**Kinetics of Dry-Batch Grinding in a Laboratory-Scale Ball Mill of Sn–Ta–Nb Minerals from the Penouta Mine (Spain)**

#### **Jenniree V. Nava 1 , Teresa Llorens <sup>1</sup> and Juan María Menéndez-Aguado 2, \***


Received: 17 November 2020; Accepted: 16 December 2020; Published: 17 December 2020

**Abstract:** The optimization of processing plants is one of the main concerns in the mining industry, since the comminution stage, a fundamental operation, accounts for up to 70% of total energy consumption. The aim of this study was to determine the effects that ball size and mill speed exert on the milling kinetics over a wide range of particle sizes. This was done through dry milling and batch grinding tests performed on two samples from the Penouta Sn–Ta–Nb mine (Galicia, Spain), and following Austin methodology. In addition, the relationships amongst Sn, Ta and Nb content, as metals of interest, the specific rate of breakage *S<sup>i</sup>* , the kinetic parameters, and the operational conditions were studied through X-Ray fluorescence (XRF) techniques. The results show that, overall, the specific rate of breakage *S<sup>i</sup>* decreases with decreasing feed particle size and increasing ball size for most of the tested conditions. A selection function, α*T*, was formulated on the basis of the ball size for both Penouta mine samples. Finally, it was found that there does exist a direct relationship amongst Sn, Ta and Nb content, as metals of interest, in the milling product, the specific rate of breakage *S<sup>i</sup>* and the operational–mineralogical variables of ball size, mill speed and feed particle size.

**Keywords:** ball mill; kinetic grinding; specific grinding rate; Sn–Ta–Nb; Penouta Mine

#### **1. Introduction**

In the mining industry, the comminution stage can represent up to 70% of the energy consumed in a mineral processing plant [1–5]. With ball-mill grinding being one of the most energy-consuming techniques, setting the optimal values of the operational and mineralogical parameters for efficient grinding is a key target in mineral processing plants [6–10]. Ball size is one of the key factors of ball-mill efficiency [11,12], and may have a significant financial impact [13]. The population balance model (PBM) has been widely used in ball mills [14]. This model is a simple mass balance to reduce size being used in fragmentation models [15]. Several methods have been implemented to determine those functions. Some were based on simple laboratory-scale grinding essays [16–21], whereas others were based on industry-scale works [22–26]. This paper focuses on studying the specific rate of breakage *S<sup>i</sup>* and its kinetic parameters based on the Austin methodology [27], which assumes that the specific rate of breakage (*S<sup>i</sup>* ) is a constant of proportionality that may or may not behave as a first-order function, whereas the function of fracture (*Bij* ) does not change with grinding time.

Tantalum and Niobium are considered critical raw material in the EU, due to their features and applications in a wide range of industrial sectors, and the strong EU import dependence [28]. This makes it of paramount importance to increase the research in the mineral deposits that contain them, and to optimize the processing plants to increase their efficiency and to minimize their energy consumption.

One of those processing plants lies in the Penouta Sn–Ta–Nb mine. Currently, it is the only working mine in Europe producing Ta and Nb concentrates as its main product. This is done by reprocessing the tailing ponds generated by the mining works up to the 1980s, and it is pending authorizations to start mining the source rock. Due to that, two types of sample have been studied: (i) unaltered rock from the Sn-, Ta- and Nb-enriched albite leucogranite (Bedrock); and (ii) material from the tailing ponds (Tailings Pond).

The aim of this work was to study the effects of ball size on milling kinetics, operating at different mill speeds and with a wide range of feed particle size. This was done through dry milling and batch grinding tests, following the methodology proposed by Austin et al. [7] and developed in [9,29]. In addition, it studied the relationships amongst the evolution of Sn, Ta and Nb content, as metals of interest, determined by XRF, the specific rate of breakage *S<sup>i</sup>* , and the operational conditions for both samples, Bedrock and Tailings Pond, from the Penouta mine.

#### **2. Theoretical Background**

The population balance model (PBM) has been widely used in ball mills. This model is based on determining the particle size distribution grouped in size classes. A mass balance for the class *i* in a well-mixed grinding process is done by means of Equation (1), where comminution is linear, and a first-order kinetic fragmentation is assumed [19].

$$\frac{dw\_i}{dt} = -S\_i w\_i(t) + \sum\_{j=1}^{i-1} b\_{ij} k\_j w\_j(t) \tag{1}$$

where *wi (t)* is the particle mass fraction of size class *i* at grinding time *t*. The first term of the right-hand side is the mass fraction of particles of the monosize *i* that break and, thus, no longer belong to that monosize. *S<sup>i</sup>* is the specific rate of breakage. The second term represents the contribution of all monosizes coarser than *i* that at breaking produce particles of monosize *i*. The fracture rate or fracture velocity of a monosize material can be expressed by Equation (2):

$$\frac{-dw\_i}{dt} = \mathcal{S}\_i w\_i(t) \tag{2}$$

where *S<sup>i</sup>* is a constant of proportionality called the specific rate of breakage or probability of fracture, whose unit is *t* −*1* . Assuming that *S<sup>i</sup>* does not change with time, the integral results in Equation (3).

