Role of reduced diameter disks in the performance of an IsaMill™

Abstract IsaMill™ is utilized mainly for ultrafine grinding of low-grade ores. Although highly efficient, it has certain operational challenges, such as frequent replacement of liners of disks and inner shells. To circumvent this challenge, a design modification of the disks, by introducing a reduced diameter disk (RDD), is commonly implemented. A discrete element method (DEM) based model is developed to investigate the effects of RDD and mill loading on media dynamics, force distribution, power draw, and collision energy. It is demonstrated that implementation of RDD results in a reduction of normal and shear stress on mill parts indicating a reduction in wear. Analysis of media dynamics shows that media mobility decreases hence decreasing total collision energy when RDDs are applied. This reduced mobility can potentially affect the grinding performance of the mill. A single-objective optimization approach based on collision energy data is employed to determine an optimal design configuration of the mill.


Introduction
The universal decline in the availability of high-grade ores, for both base and precious metals, has been necessitating effective utilization of low-grade ores for extraction of metals. Mineral-bearing ores are also becoming increasingly complex as the useful minerals are finely disseminated in the ore-posing great challenges in grinding and liberation (Young et al. 1997;Baum and Ausburn 2011;Celep et al. 2015;Chen, Jian Guo, and Li 2020;Xu et al. 2017;Santosh et al. 2021). Hence, fine or ultrafine grinding has become an essential step. Stirred media mills have been proven to be efficient for fine and ultrafine grinding as compared to the energy and time-intensive, traditional, impact-based grinding mills like ball mills and autogenous mills (Mucsi, R acz, and M adai 2013;Hacıfazlıo glu and Korkmaz 2020). IsaMill TM is one such stirred media mill that has been around for more than two decades for treating low-grade complex ores (Johnson et al. 1998;Harbort et al. 1999;Gao, Young, and Allum 2002;Pease et al. 2006;Burford and Clark 2007;Gurnett 2021).
An IsaMill TM consists of a horizontally aligned impeller that rotates at high speed and an enclosure shell. A set of disks, typically 7-8, is attached to the impeller, shown schematically in Figure 1(a). The mill is loaded with grinding media particles (hereafter referred to as media). The media loading is typically as high as 70% of mill volume (Rule 2011;Hou 2014;Larson et al. 2015;Anderson and Bandarian 2019). During operation, a slurry consisting of ore particles and a suspension medium, such as water is introduced into the mill. The stirrer operates at high speed causing vigorous mixing of the slurry. The collisions between the ore particles and the media particles cause a size reduction of ore particles. The disks have circular or kidney-shaped holes to allow the slurry to pass through them. The mill operation can be continuous or batch, depending on the requirements.
A high loading of the media particles and the ore particles ensures efficient grinding as they are closely packed, and the rotary motion of the impeller transfers kinetic energy among particles. This results in the breaking of ore particles through abrasion and impact. However, the high kinetic energy of media also causes frequent wearing of disks and the shell liner, which necessitates frequent maintenance of the equipment and associated downtime (Anderson, Smith, and Strohmayr 2011;Rule 2011;Rule and De Waal 2011;Larson et al. 2015;Larson, Lacouture, and Anderson 2019). Measures to identify regions of high wear and predict downtime in advance include using non-intrusive acoustic emission analyzers (Jackson et al. 2014) and thermal imaging (Gunda and Moys 2015;Larson, Lacouture, and Anderson 2019). To reduce the frequency of maintenance and downtime some plants have tested using disks of smaller diameter at specific locations instead of the regular disk in IsaMill (Anderson, Smith, and Strohmayr 2011;Rule and De Waal 2011;Anderson and McDonald 2016;Larson, Lacouture, and Anderson 2019). Rule and Waal reported a reduction in disk wear from 2.44 to 1.96 mm/day when the first two disks were replaced with RDDs in an IsaMill at Anglo Platinum (Rule and De Waal 2011). In the same plant, Anderson et al reported 35% reduction in power draw and a 50% reduction in wear rate in terms of disk consumption per MWh when the first seven disks were replaced with disks of smaller diameter in an M10000 IsaMill (Anderson, Smith, and Strohmayr 2011). At Kalgoorlie gold mines, the shutdown interval increased from 18 to 35 days when the small diameter disks (SDD) were installed at the first two disk positions (Anderson and McDonald 2016). These papers discuss the effectiveness of RDDs in curbing severe wear. However, a comprehensive study on the suitability and the applicability of RDDs at different mill loadings and their effect on the performance and health of the mill is still unavailable. The direct method of determining the performance of IsaMill includes measuring power drawn by the mill whereas the health of the mill is analyzed by physical examination of mill components after months of operation. This is a time and energy intensive process (Gunda and Moys 2015;Anderson and Bandarian 2019;Schons and Kwade 2019). Therefore, numerical analysis using a representative mathematical model of the process can be an important alternative in examining the effect of RDDs on the performance and health of the mill.
