laden flow in a Humphrey spiral concentrator using dust-liquid smoothed particle hydrodynamics , 2017 .

[13]  W. J. Whiten.  A matrix theory of comminution machines , 1974 .

[14]  Fengnian Shi,et al.  Modelling of vertical spindle mills. Part 1: Sub-models for comminution and classification , 2015 .

[15]  Fengnian Shi,et al.  An energy-based model for swing hammer mills , 2003 .

[16]  P. Luckie,et al.  Using Model Discrimination to Select a Mathematical Function for Generating Separation Curves , 1986 .

[17]  Paul W. Cleary,et al.  Prediction of slurry transport in SAG mills using SPH fluid flow in a dynamic DEM based porous media , 2006 .

[18]  Paul W. Cleary,et al.  Charge behaviour and power consumption in ball mills: sensitivity to mill operating conditions, liner geometry and charge composition , 2001 .

[19]  Leonard G. Austin,et al.  Methods for determination of breakage distribution parameters , 1972 .

[20]  Surya Ganguli,et al.  Identifying and attacking the saddle point problem in high-dimensional non-convex optimization , 2014, NIPS.

[21]  M. Klima,et al.  Evaluation of a laboratory rod mill when grinding bituminous coal , 2012 .

[22]  Fengnian Shi,et al.  A specific energy-based ball mill model: From batch grinding to continuous operation , 2016 .

[23]  Raj K. Rajamani,et al.  Modeling breakage rates in mills with impact energy spectra and ultra fast load cell data , 2011 .

[24]  Leonard G. Austin,et al.  A discussion of equations for the analysis of batch grinding data , 1999 .

[25]  K. J. Reid.  A solution to the batch grinding equation , 1965 .

[26]  R. Ramachandran,et al.  A multiscale DEM-PBM approach for a continuous comilling process using a mechanistically developed breakage kernel , 2018 .

[27]  L. G. Austin,et al.  RESIDENCE TIME DISTRIBUTIONS IN BALL MILLS , 1984 .

[28]  Brahmeshwar Mishra,et al.  The discrete element method for the simulation of ball mills , 1992 .

[29]  C. Cavazzoni,et al.  Investigation of particle dynamics and classification mechanism in a spiral jet mill through computational fluid dynamics and discrete element methods , 2020, Powder Technology.

[30]  Kihong Kim,et al.  Modeling of coal breakage in a double-roll crusher considering the reagglomeration phenomena , 2012 .