*Article* **Particle Size Distribution Models for Metallurgical Coke Grinding Products**

**Laura Colorado-Arango 1 , Juan M. Menéndez-Aguado 2, \* and Adriana Osorio-Correa 1**


**\*** Correspondence: maguado@uniovi.es; Tel.: +34-985458033

**Abstract:** Six different particle size distribution (Gates–Gaudin–Schuhmann (GGS), Rosin–Rammler (RR), Lognormal, Normal, Gamma, and Swebrec) models were compared under different metallurgical coke grinding conditions (ball size and grinding time). Adjusted R 2 , Akaike information criterion (AIC), and the root mean of square error (RMSE) were employed as comparison criteria. Swebrec and RR presented superior comparison criteria with the higher goodness-of-fit and the lower AIC and RMSE, containing the minimum variance values among data. The worst model fitting was GGS, with the poorest comparison criteria and a wider results variation. The undulation Swebrec parameter was ball size and grinding time-dependent, considering greater *b* values (*b* > 3) at longer grinding times. The RR α parameter does not exhibit a defined tendency related to grinding conditions, while the *k* parameter presents smaller values at longer grinding times. Both models depend on metallurgical coke grinding conditions and are hence an indication of the grinding behaviour. Finally, oversize and ultrafine particles are found with ball sizes of 4.0 cm according to grinding time. The ball size of 2.54 cm shows slight changes in particle median diameter over time, while 3.0 cm ball size requires more grinding time to reduce metallurgical coke particles.

**Keywords:** particle size distribution; metallurgical coke; comminution

#### **1. Introduction**

Metallurgical coke is a crucial raw material in the iron and steelmaking industry and is considered a critical raw material in the EU due to its high consumption volume and the strong EU import dependence [1–3]. Heat supplier, reducing agent, adequate permeability, and burden mechanical support are the features that render it a fundamental material for blast furnaces that perform metallurgical processes such as cast iron, ferroalloy, lead, and zinc production, and in kilns for lime and magnesium production [4,5]. According to particle size, metallurgical coke is used at different process stages. Coke ranging between 24–40 mm is the main form for blast furnaces; this so-called nut coke is added in ironmaking with ferrous and flux mineral layers from 6 to 24 mm, and coke breeze is considered the energy source for sintering or pelletising with particle size less than 6 mm [6].

Suitable coke selection enhances the steel production line, saves coke utilisation, minimises dust generation, reduces the significant amount of greenhouse gases discharged into the atmosphere, namely, CO2, SO2, and NOx, and optimises energy usage [7–9]. Chemical composition, mechanical strength, thermal resistance, and particle size are the most significant parameters for selecting metallurgical coke [6,10]. However, the coke particle size and shape play an essential role in blast furnace and sinter plants. Coke mean particle size determines the fluid flow resistance, the upward gases and downwards metal liquids passing efficiency, and the iron production rate. The coke bed formation and permeability are also strongly related to particle size and combustion behaviour in the sintering process.

**Citation:** Colorado-Arango, L.; Menéndez-Aguado, J.M.; Osorio-Correa, A. Particle Size Distribution Models for Metallurgical Coke Grinding Products. *Metals* **2021**, *11*, 1288. https://doi.org/10.3390/ met11081288

Academic Editors: Jürgen Eckert and Luis Norberto López De Lacalle

Received: 30 June 2021 Accepted: 13 August 2021 Published: 16 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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/).

Despite the abovementioned importance of coke particle size distribution (PSD), particle size control has not been studied enough. Many fine recycled particles from chipping in crushing processes or waste in the coke oven cause an overproduction of particulate matter and uncontrolled coke size distributions [11–22]. Poor coke without quality classification creates disturbances in the sinter plant and the blast furnace operation, producing excess dust, heat losses, inefficient reaction rates, and fluid flow obstruction. This situation has driven environmental regulation to propose eliminating or partially substituting metallurgical coke in sinter production [8,13]. Various studies [14–23] have evaluated the effect of defined ranges of coke particle size in steelmaking performance as a means of process optimisation. The thickness of combustion zone, flame front, chemical reactions kinetics, and iron-bearing phase formation (hematite, magnetite, and gangue) are broadly affected by coke PSD in sinter and blast furnace plants [15–17].

Modelling the metallurgical coke PSD allows quantitatively assessing the breakage behaviour. Several benefits in the iron and steel processes are obtained when the PSD is adequately characterised, with effective diameters (D50, D80) measured, and the effect of its variation on the processes is known. Many models have been developed to predict and describe the PSD of granulated materials. Perfect et al. [24] tested three distribution functions based on two parameters for different fertilisers. Lognormal, Rosin–Rammler and Gate–Gaudin–Schuhmann were fitted by nonlinear regression analysis. According to the goodness-of-fit of R<sup>2</sup> , they concluded the Rosin–Rammler is the more accurate model to describe material fertiliser. Botula et al. [25] evaluated ten PSD models in soils of the humid tropics. The findings demonstrated that the three and four-parameters Fredlund and three-parameter Weibull and four-parameter Anderson presented an excellent fitting correlation to soils. Bu et al. [26] characterised the coal grinding process (wet and dry ways) using PSD models, namely, GGS, Gaudin–Meloy, RR, modified RR, and Swebrec. They found that the RR and Swebrec showed outstanding fitting performance.