The discrete element method (DEM) has been used as a computational tool to simulate IsaMill operation in the past Jayasundara et al. 2006Jayasundara et al. , 2010Jayasundara, Yang, and Yu 2011;Cleary, Sinnott, and Pereira 2015). The simulation results provide microdynamic information, such as media velocity, collision energy, collision frequency, torque, and force distribution that help in analyzing power drawn by the mill and stresses on mill components. These are indirect measures of the performance and health of the mill. In the past, DEM simulations were utilized in studying the effect of different media loadings and impeller speed on IsaMill operation by Jayasundara et al. (2010). In another study, Jayasundara et al. predicted wear on a single disk face of a labscale IsaMill TM by incorporating media microdynamics information obtained from DEM simulations into Finnie wear model and validated it with experimental observations (Jayasundara et al. 2011b). Cleary et al. simulated media dynamics inside a full-scale M10000 IsaMill using DEM and analyzed wear using impact and shear energy profiles (Cleary, Sinnott, and Pereira 2015). A numerical study on the effect of different disk geometry on media dynamics has also been performed (Jayasundara, Yang, and Yu 2011). However, the role of RDD on mill performance has not been explored through numerical simulation so far. A detailed numerical study on the relevance of RDD at different media loadings and changes in media dynamics with the implementation of RDDs is not available in the literature. Moreover, the degree of reduction in disk diameter and its effect on performance indicators, such as power draw, collision energy, and stress distribution were not studied previously.
This work aims to answer these questions through numerical simulations to study the effect of implementing RDDs in a pilot scale IsaMill TM . We study the influence of RDDs on the media dynamics and force distribution on mill parts during mill operation through DEM simulations. Mill configurations based on varying disk diameters are presented and each configuration is evaluated using a DEM-based model. The model is used for calculating the micro-dynamics inside the IsaMill TM of different configurations operating under different media loadings. Analysis of the simulated collision energy data was carried out to determine the optimal combination of RDD and media loading such that the energy utilized for grinding is maximum and the energy responsible for mill wear is minimum.

Methodology for modeling an IsaMill TM
The discrete element method is used in the present work to simulate media motion inside the IsaMill (Cundall and Strack 1979). In DEM, the interaction of an individual particle with its adjacent neighbors is evaluated by using the principles of contact mechanics. More specifically, using the soft sphere approach, a small overlap of the particle with a neighbor particle is allowed. The time step of interaction is kept sufficiently low so that the velocity of the particles does not vary, and the interaction does not go beyond the immediate neighboring particle. Once, the interactions are accounted for, the new position and velocity of the particle are updated. A particle has six degrees of freedom. Therefore, both translational and rotational motions of the particle are calculated.
