Fit-for-Purpose Model of HP500 Cone Crusher in Size Reduction of Itabirite Iron Ore

Abstract

Cone crushers have a central role in the processing of quarry rocks, besides coarser ore preparation in several mineral processing plants. This is particularly true in the case of Itabirite iron ore preparation plants in Brazil, so optimizing their performance is of central importance for reaching maximum productivity of the circuit. The work presents results of modeling the HP500 cone crusher in operation in an industrial plant in Brazil (Minas Rio), from surveys carried out over a few years with different feeds and crushing conditions. A version of the Andersen–Whiten cone crusher model was implemented in the Integrated Extraction Simulator featuring a non-normalizable breakage response and a fit-for-purpose throughput model. The results demonstrate the good ability of the model to predict crusher performance when dealing with different closed-side settings and feed size distributions.

1. Introduction

Itabirite iron ores correspond to Banded Iron Formation (BIF) deposits, which are composed primarily of iron minerals such as hematite and magnetite, along with silicates and various carbonates that are widely found in the state of Minas Gerais (Brazil) [1,2]. These formations are generally classified as low-grade iron ores and exhibit varying degrees of weathering, depending on their depth relative to the topographic surface. This depth-dependent weathering results in different typologies of Itabirite, each characterized by distinct mineralogical and textural features [1]. In general, Itabirite iron ores present characteristics that affect their size reduction response, with segregation in selected size classes [1,3,4,5], often associated with some extent of preferential mineral liberation by grain-boundary breakage [1]. This fracture behavior poses an additional challenge for describing the size reduction of these iron ores within crushing stages in a circuit simulation.

Itabirites located in the Iron Quadrangle (Quadrilátero Ferrífero) province are often reduced in size through a combination of multiple stages of crushing followed by ball milling [6]. Given the important role of crushing, proper selection and assessment of crushing circuits demands a deep understanding of the ore breakage behavior since it impacts the overall productivity and cost-effectiveness of mining projects. Despite the large experience in the minerals industry in designing crushing circuits, challenges in this comminution stage are still a reality for processing Itabirite iron ores [1].

Important advances have been made in the understanding and modeling of size reduction in cone crushers in the last 50 years or so. Whiten [7] proposed a model to describe compressive crushing from the classification and breakage functions, which became the most well known in the field [8]. This modeling approach relies on fitting these functions based on data, typically obtained from industrial surveys of crushing circuits, where the key parameters of the classification function are often linearly related to the crusher closed-side setting (CSS). Such applications of the model were previously demonstrated in modeling the performance of standard Symons and short-head cone crushers in the size reduction of limestone [9] and uranium ore [10]. More recently, Neves and Tavares [11] modeled Hydrocone crushers in the size reduction of gneiss.

Andersen and Napier-Munn [12] then proposed modifications to Whiten’s model by incorporating a breakage (appearance) function that is obtained from single-particle breakage tests and a method to predict crusher power. This has the advantage of reducing the number of parameters that require fitting, besides establishing a connection between the single-particle breakage response and crusher power during operation. They used this model to describe size reduction in a Symons short-head cone crusher [12], and linear variations of the key parameters in the classification function were established not only for the CSS, but also for eccentric throw, throughput, 80% passing in the feed and liner condition. Moshgbar et al. [13] used this model incorporating the rate of wear of the concave and mantle liners to transform the classification function parameters into time-dependent functions. Magalhães and Tavares [14] then used the model to back-calculate breakage characteristics of ore samples from tests in a laboratory cone crusher. More recently, an adapted version of the model was used to describe industrial cone crushing with different values of the CSS [15].

Alternative models of cone crushers have been proposed using other population balance model approaches [16,17], besides a mechanistic model [18] and, more recently, the Discrete Element Method (DEM) [19,20,21,22]. The mechanistic model of Evertsson [18], which describes the effect of CSS, chamber geometry, eccentric throw and eccentric frequency, has been proposed as a model suitable for design but lacks fitting parameters to represent a particular operation. On the other hand, the DEM is a powerful tool for analyzing these and other variables in crushing but is not suitable for circuit simulation.

