Heterarchical modelling of comminution for rotary mills: part I—particle crushing along streamlines

Abstract

Rotary mills aim to effectively reduce the size of particles through a process called comminution. Modelling comminution in rotary mills is a challenging task due to substantial material deformation and the intricate interplay of particle kinematics of segregation, mixing, crushing, and abrasion. Existing particle-based simulations tend to provide predictions that cannot cope with the large number of particles within rotary mills, their wide range of sizes, and the physics dictating the crushing of individual particles. Similarly, there is currently no deterministic modelling means to determine the evolving population of particle sizes at any point in time and space within the mill. The aim of this two-part contribution is to address these gaps by advancing a framework for a novel stochastic comminution model for rotary mills, which has a well-defined deterministic continuum limit and can cope with arbitrarily large numbers of particles. This work describes the basic physics and structure of the new model within a heterarchical framework for ball and autogenous grinding mills. The primary focus of this Part I paper is to develop a computational model for the integration of motion of material along streamlines inside a mill. Coupled to this process is the kinetic physics dictating particle crushing. In a subsequent work, Part II, segregation and mixing will be added to this model such that realistic behaviour from the mill can be observed.

Graphical Abstract

1 Introduction

Comminution is the process of reducing the size of particles, which in mineral processing is achieved using different machinery such as crushers, rotary mills, vertical mills, and high pressure grinding rolls. These crucial unit operations are overwhelmingly energy intensive. The mineral processing industry is estimated to consume 3–4% of global energy [29, 38], where nearly half of that energy is spent in the milling operation [6, 28, 61, 62]. Here, our focus is on the rotary mills. Currently, there is no deterministic model that can explain the particle size dynamics within these mills. In this study, we introduce a novel approach based on the multiscale heterarchical modelling paradigm [43] to unveil the particle size dynamics within these mills.

The most common statistical approach to model comminution is the population balance model (PBM) [12, 14, 47, 69]. However, the major drawback of these models is that they treat the particles irrespective of their position in the mill, with the mill acting essentially as a black box. The purpose of PBM models is to track the distribution of sizes of all the particles in the mill, irrespective that at any point within it that distribution actually varies strongly. Ignoring this fact, the performance of PBM models hinges upon the empirical calibration of global breakage and selection functions. The calibration is typically made for a given set of operating parameters by curve fitting the results against the observed product size distribution, when available. Beyond the range used for their original calibration, these models tend to fail, and cannot provide sufficient insights. This is because they are not informed by either the geometry of the mill or the composition of the material. As such they cannot say much about the role of physical entities, such as the sizes of the grinding balls, the widths of the lifters, the opening of the grates from which the material flows, or the mineralogy of the particles. In summary, while PBM models may provide a first step into the problem, they cannot be used with confidence, especially when it comes to the potential optimisation of the performance of mills in terms of their vast energy consumption and wear.

Unlike PBM, in recent years the discrete element method (DEM) has been introduced to model comminution in a way that attempts to track individual particle motions and their sequential crushing [16, 18, 19, 65]. One of the problems with DEM models is that they are overly expensive computationally, and thus they cannot actually track all the many particles rotating in a typical mill. For example, considering a typical product size distribution in industrial autogenous grinding (AG) mills [11] with sizes ranging from 10 μm to ~100 mm, the number of particles in the mill may be estimated to be over $10^{14}$. Nowadays, even the most advanced DEM models supported by high-performance computers may only track up to $\sim 10^{6}$ particles over the long typical lifetimes of minerals within the mill. This means that DEM models can only handle $\sim 10^{8}$ times fewer particles than reality. While such models can still provide excellent information regarding the bulk kinematics of the particles and the typical bulk stresses that develop within the mills, they cannot be used to predict the real liberated particle surfaces during the process, which controls the energetic performance of the mill. With particle surface area scaling quadratically with particle size, DEM models may only liberate $\sim 10^{16}$ times less actual fracture surface than those liberated within real mills. In a related way, the fracture criteria implemented in DEM models tend to grossly simplify the intricate sub-particle processes that drive the crushing of individual particles, as the surfaces of the fragments created are substantially limited. Another problem with DEM models, which is linked to their high computational cost, is that unlike PBM models they tend to provide noisy predictions, which often overwhelm qualitative trends that modellers attempt to portray. In short, while aiming to resolve the shortcomings of PBM models in tracking physical processes, DEM models generate noisy results and fail to generate anywhere close to the amount of surfaces and fragments created in real mills.

In an attempt to resolve the noisiness issue of DEM models, research in geomechanics has introduced hierarchical multiscale models, which can theoretically track the distribution of particles. In particular, the finite element method and the discrete element method (FEM-DEM) [4, 34, 50] have been coupled, whereby DEM simulation replaces the constitutive model at the Gauss point in an FEM simulation. This multiscale model is hierarchical in nature, as it couples two different models for two distinguishable scales—the DEM operating at the particle scale, and the FEM at the continuum engineering scale. As described in [43], hierarchical multiscale FEM-DEM models may only capture particle size dynamics in closed systems, which do not involve segregation and mixing between the Gauss points [40, 63, 70]. However, for open systems such as grinding mills, the particles continuously advect and swap places through mixing and segregation. This preferential motion by size and density prevents hierarchical multiscale models from predicting particle size dynamics in grinding mills. When the particles move and deform, using these hierarchical multiscale models essentially means we need to transfer information (i.e., particle size in this context) from one Gauss point to another in the finite element simulation, which is not feasible due to the separation of scales in their structure.

Here we resolve the practical issues of using DEM and PBM for rotary mills by advancing an alternative multiscale approach, which unlike hierarchical DEM-FEM models is heterarchical. The heterarchical model has its origin in the stochastic lattice model of [42], which only described the open system processes of segregation and mixing. Their work illustrated the continuum limit of the corresponding stochastic open-system mechanisms. Subsequent work [43, 44] combined the closed-system stochastic mechanism of crushing heterarchically with the open-system mixing and segregation mechanisms for canonical scenarios. Finally, [35] demonstrated that the stochastic closed-system crushing mechanism also has a continuum limit.

In the following, we adopt the original ideas advanced by [43] to simultaneously describe the mechanisms of crushing, mixing, and segregation of particles. Just like PBM models, the heterarchical model is capable of accommodating an arbitrarily large number of particles. However, unlike PBM models, the heterarchical approach can further provide information regarding the particle size distributions at any point in time and space within the rotary mill domains. This new approach follows a stochastic structure with a well-defined continuum limit for open system particle size dynamics such as segregation and mixing [43] and closed system ones such as crushing [35]. This multiscale approach is an alternative to the other hierarchical multiscale approaches as here there is no scale separation between the representative volume and the continuum scale, both scales coexist within a single framework. This facilitates the transfer of information (i.e., particle size) from one scale to another as well as within the representative volume scale.