The translational motion of a particle is given by: where m is the mass of the particle, V is the translational velocity of the particle, t is time, F G is the gravitational force acting on the particle, F n and F t are normal and tangential contact forces, respectively, acting between the particle and the surrounding wall or other particles. The rotational motion of the particle is given by: where, I and x are the moment of inertia and angular velocity of the particle, respectively, and M is the resultant contact torque acting on the particle. In the present work, a non-linear model, combining Hertz theory in the normal direction and Mindlin's no-slip model in the tangential direction, is employed for modeling the contact between particles (EDEM-2.6 2014; Weerasekara, Liu, and Powell 2016). Detailed expressions for normal and tangential contact forces can be found in Section S1 of Supporting Information. The average normal and shear stress calculations are incorporated in the model to account for the relative wear due to impact and abrasion (Kalala, Bwalya, and Moys 2005; EDEM-2.6 2014). Since the surface area of each configuration is different, the average normal and tangential forces are normalized with the surface area for the given configuration. Thus, the average normal contact stress, W N is defined as: where S a is the surface area of the component for which W N is calculated and J N is defined as: t initial and t are the two subsequent time steps after a steady state is reached. The impact due to the normal force is considered only when the normal contact force at any time step is more than that in the previous time step. Thus, F i N at any time t is given by: The average tangential stress, on the other hand, is given by: where J T is defined as: The continuous tangential or shear forces on the surface are responsible for the wear due to abrasion. Thus, at any time step t, F i T ¼ F t T : A 4 L IsaMill TM with eight disks, as shown in Figure 1(b), is used in the present work to study the effect of RDD on mill performance. Since disks and shell liner close to the feed end of the mill experience maximum wear (Rule and De Waal 2011), diameters of the first two disks at the feed end are reduced (RDD) with different hypothetical combinations, resulting in five different configurations of the mill. These configurations differ in the degree of reduction in diameter of the RDD. For simplification, the configurations are denoted as a percentage diameter reduction of the regular disk. For example, RDD-20 represents the configuration in which the diameter of the first two disks is 80 mm when the diameter of the regular disk is 100 mm. The different RDD configurations with the corresponding disk diameters are reported in Table 1. The focus of the current work is on the batch operation of the mill. Hence, the separator is not included in the simulation. IsaMill TM is commonly used for wet milling processes. Thus, the influence of fluid during the milling process should be considered, especially, at low media loadings. This would require a coupled Computational Fluid Dynamics (CFD)-DEM model. A few models have been developed that can simulate wet milling in an IsaMill by combining DEM with CFD or Smooth Particle Hydrodynamics (SPH) (Jayasundara et al. 2009;Jayasundara et al. 2011a;. These models are computationally expensive, especially when simulating an IsaMill with seven or more disks. These studies focused mainly on the hydrodynamics of the slurry. However, they did not address how this affects mill wear. Moreover, dry milling models have been shown to effectively predict wear and grinding  performance in IsaMills (Jayasundara et al. 2011b;Cleary, Sinnott, and Pereira 2015;Sterling et al. 2022) as well as similar mills (Cleary et al. 2010;Sinnott, Cummins, and Cleary 2015;Esteves et al. 2021) for wet milling. Therefore, a dry milling operation considering only media dynamics is considered in this study. Mono-sized media particles were used for simulations. A stirrer speed of 1500 rpm corresponding to disk tip velocity of 15 m/s for an M4 IsaMill TM was used (Larson et al. 2012). The rotation of the stirrer was set in the clockwise direction (as seen from the feed end). The media loading was assumed to be in the range of 40-80%. Commercial software EDEM 2.6 was used for all the simulations (EDEM-2.6 2014).
The dimensions of the laboratory scale mill along with the physical properties of the grinding media used in the simulations are listed in Table 2. We have used a lower value of Young's modulus than real glass because a high value of Young's modulus requires a long simulation time. This assumption is valid if the maximum overlap is <3% of the mean diameter of the particle (Zhou et al. 1999). The simulation results are also not significantly affected due to this assumption. The material properties of the media and the mill components were assumed to be similar. The media loading is the volume percent of the media inside the mill. The initial average packing fraction of particles was kept constant at 0.52 for all the simulations. The validation of the present DEM model with experimental data from the literature (Jayasundara et al. , 2010 can be found in Section S2 of Supporting Information.