The current paper compares GGS, RR, Gamma, Normal, Lognormal, and Swebrec distributions at different metallurgical coke grinding conditions to select the best fitting models and represent the metallurgical coke PSD. Finally, the association of PSD model parameters with the grinding process was analysed for the best two models.

#### **2. Materials and Methods**

A metallurgical coke sample from Boyacá (Colombia) was used in the grinding process. The original sample was crushed in a roll mill (Denver Equipment Co., Denver, CO, USA) and sieved 100% under six mesh (3.35 mm). Product PSD is depicted in Figure 1, and the elemental and proximate analyses are shown in Table 1. A dataset of 144 PSD was collected from grinding under different dry conditions. Grinding tests were carried out in a laboratory steel ball mill with 0.20 m in diameter and 0.20 m long. Three ball sizes (2.54, 3.00, and 4.00 cm) and eight grinding times (0.5, 1, 2, 3, 4, 5, 6, and 10 min) were used to evaluate the product PSD. The operational mill conditions remained fixed: the fraction of critical speed (*ϕ*c) was 0.75; the ball filling fraction (*J*) was 0.3; the fraction of powder bed (*f* <sup>c</sup>) and void filling (*U*) were 0.12 and 1, respectively.

**Figure 1.** Metallurgical coke PSD in the feed to the ball mill.



The six PSD models assessed are shown in Table 2. Gate–Gaudin–Schuhmann [27] and Rosin–Rammler [28] models have been the most popular and oldest functions used to describe PSD in granular materials. GGS was developed in the metalliferous mining industry and is described with a size parameter (largest particle size) and a distribution parameter [27]. The RR model was defined to evaluate the coal fragmentation processes but has been broadly used in many industries. The RR size parameter corresponds to 63.21% cumulative undersize, and the shape parameter defines the spread of sizes [28]. Even though these are handy models, the fitting accuracy depends on the material nature and size ranges.

A short description of the most common powder PSD models is presented below. The Gamma distribution [29,30] presents two functional parameters related to the median and standard deviation. Yang et al. [29] compared the PSD prediction between Gamma and other models, obtaining the Gamma distribution as the best fit. Normal and Lognormal are also two-parameter models, using the mean diameter (logarithmic mean if Lognormal) and the standard deviation. According to Buchan [31], the Lognormal is more suitable in describing PSD in soils.

The three-parameter Weibull distribution is defined by fitting, size, and shape parameters. Esmaeelnejad et al. [32] compared different models to describe soil PSD, concluding that the Weibull model was the most accurate for all samples studied. Another threeparameter distribution is the Swebrec distribution, introduced by Ouchterlony [33] to predict PSD by rock blasting and crushing fragmentation. The parameters are the maximum size *xmax*, the size with 50% cumulative undersize *x*<sup>50</sup> and the undulation parameter *b*. In the work of Osorio et al. [34], the Swebrec model was evaluated in the wet grinding process of quartz ore, obtaining an excellent fitting adjustment. Menéndez-Aguado

et al. [35] presented the Swebrec distribution to fit sediment particle size distribution with a high correlation between experimental and model data.


**Table 2.** Particle size distribution models.

†—Geometric Mean Diameter.

In this study, the model comparison was carried out using three statistical indices. The adjusted *R* 2 (Equation (1)) measures the discrepancy between predicted and observed data [36]. Akaike's information criterion (Equation (2)) examines the model goodnessof-fit imposing penalties for additional fitting parameters [37]. Finally, the mean root of squared error (RMSE) presented in Equation 3 is the residual error, i.e., the information not contained in the model. The criteria selected are widely used in PSD model selection and in assessing model prediction [25,29,32,36]. The adjusted *R* 2 is a traditional goodness-of-fit measurement, but it is mainly considered in linear models' interpretation. Additionally, to assure the model selection, RMSE and AIC were used. These criteria are more appropriate to measure the goodness-of-fit in nonlinear models [28,38].

$$R\_{adj}^2 = 1 - \left(\frac{\frac{\text{RSS}}{N-P}}{\frac{\text{TSS}}{N-1}}\right) \tag{1}$$

where RSS is the residual sum of squares, *N* is the number of PSD data points, *P* is the number of independent variables in a particle size distribution model, and TSS is the total sum of squares.

$$\text{AIC} = N \cdot \ln\left(\frac{\text{RSS}}{N}\right) + 2P \tag{2}$$

$$\text{RMSE} = \left(\frac{\text{RSS}}{N}\right)^{0.5} \tag{3}$$

A custom Python script was employed in the fitting procedure, which is provided in the Supplementary Material. All models were compared with the experimental PSD data using the least-squares method to find the best fitting parameters, and the model presenting the best values of the three statistics were selected. The least-squares procedure was obtained considering a nonlinear optimisation method, and the residual sum of squares is calculated with the minimisation function established in Equation (4).

$$\text{RSS} = \sum\_{i=1}^{n} \left( P\_{i, \text{measure}} - P\_{i, \text{predicted}} \right)^2 \tag{4}$$

where *Pi,*measured and *Pi*,predicted represent experimental and model cumulative passing material, respectively. Box plots were employed as graphical representation to provide more insights into the different behaviour of PSD models. Finally, metallurgical coke's more stable grinding conditions are defined using a colour map graph about the two best model parameters.