Although significantly less computationally intensive than DEM models, the heterarchical model is more demanding than PBM models. On the other hand, when it comes to representing physical information at the finer scale, the situation is inverted — the heterarchical model cannot track the motion of individual particles in the way DEM can, but unlike PBM it can track the dynamics of the population of particle sizes in both space and time. Overall, the heterarchical model offers a sweet spot for modern computational power and physical realism. Nevertheless, so far heterarchical models were limited to simple flow systems that may be idealised using only one spatial coordinate, such as those occurring during the granular avalanches, landslides, and debris flows [7, 25, 54, 55]. Furthermore, thus far the description of crushing in the framework ignored the role of kinetic particle collisions, dealing solely with quasi-static comminution.

In this study, we extend the heterarchical framework to track comminution in higher spatial dimensions. For the first time, the approach is used to model comminution in rotary mills. To deliver these goals the heterarchical approach is here coupled with the streamline method, originally called the strain path method in [5], which has been used in the past to integrate constitutive laws for predicting penetration resistance [26] in geomechanics. Here, rather than integrating constitutive models, the streamline method would be used to integrate the heterarchical method.

In summary, the current two-part study offers three significant novel contributions. Firstly, we extend the crushing model within the heterarchical approach to account for kinetic crushing, which goes beyond quasi-static conditions in previous publications. This factor is shown to be particularly important for rotary mills, where the particles are often advected through highly turbulent regions. Secondly, we integrate the heterarchy and the streamline methods to investigate the progression of comminution in rotary mills. Lastly, by tessellating the material domain within the mill for equal mass material points along streamlines, the new streamline-heterarchical method enables us to model mixing and segregation in rotary mills. These contributions collectively advance the understanding of comminution processes and offer new insights into the intricate dynamics occurring in rotary mills.

This is the first paper of a two-part series. Here, we present the application of heterarchy in rotary mills for two cases: a ball mill¹ and an AG mill. In this Part I, we discuss comminution by solely considering the mechanism of crushing. The other two important mechanisms of mixing and segregation will be addressed in Part II [9]. The current paper is organised as follows: We begin by explaining the heterarchical model for quasi-static comminution and its extension to kinetic comminution. Next, we synthesise the kinematics of granular flow in rotary mills, as an essential step for the streamline method. Finally, we discuss the implementation of heterarchy along streamlines in rotary mills.

2 Heterarchical model for comminution

This section starts with a brief introduction to the basic concepts of the heterarchical model [43] for quasi-static comminution, with a focus on particle crushing. The capacity of the framework to handle further crushing in dynamic settings is then extended by allowing for kinetic comminution, which is expected to develop in rotary mills due to frequent particle collisions.

Fig. 1

Illustrative example for the structure of a heterarchical model. On the left, a physical one-dimensional polydisperse granular system. On the right, an equivalent heterarchical model. The physical granular system is discretized into a number of representative volume elements (RVEs) along the physical coordinate. The particles within each RVE are then further mapped along the micro-structural coordinate into an array of particle sizes $s_i^t = (s_{i,1}^t, s_{i,2}^t,\ldots , s_{i,M}^t)$ at time t with periodic boundary (PB) conditions (colour figure online)

Fig. 2

The stochastic comminution rule of the heterarchical model for a cell (i, z) under externally applied stress $\sigma _a$. At any given time t, if the external stress acting on the particles $(\sigma _a)$ is greater than or equal to the crushing strength $(\sigma _{i,z}^{s^t}(\mathbf{\vec{s}}))$ of particles, where $\mathbf{\vec{s}}=\{s_{i,z-1}, s_{i,z}, s_{i,z+1}\}$, the particle size in that cell is reduced by a factor $X_{i,z}^t$ in the next time step $t+\Delta t$. Otherwise, the particle size remains unchanged (colour figure online)

In the heterarchical framework, the physical system of a polydisperse granular medium is discretized into a number of representative volume elements (RVEs) each containing a statistically significant number of particles as illustrated for a one-dimensional column of particles in Fig. 1. The particles within each RVE are further mapped into an array of M number of cells. This mapping of particles into M cells is assumed to be randomly uncorrelated. However, in real RVEs, the positions of particles may be correlated to their relative sizes. Eventually, it should be possible to develop DEM and computed tomography (CT) investigations with which such possible correlations could be uncovered and later adopted in the heterarchical model. The internal coordinate, along which the volume within a RVE is discretized, is called the micro-structural coordinate and it is such that the position along this coordinate represents the local neighbourhood of particles. The model can be defined in a lattice in any number of spatial (external) and micro-structural (internal) dimensions. The spatial and micro-structural coordinates together form a higher dimensional framework such that each coordinate is orthogonal to one another.

In Fig. 1 the first index (i) represents the spatial coordinate of the position of the RVE and the second index ($z \in [1, M]$) represents the micro-structural coordinate along which the RVE is discretized. Furthermore, the system is deemed periodic along the heterarchical coordinate z. The number of cells (M) along the heterarchical coordinate (z) should be sufficiently large so that upon averaging it gives a smooth particle size distribution (PSD) for a RVE. Each cell contains particles of uniform size and the total mass of particles in a cell remains constant even though the particle size of a cell may evolve. This may be due to crushing, as detailed in this Part I of the paper, as well as by mixing and segregation, as explained and assessed later in Part II [9]. Moreover, the heterarchical model considers that the crushing of particles in a cell is dependent not only on the properties of the particles in the cell, but also on those in their neighbouring cells. Therefore, at any instant a given cell has to consider the information of its neighbouring cells (immediate neighbours). For cells 1 and M, i.e., first and last cells, the immediate neighbouring cells are (M, 2) and $(M-1, 1)$, respectively, owing to the periodic boundary condition.

2.1 Quasi-static comminution

To model comminution we need to define the conditions for the onset of particle crushing and the fragment size distribution for crushed particles. The onset of crushing occurs at any time t when the externally applied (bulk) stress ($\sigma _a$) is greater than or equal to the crushing strength ($\sigma _{i,z}^{s^t}(\mathbf{\vec{s}})$) of the particles. If this criteria is met, at the next time step $t+\Delta t$, the particle size at cell location (i, z) is reduced by some factor. Otherwise, the particle size in that cell is kept unchanged (see Fig. 2). The above comminution rules are applied to each cell of the RVE, and each cell of the RVE is subjected to the same external applied stress ($\sigma _a$). Even though the external stress is applied to all of the cells of the RVE equally, the stress causing crushing of individual particles within the RVE will later be shown to depend on the particle size distribution, as elaborated in [43]. Accordingly, the particle size of the $(i,z)^{th}$ cell evolves as

$$\begin{aligned} s_{i,z}^{t+\Delta t} = {\left\{ \begin{array}{ll} X_{i,z}^t s_{i,z}^t &{} \text {if}\;\;\sigma _a \ge \sigma _{i,z}^{s^t}(\mathbf{\vec{s}})\\ s_{i,z}^t &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

(1)

where $X_{i,z}^t$ is the factor determining how much the size is reduced, and is a random variable between 0 and 1 drawn from the fragment size distribution. Commonly used fragment size distributions include the Weibull and power law distributions [15, 21, 52, 56]. Here, the distribution is based on Weibull’s two-parameter model as described in [43]. We also impose the condition on the minimum size of the fragment ($s_{min}$) generated, which in this study has been set to $s_{min} = 1~\mu$m. Therefore, the probability density function of the fragment size distribution f(x) over $x\!\in \! (x_m,1]$ is given as

$$\begin{aligned} f(x) = \frac{\frac{k}{\lambda } \left( \frac{x-x_m}{\lambda } \right) ^{k-1} \exp \left[ -\left( \frac{x-x_m}{\lambda }\right) ^k\right] }{1-\exp \left[ -\left( \frac{1-x_m}{\lambda }\right) ^k \right] }, \end{aligned}$$

(2)

where $x_m = s_{min}/s$. The parameters k and $\lambda$ can be determined experimentally for the fragment size distribution of any given mineral particle [68].

