Grinding Kinetics Study of Tungsten Ore

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_i) and breakage function (B_ij) is determined. The selection function was formulated for the Barruecopardo Mine with respect to the mill speed.

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 [14,15,16,17,18,19,20] 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].

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.

$$\frac{d{w}_i}{dt}=-S_i{w}_i\left(t\right) + {\displaystyle \sum }_{j=1}^{i-1}{b}_{ij}{k}_j{w}_j\left(t\right)$$

(1)

where w_i(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_i 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):

$$\frac{-d{w}_i}{dt}=S_i{w}_i\left(t\right)$$

(2)

where S_i is the probability of fracture or specific fracture rate, whose unit is t⁻¹. Assuming that S_i does not change with time, the integral results in Equation (3).

$$Log\left(wi\left(t\right)\right)-Log\left(wi\left(0\right)\right)=\frac{-S_i\left(t\right)}{2.3}$$

(3)

where w_i is the weight fraction of mineral feed into the mill having a size 1 for time t, and S_i is the probability of fracture. According to the methodology proposed by Austin et al. [25], once S_i 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_i.

$$S_i={\alpha }_T{X}_i^{\alpha }$$

(4)

where X_i is the upper size limit of the interval in mm, and ${\alpha }_T$ 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 b_ii 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 B_ij, whose calculation is done with Equation (5).

$$B_{ij}={\displaystyle \sum }_{k=n}^i{b}_{kj}$$

(5)

That is, B_ij 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 B_ij 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 [25,26,27]. With the parameters of fracture function, B_ij can be determined graphically through an empirical function like Equation (6).

$$B_{ij}={\varphi }_j{\left(\frac{X_{i-1}}{X_j}\right)}^{\gamma } + \left(1-{\varphi }_j\right){\left(\frac{X_{i-1}}{X_j}\right)}^{\beta }n \ge i \ge j + 1$$

(6)

where ${\varphi }_j$, γ and β are parameters that depend on the material properties. The critical speed, Nc, is calculated using Equation (7).

$$Nc=\frac{42.3}{\sqrt{D-d}}$$

(7)

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

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

(8)

On the other side, Austin and Brame [28] calculated the sorting function ${\alpha }_T$ in a general way through Equation (9).

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

(9)

where ν_c 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 ½ of the acceleration due to gravity (F_r = 0.5), corresponding to ν_c = 70.7% [29].

$$F_r=\frac{\frac{D}{2}{\omega }^2}{g}=\frac{2{\pi }^2{n}^{2}D}{g}$$

(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. Mill rotation speeds used in the grinding kinetic tests.

Mill Speed	n [rpm]	Froude Number
Nc$({\upsilon }_c=1)$	112.3	1
N1 $({\upsilon }_c=0.6)$	67.4	0.43
N2 (${\upsilon }_c=0.7)$	78.6	0.58
N3 $({\upsilon }_c=0.8)$	89.9	0.76

shows the three milling speeds used in the tests.

Grinding kinetic tests were run in a laboratory mill, 17.8 cm in diameter and 4500 cm³ in capacity. Feed was of 900 cm³, 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_i), Fracture Function (B_ij), and the Kinetic Parameters (α_T, α, ${\varphi }_j$, γ, and β)

3.3.1. Fracture Probability (S_i) and Kinetic Parameters (α_T, and α)

After obtaining the oversize weights for each grinding time, and plotting the time function log (w_i(t)/w_i(0)) for each monosize, the equation for each curve, and consequently, the S_i value, were calculated through the linear fitting using Equation (3). Then, S_i, values for each monosize were plotted, and the parameters (α_T 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_T and α. The selection function α_T, 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 (B_ij) and the Kinetic Parameters (${\varphi }_j$, 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 i to j−n. Weight of oversizes i and S_j−n was determined, and fracture function values b_ij and cumulative value B_ij, were obtained using Equation (5). Finally, B_ij 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 P₈₀ and the Ratio of Reduction R_r

The evolution of some relevant parameters of the product was represented. These parameters were P₈₀ and the ratio of reduction, R_r, after a 0.5 min grinding time at several mill speeds for each feed monosize. Likewise, the evolution of the probability of fracture, S_i, and R_r 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.

F₈₀ for the Barruecopardo sample is 1690 µm.

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

Size Interval (µm)	Weight (%)	W (ppm)	STD (ppm)
>4000	1.79	ND	-
3350–4000	3.93	ND	-
2000–3350	9.71	ND	-
1000–2000	29.53	40	10
500–1000	24.01	105	13
250–500	19.94	146	13
125–250	8.23	75	12
75–125	1.52	23	10
45–75	0.78	ND	-
<45	0.55	ND	-

shows W contents obtained from a mineralogical study for each size interval.

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.

The probability of fracture (S_i) and the kinetic parameters (α, α_T), a size of 80% of product undersize (P₈₀), and the reduction ratio were obtained from this procedure (R_r):

Figure 2 shows the values (S_i) plotted to the particle size according to the mill speed.

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_r 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_i being one of the mill operational conditions more influenced by the mineralogical variability among monosize fractions.

On the other side, the evolution of S_i, P₈₀, and R_r for each feed monosize was studied at different mill speeds as summarized in

Table 3. S_i, R_r, and P₈₀ values as a function of working speed for each monosize.