#### **3. Results**

#### *3.1. Comparison of PSD Models' Goodness-of-Fit*

Figure 2 depicts the Box plots of statistical indices. The model with the better goodnessof-fit was obtained under descriptive statistics (see Table S1) considering the higher adjusted *R* 2 , the smaller RMSE and lower AIC. The adjusted *R* 2 (Figure 2a and Figure S1) provides values greater than 0.95 in all models, excluding the GGS distribution, which produced adjustments less than 0.8. The Schuhmann distribution data are widely spread out from the mean with the larger standard deviation, as depicted in the Box plot. Lognormal and Normal models explain completely well the experimental PSD with adjustments varying between 0.95–0.99. However, the Lognormal model adjusted slightly better than Normal model due to the great fitting in 4.00 cm grinding media. Gamma, Rosin-Rammler, and Swebrec exhibit values close to 1.0 and relatively narrow dispersion data; therefore, they were considered the models with superior fitting performance, providing an excellent PSD prediction for the material.

Akaike's information criteria (AIC) Box plot is shown in Figure 2b. It was used to compare the model quality fit and identify the better fitting model, for an increment in goodness-of-fit requires lower AIC values. The AIC results were consistent with *R* 2 and RMSE estimations, achieving minimum values in Gamma, Rosin–Rammler, and Swebrec distributions. However, the Swebrec model presented the least standard deviation. Lognormal and GGS models depicted poor fits with large mean and standard deviation values about AIC estimator.

**Figure 2.** Box plots to compare the particle size models: (**a**) Adjusted *R* <sup>2</sup> and (**b**) Akaike's information criterion.

Figure 3 summarises the models' criteria in a normalised bar chart. There are three bars (adjusted *R* 2 , AIC, and RMSE) for each distribution, representing the fitting results. GGS and Normal exhibit the larger RMSE values, while Lognormal function shows great AIC values. In addition, GGS presents the lowest adjusted *R* <sup>2</sup> and larger AIC (the closer this criterion to one, the smaller its actual value). Gamma, RR and Swebrec distributions illustrate the better value points according to the three selected criteria, indicating the excellent correspondence between model prediction values and observed data. Though

AIC penalised the model with additional parameters, the Swebrec model, which has three independent variables, is among the better-fitting functions.

**Figure 3.** Normalised bar chart of estimator's values for each evaluated model.

#### *3.2. Models' Prediction Ability in Grinding Conditions*

α Figure 4 shows the PSD obtained (experimental and fitted) after different grinding times in the laboratory ball mill, with 3.00 cm in diameter grinding media balls. The same behaviour was observed in the case of 2.54 cm and 4.00 cm ball diameter (See Supplementary Material, Figures S2 and S3). The GGS model shows a more significant deviation under all the evaluated grinding conditions. After one minute's grinding, the predicted value deviates from the experimental value. This model considers a linear relationship between cumulative fraction and the particle diameter in the log-log scale, where the slope is the α parameter, which does not describe the experimental metallurgical coke grinding product. Normal and Lognormal distributions exhibit, in general, high goodness-of-fit. However, as Normal PSD illustrates, the predicted results decrease at a longer grinding time and smaller grinding media diameter. The Lognormal shows excellent fitting at grinding times over 3 min with all ball sizes, especially with grinding media of 4.00 cm. The curve fitting performance of Gamma, RR, and Swebrec are highly recommended for metallurgical coke grinding products in all evaluated scenarios with an adjusted *R* 2 range between 0.98–1.0. Model parameter prediction ability for the three abovementioned models were suited correctly with the ball size and grinding time values studied.

**Figure 4.** Cumulative distribution functions with 3.0 cm ball size at different grinding times: (**a**) 0 min; (**b**) 0.5 min; (**c**) 1 min; (**d**) 2 min; (**e**) 3 min; (**f**) 4 min; (**g**) 5 min; (**h**) 6 min; (**i**) 10 min.

#### *3.3. Distribution Parameters' Assessment*

In light of the above results, Swebrec and RR models were selected to assess the distribution parameters under different grinding conditions. Figure 5a depicts a rise in the undulation parameter *b* as grinding time increases. Comparing the *b* parameter variation with different grinding media is relatively stable until 3 min, after which superior *b* values (*b* > 3) are achieved for all ball sizes. Smaller undulation parameters are obtained with 2.54 cm ball diameter and the larger ones with 4.00 cm diameter. This situation can be associated with the ball energy delivery at grinding time under 3 min; the *x*<sup>50</sup> (see Figure 5b) remains directly related to the ball size, with the *x*<sup>50</sup> at 2.54 cm being smaller than the *x*<sup>50</sup> at 4.00 cm. The undulation parameter can indicate a change in fracture behaviour, ranging from normal breakage at a shorter time and smaller grinding media, passing by chipping, and finally, achieving material pulverisation at a longer time and larger ball diameter.