Note that in the current model, the particle sizes in the neighbouring cell affect only the onset of crushing. It is acknowledged that the sizes of neighbouring particles may also affect the fragment size distribution of a crushed particle. However, here this possible phenomenon is neglected.

In the original work by Marks & Einav [43], for the $z^{th}$ cell of the $i^{th}$ RVE, the crushing strength of particles under quasi-static conditions was expressed as

$$\begin{aligned} \begin{aligned} \sigma _{i,z}^s (\mathbf{\vec{s}}) = \sigma ^m \left( \frac{s_{i,z}}{s^m} \right) ^{-3/w_s} \exp \left[ \frac{\log ^2(s_{i,z}/\overline{s}_{i,z})}{2n^2} \right] , \end{aligned} \end{aligned}$$

(3)

where $\mathbf{\vec{s}}=\{s_{i,z-1}, s_{i,z}, s_{i,z+1}\}$ represents the particle size in the considered cell as well as its nearby neighbours; $\sigma ^m$ the crushing strength of particle corresponding to the maximum particle size $s^m$ in the system; $\overline{s}_{i,z} = (s_{i,z-1} + s_{i,z+1})/2$ the average local neighbourhood particle size; $w_s$ the Weibull modulus for strength; and n is a non-dimensional scaling constant as described by [43]. In the above equation, the second term (power law term) accounts for the effect of particle size on the crushing strength primarily due to the presence of internal flaws within the particles [45]. The third term (exponential term) accounts for the cushioning effect on that strength [8, 43], where smaller particles tend to shield bigger particles from crushing. As described in [43], this effect can be related to the distribution of stress within the RVE in terms of the particle size distribution.

2.2 Kinetic comminution

The previous section describes the crushing of particles subjected to external loading under quasi-static loading conditions. However, in certain scenarios, such as in industrial mills or during rapid granular avalanches, the particles frequently collide with each other. These collisions promote further crushing due to higher impact velocities, which can happen even when the external pressure acting on them is relatively small. Figure 3 depicts such a collision between two particles, which results in their crushing into smaller fragments.

Fig. 3

Schematic representation of kinetic crushing of particles during high deformations. Each particle experiences an independent velocity that fluctuates from the local mean velocity field, as depicted by arrows over the particles. The particles inside the highlighted box (left) have a higher relative velocity at time t, which may result in crushing. The figure on the right shows the crushed state of the particles at time $t+\Delta t$. The length of the arrow (not to scale) represents the magnitude of the velocity of the particles

Considering an assembly of jiggling particles, the effect of the collisions on the crushing strength under dynamic conditions ($\sigma _{i,z}^d$) could be related to the (measurable [41]) mean fluctuating velocity ($v'$), which represents the typical difference between particle velocities and their ensemble average at any particular point in space and time:

$$\begin{aligned} \sigma _{i,z}^d(\mathbf{\vec{s}},v') = \sigma _{i,z}^s(\mathbf{\vec{s}})\cdot \mathcal {K}(v',s_{i,z}), \end{aligned}$$

(4)

where, $\sigma _{i,z}^s$ is the crushing strength of particles under quasi-static loading as specified by Eq. 3, while $\mathcal {K}$ is introduced as a kinetic strength reduction factor that is here taken to depend on the fluctuating velocity $v'$ and the particle size $s_{i,z}$.

To determine this kinetic strength reduction factor one may consider the vast experimental observations in [32, 57, 58], which show that the survival probability of the particles against crushing depends on their impact velocity. As will be shown in the following section, typical impact velocities in AG mills are within the range of the experimental observations described in [32]. Under higher impact velocities variables other than the impact velocity may affect the kinetic strength reduction factor [13].

The Weibull distribution [66] is a popular and successful representation of this survival probability, and so the kinetic strength reduction factor is here taken to follow the same statistics

$$\begin{aligned} \mathcal {K}(v',s_{i,z}) = \exp {\left[ -({v'}/{v'_{50}})^{w_v}\right] }, \end{aligned}$$

(5)

where, $v'_{50}=v'_{50}(s_{i,z})$ is the fluctuating velocity of a particle of size $s_{i,z}$ corresponding to 50% survival probability against crushing, and $w_v$ is the Weibull modulus for velocity. Both of these parameters ($v'_{50}$ and $w_v$) can be determined experimentally for a given mineral. Importantly, the work in [58] showed that $v'_{50}$ is also a function of the particle size (here $s_{i,z}$) and can be calculated as

$$\begin{aligned} v'_{50}(s_{i,z}) = a_v \exp {(-b_vs_{i,z}/s_r)}, \end{aligned}$$

(6)

where $a_v$ and $b_v$ are correlation parameters and $s_r$ is a reference particle size which in this case has been taken as equal to the maximum particle size, i.e. $s_r = s^m$.

Finally, notice that $v'$ represents the magnitude of the velocity fluctuations, and thus can only be positive. Therefore, it is important to furthermore consider the typical direction of the fluctuations, since during expansion particles tend to move apart, so we would not expect collisions and associated crushing. Provided with the field of velocities (v) in the rotary mill, one can calculate the volumetric strain rate from $\dot{\varepsilon _v}=-\nabla \cdot \mathbf{\vec{v}}$ to determine whether the medium expands or contracts. With that in mind, the particle size of a heterarchical cell in the general heterarchical comminution model for rate dependent, dynamic loading evolves according to the modified crushing criteria below

$$\begin{aligned} s_{i,z}^{t+\Delta t} = {\left\{ \begin{array}{ll} X_{i,z}^t s_{i,z}^t &{} \text {if}\;\; \sigma _a\cdot \mathcal {H}(\dot{\varepsilon _v}) \ge \sigma _{i,z}^{d^t}(\mathbf{\vec{s}},v') \\ s_{i,z}^t &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

(7)

where, $\mathcal {H}$ is the Heaviside step function, (i.e. during volumetric expansion $\dot{\varepsilon _v} < 0$ and $\mathcal {H}(\dot{\varepsilon _v}) = 0$, so the particles do not crush, otherwise $\mathcal {H}(\dot{\varepsilon _v}) = 1$, so they may crush).