Monosize (µm)	Specific Rate of Breakage, Si (1/min)			Reduction Ratio, Rr			P80 (µm)
	60% Nc	70% Nc	80% Nc	60% Nc	70% Nc	80% Nc	60% Nc	70% Nc	80% Nc
5000/4000	0.331	0.403	0.383	2.14	2.59	2.38	2244.9	1856.6	2014.7
4000/3350	0.241	0.322	0.266	1.66	2.08	1.74	2329.4	1864.2	2220.5
3350/2000	0.231	0.289	0.232	1.92	2.13	1.85	1608.1	1447.7	1665.0
2000/1000	0.143	0.159	0.135	1.67	1.84	1.69	1078.5	977.2	1063.8
1000/500	0.071	0.077	0.072	1.13	1.14	1.13	792.9	786.1	799.7
500/250	0.073	0.043	0.039	1.11	1.06	1.05	406.0	425.8	427.9
250/125	0.110	0.097	0.078	1.18	1.17	1.10	191.1	192.3	204.6
125/75	0.047	0.068	0.033	1.04	1.07	1.03	110.2	107.8	111.9
75/45	0.015	0.017	0.006	1.01	1.01	1.01	68.4	68.1	68.2

. The reduction ratio was the result of dividing the d₈₀ in the feed size (F₈₀) by the d₈₀ in the product size (P₈₀).

It must be highlighted in Table 3 that once the total grinding time was reached, a finer P₈₀ and a higher R_r, 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, α_T) obtained after linearization of Equation (4) are summarized in

Table 4. Values of the kinetic parameters α, α_T for different mill speeds.

	60% Nc	70% Nc	80% Nc
α	0.42	0.58	0.74
αT (1/min)	0.14	0.14	0.11

and the selection function is represented in Figure 3.

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_T does not vary significantly despite the increasing speed, because the mill geometry remains unchanged. Figure 4 displays α_T 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.

$${\alpha }_T=-1.775 {\upsilon }_c{}^2+2.3625{\upsilon }_c-0.6402$$

(11)

4.2. Fracture Function, Kinetic Parameters (${\varphi }_j$, γ and β)

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

The kinetic parameters of fracture function (${\varphi }_j$, γ, and β) are shown in

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

			60% Nc				70% Nc				80% Nc
Monosize (µm)	j	j/i	Bij (Test)	φj	γ	β	Bij (Test)	φj	γ	β	Bij (Test)	φj	γ	β
4000	4000	1.000	1.000	0.483	0.036	3.285	1.000	0.586	0.021	3.361	1.000	0.552	0.027	3.477
	3350	0.838	0.637				0.708				0.680
	2000	0.500	0.516				0.612				0.576
	1000	0.250	0.474				0.581				0.542
	500	0.125	0.450				0.563				0.522
	250	0.063	0.435				0.551				0.512
	125	0.031	0.426				0.545				0.503
	75	0.019	0.421				0.540				0.495
	45	0.011	0.411				0.534				0.489
3350	3350	1.000	1.000	0.805	0.022	2.732	1.000	0.828	0.015	2.507	1.000	0.861	0.015	2.734
	2000	0.597	0.844				0.869				0.888
	1000	0.299	0.799				0.833				0.856
	500	0.149	0.774				0.812				0.838
	250	0.075	0.758				0.799				0.825
	125	0.037	0.749				0.791				0.818
	75	0.022	0.743				0.785				0.812
	45	0.013	0.731				0.779				0.807
2000	2000	1	1	0.840	0.047	1.905	1	0.874	0.055	3.123	1	0.882	0.035	1.862
	1000	0.5	0.856				0.875				0.893
	500	0.25	0.793				0.818				0.845
	250	0.125	0.755				0.779				0.814
	125	0.062	0.738				0.736				0.798
	75	0.037	0.719				0.726				0.786
	45	0.022	0.707				0.721				0.774
1000	1000	1	1	0.807	0.094	2.048	1.000	0.821	0.078	1.885	1.000	0.855	0.061	1.939
	500	0.5	0.803				0.826				0.857
	250	0.25	0.708				0.738				0.785
	125	0.125	0.662				0.698				0.753
	75	0.075	0.635				0.668				0.729
	45	0.045	0.601				0.647				0.707
500	500	1	1	0.887	0.115	4.054	1.000	0.876	0.118	2.538	1.000	0.882	0.087	1.283
	250	0.5	0.826				0.829				0.879
	125	0.25	0.757				0.750				0.783
	75	0.15	0.712				0.691				0.743
	45	0.09	0.673				0.665				0.716

. According to Austin et al. (1984), ${\varphi }_j$ 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.

Figure 4, Figure 5 and Figure 6 and Table 5 show that B_ij depends on the feed grain size for parameters of 60%, 70%, and 80% of critical speed. The influence that mill speed exerts on B_ij, 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_i, but also were more prone to yield new finer particles (progeny). That meant that B_ij 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].

4.3. Chemical Characterization of the Grinding Products

Figure 8, Figure 9 and Figure 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.

As can be seen in Figure 8, Figure 9 and Figure 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_i 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_r 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:

• Austin’s methodology has allowed us to characterize the kinetic behavior of tungsten ore by determining the kinetic parameters (α, α_T, φ_j, γ, β), concluding that the values S_i and B_ij do not vary significantly with time.
• The probability of fracture, S_i, is highest at 70% critical speed. Fracture function B_ij, does not vary significantly with mill speed. Nevertheless, it is affected by the feed particle size, becoming higher for coarser sizes.
• Equation (10) is proposed as the best α_T fit, specifically for the studied ore.
• Values of parameter γ are influenced by both mill speed and feed particle size. Coarser particles yield smaller γ values, i.e., they produce more fines, whereas finer particles produce lesser quantities. The parameters ϕ_j and β depend on the features of the ore.
• The chemical characterization and the study of the evolution of the relevant grinding parameters, such as P₈₀ and R_r, to the grinding time have demonstrated that, first, the highest probability of fracture occurs at 70% of the critical speed and; second, the effect of differential grinding is evidenced between the W-bearing species and the gangue. The latter results in an increase of the W grade in the monosize 250/125 µm.