*λ λ* The *λ* and *k* RR parameters are illustrated in Figure 6. The particular behaviour of the shape parameter *λ* (Figure 6a) between 0 to 8 min grinding time is evidenced. The smaller parameter value, linked with the larger fines quantity, is formed using the ball size range 2.54–3.00 cm. A widening is noticed at grinding times shorter than 2 min, with 3.00 and 4.00 cm ball sizes. Scale parameter *k* (Figure 6b) presents smaller values at longer grinding times. The value increases from 2.54 to 3.00 cm, decreasing afterwards to 4.00 cm ball size. Larger ball sizes (3.00 and 4.00 cm) lead to greater *k* values' variation with the grinding time.

**Figure 5.** Swebrec parameters' variation under different grinding conditions: (**a**) *b*, undulation parameter and (**b**) *x*<sup>50</sup> parameter.

*λ λ* **Figure 6.** RR parameters' variation under different grinding conditions: (**a**) *λ* parameter and (**b**) *k* parameter.

Table 3 shows the *x*<sup>50</sup> values measured and predicted by the Swebrec model under different grinding conditions. The median diameter decreases when increasing the grinding time in all cases (as expected, due to the comminution action). The variation is different for each ball diameter; higher grinding kinetics is observed in the case of 4.00 cm ball diameter at grinding times longer than 2 min.



#### **4. Discussion**

Grinding conditions influence the product PSD, and the accuracy of the fitting distribution is highly dependent on the selected model. Statistical indexes used in this study, namely adjusted *R* 2 , AIC, and RMSE, have been widely used to the goodness-of-fit assessing of different PSD and have presented advantages in the models' calibration by least-squares method [25,29,32,35].

Both RR and Swebrec functions show excellent fitting performance for all grinding conditions studied, whereas GGS indicated the poorest yield fitting. These results agree with previous research, which reported better fitting of RR and Swebrec to grinding products and an accurate description of PSD [26,34,39]. Menéndez-Aguado et al. [35] compared different distribution models to sediments and found that the Swebrec model had better performance than Normal, Lognormal, Weibull, and Gamma functions. The undulation Swebrec parameter related to grinding conditions showed dependence on the grinding time and the ball size but only after 4 min for the last mentioned condition. In the case of grinding times less than 4 min, the *b* parameter seems to be independent of grinding media. As the particle size decreases with the grinding process (Figures 4 and 5), the undulation parameter value increases with high values when chipping action predominates in the fracture process, resulting in a fines overproduction.

Regarding RR parameters, the shape parameter *λ* (Figure 6a) shows more stability in the central region, which means that the distribution is highly affected by conditions at lower and greater times. Meanwhile, at grinding times of 3–8 min, *λ* remains between 0.9– 1.2, indicating a more wide size interval and a lower slope in the PSD. The scale parameter *k* (Figure 6b) presents a sensible change in the stability at 4 min grinding time. At grinding time less than 4 min, 3.00 and 4.00 cm ball sizes produce a limited size range with the feature that as ball size increases, the larger fragments do not break, resulting in larger values of *k* and *λ*, and indicating an inefficient ball-particle interaction. This situation could indicate that when using 3.00 and 4.00 cm grinding media, the dominant fragmentation mechanism is the chipping abrasion instead of impact breakage.

Although GGS is a popular model used in many sectors, it does not present sound goodness-of-fit in metallurgical coke PSD grinding products under the considered conditions. The GGS predicts distribution with a linear relationship between cumulative weight and particle size in the log-log scale. However, the actual metallurgical coke grinding product shows higher percentage of fines than the model prediction. This result is aligned with previously reported results, which found that the RR gets better PSD fitting than the GGS model [24,40,41].

The Gamma distribution offers an excellent approximation to predict metallurgical coke PSD, but the statistical parameters were more lacking than in the Swebrec and RR models. As shown in Figures 2 and 4, the Gamma function has some outliers in the adjusted *R* <sup>2</sup> due to shorter grinding times.

Under the grinding test conditions, the production of fines accelerates with all grinding media tested after 3–4 min, producing a widening of the PSD. It has been established in previous studies that the grinding efficiency is related to ball size selection [42–44]. Over time, the metallurgical coke breakage in 4.0 cm balls presents significant variation with undergrinding for 0.5–1 min and overgrinding to grinding time from 3–10 min; 4.00 cm grinding media is perhaps too large, creating voids inside the ball charge and generating less normal forces into particles. These results are consistent with Austin et al. [45] and Khumalo et al. [46] results, which established that the larger ball size action is directed to larger particles whereas small grinding media action is preferent on finer particles. Additionally, Austin et al. [45] proposed that the impact force of collision involving large ball sizes gives larger quantities of fines and more catastrophic fracture behaviour. Ball sizes of 2.54 and 3.00 cm evidence small sudden changes in median diameters. However, lower ball size produces lesser *x*<sup>50</sup> sizes, considering balls of 3.00 cm show more grinding to achieve a defined particle size than balls of 2.54 cm. The relationship between ballparticle size for breaking metallurgical coke improves with small ball sizes, possibly due to increased collision frequency.