Media dynamics
The media velocity distribution and the flow pattern inside the IsaMill TM for three different disk configurations at a constant media loading are presented in Figure 2. The steady state was determined by monitoring impeller torque. A steady state is usually reached within 2 s after the start of rotation (Jayasundara et al. 2011b). Simulation results that show attainment of a steady state are provided in Section S3 of Supporting Information. The axial velocity distribution of media along the length of the mill for three mill configurations, such as regular disk, RDD-20, and RDD-40 at a media loading of 80% are shown in Figure 2(a). The distribution is captured after 10 seconds from the start of the rotation. It can be observed that the average velocity of media particles between disks in a regular disk mill is typically 2.5 m/s, which is 16% of the disk tip velocity (15 m/s). The media particles at either end of the shell have a low/near zero  velocity indicating stagnant zones. A weak recirculation of media particles is evident for the RDD configurations. Figure 2(b) shows the flow contours of media near the feed end for regular, RDD-20, and RDD-40 mill configurations.
In the case of a regular disk mill, recirculation of media particles from top and bottom regions toward the disk hole is prominent. In contrast, recirculation of media toward the disk holes decreases as disk diameter decreases. The average velocity of media drops to 0.9 m/s in the RDD-20 configuration from 1.6 m/s in the regular disk configuration. The average media velocity further decreases to 0.7 m/s upon the further decrease in the diameter of RDD from 80 to 60 mm. This means the mobility of media particles decreases with the implementation of RDDs indicating reduced energy among media particles. This reduction in energy can negatively affect grinding performance in actual milling operations. Additionally, media dynamics at different media loadings, mean media velocity, and force distribution on mill components are discussed in Sections S4-S6 of Supporting Information.

Force distribution
Tangential and normal forces acting on the mill internals (disks and inner shell), due to media and device collisions, can contribute toward shear and impact-based wear, respectively. Figure 3 shows the distribution of cumulative tangential force on disks of regular and RDD-10 configurations when media loading is 80%, taken seven seconds from the start of the rotation. The magnitude of cumulative shear force is higher at the edge of the disk and the leading edge of disk holes in the regular IsaMill. The high tangential forces in these regions indicate a high chance of wear. These findings are consistent with the experimental observations reported by Jayasundara et al. (2011b). Moreover, a significant reduction in cumulative tangential force, especially at the edges of disks, can be observed in the RDD-10 configuration. The reduction in cumulative tangential force is not just limited to the first two disks where RDDs are employed but it is also evident in the remaining disks. One can also observe a slight increase in the cumulative tangential force on the face of the first two disks in the case of RDD-10. A similar force distribution can be observed in the case of cumulative normal force as shown in Figure S5 of Supporting Information.

Tangential and normal stress
The average normal (W N ) and shear stresses (W T ) account for the relative wear due to impact and abrasion (Kalala, Bwalya, and Moys 2005;EDEM-2.6 2014). Figure 4 shows W N and W T on disks and inner shells for different disk configurations. The value of W N and W T on both the disk and the inner shell is maximum for a regular configuration indicating that it suffers more wear compared to other configurations. However, both normal and shear stresses (W N and W T ) on disks and inner shells significantly decrease with the implementation of RDD at a mill loading of 80%. The effect of RDD on wear rate indicated by normal and shear stresses is more pronounced in RDD-10 and RDD-20 configurations, whereas it is negligible for higher RDD configurations (RDD-30 and RDD-40). For both ML ¼ 80% and ML ¼ 60%, the average normal and shear stresses are more on the disks than on the inner shell, irrespective of configuration. Therefore, disks suffer more wear compared to the inner shell, which is evident from the ratios of (W N ) inner shell to (W N ) disk and (W T ) inner shell to (W T ) disk , $0.6 and 0.2, respectively for both loadings.