Another important aspect of cone crusher simulation is the prediction of throughput when operating with a filled chamber (choke-feeding). Gauldie [23] proposed a theoretical throughput model for cone crushers, which accounts for crusher geometry, speed and liner configuration, which was later improved [24]. Karra [9] proposed an empirical model that described the variation in crusher capacity as a function of eccentric throw, CSS and head angle.

The present work initially proposes minor modifications to Whiten’s crusher model to account for non-normalizable breakage behavior that is found in cone crushing of Itabirite iron ores, besides expressions to identify maximum crusher capacity. Fitting of model parameters from data collected over several years of operation of HP500 cone crushers in the Minas Rio crushing plant in a secondary crushing role is then carried out.

2. Modelling Background

2.1. Andersen–Whiten Crusher Model

Whiten’s model [7] describes the compressive crushing of ore particles as a size–mass balance in different size classes by successive stages of classification and particle breakage. The overall product size distribution in distributed form (p) can be predicted as follows [25]:

$$p=\left(I-C\right){\left(I-BC\right)}^{-1}f$$

(1)

where $I$ is the identity matrix, $f$ is the feed size distribution vector, $B$ is the lower-triangular breakage function matrix, and $C$ is the classification function, represented by a diagonal matrix.

The classification function defines the probability of a particle in a given size range being classified for breakage and is given as follows [25]:

$$\begin{array}{cc}C\left(\overline{x}\right)=0& \overline{x}\le K_1\\ C(\overline{x})=1-{\left({\displaystyle \frac{K_2-\overline{x}}{K_2-K_1}}\right)}^{K_3}& K_1<\overline{x}<K_2\\ C\left(\overline{x}\right)=1& \overline{x}\ge K_2\end{array}$$

(2)

where model parameter $\overline{x}$ is the representative size given by $\overline{x}=\sqrt{x_i{x}_{i-1}}$, where x_i represents the sieve size i. Parameter K₁ defines the bottom limit of the middle portion of the distribution curve described by Equation (2), which defines the particle size below which particles are not nipped by the crusher. Parameter K₂ defines the upper limit of the distribution curve, which determines the size above which all particles are nipped. Parameter K₃ defines the slope of the curve. The optimal values of $K_1$ and $K_2$ depend on the operating conditions of the crusher, feed characteristics and machine settings. Empirical relationships have been proposed between them and the cone crusher variables, giving [25]

$$K_1=A_0+A_{1}CSS+A_{2}Q+A_3{f}_{80}+A_{4}LLEN$$

(3)

$$K_2=B_0+B_{1}CSS+B_{2}Q+B_3{f}_{80}+B_{4}LHR+B_{5}ET$$

(4)

where $CSS$ is the closed-side setting, Q the crusher throughput, $f_{80}$ the 80% passing size in the feed, $LLEN$ the thickness of the mantle liner, $LHR$ the mantle operating time, $ET$ the eccentric throw and $A_0$ to $A_4$ and $B_0$ to $B_5$ are model fitting parameters. $K_3$ is often assumed as a constant value of 2.3 [25], although Karra [9] found that the optimal value was 3.0.

Particles that are classified for breakage in Equation (2) will be compressed down to the CSS and eventually break. Whiten’s model was later modified to use the appearance function obtained from single-particle breakage tests [12], using the t₁₀-t_n relationship [26]. The appearance function method is well known for its use in energy-based modeling, but it is assumed to be a function of the crushing operating conditions as [25]

$${\mathrm{t}}_{10}={\mathrm{D}}_0-{\mathrm{D}}_1\mathrm{C}\mathrm{S}\mathrm{S}+{\mathrm{D}}_2\mathrm{Q}+{\mathrm{D}}_3{\mathrm{f}}_{80}$$

(5)

$${B_{ij}}^{\prime }=\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\left[t_{10},t_n,x_i/x_j\right]$$

(6)

where D₀ to D₃ are fitting parameters and $t_{10}$ and $t_n$ are, respectively, the percent passing in 1/10th and 1/nth of the parent particle size. The effect of size on breakage resistance can be taken into account by using drop-weight data for particles contained in different size ranges.