The assessment of the coke grinding product comparing different particle size functions was carried out. The Swebrec distribution function presented outstanding fitting conditions in grinding products compared to traditional distributions. It is interesting to note that Swebrec parameters, such as GGS and RR models, are related to the fine particles' produced quantities. Therefore, in agreement with other authors [26,34,35], Swebrec function's employment could pose a good alternative to evaluate and control PSD in grinding processes and small particles. Additionally, the metallurgical coke particle range addressed in this study is considered a critical point in steelmaking, especially in sinter plants. The superior sinter properties have been obtained with 3.35 mm undersize [15–17]. As mentioned above, coke particle size influence the sinter porosity, microstructural phases and thermal properties of the sinter bed. The metallurgical coke PSD evaluation in the range between 3.35 mm to 0.212 mm is consistent with Umadevi et al. [15], which found that the use of this size range increases the calcium ferrite phase and decreases the number of the bigger pore size, thus decreasing coke quality and the coke strength index. On the other hand, Dabbagh et al. [16] evaluated the coke PSD effect on the maximum temperature of the sinter bed and the flame front velocity, finding that the particles ranging from 3.35 mm to 0.212 mm increase the heat production and favour the diffusive processes of the sinter bed.

#### **5. Conclusions**

Several PSD models were evaluated on metallurgical coke grinding products using adjusted *R* 2 , root means of square error (RMSE) and Akaike's information criterion (AIC) as statistical indices. Variety of grinding media size and grinding time were employed to investigate the goodness-of-fit to PSD models based on different grinding conditions.


**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/met11081288/s1, Figure S1: Box plot to compare the particle size models base on adjusted R2 criterion, excluding Gates Gaudin Schuhmann distribution, Figure S2: Cumulative distribution functions with 2.54 cm ball size, at different grinding times, Figure S3: Cumulative distribution functions with 4.0 cm ball size, at different grinding times, Table S1: Statistical descriptors for three criteria, Table S2: Custom script to determine fitting parameter by least square method.

**Author Contributions:** Conceptualisation, L.C.-A., J.M.M.-A. and A.O.-C.; methodology, L.C.-A., J.M.M.-A. and A.O.-C.; software, L.C.-A.; validation, L.C.-A., J.M.M.-A. and A.O.-C.; formal analysis, J.M.M.-A. and A.O.-C.; investigation, L.C.-A.; resources, A.O.-C.; writing—original draft preparation, L.C.-A., J.M.M.-A. and A.O.-C.; writing—review and editing, L.C.-A., J.M.M.-A. and A.O.-C.; visualisation, L.C.-A.; supervision, A.O.-C.; project administration, A.O.-C.; funding acquisition, A.O.-C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are thankful to the Faculty of Engineering of the Universidad de Antioquia, to the Committee for Research Development, and CODI for economic support to conduct this work in the project framework: Study of alternatives for the improvement of specific rate of breakage through the intensification of the grinding process in ball mill PVR2018 21371.

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

#### **References**


**Angel R. Llera 1 , Ana Díaz 1 , Francisco J. Pedrayes 2 , Juan M. Menéndez-Aguado 1, \* and Manuel G. Melero 2**


**Abstract:** A significant challenge in mineral raw materials comminution is the improvement of process energy efficiency. Conventional comminution techniques, although globally used, are far from being considered power-efficient. The use of high-voltage electric pulses in comminution is a concept that is worthy of study; despite its lack of industrial-scale validation after several decades of lab-scale research, it seems promising as a pretreatment leading to energy savings. In this article, the Cumulative Kinetic Model methodology is adapted to model the comminution effect in an electrofragmentation device, and study a dunite rock ore. The results show that product particle size distribution (PSD) can be predicted with reasonable accuracy using the proposed model.

**Keywords:** electrofragmentation; comminution; Marx generator; modeling

#### **1. Introduction**

Comminution operations are essential in mineral raw materials industries, and estimations of their share in global energy consumption range from 3 to 5% [1–4], so the improvement of process energy efficiency poses a significant challenge in mineral processing technology. Conventional comminution techniques, although globally used, are far from being considered power-efficient. The use of high-voltage electric pulses (HVEP) in comminution is a concept worth studying; despite its lack of industrial-scale validation after several decades of lab-scale research, it seems promising as a pretreatment leading to energy savings. Moreover, it is probably the only known comminution technology capable of maintaining its efficiency in a zero-gravity environment.

Initial research into HVEP use in comminution started in the mid-20th century to produce rock weakening and selective mineral fragmentation [5,6]. Some studies performed comparisons with conventional technologies on such issues as size reduction capability and energy consumption [7–10], while other studies focused on improving mineral liberation [11–15].

This study proposes a mathematical model to predict product PSD in an HVEP device after one or more electric pulses under specific working conditions. Preliminary tests showed the particular influence of pulse polarity on breakage results, so this effect will also be analyzed.