Fig. 4

Kinematics for an idealised slowly rotating ball mill using the analytical solution by Ding et al. [24]. a Flow profile for the rolling regime in a ball mill. Contour plots: b Velocity magnitude (streamlines superimposed in white colour), c Volumetric strain rate (positive in compression), d Shear strain rate, and e Fluctuating velocity (colour figure online)

Fig. 5

Coarse-grained kinematics from DEM for an autogenous grinding mill (AG). a Snapshot of particle distribution and their sizes for the cataracting regime in an AG mill. Contour plots: b Velocity magnitude (streamlines superimposed in white colour), c Volumetric strain rate (positive in compression), d Shear strain rate, and e Fluctuating velocity (colour figure online)

3 Continuum fields

As shown in the previous section, the crushing of particles within rotary mills depends on local continuum information regarding kinematic entities such as the velocities and strain rates, as well as stress variables. Therefore, in the following, we describe how these fields could be obtained for the granular material within rotary mills.

The methodology in this paper will follow a one-way coupling scheme. Specifically, the use of the kinematic and stress fields will be based on either analytical or numerical proxies for the flow of non-crushable particles. This has certain limitations since the fields retrieved ignore the fact that the particles actually crush in those mills. The tacit assumption here is that crushing would not substantially affect the measured continuum velocity and stress fields. In other words, for simplicity, the fields determined below are decoupled from the crushing that happens in the mill. However, the extent of crushing that will be modelled later is determined based on those fields. In reality, we might expect some consolidation of the pack within the mill, with additional effects potentially arising due to frictional lubrication by the small fragments produced, which might alter their rheological behaviour and impact the adopted fields. However, these effects are here considered negligible for all practical purposes.

The kinematics of the bulk material flow within rotary mills can be estimated through a variety of methods, including experimental, numerical, or analytical calculations. Experimentally, this may include imaging techniques such as particle image velocimetry [37], positron emission particle tracking [53], and radioactive particle tracking [3]. Numerically, the kinematic fields could be evaluated using the discrete element method (DEM), or using large-deformation finite element simulations [51, 59] equipped with constitutive granular rheology models [31]. Analytically, the bulk velocity of the particles inside the mills are often established by considering the aforementioned experimental and numerical observations, with additional input from either engineering intuition or fundamental imposition of continuum compatibility requirements  [3, 24].

For demonstration purposes, the current study will explore the use of kinematics from both analytic and numerical sources. Specifically, we will demonstrate the use of the analytical velocity solution by Ding et al. [24] for ball mills operating under slow rotation rates and the use of a numerical velocity field for the case of an AG mill.

3.1 Slowly rotating ball mill

The details of the analytical solution for the ball mill are discussed in the Appendix 1. Ball mills are typically operated under a Froude number $F_r$ (the ratio of centrifugal force to gravity) that falls within the range $10^{-3}<F_r< 1$ [46]. Note that one of the primary purposes of this study is to demonstrate the application of the heterarchical approach using an existing analytical velocity solution. In this case, we consider the available and relatively simple analytic solution by [24]. However, it should be noted that this solution is applicable only for rotation speeds $(10^{-4}<F_r< 10^{-2}$) lower than the usual operating speeds of ball mills. Under such a rotational speed, the bulk of the material predominantly rotates as a rigid body. As the material reaches its dynamic angle of repose, it rolls down along the surface. The flowing layer is very thin as compared to the pseudo-rigidly rotating bulk (see Fig. 4a). The velocity varies linearly along the depth of this flowing layer with a maximal speed achieved at the free surface. Additionally, the velocity also varies along the length of this layer with a maximum residing at the centre of the layer. Based on this analytical solution the velocity and strain rate fields for the granular flow in a ball mill can be calculated, as shown in Fig. 4b–d, respectively.

Furthermore, in the case of the ball mill, the fluctuating velocity ($v'$) of the particles (see Fig. 4e), as required from Sect. 2, can be derived as

$$\begin{aligned} v' = \sqrt{{P_k}/{\rho _b}}, \end{aligned}$$

(8)

where $P_k$ is the kinetic pressure and $\rho _b$ is the bulk density.

The kinetic pressure can be estimated for shear-induced flow in terms of the dimensionless kinetic number ($I_k$), as introduced by [2], which can be expressed as $P_k = P_s I_k$, with $P_s$ being the static pressure. The calculation of bulk stresses acting on particles is discussed in the following subsection. Furthermore, the kinetic number can be approximated from [2] as $I_k = I$, where $I = \dot{\varepsilon _s}\overline{s}\sqrt{\rho _m/P}$ is the inertial number [22], $\dot{\varepsilon _s}$ the shear strain rate, $\overline{s}$ the mean particle size, $\rho _m$ the material density, and P the total pressure (sum of static pressure and kinetic pressure). Given these relations, the kinetic pressure may be estimated analytically from:

$$\begin{aligned} P_k = \dot{\varepsilon _s}\overline{s}\sqrt{\rho _m P}. \end{aligned}$$

(9)

Finally, the bulk static pressure ($P_{s}$) is assumed to be a linear function of depth ($y \cos \varphi$) measured from the free surface, i.e.

$$\begin{aligned} P_{s} = \rho _b gy\cos \varphi , \end{aligned}$$

(10)

where $\rho _b$ is the bulk density, g is the acceleration due to gravity and $\varphi$ is the dynamic angle of repose of material. The calculation of bulk density will be discussed in Sect. 4.3. The modelled static pressure field for the idealised slowly rotating ball mill is shown in Fig. 6a.

Fig. 6

Static pressure field for: a Slowly rotating ball mill using Eq. 10, and b AG mill using coarse-graining (see Appendix 2) from DEM simulation (colour figure online)

3.2 AG mill

The numerical simulations from which we retrieve the above quantities for the AG mill are based on a standard DEM model, whose technical details are provided and discussed in Appendix 2. The DEM simulations have been used to extract the required continuum field variables using coarse-graining [30, 67] for the AG mill case. These continuum fields are consistent with previous results by others for rotary mills [14, 18, 19]. In this case, the mill is operated at a high rotational speed ($0.1<F_r<1$) [46] such that we generally observe a cataracting flow [18, 20, 46] which is characterized by the individual particles detaching from the bed and being thrown off into free space inside the mill as shown in Fig. 5a. The obtained continuum velocity, strain rate, and fluctuating velocity fields from the DEM simulation are shown in Fig. 5b–e, respectively.

In addition to information on the kinematics of the particles, we also need to have information about the static stresses acting on them at any point in space and time inside the mill. For the case of the AG mill, the corresponding static pressure field is extracted from the DEM simulation using coarse-graining, as described again in Appendix 2. The static pressure field for the AG mill is shown in Fig. 6b.

4 Integrating heterarchy along the streamlines

The preceding sections described the essential components of our comminution model, i.e., the heterarchical model of kinetic crushing of particles and the kinematics of the particles within rotary mills. In this section, we delve into the implementation of the heterarchical model for the rotary mills, marking the second novel contribution of this study.

The concept of heterarchy is extended to the rotary mills by leveraging the potential of the streamline method, previously known in geomechanics as the strain path method [5]. Specifically, past work by [26] had demonstrated the effectiveness of the streamline method by integrating advanced constitutive assumptions along streamlines to determine the penetration resistance of natural soils subjected to large deformations. Rather than integrating constitutive assumptions, the novelty here is in integrating instead the heterarchical model along the streamlines to form a new powerful tool with which we could investigate the evolution of the particle size distributions at any point in time and space within the rotary mills.