$$
\log(w\_i(t)) - \log(w\_i(0)) = \frac{-S\_i(t)}{2.3} \tag{3}
$$

where *w<sup>i</sup> (t)* and *w<sup>i</sup> (0)* are the mass fractions for size class *i*, at grinding times t and 0, respectively. *S<sup>i</sup>* is the specific rate of breakage. Following the methodology proposed by Austin et al. [7] once *S<sup>i</sup>* values have been obtained through slope determination, they are plotted to the particle size, and Equation (4) is proposed to study the behavior of the specific rate of breakage *S<sup>i</sup>* .

$$S\_i = \alpha\_T \cdot X\_i^a \cdot Q\_i \tag{4}$$

where *X<sup>i</sup>* is the upper size limit of the interval (in mm), and α*T*, is a parameter that depends on milling conditions and, is the breakage rate for size x<sup>i</sup> = 1 mm, while α is a characteristic parameter depending on material properties; *Q<sup>i</sup>* is a correction factor, which is 1 for small particles (normal breakage, which was assumed in this case) and less than 1 for large particles that need to be nipped and fractured by the grinding media (abnormal breakage); *S<sup>i</sup>* increases up to a specified size x<sup>m</sup> (optimum feed size), but above this size breakage rates decrease sharply [9].

Rotating critical speed of the mill, *Nc*, is calculated with Equation (5).

$$N\mathcal{c} = \frac{42.3}{\sqrt{D-d}}\tag{5}$$

where *D* is the mill diameter and *d* is the ball diameter (in m). Ball mill filling volume is calculated using Equation (6), assuming that the bed porosity of balls is 40%.

$$J = (\frac{\text{mass of balls}}{\text{ball density} \times \text{mill volume}}) \times \frac{1.0}{0.6} \tag{6}$$

On the other hand, Austin and Brame [25] calculated the selection function α*<sup>T</sup>* in a general way through Equation (7).

$$\alpha\_T = \frac{\upsilon\_\text{c} - 0.1}{1 + e^{\left[15.7(\upsilon\_\text{c} - 0.94)\right]}} \tag{7}$$

where υ*<sup>c</sup>* is the mill speed expressed as the fraction of critical speed.

#### **3. Methodology**

#### *3.1. Sample Characterization*

First, a representative sample of a metric tonne from each of both areas of interest of the Penouta mine, Bedrock and Tailings Pond, was crushed at a size of −4 mm using a jaw crusher. Working samples were obtained after homogenization and quartering using a Jones splitter. Feed monosizes of 3350/2000, 2000/1000, 1000/500, 500/250, 250/125, 125/75 and 75/45 µm were obtained in a sieve shaker using a series of sieves with the openings of above.

Next, feed was characterized by means of grain-size analysis of the above-mentioned size fractions and by means of XRF analysis of fused bead samples using a 4 kW BRUKER spectrometer (Leipzig, Germany), specifically calibrated for this mineralogy, installed in the ALS laboratory at the Penouta mine.

#### *3.2. Calculation of the Critical Speed and Initial Conditions for the Grinding Kinetics Tests*

Critical speed was calculated using Equation (5). Table 1 displays mill rotational speeds as a function of ball monosizes for each test.


**Table 1.** Working speeds for the grinding kinetic tests.

Dry batch milling kinetics tests were done in a lab-scale mill, 17.8 cm in diameter and 4.5 L in capacity, on a 600 cm<sup>3</sup> representative volume of each Penouta mine sample. The mill charge consisted of 5.0 kg of steel balls, of 19.0 mm, 22.0 mm and 31.0 mm monosizes. Fill fraction was calculated from Equation (6). Seven feed size fractions (3350/2000, 2000/1000, 1000/500, 500/250, 250/125, 125/75, 75/45 µm) were used to evaluate the influence of this mineralogical variable in the kinetic parameters. Mill discharges were marked through 5 grinding times (0.5; 1; 1.5; 3.5; 7.5 min). This way, each sample was dumped from the mill, and then it went through a grain size analysis by means of dry sieving. In addition, after completing the grinding time, Sn, Ta and Nb content was determined for the undersize to grid *i* in order to evaluate the evolution of Sn, Ta and Nb grades, with respect to the specific rate of breakage.

#### *3.3. Determination of the Specific Rate of Breakage (S<sup>i</sup> ) and the Kinetic Parameters (*α*T,* α*) α α*

Following the BII methodology introduced in [27], after measuring the oversize weight for each grinding time, the graph log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) vs. time is plotted for each monosize. The equation of each curve and thus the *S<sup>i</sup>* value are obtained through linear fitting using Equation (3). Then, the *S<sup>i</sup>* values for each monosize are plotted and using Equation (4) the parameters (α*<sup>T</sup>* and α) are calculated for each condition of mill speed and ball size. This allows studying the influence of these two operational variables on the specific rate of breakage and the kinetic parameters α*<sup>T</sup>* and α. The selection function α<sup>T</sup> was formulated by means of Equation (9). Nevertheless, this is a general equation, so a specific formula was generated to characterize the samples Bedrock and Tailings Pond from Penouta mine. α α α α α

#### **4. Results and Discussion**

#### *4.1. Chemical Characterisation of the Feed*

Both the Tailings Pond and Bedrock head samples display the grain size distribution shown in Figure 1.

