XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 871
of the plant (de Bakkar 2014 Somani et al., 2017). Even
small improvements in grinding energy efficiency can pro-
vide major financial dividends.
The quality of ores has been degrading over the decades,
resulting in lower concentration of minerals per unit mass
of ores (Li et al., 2021). As a result, extraction occurs at
smaller liberation sizes and fine or even ultrafine grinding
becomes necessary for extracting the minerals (Perez-Garcia
et al., 2020). New grinding technologies such as Isamill
or Stirred Media Detritor can achieve ultrafine grinding
and particle sizes as low as 7 microns (de Bakker 2014).
However, many existing industrial plants cannot feasibly
transition to such new technologies due to high capital
investment and operational costs (Farjana et al., 2019). Ball
mills remain a prevalent equipment in many existing min-
eral processing plants. An alternate approach to meet the
requirements of efficient fine grinding is to have accurate
prediction of ball mill performance and real-time optimiza-
tion of grinding circuits.
The milling performance can be characterized by prod-
uct particle size distribution (PSD), specific power con-
sumption, volumetric filling of the ball mill, etc. The mass
fraction of total solids that pass through the middle sieve
size is known as midsize passing percentage and is typically
used as one of the alternative metrics to the entire PSD
(Allen, T. 2003). Various operating parameters such as ball
diameter, ball filling (J), mill speed, slurry concentration,
grinding time and ore quality influence mill performance.
Abnormal variations in these parameters may lead to infe-
rior product quality or overgrinding with higher power
consumption. Thus, it is necessary to implement real-
time optimization of grinding circuits with the objectives
of minimizing particle size, maximizing throughput, and
minimizing specific power consumption. However, sen-
sors for measuring some of the key variables accurately are
not available. It is difficult to measure ball mill hold-up
with physical sensors. Inline particle size analyzers are used
for measuring the PSD but they require frequent calibra-
tion. The lab-scale measurement of these variables results in
time delay in controlling the process variables. To circum-
vent these drawbacks, development of soft sensors for key
performance indicators (KPI) using mathematical models
and employing them to monitor the process will pave the
way for enhanced optimization. Several studies have been
devoted for developing mathematical models of KPIs such
as PSD, mill holdup and power draw as functions of mill
design and operating variables (Austin et al., 2007 Klimpel
et al., 1989 Latchireddi and Morrell 2003a Makokha et
al., 2011 Morrell 1996a, b). Although population bal-
ance models (PBMs) describe the PSD accurately, there is
an uncertainty in scale-up of the mill and control over the
time increment. The mill holdup estimation models have
shown discrepancy in terms of effect of flow rate on hold-
up. In some studies, the holdup is found to vary linearly
with flow rate (Kinneberg and Herbst 1984 Songfack and
Rajamani 1999) while it has been shown to vary as a square
root of flow rate in others (Rogers and Austin 1984 Weller
1980 Marchand et al., 1980 Gardner et al., 1982). In
power draw models, the role of charge motion or mate-
rial filling on power consumed during milling is lacking.
Besides, very few power draw models have been tested with
industrial data (Austin 1990 Morrell 1996a). Both PSD
and throughput depend strongly on mill holdup and it is
important to build grinding models incorporating the mill
holdup.
To address the above need, we have developed a novel
grinding model combining the population balance model
for PSD with semi-empirical models mill holdup and
power draw. The models are first tested with published data
from laboratory or pilot-scale or industrial scale ball mills.
These models are employed for generating synthetic data
for a closed-circuit grinding operation through simula-
tions. Machine learning models are developed using this
simulated data and incorporated in a multi-objective opti-
mization framework for optimizing the performance of a
grinding circuit.
METHODOLOGY FOR MODEL-BASED
OPTIMIZATION
The primary focus of this work is the development of a
framework for model-based optimization of the perfor-
mance of a grinding circuit. The key component of this
framework is a hybrid physics-based plus semi-empirical
model for grinding, consisting of three sub-models for
PSD, mill holdup (H) and power draw. The framework
described in Figure 1 is applicable for any size reduction
equipment by using a suitable model for particle size dis-
tribution prediction. The models developed in the current
work are specifically for optimization of a grinding circuit
constituting of a ball mill as the size reduction equipment.
Novel Grinding Model
The modeling framework for grinding is shown in Figure 1.
The three sub-models are combined using slurry filling (U)
and mean residence time (τ), estimated using the Holdup
(H) model.
