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

Abstract

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.

Population Balance Modeling of Milling Processes: Are We Falsifying Breakage Kinetics and Distribution via Back-Calculation Methods?

“…In the back-calculation method, the non-uniqueness of the estimated parameters and their statistical insignificance originate from the high number of breakage parameters and lack of sufficiently dense data sets with acceptable precision [8,31]; the latter aspect will be elaborated in Section 3.3. For a PBM with N size classes, there exist N S i values, N d * i values, and N(N-1)/2 values of B ij.…”

“…If these factors are multiplicatively added to Equation ( 7) in a power-law fashion to make the PBM more useful and predictive for practical applications (see Section 3.6), the number of parameters will increase to 13, at a minimum. An increase in the number of parameters causes more inaccurate estimation of the parameters, and the parameters may not be statistically significant besides the fact that a probable globally optimum solution will not be reached by optimization [8][9][10]31].…”

“…Since the early inception of the full back-calculation method, local optimization has been commonly used to estimate the parameters [9,10,17,35]. Recent research [8,31] clearly that when the number of PBM parameters is above a few, it is very difficult, if not impossible, to find a unique set of parameters via local optimization. The obtained solution may correspond to a local optimum [36] or a flat surface around an objective function [31].…”

“…Recent research [8,31] clearly that when the number of PBM parameters is above a few, it is very difficult, if not impossible, to find a unique set of parameters via local optimization. The obtained solution may correspond to a local optimum [36] or a flat surface around an objective function [31]. The significant dependence of the optimization solution on the initial guess values of the parameters is a signature of its non-uniqueness.…”

On the Similarity of Austin Model and Kotake–Kanda Model and Implications for Tumbling Ball Mill Scale-Up

“…(10), is appealing as it has 5 parameters as opposed to the Austin model with 6 parameters, when ball sizes are explicitly considered. The milling studies, where the back-calculation approach involved more than a few breakage parameters to be estimated, have shown that the accuracy decreases as the number of parameters to be estimated increases (Klimpel and Austin, 1977;Kwon and Cho, 2021). However, to the best knowledge of the author, the KK model has not been used for the PBM simulation of ball milling of a natural feed with a wide size particle size distribution (PSD), and no scale-up has been performed using it.…”