# The Evolution of Grinding Mill Power Models

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## Abstract

Mill power models have been used in a variety of ways in industrial practice since power directly equates to throughput and fineness of ground product. We first start with Hogg-Fuerstenau Power Model and show how this model successfully predicted the power draw of many grinding mills in several mining operations. Then, we show how this model was on the verge of being able to predict the influence of lifter design on power draw. Next, we describe the discrete element model and how it overcame the issues faced by the previous power model. Using a DEM software known as Millsoft, we show the influence of lifter design geometry on power draw and analyze the power draw of rubber lifters versus the steel lifters via several case studies. As years passed, the two-dimensional discrete element model imbedded in Millsoft is superseded by three-dimensional discrete element method. Due to the gigantic computational power of graphic processing units, new computational codes that can do the tumbling motion along the entire length of the mill has come about. Here, we show the predictive capability of Blaze-DEM for ball and SAG mills.

## Keywords

Grinding Mill Power Discrete element method## 1 Introduction

In a mining operation grinding mill power draw is an essential operating parameter since the energy spent in milling the ore contributes significantly to the cost metal production. Besides, the energy spent per ton of processed in the mill influences the degree of size reduction in the mill. As a result, several mill power models have been published in the last 50 years. Of notable interest is Austin’s mill power equation [1] and Morrell’s more detailed yet semi-empirical power model [2]. Daniel et al. have described the consolidation of several mill power models [3]. In this manuscript, we focus on one such power model published by Hogg and Fuerstenau [4]. The extension of this model to autogenous and semi-autogenous model is found in Sepulveda [5].

## 2 Hogg and Fuerstenau Mill Power Model

The balls are carried around with the drum until a point is reached where gravitational forces are just balanced by the centrifugal forces. At this point, balls are released from the shell and they follow a parabolic “free-flight” trajectory and then reenter the ball charge. The point of projection into free-flight starts on the circumference of a circle of radius *g*/2*ω*^{2}, known as Davis circle.

*α*in Fig. 2).

*ρ*

_{app}is the apparent charge density (including voids filled with slurry),

*g*is the gravitational constant,

*L*is the mill length,

*N*

_{c}is the fractional critical speed,

*D*

_{m}is the mill diameter,

*α*is the angle of repose of the ball charge, and

*b*is the distance of the equilibrium surface from the mill center. The angle

*θ*

_{0}is related to the mill filling

*J*by

### Rate of Breakage in Mills

The size-discretized population balance model has been the subject of evolution in the 1970s, and by 1990s, this model has been imbedded in flowsheet simulation software. For instance, MODSIM, USIMPACK, JKSIMMET, and Moly-Cop Tools incorporate the population balance model in a variety of ways to compute mill discharge distribution of ball and autogenous and semi-autogenous grinding mills. Herbst and Fuerstenau [6] made a seminal contribution to the “rate of breakage” parameter used in this model. This key relationship between time rate of breakage and energy input enabled the use of experiments done in a small laboratory mill to study the behavior of plant scale mills.

where *m*_{i}(*t*) is the mass fraction of material in the *i*th size interval at time *t*, *S*_{i} is the size-discretized breakage rate function for the *i*th interval, and *b*_{ij} is the size-discretized breakage function, representing the fraction primary breakage product from the *j*th size interval which appears in the *i*th size interval.

*i*with power

*P*and particle holdup

*H*,

*S*

_{i}

^{E}is a constant dependent only on the hardness or softness of the material in the mill. Since the product of specific power,

*P*/

*H*, and time is equal to the specific energy input to the mill,

*E*, the population balance model is written in “energy normalized form” as

Equation (4) implies that the evolution of size distribution within the mill is only dependent upon the specific energy in put to the mill. In other words, a large ball mill operating with two orders of magnitude in power and feed rate to the mill exhibits the same breakage regime as that of a laboratory size mill operating at the same energy input. This result is another seminal contribution to come out of Fuerstenau’s studies that led to successful scale-up studies. Herbst et al. (1982) proved the scale-up from 10 × 11.5-in. ball mill to 30 × 18-in. continuous flow mill. In a similar study, Herbst et al. [8] demonstrated successful scale-up from the 10 × 11.5-in. ball mill to 14 × 22-ft. plant-scale mill. Because of number of such studies, this scale-up procedure is imbedded in Moly-Cop Tool’s “Ball Mill Simulation” worksheet.