**Figure 1.** Grain size distribution curves for the Penouta mine head samples.

Bedrock and Tailings Pond samples display an *F<sup>80</sup>* of 2110 µm and 1369 µm, respectively. The smaller *F<sup>80</sup>* value of Tailings Pond sample results from this material having been previously processed during the mining activities throughout the 20th century, until the 1980s.

The representative chemical composition for both the Tailings Pond and Bedrock head samples is shown in Table 2 and has been obtained through XRF analysis in the ALS-Penouta lab.

**Table 2.** Chemical composition of Bedrock and Tailings Pond head samples obtained through XRF.


The obtained values are consistent with Polonio [30], taking into account that the tailings pond contains 4,815,307 metric tonnes of material, which, as occurs in this kind of deposits, displays a highly heterogeneous distribution of the metals of interest, in contrast to the homogeneous distribution displayed by the source rock. Furthermore, Sn, Ta and Nb values obtained for the Bedrock sample are within the range reported by [30–32].

#### *4.2. Obtaining the Specific Rate of Breakage (S<sup>i</sup> )*

Figures 2–5 display the relationship between log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) and time for 75% and 85% critical speed and ball size *d* = 1.9 cm, for Bedrock and Tailings Pond samples.

**Figure 2.** (**a**) Plot of log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) vs. time for 75% critical speed and *d* = 1.9 cm (Penouta-Bedrock), (**b**) linear least square fitting performed.

**Figure 3.** (**a**) Plot of log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) vs. time; for 75% critical speed and *d* = 1.9 cm (Penouta-Tailings Pond), (**b**) linear least square fitting performed.

**Figure 4.** (**a**) Plot of log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) vs. time for 85% critical speed and d = 1.9 cm (Penouta Bedrock), (**b**) linear least square fitting performed.

**Figure 5.** (**a**) Plot of log (*w<sup>i</sup> (t)*/*w<sup>i</sup>* (0)) vs. time for 85% critical speed and *d* = 1.9 cm (Penouta Tailings Pond), (**b**) linear least square fitting performed.

Figures 2–5 show a deviation from the straight lines at initial grinding stages. This is probably due to abnormal breakage and, according to [8], it should be performed a pre-grinding stage in the mill for about 2 min in order to avoid abnormal breakage behavior, which was not considered in this study.

Overall, fracture velocity of the feed monosizes fits a first order kinetic behavior, thus, being independent from time. *S<sup>i</sup>* was obtained for each sample using Equation (3), and the slope calculated from Figures 2–5 for each ball-size and mill-speed condition. The relation between the specific rate of breakage *S<sup>i</sup>* , and feed grain size was plotted in Figures 6–9 for each condition to visualize the behavior of *S<sup>i</sup>* , as operating parameters varied for each sample.

*Metals* **2020**, *10*, 1687

**Figure 6.** The specific rate of breakage vs. particle size for selected ball sizes at 75% of working speed (Penouta Bedrock).

**Figure 7.** The specific rate of breakage vs. particle size for selected ball sizes at 85% of working speed (Penouta Bedrock).

**Figure 8.** The specific rate of breakage vs. particle size for selected ball sizes at 75% of working speed (Penouta Tailings Pond).

**Figure 9.** The specific rate of breakage vs. particle size for selected ball sizes at 85% of working speed (Penouta Tailings Pond).

In Figures 6–9, the specific rate of breakage *S<sup>i</sup>* in the usual operational range increases as ball size diminishes [8,10,11,26,33–35], as happens for most of the feed grain sizes at 75% of critical mill speed. Nevertheless, at 85% critical speed, the opposite seems to happen for the Tailings Pond sample shown in Figure 9. This is probably due to better behavior under a greater influence of mill speed and ball size, mainly for the coarse feed particles size as a consequence of a greater influence of the impact breakdown and the cascading effect [36,37]. In addition, the harder ores, such as Tailings Pond samples and the coarser feeds, require high impact energy and large grinding media, and, on the other hand, very fine grind sizes require substantial grinding media surface area and small grinding media [38–40]. As a consequence, medium size balls (d = 2.23 cm) seem to have a better performance for most feed sizes, mill speeds, and samples tested [10,34,41,42].

### *4.3. Kinetic Parameters (*α*,* α*T)*

*α α*

The grinding kinetic parameters for Bedrock and Tailings Pond samples from Penouta mine are shown in Table 3 to study the influence of ball size and mill speed in those parameters.


**Table 3.** Kinetic parameters for several ball sizes and mil speed (Penouta Bedrock and Tailings Pond).

It can be seen that α values fall within the reported normal values [26], and that the selection function α*<sup>T</sup>* varies little with mill speed. From this data, the graph of Figure 10 was constructed. It plots the selection function, α*T,* vs. the ball size, at constant working speed, for the studied samples. *α α α*

**Figure 10.** Graph showing the selection function vs. ball size for Penouta Bedrock and Tailings Pond.

 α<sup>Τ</sup> α From Table 3, the Bedrock sample yields higher α<sup>T</sup> values than Tailings Pond sample, thus, being ground more rapidly than the latter. It must be highlighted that the Bedrock sample was taken from a slightly altered leucogranite, which results in low hardness and fracture strength. On the other hand, and due to its origin, the sandy Tailings Pond sample is heterogeneous, with a higher quartz content. It is a previously processed material and, consequently, with a higher fracture strength. In his study focused on the parameter αT, Teke et al. [33] found a linear trend between that parameter and the ball size, characterizing the mineral calcite in this way. A good approach to determine the selection function from ball diameter in the studied samples is shown in Figure 10 with the Bedrock and Tailings Pond samples characterized through Equations (8) and (9), respectively.

$$
\mathcal{K}\_{T=d\_b+0.1453} \tag{8}
$$

$$\mathcal{A}^{\prime}T = d\_b + 0.0128 \tag{9}$$

்ୀௗ್ା.ଵଶ଼ where α*<sup>T</sup>* is the selection function and *d<sup>b</sup>* is ball size in cm.