#### **2. Experimental**

#### *2.1. Materials*

Samples were supplied by the mineral processing plant at Mina David (Pasek Minerales), located in Landoi (Spain). This is the only dunite producer in Spain; despite the olivine content being too low (20–30%) to classify it as a dunite rock, it keeps this commercial denomination. Along with olivine, it is usually accompanied by orthopyroxene (8–16%), amphibole (14–20%) and chrysotile (0–33%). Moreover, other minerals can appear

**Citation:** Llera, A.R.; Díaz, A.; Pedrayes, F.J.; Menéndez-Aguado, J.M.; Melero, M.G. Study of Comminution Kinetics in an Electrofragmentation Lab-Scale Device. *Metals* **2022**, *12*, 494. https:// doi.org/10.3390/met12030494

Academic Editors: Gunter Gerbeth and Petros E. Tsakiridis

Received: 10 February 2022 Accepted: 11 March 2022 Published: 14 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 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/).

in the open pit due to hydrothermal alterations, such as chlorite, serpentinite and clay group minerals. Table 1 shows the X-ray fluorescence (XRF) results. Further characterizations of this ore can be found in [16].

**Table 1.** XRF ore results (%) (L.O.I. = lost on ignition).


Due to the high Mg content shown above, Pasek Minerales is currently developing an extraction process, aimed at producing high-quality magnesium oxide from dunite fines; any step towards a reduction in the specific energy consumption in the fines production process would be desirable.

To provide comminution characterization, a Bond ball mill standard test was performed on a representative sample, with a result of 11.6 kWh/t at 100 microns.

A sufficient amount of sample was prepared within narrow size intervals via sieving. These fractions can be considered monosizes, and they were tested separately to determine the influence of particle size. The selected intervals were (in microns): 5000/3350; 3350/2000; 2000/1000; 1000/500; 500/125 and 125/0. Table 2 shows the total weights of each monosize after sieving. Aliquots of 500 g were prepared for each monosize using a Jones sample divider (RETSCH, Haan, Germany).

**Table 2.** Sample weight after preparation.


#### *2.2. Methods*

2.2.1. HVEP Test Rig

The test rig (see Figure 1)is based on a Marx pulse generator SGSA 400-20 (HAEFELY, Basel, Switzerland), located at the Electrical Engineering Department facilities in Gijon (University of Oviedo, Spain). The main characteristics of this HVEP test rig are depicted in Table 3.

**Table 3.** General specifications of the HVEP generator.


Figure 2 shows the diagram of a Marx impulse generator. The depicted C and Cs correspond to the test cell and the impulse capacitance, respectively. Rs and Rp are the resistances that define the pulse leading edge time and trailing edge time, respectively. The element SF represents the spark gap that starts the discharge of the impulse capacitance into the test cell, thus generating the requested pulse.

**Figure 1.** Test rig (1) pulse generator, (2) charge unit, (3) capacitive divider, (4) compensation circuit.

**Figure 2.** Marx generator diagram.

The pulse generation and measurement process are represented in Figure 3. Firstly, the desired impulse specifications are set in the generator control unit, including the number of work stages, peak impulse value, capacitance charging time and impulse polarity. Secondly, the charging unit raises the voltage to the specified peak value and the charging rectifier converts this to direct current, which is used to charge the generator capacitors. Afterwards, when the capacitors reach the pre-set voltage, the control unit orders the impulse to discharge on the sample within the test cell. Finally, the impulse is registered using a voltage divider in parallel, which permits the signal's digitalization and treatment.

In contrast to the devices used in previous studies [17–20], this test rig has the option of changing impulse polarity. This feature can be achieved by changing the positions of the charge unit diodes (Figure 4), inverting the voltage discharge polarity and thus getting positive or negative discharge impulses on the test sample. Figure 5 shows two examples of no-load impulse curves of different polarities (X axis time in microseconds; Y axis voltage in kV).

**Figure 3.** HVEP test rig block diagram.

**Figure 4.** Charge unit diodes.

A relevant parameter in the electrofragmentation tests is the pulse rise time, for this must be short enough to produce a successful fragmentation [21]. Impulse discharge through a mineral sample requires enough voltage to overcome the sample dielectric strength, but the voltage achieved should not surpass the surrounding material's dielectric strength, because, in that case, the discharge would concentrate in the surrounding medium. Additionally, if a medium with higher electric permittivity surrounds the mineral sample, a very uneven distribution of the applied electric field occurs, with a high concentration in the mineral and a much lower concentration in the surrounding medium.

Both effects can be achieved by soaking the mineral sample in distilled water; at a very short pulse rise time, water's dielectric strength and permittivity are higher than rock's [21,22], as shown in Figure 6, which shows that the pulse rise time should be less than 500 ns.