To calculate the crusher power (P_c) during operation, Andersen and Napier-Munn [12] proposed a model that is based on the no-load power of the crusher and the power estimated from the impact test, originally conducted using the pendulum test and more recently using the drop-weight test, as follows:

$$P_c=\mathsf{\zeta }{P}_d+P_n$$

(7)

where $P_n$ is the no-load power (kW), $\zeta$ is a dimensionless factor for a specific crusher and $P_d$ is the power estimated based on drop-weight tests given as follows [25]:

$$P_d=Q\sum _{i=1}^N{Ecs}_{{t10}_i}{C}_i{w}_i$$

(8)

where $w_i$ is the mass fraction of material in each size in the crusher, N is the number of size classes and ${Ecs}_{{t10}_i}$ is the energy matrix compiling the specific energy required to generate a t₁₀ equal to 10%, 20% or 30%, which allows describing the effect of particle size on impact strength [25]. The DWT power from Equation (8) assumes that the work performed by the crusher is a function of the energy required to break the particles from the feed that were classified for breakage, with a given efficiency (1/$\zeta$), besides the no-load power.

2.2. Adaptations to the Crusher Model

With the aim of further improving the applicability of the Andersen–Whiten model [7,12] for the Itabirite iron ore in question, some modifications were implemented in the present work. At first, a modification was introduced to account for the non-normalizable character of the breakage (appearance) function with respect to size. A second modification consisted in the estimation of the maximum throughput of the crusher when operating with a full chamber as a function of feed and operating conditions.

As extensively reported for processing Brazilian Itabirite iron ores [1,3,4,5] and already discussed, the breakage behavior of the ore may follow a size distribution that is non-normalizable with respect to the original particle size. A simple and effective way to account for this feature was proposed elsewhere [27] and was recently used by population balance modeling [3,4,5,28,29] and mechanistic breakage modeling [30]. The approach is given as follows [27]:

$$B_{ij}={B_{ij}}^{\prime }{\left({\displaystyle \frac{x_i}{y_0}}\right)}^{n_3}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{1em}{x}_i<y_0$$

(9)

where ${B_{ij}}^{\prime }$ is the cumulative breakage function given by Equation (6). By considering $n_3$ and $y_0$ equal zero, the breakage function in Equation (9) will rely on the breakage function predicted from Equation (6).

Cone crushers can operate at different feed throughputs, and these values directly affect crusher performance, as suggested in Equations (3)–(5). However, a maximum throughput is reached when the crusher operates with a filled chamber. Knowledge of this value is important when simulating the cone crusher in either dynamic or steady-state conditions since a mismatch between it and the feed throughput would define either the need to interrupt, at times, crusher feed to allow a minimum hopper level to be reached for choke-feeding operation or to divert part of the crusher feed to a crusher next to it, when that is the case. Empirical evidence has been compiled from data collected by cone crusher manufacturers over a century of operation of these machines regarding the effect of variables on this maximum throughput. Such evidence has been compiled in the past in crushing manuals [31], but it is now built into simulators such as Bruno^© from Metso [32] and PlantDesigner^© from Sandvik. They show that maximum cone crusher throughput (Q_max) is a function of closed-side setting (CSS), feed size ($f_{80}$), bulk ore density (${\rho }_a$), ore competence (impact work index—$Wi$) and feed moisture ($\omega$).

A reverse engineering approach was adopted to define relationships between some of the variables listed above and the maximum throughput of the crusher. Simulations were run with Bruno^© (Version 4.2.2.0, exe v4.1.2.9, DB v9.50, Metso Corporation), while crusher manufacturer´s tables were analyzed [31] by investigating the effect of one variable at a time. A base case was defined following the default material as suggested in Bruno^©, which is a granite with specific gravity (${\rho }_{sp}$) of 2700 kg/m³, with bulk density (${\rho }_a$) of 1600 kg/m³ and a $Wi$ value of 16 kWh/t. The feed was considered with a moisture content of 1% and the 80% passing size of 100 mm, while the HP500 crusher was set to a nominal CSS of 38 mm with a medium cavity. The maximum crusher throughput can be defined through an empirical relationship, which is modified from a preliminary version presented in the crusher manuals [31]:

$$Q_{max}={\lambda Q}_T\alpha \beta \gamma \delta \mathsf{\Psi }$$

(10)

where $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mathsf{\Psi }$ are correction factors, equal to 1 for the base case, varying as a function of ore bulk density, resistance to crushing, feed size, moisture content and closed-side setting, respectively. Q_T is the crusher capacity of the base case, and $\lambda$ is a dimensionless fitting parameter that should be fitted to the industrial data available.