*α*

In this sense, the results shown in Figure 10 are sound and agree with the Bond index trends previously reported for the same samples [43]. Other authors [9,44] also compared the features of other rocks like quartzite and metasandstone through the selection function, α*T*. *αΤ αΤ*

#### *4.4. Chemical Characterisation of the Grinding Products*

The results depicted in Figures 11–14 show the relationship between the Sn yield trends and the specific rate of breakage, *S<sup>i</sup>* , for each mill-speed and ball-size condition employed. Tables 4–7 include the Pearson coefficient in each case, showing a better correlation in the case of medium size balls in all cases. ‐ ‐ ‐ ‐

**Figure 12.** Plot of Sn yield and the specific rate of breakage, *S<sup>i</sup>* , vs. feed size at 85% Nc for several ball sizes (Penouta Bedrock).

**Figure 13.** Plot of Sn yield and the specific rate of breakage, *S<sup>i</sup>* , vs. feed size at 75% Nc for several ball sizes (Penouta Tailings Pond).

**Figure 14.** Plot of Sn yield and the specific rate of breakage, *S<sup>i</sup>* , vs. feed size at 85% Nc for several ball sizes (Penouta Tailings Pond).


**Table 4.** Correlation between the specific rate of breakage *S<sup>i</sup>* and Sn, Ta, Nb yields (%) for each ball size (Penouta Bedrock, 75% Nc).

**Table 5.** Correlation between the specific rate of breakage *S<sup>i</sup>* and Sn, Ta, Nb yields (%) for each ball size (Penouta Bedrock, 85% Nc).


**Table 6.** Correlation between the specific rate of breakage *S<sup>i</sup>* and Sn, Ta, Nb yields (%) for each ball size (Penouta Tailings Pond, 75% Nc).


**μ**


**Table 7.** Correlation between the specific rate of breakage *S<sup>i</sup>* and Sn, Ta, Nb yields (%) for each ball size (Penouta Tailings Pond, 85% Nc).

Finally, Figures 15 and 16 depict the plot of Sn, Ta and Nb yield in the undersize product, vs. the specific rate of breakage *S<sup>i</sup>* , at 75% mill critical speed and ball size of 2.23 cm for both studied samples.

**−**

**− −**

The results depicted in Figures 11–16 demonstrate that direct relationships exist amongst Sn, Ta and Nb yield in the undersize product, as elements of interest in the product, the specific rate of breakage and the operational variables mill speed, ball size and feed size. Consequently, it can be stated that, at 75% of critical speed, grinding is more efficient with medium to small ball sizes, whereas, at 85% of critical speed, better results occur with larger ball sizes. These conditions would represent the optimal working parameters to enhance the specific rate of breakage, thus, guaranteeing a proper mineral liberation and concomitantly a higher mineral recovery and product grade. 

**Figure 15.** Plot of Sn, Ta and Nb yield in the undersize product and the specific rate of breakage, *S<sup>i</sup>* , vs. feed size for 75% N<sup>c</sup> and ball size = 2.23 cm (Penouta Bedrock).

**Figure 16.** Plot of Sn, Ta and Nb yield in the undersize product and the specific rate of breakage, *S<sup>i</sup>* , vs. feed size for 75% N<sup>c</sup> and ball size = 2.23 cm (Penouta Tailings Pond).

#### **5. Conclusions**

The experimental work done and its further analysis permit to draw the following conclusions:


 ‐ *α αΤ* **Author Contributions:** Conceptualization and execution of experiments, J.V.N.; methodology, J.V.N.; formal analysis, J.V.N., T.L., J.M.M.-A.; investigation, J.V.N.; data curation, J.V.N.; writing–original draft preparation, J.V.N.; writing–review and editing, J.V.N., T.L. and J.M.M.-A.; supervision, T.L. and J.M.M.-A.; and project administration and funding acquisition, J.M.M.-A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is part of the OptimOre project funded by the European Union Horizon 2020 Research and Innovation Programme under grant agreement No 642201.

‐

**Acknowledgments:** The authors thank Strategic Minerals Spain, S.L. for their support providing the samples. **Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


44. Petrakis, E.; Stamboliadis, E.; Komnitsas, K. Identification of Optimal Mill Operating Parameters during Grinding of Quartz with the Use of Population Balance Modeling. *KONA Powder Part. J.* **2017**, *34*, 213–223. [CrossRef]

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## *Article* **Grinding Kinetics Study of Tungsten Ore**

**Jennire V. Nava 1 , Alfredo L. Coello-Velázquez <sup>2</sup> and Juan M. Menéndez-Aguado 1, \***


**Abstract:** The European Commission (EC) maintains the consideration of tungsten as a critical raw material for the European industry, being the comminution stage of tungsten-bearing minerals an essential step in the tungsten concentration process. Comminution operations involve approximately 3–4% of worldwide energy consumption; therefore, grinding optimization should be a priority. In this study, the grinding behavior of tungsten ore from Barruecopardo Mine (Salamanca, Spain) is analyzed. A protocol based on Austin's methodology and PBM is developed in order to study the influence of operational and geometallurgical variables on grinding kinetics. In addition to the kinetic parameters, the breakage probability (*S<sup>i</sup>* ) and breakage function (*Bij* ) is determined. The selection function was formulated for the Barruecopardo Mine with respect to the mill speed.