Now, this energy-based rate of breakage has an implication on grinding regimes within ball mills. In ball mill grinding, it has been well established that grinding of particles occurs within the ball charge mainly due to shearing action of the cascading ball charge. The successful scale-up from a mill diameter as small as 10 in. to as large as 14 ft. and higher implies the following: the shear energy per unit mass of particles present in the lab mill is similar in magnitude to the shear energy in the cascading regime of a plant mill. This is reasonably possible for medium critical speed of the mill; the static region of the ball mass moves along with the mill shell (in motion) and then after reaching the shoulder region the bed of balls cascade freely and return to the static region at the toe region. Therefore, the shearing action in the cascading region can be similar in both the lab mill and plant mill. It is just that the shear volume and particle mass is just proportional to mill size and hence the energy normalized breakage rate is nearly the same in both mills. This idea was the point of careful examination for one of the authors of the current manuscript, and that was the seed for the birth of the discrete element simulation of charge motion in ball mills [9, 10].

In adapting the power formula for ease of use, the Moly-Cop Tool designers have converted the Sin *θ*^{3}_{0} term in terms of mill filling, *J*, including voids. Because of this tool box, the power equation has seen extensive use in the hands of authors as well as many mines which use this tool package routinely.

## 3 The Discrete Element Method

The mill power models were undergoing several refinements in the 1980s. However, a new method known as the discrete element method, due to Cundall and Strack [11], was sweeping the landscape for the simulation of an assemblage of particles in motion due to external forces. The first application of this method to the prediction of power in tumbling mills is due to Mishra and Rajamani [9]. In this method, the contents of the mill charge, instead of being modeled as a single mass, is given individual identity to each and every ball. The grinding balls are modeled as spheres and the mill shell is modeled as a long cylinder and the exact geometry of the lifter bars is modeled with exact planar dimensions. The key simulation feature is that as the mill shell rotates, it imparts momentum to the spheres that are in immediate contact with the shell, and these spheres in turn impart momentum to the spheres that they are in contact with and so on. Hence, the discrete element method (DEM) simulation divides the simulation into two parts: (i) contact or collision between sphere elements, cylinder element and lifter element, and (ii) the movement of the sphere elements because of collisions. In the DEM scheme, the collision is modeled via a contact-force model known as spring-dashpot model and then the Newton Laws of Motion is carefully implemented to follow the motion of the discrete elements. Hence, this physics-based model offers a significant advancement over the empirical models. Since this manuscript is focused on mill power draw, the contact-force model and the calculation of power is briefly described in the following.

**F**_{n},between two particles is given by

*δ*is the overlap between the elements,

**V**_{R}is the relative velocity between the two colliding spheres,

*k*

_{n}is the spring stiffness,

*C*

_{n}is the viscous damping coefficient, and \( \overline{\boldsymbol{n}} \) is the normal vector at contact. The tangential force is modeled by the spring-dashpot in the tangential direction as shown in Fig. 3. The tangential force,

**F**_{T}, is given by

**V**_{T}is the relative tangential velocity,

*μ*is the friction coefficient,

*k*

_{t}is the tangential spring stiffness, and

*C*

_{t}is the tangential damping coefficient. The spring in both the directions represents the elastic deformation of the colliding spheres, and the dashpot dissipates a proportion of the collision energy. In tumbling mill simulation, the energy expended in all the collisions occurring over a specified time (division by time) represents the net power draw of the mill and hence the damping coefficients are carefully chosen. In fact, the damping coefficient is related to the coefficient of restitution,

*ε*, between the two colliding particles as

*m*

_{eff}= (

*m*

_{1}

*m*

_{2}∕

*m*

_{1}+

*m*

_{2}) is obtained from the mass of the two colliding particles (

*m*

_{1},

*m*

_{2}).

In the discrete element simulation of tumbling mills, the energy expended in the collisions between ball to ball and ball to mill shell is summed over one or two full revolutions of the mill to predict net power draw. Correspondingly, in the case of SAG mills, the collision pair between balls, ore particles, mill shell, and liner is summed.

The advantage of DEM simulation over the empirical models is that besides predicting power, it gives a realistic picture of the cascading and cataracting balls and ore particles within the mill. More importantly, DEM can delineate the differences in power draw due to different designs of lifters. The difference in power due to widely different lifter design can be as high as 15% of total net power draw.