The correction factor for the effect of ore bulk density ($\alpha$) is simply given by

$$\alpha ={\displaystyle \frac{{\rho }_a}{1600}}$$

(11)

where ${\rho }_a$ is the ore bulk density, given in kg/m³. The correction factor for resistance to crushing is calculated by

$$\beta ={\left({\displaystyle \frac{16}{Wi}}\right)}^{\eta }$$

(12)

where $Wi$ is the impact work index [33,34], given in kWh/t, and $\eta$ is a fitting parameter. The contribution of the feed coarseness is given as

$$\gamma ={\left({\displaystyle \frac{100}{f_{80}}}\right)}^{\ni }$$

(13)

where $f_{80}$ is the 80% passing size in the feed, given in mm, and $\ni$ is a fitting parameter. The correction factor that accounts for the moisture content is

$$\begin{array}{cc}\delta =1& \mathrm{f}\mathrm{o}\mathrm{r}\mathsf{\omega }<1\mathrm{\%}\\ \delta =\mathrm{e}\mathrm{x}\mathrm{p}\left[-\mathsf{\xi }(\mathsf{\omega }-1)\right]& \mathrm{f}\mathrm{o}\mathrm{r}\mathsf{\omega }\ge 1\mathrm{\%}\end{array}$$

(14)

where $\mathsf{\omega }$ is the feed moisture content (%) and $\mathsf{\xi }$ is a fitting parameter. Care should be taken when using Equation (14) since the relevant moisture is the one contained in the crusher feed, which is often lower than that of the ROM ore.

Finally, the contribution of the closed-side setting is given as

$$\mathsf{\Psi }={\left({\displaystyle \frac{CSS}{{CSS}_{BC}}}\right)}^{\nu }$$

(15)

where CSS and ${CSS}_{BC}$ are, respectively, the closed-side setting for the crusher in operation and the closed-side setting of the base case, given in mm, while $\nu$ is a fitting parameter.

In the case of the HP500 crusher, the capacity of the base case (Q_T) becomes equal to 647 t/h according to Bruno^© [32]. The final parameter to be calibrated is λ, which is estimated on the basis of at least one relevant measurement of throughput when the crusher is operating under choke-fed conditions. Other effects, such as chamber type, eccentric throw, clay content and wear condition of the liner, are also known to affect the maximum throughput of the cone crusher [21,24,33]. The model corresponding to Equations (10)–(15) assumes that these need to be accounted for in the fitting parameter λ since they are not explicitly represented in the equations.

3. Methodology

3.1. Experimental Section

The work was focused on the secondary crushing stage of the Minas Rio Plant, operated by Anglo American and located in Conceição do Mato Dentro (Minas Gerais, Brazil). The Run-of-Mine ore is composed of four different iron ore lithologies, named Friable, Compact and Supercompact Itabirites, and Canga. Such lithologies were previously characterized according to their liberation and breakage response, being composed mostly of hematite and quartz [1]. While the feed is represented by a blend of these ores, the significantly finer feed size of Friable Itabirite, besides the small amount of Canga in the Run-of-Mine (<5%), allows us to consider that the cone crusher is fed by a blend of Compact and Supercompact Itabirites. The mean specific gravity of these, measured by pycnometry, was estimated as 3480 kg/m³. The bulk (apparent) density was measured by placing each weighed sample in a beaker and measuring the bulk volume after vibration, being given as 2088 kg/m³.

The secondary crushing stage, represented by four HP500 crushers, is fed by two primary jaw crushers. The cone crushers operate in a closed circuit with vibrating screens with 25 mm openings, whose undersize is fed to HPGRs.