**Keywords:** critical raw materials; tungsten ore; grinding kinetics

#### **1. Introduction**

The European Union has recently published the updated list of critical raw materials, in which tungsten (W) is included. This critical condition is defined by both the supply risk to the EU and the economic importance developed on the industrial value chains of the European Union [1].

Tungsten presents strategic applications on high strength alloys for machining tools, automotive and mobile phone sectors, among others [2]. Currently, there are several tungsten mines in Europe, some active and others on the exploration stage [3]. This is the case of the Barruecopardo mine in Salamanca (Spain), owned by Ormonde Mining PLC and currently administered by Saloro S.L., which is estimated to provide 11% of the non-Chinese global supply of tungsten [4]. The main minerals present in Barruecopardo are scheelite (CaWO4) and wolframite ((Fe, Mn)WO4), which constitute the ore. Arsenopyrite (FeAsS), pyrite (FeS2), chalcopyrite (CuFeS2), and ilmenite (FeTiO3) are also present as primary minerals of the gangue [5,6].

Regarding mineral benefit, the comminution stage represents 3–4% of the energy consumption worldwide and 40–70% of the energy consumed in a mineral processing plant [7,8]. In fact, ball-mill grinding is one of the most energy-consuming techniques. Therefore, setting the optimal values of the operational and mineralogical parameters both for the initial design and the process adaptation to ore variations [9].

Several researchers have investigated the influence exerted on kinetic conditions by operational parameters such as mill speed [10,11] and filling volume [12,13]. Other researchers devoted their work to study geometallurgical variables such as grain size, shape and roughness, specific surface area, orientation, hardness, fracture strength, feed particle size distribution, and mineralogy [14–20] using optical microscopy or more advanced techniques, such as Quantitative Microstructural Analysis (QMA) [21]. Consequently, a small improvement in machinery efficiency and an optimal design in the grinding system, taking into account the optimization of the above-mentioned parameters, would greatly cut down plant operational costs, impacting environmental issues and resource management [22,23].

**Citation:** Nava, J.V.; Coello-Velázquez, A.L.; Menéndez-Aguado, J.M. Grinding Kinetics Study of Tungsten Ore. *Metals* **2021**, *11*, 71. https://doi. org/10.3390/met11010071

Received: 28 November 2020 Accepted: 29 December 2020 Published: 31 December 2020

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**Copyright:** © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

<sup>1</sup> Escuela Politécnica de Mieres, Universidad de Oviedo, C/Gonzalo Gutiérrez Quirós, 33600 Mieres, Asturias, Spain; jvanessanavar@gmail.com

This work aims to characterize the grinding kinetic behavior at a lab-scale of tungsten ore, due to its importance as a critical raw material to the EU, by determining the kinetic parameters following the Austin model. A second objective is to assess the influence of mill speed on the kinetic and geometallurgical parameters, being mill speed the more easily adjustable operational parameter at an industrial scale.

#### **2. Theoretical Background**

The population balance model (PBM) has been widely used in ball mills since its proposal by Austin [24]. This model is based on determining the particle size distribution grouped in size classes. A mass balance for the class *i* in a well-mixed grinding process is done by employing Equation (1), where a first-order kinetic fragmentation is assumed.

$$\frac{dw\_i}{dt} = -S\_i w\_i(t) \ + \sum\_{j=1}^{i-1} b\_{ij} k\_j w\_j(t) \tag{1}$$

where *w<sup>i</sup> (t)* is the remnant mass fraction of particle size class *i* at grinding time *t*. The first term of the right-hand side is the mass fraction of the monosize *i* particles that break and thus no longer belong to that monosize, being *S<sup>i</sup>* the probability of fracture. The second term represents the contribution of all monosizes coarser than *i* that break to produce particles of monosize *i*. The fracture rate of a monosize material can be expressed by Equation (2):

$$\frac{-dw\_i}{dt} = S\_i w\_i(t) \tag{2}$$

where *S<sup>i</sup>* is the probability of fracture or specific fracture rate, whose unit is *t* −1 . Assuming that *S<sup>i</sup>* does not change with time, the integral results in Equation (3).

$$\log(wi(t)) - \log(wi(0)) = \frac{-S\_i(t)}{2.3} \tag{3}$$

where *w<sup>i</sup>* is the weight fraction of mineral feed into the mill having a size 1 for time *t*, and *S<sup>i</sup>* is the probability of fracture. According to the methodology proposed by Austin et al. [25], once *S<sup>i</sup>* values have been obtained through slope determination, they are plotted to the particle size, and Equation (4) is proposed to study the behavior of the probability of fracture *S<sup>i</sup>* .