4.1 Streamlines

Using the velocity fields in Sect. 3 for both the ball and the AG mills, the first step is to obtain a corresponding set of streamlines. As described earlier, the flow is assumed to follow steady state kinematics, and thus the streamlines should form a closed loop in order to allow the material to flow continuously along them as the mill rotates. The following section explains our methodology for obtaining these closed sets of streamlines, which play a crucial role in the new comminution model.

Fig. 7

Streamlines in: a the slowly rotating ball mill obtained using the analytical velocity solution by Ding et al. [24], and b the AG mill obtained using Spline interpolation from the coarse-grained DEM velocity field. The streamlines are superimposed over the velocity vector data (shown in red colour) from DEM. The streamlines represented by dotted lines are the master streamlines, while the rest are interpolated using them. The number of streamlines in panels a and b are just for visualisation, while in the actual simulations the number of streamlines is much higher

4.1.1 Slowly rotating ball mill

The streamlines in the ball mill are established by considering the velocity fields in both the active and passive regions of the flow (see Appendix 1). In the active region, the velocities are distributed linearly along the depth of flow in the streamwise direction (X-axis) and the velocity along the other direction (Y-axis) is negligible. Consequently, the particles can be assumed to move in a linear path within the active region, from which we define in that region a set of streamlines parallel to the X-axis, with their endpoints lying on the zero-velocity ($\alpha$) line.

Fig. 8

Schematic allocation of material points along streamlines for the flow within an example rotary mill. The material points are distributed by marching along the length of the streamlines in discrete time-step $\Delta t$. The material points shown here are just for visualisation, while in the actual simulations they are distributed by adopting a very small time step. The granular mass within a material point is then further discretized along the micro-structural coordinate. Such a scheme is applied to both the ball and the AG mills (colour figure online)

In the passive region, the particles move along the drum as a rigid body, allowing us to define the streamlines as circular arcs, with their endpoints positioned at the interface of the active and passive regions ($\delta$-line). Alternatively, one could also derive a stream function using the analytical solution for the velocity and determine the streamlines for the flow in the active and passive regions.

To ensure that the streamlines form closed loops, the streamlines are assumed to follow circular arcs between the active-passive interface ($\delta$) and the zero-velocity ($\alpha$) line, as defined in Appendix 1. This approach indeed produces closed streamlines, as illustrated in Fig. 7a.

Notice, however, that the consequence of the analytic solution is that we must introduce a velocity component in the Y-direction for the portion of the streamlines between the $\delta$-line and the $\alpha$-line. This addition is necessary to enable the material to flow along the circular curve since the particles initially possess velocity solely in the X-direction within this region.

4.1.2 AG mill

The scheme for constructing closed streamlines for the AG mill is different since the kinematics have been established thanks to the coarse-grained velocity field derived from the DEM simulation. Given this velocity field, the streamlines for the granular flow can be directly generated using MATLAB built-in function streamline. However, given the inevitable numerical noise, the streamlines that are generated from MATLAB would not necessarily form the required set of closed streamlines.

To obtain closed streamlines, we adopt the following alternative approach. We start by selecting a subset of streamlines generated from MATLAB, ensuring they cover the entire flow region. In order to close these streamlines, we strategically add additional data points where needed and then generate a Spline function [27]. Future work could be incorporated to replace the method of streamline determination with a more objective automatic approach. These carefully selected streamlines are referred to as master streamlines for the bulk flow within the AG mill. By employing the Spline function, we can effectively interpolate any number of streamlines between any of these master streamlines, as illustrated in Fig. 7b. This technique allows us to obtain a set of closed streamlines that accurately describe the granular flow inside the AG mill.

4.2 Material points on streamlines

Once the closed streamlines for the bulk material flow are obtained, the subsequent step is to allocate discrete set of material points on the streamlines. Each material point is defined as a representative volume element (RVE) in space, which holds a fixed mass of particles. To achieve this, we march along the length of a streamline in discrete time-steps $\Delta t$, as illustrated in Fig. 8. The time-step $\Delta t$ is chosen to be sufficiently small, ensuring that the evolution of the particle sizes within a volume element remains independent of its value. Also, the total number of streamlines considered for the bulk material flow should be sufficiently high such that the evolution of the overall particle size distribution for the whole material inside the mill is independent of the number of streamlines.

The position of a material point along the streamline at any time-step can be calculated using the Verlet integration scheme [36] as

$$\begin{aligned} X(t+\Delta t) = X(t) + v(t)\Delta t + \tfrac{1}{2}a(t) \Delta t^2, \end{aligned}$$

(11)

where $\Delta t$ is the time-step, v and a are the bulk velocity and acceleration, respectively, obtained from the velocity field. The term $\tfrac{1}{2}a(t)\Delta t^2 \approx 0$ in Eq. 11 for very small accelerations, as long as the time-step ($\Delta t$) chosen is very small. Therefore, $X(t+\Delta t ) = X(t) + v(t)\Delta t$ is also a good approximation to determine the position of a material point in subsequent time-steps. Considering the above, it is important to make a couple of observations with respect to the distribution of the material points along the streamlines:

• 1. For the slowly rotating ball mill case, near the vicinity of the zero-velocity line ($\alpha$-line) the accelerations are actually significant since the velocities transform rapidly from $v=0$ on the $\alpha$-line to some non-zero value on the $\delta$-line, given by the analytic velocity solution. Therefore, in Eq. 11 the term $\tfrac{1}{2}a(t)\Delta t^2 \ne 0$, and does actually have a significant role. In fact, it is the acceleration that drives the material when it lies on the $\alpha$-line, since $v = 0$ at that location.

• 2. For the AG mill case, the streamlines are obtained using Spline interpolation. It was observed that the gradient of a streamline may not always be precisely oriented along the velocity vector obtained from the DEM simulation, as depicted in Fig. 7b, where the interpolated streamlines are superimposed over the velocity vectors obtained from DEM simulation. However, while distributing the material points along the streamlines, we neglect this second-order effect and assume that the velocity vectors are parallel to the streamlines.

The positions of the material points along the streamlines remain as such and do not need to be updated further during the simulation.

Table 1 Simulation parameters

4.3 Tessellation into subvolumes

Given a set of material points along the streamlines, the material domain within the mill is tessellated into subvolumes. To compute the subvolume occupied by a material point in space we have used the Voronoi neighborhood method [1] as shown in Fig. 9. The volume will therefore be equal to the area of the Voronoi polygon assuming unit length along the mill axis. These Voronoi subvolumes are here required for the purpose of mass calculation, while in the following paper (Part II) [9] they will be further required for implementing the segregation and mixing mechanisms.

Each of the Voronoi subvolumes are then further discretised along the micro-structural coordinate (z) into M cells as shown in Fig. 8. Therefore, the whole volume of a material point is divided into M subvolumes, each containing particles of different sizes which evolve as the system undergoes crushing.