**Figure 5.** No-load impulses: (**a**) positive polarity; (**b**) negative polarity.


**Dielectric strength** 

**Impulse strength (kV/mm)** 

With the aim of a more significant reduction in the pulse rise time, we substituted the resistance Rs (Figure 2) for a short-circuit; thus, a pulse rise time around 300 ns can be achieved, with a peak voltage of 150 kV (this value was set in all tests performed), plus an additional value due to overshooting. Under these conditions, the discharge effect will concentrate in the mineral sample; the wave shapes obtained when applying these pulses (both with positive and negative polarity) are shown in Figure 7. **time (ns) ~500 ~**

**rock water impulse** 

**Figure 7.** Wave shapes after impulse discharge on dunite sample: (**a**) positive polarity; (**b**) negative polarity.

#### 2.2.2. HVEP Test Cell

In order to correctly apply generated pulses to the mineral sample, a test cell was developed that was to be attached to the Marx pulse generator, following the scheme proposed in [21]. Because the peak voltage values could reach hundreds of kV, the insulator definition, electrode configuration and distances among live elements and grounded elements were critical.

The basis of the test cell was an inox steel vessel acting as the grounded electrode. This vessel has a high-density polyethylene (HDPE) shell inside it that acts as an insulator. The active electrode is also embedded in HDPE and is supported by 3D printed parts that stabilize the whole (Figures 8 and 9), so a flat-tip electrode configuration is defined.

**Figure 8.** (**Left**): Test cell diagram. (**Right**): Test cell connected to the impulse generator.

**Figure 9.** Diagram of the pulse generator and coupled test cell.

The mineral sample and the dielectric liquid are placed at the bottom of the steel vessel, which, in turn, rests on a grounded copper sheet. The active electrode, connected to the pulse generator output, comprises a copper rod that comes into contact with the sample. The HDPE cylindrical pieces guarantee that no electric arcs are formed outside the sample volume. With this electrodes configuration and the expected voltage values, the electrode distance was estimated at 25 mm; this value is in line with the values reported in [11,20–22], within the interval 20–40 mm.

#### 2.2.3. HVEP Test Procedure

The tests were carried out on the pulse generator, applying high-voltage electrical pulses. At each monosize, a total of fourteen tests was performed, seven tests with positive polarity and seven more with negative polarity, in order to establish the possible influence of polarity on the degree of fragmentation of the sample. At each polarity, four samples were tested with one, two, three and four pulses, respectively. The three remaining samples were tested using five pulses to determine the test's repeatability on the final PSD.

After each test, the collected sample was dried to remove the distilled water used as a dielectric medium and sieved to obtain the PSD.

#### 2.2.4. Mathematical Model

A mathematical model that describes the effect of electrofragmentation on PSD is proposed, based on an adaptation of the Cumulative Kinetic Model [23,24] into a discontinuous process, as expressed in Equation (1).

$$\mathcal{W}\_{(\mathbf{x},i)} = \mathcal{W}\_{(\mathbf{x},f)} \cdot e^{-k \cdot i} \tag{1}$$

wherein:

*W*(*x,i*) is the cumulative oversize of size class *x* after *i* pulses;

*W*(*x,f*) is the cumulative oversize of size class *x* in the feed;

*k* is the breakage rate parameter.

The relationship between the breakage rate parameter and the particle size is shown in Equation (2):

$$k = a \cdot \mathbf{x}^b \tag{2}$$

where *a* and *b* can be determined experimentally. Accordingly, once one has defined the model parameters, the electrofragmentation product PSD after *i* pulses can be obtained from the feed PSD using Equation (3).

$$\mathcal{W}\_{(\mathbf{x},\mathbf{i})} = \mathcal{W}\_{(\mathbf{x},\mathbf{0})} \cdot e^{-a \cdot \mathbf{x}^{\mathbf{b}} \cdot \mathbf{i}} \tag{3}$$

The *k* value is determined for each monosize after taking logarithms at Equation (1):

$$\ln(\mathcal{W}\_{(\mathbf{x},i)}) = \ln(\mathcal{W}\_{(\mathbf{x},0)}) - k \cdot i \Rightarrow \ln(\mathcal{W}\_{(\mathbf{x},i)}) - \ln(\mathcal{W}\_{(\mathbf{x},0)}) = k \cdot i \tag{4}$$

Once one has obtained *k* values for each monosize, an additional linear regression can be performed to calculate *a* and *b*, according to Equation (5).

$$
\ln(k) = \ln(a) + b \cdot \ln(x) \tag{5}
$$

#### **3. Results and Discussion**

Tables S1–S10 show the results of the 70 impulse tests performed on different monosizes, with positive and negative polarity, including the three five-pulse replicas.

Figure 10 compares the PSD values (cumulative oversize) in the monosize 5000/3350 µm case when using different polarities. In the case of no influence of the polarity, values should be randomly spread following the diagonal line. However, in this case, plotted points are located above the diagonal line due to the cumulative oversize value being higher in the case of positive polarity; this means that the comminution effect is higher in the case of negative polarity. This monosize shows the same behavior in the case of one to four pulses, while in the case of five pulses, values almost fit the diagonal, thus meaning that polarity does not influence the PSD after five pulses.