The crushers are installed in parallel, making it possible to alternate their use according to different operational demands. All crushers were designed with a nominal closed-side setting (CSS) of 38 mm, installed power of 355 kW and maximum feed particle size of 204 mm, owing to the medium cavity of the crusher used [32]. A total of ten industrial surveys were carried out over three years of operation, and a summary of operating conditions and feed characteristics of each survey is presented in Appendix A (

Table A1. Summary of operating conditions and feed characteristics for the different industrial surveys performed on HP500 cone crushers.

Industrial Survey #	$80\%\mathbf{Passing}\mathbf{Size}\text{-}{\mathit{f}}_{80}$ (mm)	Closed-Side Setting (mm)	Solid Feed Rate (t/h)	Power (kW) *
1	88	54	1639	209
2	84	54	1267	212
3	47	48	1240	262
4	88	48	1034	349
5	62	48	1124	325
6	79	40	813	355
7	140	38	1050	306
8	123	44	1138	325
9	81	40	820	350
10	81	55	1334	187

* Estimated based on electric current values.

). The range of operating conditions varied from a CSS of 38 to 54 mm and from an 80% passing size in the feed (f₈₀) of 47 to 140 mm. All surveys were considered to be conducted with the crushers operating with a full chamber.

Representative samples of the feed and product of the cone crusher were collected from belt cuts in the surveys after the crash-stopping of the operation. Operating conditions, including CSS, solid feed rate and crusher power were recorded from the supervisory system for each survey. All samples were split, and their particle size distribution was analyzed by dry sieving in a RoTap^® sieve shaker.

The ore breakage characterization of representative samples of Compact and Supercompact Itabirites was conducted through drop-weight tests (DWTs) [25] and the Bond impact crushability (Wi) test [33,34]. In the first, narrow size samples contained in size ranges of 13.2–16.0 mm to 53.0–63.0 mm were impacted at stressing energies varying from 0.05 to 2.5 kWh/t, and their size distribution was analyzed by sieving. Markers from the size distributions, representing the proportions passing 1/2, 1/4, 1/10, 1/25, 1/50 and 1/75th of the initial particle sizes, called t_n markers [26], were calculated. Relationships between the stressing energy and the percent passing 1/10th of the initial size (t₁₀), and between this parameter and the various t_n values were then recorded. Additionally, the Bond impact work index was estimated based on the standard test from a selection of 70 particles contained in the size range of 53.0–75.0 mm for each ore type [34].

3.2. Model Implementation and Calibration

The crusher model presented in detail in Section 2 was implemented in the commercial software Integrated Extraction Simulator (IES) (Orica Inc., Victoria, Australia) to perform steady-state simulations.

The appearance or breakage function was described using the t₁₀-t_n relationship. Data were interpolated using the incomplete beta function [35], with the non-normalizable breakage correction term (Equation (9)) used to describe the breakage (appearance) function. On the other hand, parameters of the classification function (Equation (2)) were, at first, fitted based on results from each industrial survey. Thus, based on the appearance function (t₁₀-t_n) from the DWT tests, the present work calibrated the classification function parameters (Equations (2)–(4)) and the t₁₀ (Equation (5)) to data from each industrial survey. Parameters of these equations were fitted separately to data from each industrial survey using the IES platform.

The ultimate goal of the work was to obtain a single set of parameters to describe all industrial surveys, which would enable the simulation of the plant under different operating conditions and feed characteristics. As there was a significant difference in the CSS and 80% passing sizes of the feed in the industrial surveys, the present work fitted Equations (2)–(5) considering these two operating variables. Information on the wear condition of the liners was not available, so it could not be included in the model, besides minor variations in ore composition. The optimal parameters fitted for each industrial survey were then used as part of the multiple regression to obtain the final set of parameters that would allow representing a compromise between simplicity in modeling and fidelity in representing the survey data. Finally, all parameters were used to simulate the industrial campaigns.