Fig. 9

Volume of material points for the granular flow in AG mill calculated using the area obtained from Voronoi polygons [1]. The length along the axis of the mill is assumed to be unity. The polygons are coloured randomly (colour figure online)

The mass of particles in a given material point can therefore be obtained as $\mathcal {M} = \rho _b V$, where $\rho _b$ is the bulk density and V is the volume of a material point calculated using the Voronoi polygon method.

For the AG mill, the bulk density field is obtained from the DEM simulations, where the resulting spacing of the material points is determined using the associated velocity field. This purely numerical approach leads to a certain level of variation between the masses of the material points, which in theory needs to be constant. In the case of the current simulation, this variation/error is at the order of $13.8\%$. In the following, we consider the effects of this variation to be negligible. In the future, a correction scheme may be developed to alleviate this issue so as to ensure the constancy of the material point masses.

For the slowly rotating ball mill, the bulk density field is obtained by integrating the bulk density along the material points of a streamline using a suitable integrating scheme. Correspondingly, the material points are spaced using the analytical velocity field. As such, this method does in fact ensure the constancy of the material point masses. Here, using the forward Euler method, the bulk density field can be calculated as:

$$\begin{aligned} \rho _{b_{\{p,q \}}} = \rho _{b_{\{p,q-1 \}}} + \rho _{b_{\{p,q-1 \}}}\dot{\varepsilon _v}_{\{p,q-1 \}}\Delta t, \end{aligned}$$

(12)

where the indices $p\in [1, N_s]$, and $q\in [1, N_p]$ represent the streamline and the corresponding material point along it. $N_s$ is the total number of streamlines and $N_p$ is the total number of material points along a given streamline. $\dot{\varepsilon _v}$ is the volumetric strain rate and $\Delta t$ is the time step that we chose to discretise the streamlines into discrete material points.

5 Simulations and results

It is now possible to demonstrate how the newly defined ‘heterarchical streamline method’ may be used to analyse the process of comminution within rotary mills. So far the method can only accommodate for the physics of kinetic particle crushing, leaving aside the treatment of further effects of segregation and mixing to the next Part II [9] of this work. Here, we show the application of this model in predicting the particle size distribution at any point in space and time in rotary mills, for the examples of slowly rotating ball and AG mills using the previously defined fields.

Fig. 10

Cumulative distribution function (CDF) of product particle sizes along different streamlines after one mill revolution. a Slowly rotating ball mill, and b AG mill. The insets show the actual location of the corresponding streamlines in the flow region over the field of magnitude of the fluctuating velocity. An ultimate product size distribution for the entire minerals within the mills is also plotted. The number of cells used along the micro-structural coordinate for each simulation are $M = 10^3$ (colour figure online)

Fig. 11

Cumulative distribution function (CDF) of ultimate product particle sizes after one and two revolutions. a Slowly rotating ball mill, and b AG mill. This shows that when only the crushing mechanism is activated in the simulation, there is no significant crushing after one revolution of the mill. (As will be shown in Part II [9], this phenomenon disappears when considering further particle size dynamics through the mechanism of segregation and mixing)

Fig. 12

Evolution of mean particle size ($d_{50}$) normalized with respect to the feed mean particle size ($d_{50}^0$) for variation in values of impact velocity parameters $a_v$, $b_v$ and $w_v$. The base values of the parameters are listed in Table 1. Only one of the parameters is varied at a time keeping the other two parameters equal to their base values. [Top - slowly rotating ball mill] The horizontal axis represents the depth of the flow measured from the free surface and normalised with respect to the maximum depth of zero velocity line ($\alpha _0$). [Bottom - AG mill] The horizontal axis corresponds to the number of streamlines denoted from 1 to $N_s$ as shown in the inset at the bottom figure e (colour figure online)

The geometrical and mechanical parameters used for the simulations are listed in Table 1. All these parameters have a clear statistical meaning and can be determined experimentally. The crushing strength parameters ($\sigma ^m$ and $w_s$) can be determined experimentally for any mineral using several single particle crushing tests [48, 49]. The impact velocity parameters ($a_v$, $b_v$ and $w_v$) can also be determined experimentally for any given mineral using impact crushing tests [15, 60, 64].

Note that for this study, we use the geometrical and mechanical parameters which are typical for an industrial-scale AG mill. For the case of our conceptual ball mill, we use scaled parameters that are obtained for the AG mill, specifically the mineral strength parameters. At this stage, we have not explicitly included the grinding media in the heterarchical model, which plays a crucial role in the comminution for ball mills. We assume the crushing in the ball mill occurs only due to impacts between the particles and therefore, the scaled mineral strength parameters in the simulation represent a weak crushable mineral.

For the AG mill simulation, we considered a mill with a diameter (d) of 6 m rotating at 13.5 rpm, and the feed consisting of particles uniformly distributed between 20 and 100 mm in size. These operating parameters lie within the range that is typically associated with actual AG and SAG mills [11, 47]. The crushing strength parameters for this simulation correspond to the mineral Feldspar and are based on single particle crushing tests by  [48]. The fragment size distribution parameters are also for the mineral Feldspar derived from the single particle crushing results by [68]. The impact velocity parameters could be determined experimentally for the mineral chosen in this study. However, we rather estimate these parameters based on the impact crushing results of [32] for cement-mortar balls. Here, our assumption is that these values would not differ much for the minerals. In order to check the sensitivity of these impact velocity parameters on crushing, we do a parametric study towards the end of this paper.

For the conceptual ball mill simulation, we utilised a mill of diameter (d) of 30 cm rotating at 6.3 rpm speed, and the feed consisted of particles with a uniform size distribution ranging between 2 and 3 mm. The value of the crushing strength parameter ($\sigma ^m$) was chosen to represent a weak, crushable mineral and its value is taken two orders of magnitude lower than the crushing strength of the mineral used in the AG mill simulations. Similarly, we also scale the impact velocity parameters for the ball mill based on the corresponding value obtained for the AG mill.

Given the parameters above, and the heterarchical streamline method, we can track the evolution of particle size at any point in space and time inside the ball mill. Figures 10a, b illustrate the cumulative distribution functions (CDFs) of particle size for different streamlines in the flow region after one revolution of the ball mill and the AG mill respectively.

The CDF plots provide clear insights into the influence of the fluctuating velocities on the crushing of particles in the granular flow. Notably, streamlines passing through regions of higher fluctuating velocity experience more crushing. In addition to generating CDF plots for individual streamlines, it is instructive to plot the overall or ultimate CDF that represents the entire material within each given mill, after one mill revolution. This product CDF is the weighted average by mass of the CDFs for each streamline, which is specified by

$$\begin{aligned} F_u(s) = \frac{1}{\mathcal {M}_{total}} \sum \limits _{p=1}^{N_s}\sum \limits _{q=1}^{N_p} F_{p,q} (s) \times \mathcal {M}_{p,q}, \end{aligned}$$

(13)

where $F_{p,q}(s)$ is the cumulative mass distribution function, $\mathcal {M}_{p,q}$ is the mass of particles in a material point for a given streamline and $\mathcal {M}_{total}$ is the total mass of particles in the system which can be calculated as

$$\begin{aligned} \mathcal {M}_{total} = \sum \limits _{p=1}^{N_s}\sum \limits _{q=1}^{N_p} \mathcal {M}_{p,q}. \end{aligned}$$

(14)

The results of the ultimate CDF of the particle sizes are shown in Figs. 10a, b for the ball and the AG mills, respectively. These ultimate CDF plots give novel information about the overall extent of crushing in the mill at any given point in time.