**Figure 10.** Product PSD after a different number of pulses (feed monosize 5000/3350 µm) and different polarity.

60% 65% 70% 75% 80% 85% 90% 95% 100%

1 2 3 4 5

Negative polarity

60%

65%

70%

75%

80%

Positive polarity

85%

90%

95%

100%

75%

80%

Positive polarity

85%

90%

95%

100%

The same analysis was performed with the rest of the monosizes. Figure 11 shows the result in the 3350/2000 µm size interval case, which shows an opposite behaviour from the previous monosize. In this case, the positive polarity seems to produce a more intense comminution effect in the case of one to four pulses, while again, in the case of five pulses, the polarity seems not to influence the PSD. On the other hand, with monosizes 2000/1000 µm, 1000/500 µm, and 500/125 µm (Figures 12–14), the results suggest that the polarity does not influence the comminution effect. From these results, the influence of the polarity cannot be concluded; however, under certain conditions, the results show that a specific polarity could improve the comminution effect in the electrofragmentation device. 60% 65% 70% 60% 65% 70% 75% 80% 85% 90% 95% 100% Negative polarity 1 2 3 4 5

**Figure 11.** Product PSD after a different number of pulses (feed monosize 3350/2000 µm) and different polarity.

**Figure 12.** Product PSD after a different number of pulses (feed monosize 2000/1000 µm) and different polarity.

60% 65% 70% 75% 80% 85% 90% 95% 100%

1 2 3 4 5

Negative polarity

60%

65%

70%

75%

80%

Positive polarity

85%

90%

95%

100%

60%

65%

70%

75%

80%

Positive polarity

85%

90%

95%

100%

60% 65% 70% 75% 80% 85% 90% 95% 100%

1 2 3 4 5

Negative polarity

**Figure 13.** Product PSD after a different number of pulses (feed monosize 1000/500 µm) and different polarity.

Regarding the comminution modeling, from data gathered in Tables S1–S10 and Equations (4) and (5), the proposed model parameters can be calculated, again for each polarity. Table 4 shows the results of *a* and *b* parameters and the correlation coefficient value

**μ**

**μ**

obtained in Equation (5) for linear regression. According to the R<sup>2</sup> values, both polarities show a better fit at coarser monosizes, with very similar values.


**Table 4.** Model parameter values calculated.

The parameter values shown in Table 4 were calculated by considering replica 1 at five pulses, in order to compare the model's estimated PSD with the remaining replicas. Table 5 gathers the results obtained with both polarities, in the case of the 5000/3350 µm monosize; these results are also plotted in Figure 15. Tables S11–S13 in the Supplementary Material gather the results of the other monosizes.

**Table 5.** PSD values (modeled and real), feed 5000/3350 µm monosize, five pulses.


**Figure 15.** Product PSD after five pulses, monosize 5000/3350 µm.

In order to analyses the results, a first comparison was made between replicas 2 and 3. Subsequently, a second comparison was performed between the modeled PSD values and the average distribution obtained from replicas 2 and 3 (labeled as real). Model deviation had a relative error lower than 2%, which was even lower than 0.5% at finer monosizes. The F-test values are shown in Table 6 for all monosizes and both polarities.

**μ**


**Table 6.** F-test values obtained in the comparisons performed.

According to the results shown in Figure 15, in general terms, the proposed model achieves a good fitting of PSD after five pulses, with a slightly better result in the case of positive polarity; this can also be deduced from the F-values shown in Table 6, obtaining a value of 0.9777 in the case of positive polarity, which is higher than the value obtained in the case of negative polarity, 0.8724. Further research must be performed with different ores and pulse conditions to define the influence of pulse polarity.

#### **4. Conclusions**

From the results obtained in this research, the following conclusions can be highlighted:


**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/met12030494/s1, Table S1: Results obtained with monosize 5000/3350, negative polarity; Table S2. Results obtained with monosize 5000/3350, positive polarity; Table S3. Results obtained with monosize 3350/2000, negative polarity; Table S4. Results obtained with monosize 3350/2000, positive polarity; Table S5. Results obtained with monosize 2000/1000, negative polarity; Table S6. Results obtained with monosize 2000/1000, positive polarity; Table S7. Results obtained with monosize 1000/500, negative polarity; Table S8. Results obtained with monosize 1000/500, positive polarity; Table S9. Results obtained with monosize 500/125, negative polarity; Table S10. Results obtained with monosize 500/125, positive polarity; Table S11: PSD values (modeled and real), feed 3350/2000 monosize; Table S12: PSD values (modeled and real), feed 2000/1000 monosize; Table S13: PSD values (modeled and real), feed 1000/500 monosize.

**Author Contributions:** Conceptualization, A.R.L., J.M.M.-A. and M.G.M.; methodology, J.M.M.-A., M.G.M. and F.J.P.; software, A.R.L. and A.D.; validation, F.J.P. and M.G.M.; formal analysis and investigation, A.R.L., F.J.P. and M.G.M.; resources, J.M.M.-A. and M.G.M.; writing—original draft preparation, A.R.L., A.D. and F.J.P.; writing—review and editing, J.M.M.-A. and M.G.M.; supervision, J.M.M.-A. and M.G.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by the Spanish Ministry of Economy and Competitiveness, under project DPI2017-83804-R.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

#### **References**


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