$$S\_{\bar{i}} = \mathfrak{a}\_T \mathbf{X}\_{\bar{i}}^{\mathfrak{a}} \tag{4}$$

where *X<sup>i</sup>* is the upper size limit of the interval in mm, and *α<sup>T</sup>* and *α* are model parameters that depend on the material properties and the grinding conditions. To find the second term of Equation (1), the fracture function *bii* is defined. This function represents the particle fraction that belongs initially to interval *j*, after fracture falls in interval *i*. It is recommended to represent this value in cumulative form *Bij*, whose calculation is done with Equation (5).

$$B\_{ij} = \sum\_{k=n}^{i} b\_{kj} \tag{5}$$

That is, *Bij* is the sum of the mineral fractions finer than the upper limit of interval *i* as a result of the primary break of the size interval *j*. Austin et al. [25] showed that *Bij* values could be estimated from a size analysis of the products over short grinding times of an initial feed chiefly of size *j* through the method *BII* [25–27]. With the parameters of fracture function, *Bij* can be determined graphically through an empirical function like Equation (6).

$$B\_{ij} = \phi\_j \left(\frac{X\_{i-1}}{X\_j}\right)^\gamma + (1 - \phi\_j) \left(\frac{X\_{i-1}}{X\_j}\right)^\beta n \ge i \ge j + 1 \tag{6}$$

where *φ<sup>j</sup>* , *γ* and *β* are parameters that depend on the material properties. The critical speed, *Nc*, is calculated using Equation (7).

$$N\mathcal{c} = \frac{42.3}{\sqrt{D-d}}\tag{7}$$

where *D* is the mill diameter and *d* is the ball diameter [m]. Ball mill filling volume is calculated using Equation (8).

$$J = \left(\frac{mass\ of\ balls}{ball\ density \times mill\ volume}\right) \times \frac{1.0}{0.6} \tag{8}$$

On the other side, Austin and Brame [28] calculated the sorting function *α<sup>T</sup>* in a general way through Equation (9).

$$\alpha\_T = \frac{v\_c - 0.1}{1 + \ \left[^{15.7(v\_c - 0.94)}\right]} \tag{9}$$

where *ν<sup>c</sup>* is the mill speed expressed as a fraction of the critical speed. Finally, according to [29], the Froude number expresses the ratio of centrifugal acceleration to gravity acceleration at the perimeter of the mill chamber (Equation (10)). This number can be used to characterize the charge motion in the mill and the ball regime. Thus, in laboratory ball mills, it is recommended to define work conditions with centrifugal acceleration at the shell equalling <sup>1</sup>/<sup>2</sup> of the acceleration due to gravity (*F<sup>r</sup>* = 0.5), corresponding to *ν<sup>c</sup>* = 70.7% [29].

$$F\_r = \frac{\frac{D}{2}\omega^2}{\mathcal{g}} = \frac{2\pi^2 n^2 D}{\mathcal{g}}\tag{10}$$

#### **3. Materials and Methods**

#### *3.1. Sample Preparation and Feed Characterization*

The sample mineralogy was extensively characterized by Alfonso et al. [6]. A representative sample from an old waste dump of the Barruecopardo mine was prepared in a 4 mm jaw crusher. After homogenization and quartering using a riffle splitter, sieving provided an adequate quantity of the following size intervals, which will be considered as monosizes in this work: 5000/4000, 4000/3350, 3000/2000, 2000/1000, 1000/500, 500/250, 250/125, 125/75, 75/45 µm.

A representative sample was characterized chemically through XRF, using a Bruker XRF S-4 Pioneer Advance, with sample preparation in a Claisse Perler, model M-4.

#### *3.2. Calculation of Critical Speed and Initial Conditions for the Grinding Kinetic Tests*

Mill critical speed was calculated using Equation (7). Table 1 shows the three milling speeds used in the tests.


**Table 1.** Mill rotation speeds used in the grinding kinetic tests.

Grinding kinetic tests were run in a laboratory mill, 17.8 cm in diameter and 4500 cm<sup>3</sup> in capacity. Feed was of 900 cm<sup>3</sup> , and milling load consisted of 6.6 kg of steel balls with the following ball size distribution: 45 balls 19 mm in diameter, 23 balls 29.7 mm in diameter, and 17 balls 36.8 mm in diameter. Fill fraction was calculated using Equation (8). The feed consisted of samples of the 9 monosizes selected (5000/4000, 4000/3000, 3000/2000, 2000/1000, 1000/500, 500/250, 250/125, 125/75, 75/45 µm). Grinding times were 0.5, 1, 1.5, 3.5, 6.5, and 10.5 min. Each sample was dumped and, after performing a grain size analysis, W content was measured to assess the evolution of the W grade with respect to the kinetic parameters.

#### *3.3. Determination of Fracture Probability (S<sup>i</sup> ), Fracture Function (Bij), and the Kinetic Parameters (αT, α, φ<sup>j</sup> , γ, and β)*

3.3.1. Fracture Probability (*S<sup>i</sup>* ) and Kinetic Parameters (*αT*, and *α*)

After obtaining the oversize weights for each grinding time, and plotting the time function log (*w<sup>i</sup>* (*t*)/*w<sup>i</sup>* (0)) for each monosize, the equation for each curve, and consequently, the *S<sup>i</sup>* value, were calculated through the linear fitting using Equation (3). Then, *S<sup>i</sup>* , values for each monosize were plotted, and the parameters (*α<sup>T</sup>* and *α*) were calculated using Equation (4) for each mill speed condition to study the influence of this operational variable on the probability of fracture and on the kinetic parameters *α<sup>T</sup>* and *α*. The selection function *αT*, obtained through Equation (9) was calculated using an equation designed for this particular ore, as detailed in Section 4.2.