Power draw and collision energy
The quantification of variables like power draw and collision energy help in understanding the role of RDD on the overall mill performance. Figure 5 shows the power drawn (P) by the mill for each configuration at different media loadings. The power drawn does not vary with mill configuration for media loading up to 60%. However, it increases exponentially at higher loadings. As RDD increases, there is a drop in power drawn by the mill. This is more pronounced at media loading above 70% for RDD-20, RDD-10, and regular configuration. For media loadings, 70, 75, and 80%, the rate of decrease in power draw is higher when disk diameter is reduced from 100 mm (Regular) to 80 mm (RDD-20) compared to the decrease in power draw when the diameter is reduced from 80 mm (RDD-20) to 60 mm (RDD-40). The performance of the mill varies significantly with a slight change in diameter at high loadings. The collisions inside the mill can be categorized as media-media and media-device. The media-media collisions contribute to the useful energy of milling, that is, energy utilized in the size reduction of ore, whereas media-device collisions are responsible for wear on the disk and the shell. Figure 6 shows the effect of disk diameter reduction on collision energy for media-media collisions (E m ) (Figure 6(a)) and media-device collisions (E d ) (Figure 6(b)) at mill loadings of 70-80%.
The collision energies, both media-media and mediadevice, are high for a regular disk configuration as compared to RDDs, indicating a high conversion of power drawn for grinding as well as for wear. However, both E m and E d decreases in the case of RDDs. Thus, RDD configuration can lower the wearing of components at the expense of a decrease in grinding rate. The decrease in E m and E d with the implementation of RDD is more significant at higher media loadings, i.e., 75 and 80%, whereas it is relatively low in the case of 70% loading. Further, an analysis of the grinding rate through collision energy can be found in Section S7 of Supporting Information.

Predictive model and optimization
Based on the results so far, it can be observed that mill performance in terms of grinding efficiency and wear rate changes due to the implementation of RDDs. However, this  change is significant only when mill loading is >70%. To determine the optimum settings of two key variables, disk diameter, and mill loading, single objective optimization was carried out. Since optimization requires a certain number of function estimations, simulations were first conducted through DEM for a limited number of settings, and the following statistical models were derived, using the simulation data for E m and E d as functions of RDD and media loading.
Here, E p m and E p d are predicted media-media collision energy and media-device collision energy, respectively. x a is the disk diameter of the first two disks in mm, x b is media loading in percentage. The model is limited to 80 mm x a 100 mm and 70% x b 80%. The last terms on the RHS of both equations represent the interaction between media loading and disk diameter. The statistical metrics, such as the adjusted R 2 value and the p-value for the E p m model are 0.9446 and 0.0004495, respectively, while those values for of E p d model are 0.9592 and 0.000209, respectively. Please refer to Section S8 of Supporting Information for more details.
The objectives of single objective optimization involve a minimal reduction in grinding efficiency (as represented by maximizing the media-media collision energy) and maximum reduction in wear rate (as represented by minimizing the media-device collision). Several techniques are available in the literature to solve the single objective optimization problem (Yang 2020). In the present work, the particle swarm optimization (PSO), proposed by Kennedy and Eberhart (Kennedy and Eberhart 1995) was employed.
Here, Equations (8) and (9) are used in the objective function, wherein the decision variable is x a . These two equations are converted into a single objective function by multiplying each function with a weightage factor and adding these weighted functions as formulated below: Decision variables: 80 mm x a 100 mm (10) In Equation 10, x 1 represents the weightage factor that can assume values between 0 and 1. The weightage factors for E m and E d are x 1 and (1 À x 1 ). This ensures that the sum of the weightage factors is always 1 and increasing the weightage factor for E m decreases the weightage for E d and vice versa. The numerical values of E m and E d are normalized to vary between 0 and 1 such that large numerical values of E m do not bias and affect the optimization results.