Another significant finding from the simulations is that without segregation and mixing, no crushing could happen within the system after one revolution of the mill, as indicated by the CDF plots. Figures 11a, b show the evolution of the CDF of the ultimate product particle sizes for the ball and AG mills, respectively. The CDFs do not evolve with any further rotation of the mill beyond one revolution. This behaviour is attributed to the current simulation setup, where the particles are only undergoing crushing without any mixing and segregation. As the particles crush, they attain a relatively stable configuration along the material points such that they are cushioned against further crushing. With the heterarchical stochastic physics of mixing and segregation (developed in Part II) [9] not being activated at this stage, there is no mechanism that will disturb this cushioned state of particles after one revolution. However, with the introduction of mixing and segregation, as we will see in the next Part II [9] of this work, the particles would no longer remain protected from further crushing after one mill revolution, allowing the particle size distribution to continuously evolve as the mill undergoes further revolutions.

The heterarchical model introduced in this study includes several parameters, namely $a_v$, $b_v$, and $w_v$, which are dependent on the impact velocity (or fluctuating velocity) of particles. Here we provide a short demonstration of the effects of these parameters on the crushing simulations. In Fig. 12, we illustrate the variation of the mean particle size ($d_{50}$) with changes to these parameters for both the ball and the AG mills. It can be seen from the plots that for both the ball mill and the AG mill, the mean particle size varies significantly with the variation of the parameters $a_v$ and $w_v$ while there is no significant change in mean particle size with variation of the parameter $b_v$. Overall, it can be concluded that the mean grain size increases with an increase in the value of parameter $a_v$ while it decreases with an increase in the value of parameter $w_v$.

Lastly, we present the numerical robustness of the heterarchical model. This robustness is demonstrated by varying the number of cells (M) along the microstructural coordinate (z) and monitoring the resulting evolution of the product size for any streamline. As shown in Fig. 13 for the AG mill simulation case, the results indicate that even with $M=10^2$, meaning ten times fewer cells than what is used in the present study, we obtain stable results that are independent of the number of heterarchical cells. In addition to the numerical robustness, the results show that for a very high number of heterarchical cells, we get a smooth deterministic particle size distribution. This reaffirms the fact that the heterarchical model has a well-defined continuum limit as previously demonstrated by [35].

Fig. 13

Cumulative distribution function (CDF) of the product particle sizes for a streamline of the AG mill as a function of the number of cells (M) along the heterarchical coordinate (colour figure online)

6 Discussion

We have introduced a new methodology to model industrial comminution, utilising the innovative concept of heterarchical multiscale modelling combined with the streamline method. This work (Part I and II) serves as the foundational basis for the heterarchical comminution model, uncovering the particle size dynamics occurring inside rotary mills. However, at this stage, the model does not capture the entire particle size dynamics of the rotary mills.

This Part covers the particle size dynamics of kinetic crushing occurring due to impacts and the second Part covers the particle size dynamics of segregation and mixing. Other than impact crushing, there are other modes of particle breakage in rotary mills such as abrasion or attrition, chipping, and incremental breakage [17]. These modes of breakage are not incorporated yet into the heterarchical model and require further investigation.

In addition to these particle size dynamics, further considerations remain to be explored for incorporation into the heterarchical model. Firstly, including grinding media into the mill charge, which plays a crucial role in particle crushing for ball and SAG mills. Secondly, addressing the input and output material flux. In actual mill operation, there is a continuous flux of material with finer particles exiting the mill from the discharge end and fresh feed material being fed from the other end. This rate of mass flux will influence the grinding of the material. Thirdly, considering the influence of water in the grinding operation. The water is expected to help reducing material friction while adding viscosity, maintain charge density, and control the discharge of fine materials through the mill. Lastly, considering the effect of lifter design on mill performance, and controlling the wear of lifters and grinding balls. These objectives form part of this ongoing project, and further developments will contribute to a more comprehensive and versatile heterarchical comminution model for the mineral processing industry.

7 Conclusions

This study presents a new method based on stochastic heterarchical multiscale modelling for studying comminution in rotary mills. This new approach allows us to obtain the particle size distribution at any point in space and time within the mill domain and it can deal with arbitrarily large numbers of particles, allowing us to accurately account for the polydispersity of particle sizes in rotary mills. This method has a well-defined continuum limit as it gives smooth deterministic results for a large number of heterarchical cells.

The highlights of this paper are: Firstly, we derive the heterarchical rule governing the kinetic crushing of particles, which is predominant in rotary mills due to frequent particle collisions. Secondly, we build up the comminution model for rotary mills by integrating the concept of heterarchy and the streamline method.

Here, we uncover the physics of particle size dynamics occurring inside the mill with a focus exclusively on the crushing mechanism. The investigation of the other two important particle size mechanisms of mixing and segregation is reserved for Part II [9]. Finally, We demonstrate the application of the heterarchical model in predicting the time evolution of particle size distribution for a slowly rotating ball mill and an AG mill.

Using this model, we want to empower operators with the ability to understand the effect of mill operating parameters such as degree of filling, mill speed, feed size distribution, and the fraction of grinding media in the mill charge. By providing this understanding the operator can decide on the optimum value of these parameters, leading to enhanced mill performance.

Appendices

Appendix 1: Analytical solution for velocity in slowly rotating ball mills

The analytic solution [24] for the bulk flow in ball mills is suitable for low rotational speeds in the so-called rolling flow regime ($10^{-4}<F_r<10^{-2}$, with $F_r$ being the Froude number [46]). The rolling flow regime [3, 23, 46] is characterized by a near levelled free surface inclined at the dynamic angle of repose $(\varphi )$ of the material. The flow can be divided into two distinct regions, i.e., an active region and a passive region as shown in Fig. 4a. The passive region occupies the bulk of the material. In this region, the particles move as a rigid body and are carried up along the ball mill with a velocity equal to the angular velocity of the mill. As the particles reach the free surface, they start to cascade down into the active region which is characterized by shear and volumetric deformations. The active region of the flow is where we would expect most of the particle size dynamics from crushing, mixing, and segregation of particles to occur. The interface between the active and the passive region is defined by the $\delta$-profile. In the active region, the velocity is maximum at the free surface and it decreases along the depth of flow. The point where the velocity becomes zero is defined by the $\alpha$-profile.