Particle Size Distribution Model
For a fully mixed batch grinding operation, the population
balance equation (PBE) for a continuous dynamic system
of the plant (de Bakkar 2014 Somani et al., 2017). Even
small improvements in grinding energy efficiency can pro-
vide major financial dividends.
The quality of ores has been degrading over the decades,
resulting in lower concentration of minerals per unit mass
of ores (Li et al., 2021). As a result, extraction occurs at
smaller liberation sizes and fine or even ultrafine grinding
becomes necessary for extracting the minerals (Perez-Garcia
et al., 2020). New grinding technologies such as Isamill
or Stirred Media Detritor can achieve ultrafine grinding
and particle sizes as low as 7 microns (de Bakker 2014).
However, many existing industrial plants cannot feasibly
transition to such new technologies due to high capital
investment and operational costs (Farjana et al., 2019). Ball
mills remain a prevalent equipment in many existing min-
eral processing plants. An alternate approach to meet the
requirements of efficient fine grinding is to have accurate
prediction of ball mill performance and real-time optimiza-
tion of grinding circuits.
The milling performance can be characterized by prod-
uct particle size distribution (PSD), specific power con-
sumption, volumetric filling of the ball mill, etc. The mass
fraction of total solids that pass through the middle sieve
size is known as midsize passing percentage and is typically
used as one of the alternative metrics to the entire PSD
(Allen, T. 2003). Various operating parameters such as ball
diameter, ball filling (J), mill speed, slurry concentration,
grinding time and ore quality influence mill performance.
Abnormal variations in these parameters may lead to infe-
rior product quality or overgrinding with higher power
consumption. Thus, it is necessary to implement real-
time optimization of grinding circuits with the objectives
of minimizing particle size, maximizing throughput, and
minimizing specific power consumption. However, sen-
sors for measuring some of the key variables accurately are
not available. It is difficult to measure ball mill hold-up
with physical sensors. Inline particle size analyzers are used
for measuring the PSD but they require frequent calibra-
tion. The lab-scale measurement of these variables results in
time delay in controlling the process variables. To circum-
vent these drawbacks, development of soft sensors for key
performance indicators (KPI) using mathematical models
and employing them to monitor the process will pave the
way for enhanced optimization. Several studies have been
devoted for developing mathematical models of KPIs such
as PSD, mill holdup and power draw as functions of mill
design and operating variables (Austin et al., 2007 Klimpel
et al., 1989 Latchireddi and Morrell 2003a Makokha et
al., 2011 Morrell 1996a, b). Although population bal-
ance models (PBMs) describe the PSD accurately, there is
an uncertainty in scale-up of the mill and control over the
time increment. The mill holdup estimation models have
shown discrepancy in terms of effect of flow rate on hold-
up. In some studies, the holdup is found to vary linearly
with flow rate (Kinneberg and Herbst 1984 Songfack and
Rajamani 1999) while it has been shown to vary as a square
root of flow rate in others (Rogers and Austin 1984 Weller
1980 Marchand et al., 1980 Gardner et al., 1982). In
power draw models, the role of charge motion or mate-
rial filling on power consumed during milling is lacking.
Besides, very few power draw models have been tested with
industrial data (Austin 1990 Morrell 1996a). Both PSD
and throughput depend strongly on mill holdup and it is
important to build grinding models incorporating the mill
holdup.
To address the above need, we have developed a novel
grinding model combining the population balance model
for PSD with semi-empirical models mill holdup and
power draw. The models are first tested with published data
from laboratory or pilot-scale or industrial scale ball mills.
These models are employed for generating synthetic data
for a closed-circuit grinding operation through simula-
tions. Machine learning models are developed using this
simulated data and incorporated in a multi-objective opti-
mization framework for optimizing the performance of a
grinding circuit.
METHODOLOGY FOR MODEL-BASED
OPTIMIZATION
The primary focus of this work is the development of a
framework for model-based optimization of the perfor-
mance of a grinding circuit. The key component of this
framework is a hybrid physics-based plus semi-empirical
model for grinding, consisting of three sub-models for
PSD, mill holdup (H) and power draw. The framework
described in Figure 1 is applicable for any size reduction
equipment by using a suitable model for particle size dis-
tribution prediction. The models developed in the current
work are specifically for optimization of a grinding circuit
constituting of a ball mill as the size reduction equipment.
Novel Grinding Model
The modeling framework for grinding is shown in Figure 1.
The three sub-models are combined using slurry filling (U)
and mean residence time (τ), estimated using the Holdup
(H) model.
Particle Size Distribution Model
For a fully mixed batch grinding operation, the population
balance equation (PBE) for a continuous dynamic system