The normalized values of E m and E d are evaluated as: The E l m and E u m in Equation (11) represent the least possible and the highest possible values of E m , respectively for the range of x a and x b . We aim to minimize E d and maximize E m , simultaneously. Therefore, to formulate Equation (10) as an optimization problem that minimizes a single objective function, the normalized E m ð Þ is multiplied by À1. Solving the optimization problem formulated as shown in Equation (10) provides the optimal values of x a that minimizes E d and maximizes E m simultaneously, for given values of media loading.
To understand and analyze the solution of optimization problem, multiple single objective optimization problems were solved with different weightage factors (x 1 ¼ 0.2, x 1 ¼ 0.5, and x 1 ¼ 0.8) and different media loadings % (70, 71, … , 80). Hence, a total of 33 optimization problems were solved with the above combinations of weightage factors and media loadings.
The function evaluations were studied to understand the results of all the 33 optimization problems. Figure 7 demonstrates the effect of RDD and mass loading on the objective function. In the case of x 1 ¼ 0.2, the objective function is always minimum for RDD-20. When x 1 ¼ 0.8 the value of the objective function is minimum for regular configuration. However, in the case of x 1 ¼ 0.5, the evaluated values of the objective function are at the minimum value for RDD-20 for mass loadings ranging from 70 to 75% and at the minimum value for regular configuration for mass loadings ranging from 75 to 80%. Table 3 summarizes the optimization result in different cases.
The results presented here are useful from the point of identifying optimum operating conditions to minimize wear of the disks and the shell lining. One of the key improvements that need to be addressed in the future is the influence of fluid on the media dynamics. The current model needs to be extended by coupling the DEM with a CFD model for hydrodynamics and updating the data-driven models to arrive at more accurate optimization results.

Conclusions
A discrete element method (DEM) based mathematical model was developed for a laboratory scale IsaMill TM and tested with published experimental data (Jayasundara et al. 2010). The role of RDD, a design strategy adopted by the plant engineers to overcome the frequent wear of the mill, on power draw, collision energy, normal and shear stresses on mill parts was investigated through numerical simulations. Implementation of RDDs results in a change in flow profiles of media inside the mill. The normal and shear stresses on the mill components decrease upon implementation of RDD, which indicates a reduction in wear loss due to impact and abrasion. Analysis of collision energy to indicate a reduction in wear when RDDs are employed as a decrease in media-device collision energy is observed. There is a significant reduction in the power draw, especially at high media loadings, when RDDs are used. The mediamedia collision energy, which is an indirect measure of grinding performance, also decreases with the implementation of RDD. The interplay between grinding performance and wear loss determines the choice of RDD and media loading. A single-objective optimization approach using PSO Figure 7. Value of objective function with respect to the media loadings for regular and RDD-20 configurations evaluated using the particle swarm optimization algorithm with weights (a) x 1 ¼ 0.2, (b) x 1 ¼ 0.5, and (c) x 1 ¼ 0.8. Table 3. Optimization cases and solutions.

Case
Description Weightage factor (x 1 ) Media loading (%) Preferred configuration 1 When wear reduction is prioritized over grinding performance 0.2 70-80 RDD-20 2 When grinding performance is prioritized over wear reduction 0.8 70-80 Regular 3 When both grinding performance and wear reduction are given equal weightage 0.5 70-75 RDD-20 75-80 Regular was implemented to identify optimal configurations according to media loading. Based on the weightage factor used in formulating the objective function, three cases were identified. In the first case, when wear reduction is prioritized over grinding performance, the optimization result suggests using a configuration with RDD, preferably with more diameter reduction, for all media loadings. In the second case, when grinding performance is prioritized over wear reduction, the optimization result suggests using regular configuration for all media loadings. However, in the third case, when grinding performance and wear reduction are equally prioritized, the optimization result suggests using a configuration with a lower diameter for media loadings <75% and a regular configuration for media loadings above 75%. The results obtained from the optimization algorithm are further explained through the value of the objective function obtained in each case. The present study not only elucidates the implications of using RDD instead of the regular disks in the IsaMill TM but also guides in deciding the configuration of RDD according to the operational needs.