The velocity solution is derived using the momentum and mass balance equations. Here, we present the solution for the velocity. The stream-wise velocity (along the x-direction) in the active layer can be calculated as

$$\begin{aligned} u = u_s - \left[ \frac{1+\Lambda }{\Lambda }u_s - \frac{\Lambda }{1-\Lambda }\omega (H+\delta ) \right] \left( \frac{y}{\delta } \right) + \left[ \frac{u_s}{\Lambda } - \frac{\omega (H+\delta )}{1-\Lambda } \right] \left( \frac{y}{\delta } \right) ^2, \end{aligned}$$

(15)

where H is the depth of the free surface from the centre of the ball mill, $\delta$ is the depth of the active-passive interface measured from the free surface, $\omega$ is the angular velocity, $u_s$ is the free surface velocity, $\Lambda = \alpha /\delta <1$ is the parameter defining the rheological properties of the flowing material, and $\alpha$ is the depth measured from free surface to the point of zero streamwise velocity (see Fig. 4a).

The active-passive interface ($\delta$) profile is given as

$$\begin{aligned} \delta= & {} \Bigg (-\left[ \frac{3\Lambda -1}{\Lambda }u_s + \frac{4-3\Lambda }{1-\Lambda }\omega H \right] \nonumber \\{} & {} \quad + \sqrt{\left[ \frac{3\Lambda -1}{\Lambda }u_s + \frac{4-3\Lambda }{1-\Lambda }\omega H \right] ^2 + \frac{12\omega ^2(2Lx - x^2)}{1-\Lambda }} \Bigg )\nonumber \\{} & {} \quad \bigg /2\left( \frac{\omega }{1-\Lambda } \right) , \end{aligned}$$

(16)

where we have used the parabolic free surface velocity profile given by [3]:

$$\begin{aligned} u_s = \frac{\omega L^2}{\alpha _0}\left[ 1 - \left( \frac{x-L}{L} \right) ^2 \right] , \end{aligned}$$

(17)

in which L is the half chord length of free surface and $\alpha _0 = \alpha _{x=L}$

The transverse velocity (along the y-direction) is negligible [3] and is thus ignored. The passive region is typified by rigid body rotation, where the velocity can simply be calculated as

$$\begin{aligned} v = \omega (y+H). \end{aligned}$$

(18)

Figure 14a shows the variation of the free surface velocity ($u_s$) normalized with respect to the maximum free surface velocity ($u_m$). The velocity variation along the depth at the centre of the flow is shown in Fig. 14b. It can be seen from the figure that for the depth of the flow between $y\ge \alpha$ and $y<\delta$ in the active region, the mass moves along with the drum but the flow is not a rigid body motion.

Fig. 14

Velocity distribution obtained using the analytical solution of Ding et al. [24]. a Free surface velocity normalized with respect to its maximum value, and b Velocity variation along the depth as measured at the midpoint of the free surface

Appendix 2: DEM simulation for AG mill

The Discrete Element Method (DEM) simulations were carried out using the open source software Liggghts [39]. For simplicity and comparison with industrial conditions, the parameters of the simulation are given with dimensions in the following. A total of 45,000 particles are simulated with a uniform diameter distribution by number between 0.02 and 0.1 m, and density 2500 kg m$^{-3}$. The contact force between the particles follows a non-linear (hertzian) inelastic frictional law with Young’s modulus of 0.5 GPa, Poisson’s ratio of 0.3, restitution coefficient of 0.6, and friction coefficient of 0.5 [10]. The particles are integrated in time using a Verlet algorithm with timestep 10 μs, low enough to ensure proper resolution of tracking collisions between particles.

The particles are placed in a cylindrical mill of diameter 6 m and length 2 m rotating around its horizontal axis at a speed of 13.5 rpm, leading to a typical Froude number of 1.4. The perimeter of the mill is lined with 40 lifters of height 10 cm to prevent slippage between the particles and the mill and to mimic industrial equipment. Contact properties between walls and particles are the same as between particles.

After the particles are inserted and the rotation started, the flow eventually reaches a steady state within less than two mill revolutions, from which point onwards all the particle properties (location, velocity, contact forces, etc.) are extracted every 0.01 s for around 10 s (till coarse-grained fields become stationary and smooth for the following analysis), which corresponds to 2.25 mill revolutions at the given mill speed of 13.5 rpm. This avoids correlation between the different snapshots of the simulation and allows to perform time averaging. The spatial averaging is performed using the coarse-graining method indicated in [67], by specifically following the implementation detailed in [33]. The fields are calculated on a grid in space using a Lucy function as the coarse-graining window $w$, with a width $c=0.4$ m:

$$\begin{aligned} w(x) = \frac{105}{16\pi c^3} \left( -3 \left( \frac{x}{c}\right) ^4+8 \left( \frac{x}{c}\right) ^3 - 6 \left( \frac{x}{c}\right) ^2 +1\right) {\mathcal {H}}(c-x). \end{aligned}$$

(19)

The bulk density $\rho _b$ and velocity $\textbf{v}$ fields are obtained using:

$$\begin{aligned} \rho _b(\textbf{x})= & {} \sum _i m_i w(|\textbf{x}-\textbf{x}_i|) \end{aligned}$$

(20)

$$\begin{aligned} \rho _b \textbf{v}(\textbf{x})= & {} \sum _i m_i \textbf{v}_i w(|\textbf{x}-\textbf{x}_i|) \end{aligned}$$

(21)

where $\textbf{x}_i$, $\textbf{v}_i$, $m_i$ are the position, velocity, and mass of particle i, respectively, and $\textbf{x}$ is the position at which the field is estimated. The velocity field is then time-averaged to obtain $\langle \textbf{v}\rangle$, and the spatial gradients are taken to obtain the strainrate tensor ${\dot{\varvec{\epsilon }}} = \frac{1}{2}(\nabla \langle \textbf{v}\rangle + \nabla ^\text {T} \langle \textbf{v}\rangle )$. The volumetric strain rate is then calculated with ${\dot{\varepsilon _v}} = \text {Tr}\ {\dot{\varvec{\epsilon }}}$, and we define the shear strain rate as ${\dot{\varepsilon _s}} = \sqrt{ ({\dot{\varvec{\epsilon }}} - \frac{\dot{\varepsilon _v}}{3} \mathbb {1}):({\dot{\varvec{\epsilon }}} - \frac{\dot{\varepsilon _v}}{3} \mathbb {1})}$, with $\mathbb {1}$ the identity tensor.

The fluctuating velocity $\textbf{v}'$ is obtained using

$$\begin{aligned} \rho _b{\textbf{v}'} = \sum _i m_i \left( \textbf{v}_i - \langle \textbf{v}(\textbf{x}_i)\rangle \right) w(|\textbf{x}-\textbf{x}_i|), \end{aligned}$$

(22)

where $\langle \textbf{v}\rangle$ is linearly interpolated at the location of particle i from the field calculated at discrete grid points.

Finally, the static pressure $P_s=\text {Tr}\ {\varvec{\sigma }}$ is calculated from the contact forces $\textbf{f}_{ij}$ between particles i and j by

$$\begin{aligned} \varvec{\sigma }= \sum _\text {contacts} \mathbf{{f}}_{ij} \otimes (\textbf{x}_j-\textbf{x}_i) \int _0^1 w(|\textbf{x}_i + s (\textbf{x}_j-\textbf{x}_i)|) ds, \end{aligned}$$

(23)

with the integration performed numerically.