3.3.2. Determination of the Fracture Function (*Bij*) and the Kinetic Parameters (*φ<sup>j</sup>* , *γ* and *β*)

These calculations were done for each feed monosize, *i*, at each mill speed condition and after the grinding time. Product particle size analysis was done by sieving with mesh sizes *<sup>i</sup>* to *<sup>j</sup>*−*n*. Weight of oversizes *<sup>i</sup>* and S*j*−*<sup>n</sup>* was determined, and fracture function values *bij* and cumulative value *Bij,* were obtained using Equation (5). Finally, *Bij* data were plotted to relative size *j/i* for each monosize and mill speed conditions. The rest of the kinetic parameters were calculated through Equation (6).

#### *3.4. Study of P<sup>80</sup> and the Ratio of Reduction R<sup>r</sup>*

The evolution of some relevant parameters of the product was represented. These parameters were *P*<sup>80</sup> and the ratio of reduction, *R<sup>r</sup>* , after a 0.5 min grinding time at several mill speeds for each feed monosize. Likewise, the evolution of the probability of fracture, *Si* , and *R<sup>r</sup>* with mill speed was studied, for each feed monosize.

#### *3.5. Chemical Characterisation of the Product*

Milling products were chemically characterized with the equipment specified in Section 3.1.

#### **4. Results and Discussion**

#### *4.1. Feed Characterzsation*

The sample mechanically prepared displays a particle size distribution (PSD), as shown in Figure 1.

*F*<sup>80</sup> for the Barruecopardo sample is 1690 µm. Table 2 shows W contents obtained from a mineralogical study for each size interval.


**Table 2.** Fractional results of XRF (ND = not detected).

**Figure 1.** PSD for the Barruecopardo ore head samples.

W contents shown in Table 2 reflect that the sample comes from a waste dump of the Barruecopardo mine, and therefore from an area with low W contents.

**m) STD (ppm** The probability of fracture (*S<sup>i</sup>* ) and the kinetic parameters (*α, αT*), a size of 80% of product undersize (*P*80), and the reduction ratio were obtained from this procedure (*Rr*): Figure 2 shows the values (*S<sup>i</sup>* ) plotted to the particle size according to the mill speed.

**Figure 2.** Evolution of the probability of fracture (*S<sup>i</sup>* ) with particle size, in relation to mill speed.

's

*Nc Nc Nc Nc Nc Nc Nc* **%** *Nc*

000 0.331 0.403 0.383 2244.9 1856.6 350 0.241 0.322 0.266 2329.4 1864.2

Figure 2 shows that comparing the three speeds tested, the highest probability of fracture occurred at 70% critical speed, agreeing with Gupta amd Sharma [9], Gupta [10], and Herbst and Fuerstenau [30], who did their experiments at lab scale, with several materials, simulations, and finally upscaling. This also verified the Steiner [29] recommendation of a configuration with *F<sup>r</sup>* value close to 0.5. For intermediate and coarse grain sizes, a rather linear trend was observed using a logarithmic scale. This coincided with Deniz [22] for this grain size interval.

Noteworthy was a sharp break in slope that occurred at around 250 µm for the three tested speeds, coinciding with the size fractions more enriched in W, as shown in Table 2. Moreover, the results backed Gupta and Sharma's [9] statements, pointing to the probability of fracture *S<sup>i</sup>* being one of the mill operational conditions more influenced by the mineralogical variability among monosize fractions.

On the other side, the evolution of *S<sup>i</sup>* , *P80,* and *R<sup>r</sup>* for each feed monosize was studied at different mill speeds as summarized in Table 3. The reduction ratio was the result of dividing the d<sup>80</sup> in the feed size (F80) by the d<sup>80</sup> in the product size (P80).

**Table 3.** *S<sup>i</sup>* , *Rr*, and *P*<sup>80</sup> values as a function of working speed for each monosize.


It must be highlighted in Table 3 that once the total grinding time was reached, a finer *P*<sup>80</sup> and a higher *Rr,* were obtained at 70% working speed for most of the monosizes. This confirmed that the mill speed affected the grinding product [11,31].

The kinetic parameters (*α*, *αT*) obtained after linearization of Equation (4) are summarized in Table 4 and the selection function is represented in Figure 3.

**Table 4.** Values of the kinetic parameters *α*, *α<sup>T</sup>* for different mill speeds.


Table 4 shows that the parameter *α* is coherent with what Austin et al. [25] reported. These authors pointed out that it usually ranges between 0.5 and 1.5 and that it depends only on the mineral. On the other side, Table 4 and Figure 3 show that the value of the selection function *α<sup>T</sup>* does not vary significantly despite the increasing speed, because the mill geometry remains unchanged. Figure 4 displays *α<sup>T</sup>* values calculated from Equation (9), as proposed by Austin and Brame [28]. It can be seen that this expression does not fit this case. This led to a polynomial adjustment using the experiment values, as it was shown in Equation (11) that it fitted better with the studied sample.

$$u\_T = -1.775 \, v\_\varepsilon \, ^2 + 2.3625 v\_\varepsilon - 0.6402 \, \tag{11}$$

*α α*

*α*

*α*

*α α*

α **Figure 3.** Variation of α<sup>T</sup> with the working speed fraction.