## 4 Millsoft—Two-dimensional DEM Simulation of Charge Motion

The very first software for two-dimensional simulation of tumbling mills incorporated the physics of collision expressed in Eqs. 7–9 [9]. The mill was modeled as a two-dimensional circular slice of width equal to the diameter of the largest particle in the charge mass. Then, the lifter geometry in its exact dimension is imposed on the mill circle followed by the insertion of spherical particles. For two full revolutions of the mill, the simulated collisions and trajectory of particles are computed. Then, the power draw of the two-dimensional slice was computed. The power draw of the full length of the mill was computed by multiplying the computed power by the ratio of the mass of charge in the full length to the mass in the slice. Surprisingly, even though the collisions in the length direction have been suppressed in the simulation, the power predictions are extremely close to the plant data. This software continues to provide valuable insight into the charge motion and power draw to the mining industry [9]. Furthermore, the charge animation provides clear picture of direct collisions against mill shell and the shape of the cascading charge. Hence, this software led to the design of lifters for ball mills and SAG mills.

## 5 DEM Computations with Graphic Processing Units

In the last 5 years, the three-dimensional simulation of tumbling mills is a major step forward. A full 3D simulation of a mill gives much more valuable insights into the dynamics within the mill. To begin with, such a simulation incorporates the collisions in the length direction of the mill besides the planar direction of the mill cross section. It can include the influence of feed and discharge-end lifters on the charge motion. Therefore, the mill power prediction is much more reliable with such simulation. Furthermore, the 3D simulation can also simulate the flow of charge through the grate-plate into the pulp lifter, as well as the dynamics of flow within the pulp lifter. Hence, one can visualize the back flow and the carry-over flow. As a result, the 3D simulation offers the pathway to study liner wear. More importantly, it can quantify the carry-over flow and the back flow inherent in pulp lifters in SAG mills. In this manuscript, we present only the power draw prediction of the 3D simulation.

An emerging trend in the past few years is the implementation of graphic processor units (GPU) for large-scale computations, such as computing with millions of particles in a plant-scale mill. The GPU offers computing performance like that of cluster computing with multiple CPUs, except at a fraction of the cost. GPU computing has been proven for a speed-up of 50× [12] and even 132× [13] over CPU implementation, for mill charge motion.

## 6 Results and Discussion

## 7 Millsoft and Blaze-DEM Mill Simulation Results

## 8 Conclusion

The evolution of mill power equations by empirical reasoning and the discrete element method was described in this manuscript. The empirical equation by Hogg and Fuerstenau, although formulated on a simple concept of charge being lifted from the toe to the shoulder of the mill, predicts power very close to operating plant data. It is shown that this equation ball mill power draws very reasonably. However, SAG mill power prediction requires additional development.

The next step is power prediction by the discrete element method–based tumbling mill simulations. Since the density of grinding balls and ore particles is considered in this method, it is equally applicable to ball mills as well as SAG mills. The two-dimensional DEM code is easy to use, and it does not require extra computational resources. These codes, when simulating mills with under 10,000 particles, can predict mill power with greater accuracy. For an operating mill, after confirming the power with simulation, the simulation tool can be used for examining the power draw for a variety of lifter designs. Today, such codes are used in North and South America and Australia in the mining industry.

The three-dimensional DEM simulations codes hold great promise. In such simulations, millions of particles are followed in the collision calculations and hence the code execution is computationally intensive. The advances made in GPU computing have made the price of computing hardware for such tasks at par with the price of a laptop computer. The first of such GPU codes, called Blaze-DEM, was shown to produce accurate power predictions in this manuscript. The promise of three-dimensional computing is that the transport of material through the grate and pulp lifter to can be simulated. The carry-over flow and back flow inherent to pulp lifter dynamics can be simulated for gaining knowledge for its design. It should be mentioned that these flow regimes are internal to the mill and hence there are no observable measurements in practice.

The future of DEM is in its ability to include the breakage of ore particles in the simulation. However, this is an extreme degree of challenge even for GPU computing. The particle numbers increase by 100-fold after tens of breakage events. Further, modeling ore bodies whose hardness vary unpredictably across the size spectrum is cumbersome. Yet, such efforts are underway in the past 5 years.

## Notes

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that there is no conflict of interest.

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