–

**Figure 4.** Behavior of the fracture function (*Bij* ), with respect to particle size. 60% working speed *Nc* (Barruecopardo ore).

#### *4.2. Fracture Function, Kinetic Parameters (φ<sup>j</sup> , γ and β)*

The values of fracture function *Bij* in relation to the particle size for each monosize as the mill speed varied were determined through Equation (5) and are shown in Figures 4–6 for 60%, 70%, and 80% mill speed, respectively.

**Figure 5.** Behavior of the fracture function (*Bij* ), with respect to particle size. 70% working speed *Nc* (Barruecopardo ore).

**Figure 6.** Behavior of the fracture function (*Bij* ), with respect to particle size. 80% working speed *Nc* (Barruecopardo ore).

 *γ β β γ β* – – The kinetic parameters of fracture function (*φ<sup>j</sup>* , *γ*, and *β*) are shown in Table 5. According to Austin et al. (1984), *φ<sup>j</sup>* and *β* are parameters that depend on the material. Regarding *γ* and *β*, these authors propose that their values were usually in the range of 0.5–1.5 and 2.5–5.0, respectively.

*φ ϒ β φ ϒ β φ ϒ β*

**80%** *Nc*

*φ ϒ β*

–


**Table 5.** Values of the kinetic parameters (*ϕj*, *γ*, *β*) for each monosize and for different mill speed conditions.

Figures 4–6 and Table 5 show that *Bij* depends on the feed grain size for parameters of 60%, 70%, and 80% of critical speed. The influence that mill speed exerts on *Bij,* can also be noticed by comparing the different monosizes: a greater difference existed for coarser sizes, whereas it was lesser for finer sizes. This was due to the fact that coarse sizes not only possessed a higher *S<sup>i</sup>* , but also were more prone to yield new finer particles (progeny). That meant that *Bij* depended on the feed particle size, as Ipek and Goktepe [32] observed, which was also influenced by the mill speed, and concurred with results by Deniz [22]. Nevertheless, this variation was not as significant as reported by Austin et al. [25].

Table 5 shows parameter *γ*, which represents the fineness factor. In Figure 7, the *γ* values are plotted against mill speed for two feed particle sizes (4000 and 500 µm, respectively).

Figure 7 depicts that *γ* values are influenced by both mill speed and feed particle size. Smaller *γ* values were related to coarse particles (4000 µm), which meant that more fine particles were generated. Conversely, finer particles (500 µm) generated a lesser proportion of fine particles, agreeing well with results by Ipek and Goktepe [32] and Austin et al. [25].

*γ* **Figure 7.** Variation of *γ* with mill speed.

#### *γ 4.3. Chemical Characterization of the Grinding Products*

*γ* Figures 8–10 illustrate the evolution of W content in the product in relation to the feed monosizes and their grain size fractions for each mill speed. –

**Figure 8.** Evolution of the W grade in the product with respect to the monosizes and their grain size fractions for 60% *Nc*.

**Figure 9.** Evolution of the W grade in the product with respect to the monosizes and their grain size fractions for 70% *Nc*.

**Figure 10.** Evolution of the W grade in the product with respect to the monosizes and their grain size fractions for 80% *Nc*.

As can be seen in Figures 8–10, for most monosizes, W content increased with decreasing particle size. This could be partly explained because feed grain sizes around 250 µm already yielded higher W content, as shown in Table 2. Nevertheless, maximum values after grinding were 6–7 times higher than original values pointing undoubtedly to a differential grinding effect leading to W mineral grains mainly falling in the 250–125 µm interval. The accumulation of these W-enriched particles, which were more difficult to grind, supports the mineralogical explanation of the aforementioned breakage probability reduction at 250 µm size. In addition, this interval always presented *S<sup>i</sup>* values higher than the coarser and finer intervals. This would suggest that particles of this size would have a higher probability of fracture compared to the adjacent size intervals. Indeed, in all cases, a decrease in W content could be observed down to 75 µm size, followed by an increase of further finer particles. It must be highlighted that the highest W content was yielded for grinding at 70% *Nc*. This could be explained because, under these grinding conditions, the mill performance was more efficient due to a more adequate charge regime (*F<sup>r</sup>* close to 0.5), leading to better grinding kinetics.

#### **5. Conclusions**

The experimental work here presented and its further analysis permits to draw the following conclusions:


**Author Contributions:** Conceptualization and execution of experiments, J.V.N. and J.M.M.-A.; methodology and investigation, J.V.N. and A.L.C.-V.; formal analysis and data curation, J.V.N. and J.M.M.-A.; writing–original draft preparation, J.V.N.; writing–review and editing, J.V.N. and J.M.M.-A.; supervision, J.M.M.-A. and A.L.C.-V.; project administration and funding acquisition, J.M.M.-A. All authors have read and agreed to the submitted version of the manuscript.

**Funding:** This work is part of the OPTIMORE project funded by the European Union Horizon 2020 Research and Innovation Programme under grant agreement No 642201.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