4. Results

4.1. Breakage Characterization

As mentioned in Section 3.1, the two ore types that predominate in the feed to the secondary crushing stage, namely the Compact and the Supercompact Itabirites, have been subjected to testing separately in order to assess their breakage response. The amenability of the iron ore lithologies to impact breakage was analyzed by inspecting the relationship between the $t_{10}$ and the specific impact energy. The results are summarized in Figure 1 and clearly present a steep reduction in breakage intensity for specific energies below 1 kWh/t, while breakage saturation is reached above this point. The results also show the higher values of t₁₀ for comparative impact energies for the Compact Itabirite when compared to the Supercompact Itabirite, which highlights the lower amenability for breakage of the latter.

The results from Figure 1 may be described mathematically as follows [26]:

$$t_{10}=A\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-b{E}_{sp}\right)\right]$$

(16)

where $E_{sp}$ is the specific energy and A and b are fitting parameters. The optimal values of parameters from Equation (16) used to describe the data presented in Figure 1 are listed in

Table 1. DWT breakage model parameters and impact work index of the ore types.

Ore Type	A	b	A × b	Wi (kWh/t)
Compact Itabirite	64.5	10.5	677.3	7.2
Supercompact Itabirite	56.1	2.20	123.4	10.6

. The product of the values (A × b), which represents the derivative of Equation (16) with respect to the specific impact energy (E_sp) when its value is equal to zero, is often used to characterize the amenability for breakage of different materials [25]. The values found for the ore types demonstrate their higher amenability to breakage when compared to other materials [36].

A more detailed examination of results from drop-weight tests for selected size ranges is presented in Figure 2, which shows the product size distributions from DWTs for the narrow size ranges of 37.5–45.0 mm (a) and 26.5–31.0 mm (b) for Compact Itabirite under different specific energies. The graphs highlight the significant non-normalizable breakage behavior of this Itabirite iron ore, which is identified by the inflection point, highlighted by the red arrow, at around 240 µm for the different narrow size ranges tested.

In addition, the relationship between the values of t₁₀ and the various values of t_n [26] has also been obtained. The results from all narrow size ranges tested are presented in Figure 3 for the two ore types, showing that they were reasonably well grouped for mapping the breakage behavior of the material investigated in the present work, irrespective of the ore blend fed to the crusher. The construction of this graph only included data for particle sizes above 240 µm, which was identified as the lower limit for the validity of the normalizable response (Figure 2).

The results from Figure 3 are well described by the incomplete beta function [34], given by

$$t_n\left(t_{10}\right)={\displaystyle \frac{100}{{\int }_0^1{x}^{{\alpha }_n-1}{\left(1-x\right)}^{{\beta }_n-1}dx}}{\int }_0^{t_{10}}{x}^{{\alpha }_n-1}{\left(1-x\right)}^{{\beta }_n-1}dx$$

(17)

where ${\alpha }_n$ and ${\beta }_n$ are fitting parameters. The optimal parameters from Equation (17) are listed in

Table 2. Summary of the incomplete beta function parameters fitted for each t_n for the Itabirite iron ore samples for particle sizes above 240 µm.

${\mathit{t}}_{\mathit{n}}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{\beta }}_{\mathit{n}}$
t2	0.73	3.20
t4	1.03	1.83
t25	0.99	0.73
t50	0.94	0.60
t75	0.88	0.52

, while Figure 3 demonstrates the reasonable agreement between experimental and fitted values for each t_n analyzed.

The Whiten crusher model uses tabulated values relating selected t₁₀ values to the various t_n values, and the data are listed in

Table 3. Summary of the appearance function (t₁₀-$t_n$ relationship) for selected values of $t_{10}$ used as an input for the Whiten crusher model.

tn	t75 (%)	t50 (%)	t25 (%)	t4 (%)	t2 (%)
10	7.69	7.93	7.98	16.16	27.96
20	14.44	15.54	15.90	31.82	51.63
30	21.13	23.24	23.97	46.27	69.54

. Standard values from Table 3 were obtained by interpolation using the incomplete beta function (Equation (17)) with optimal parameters from Table 2. Values from Table 3 are used internally by the model to estimate the breakage function from Equation (6), which is performed by spline interpolation.

The Bond impact work index results, measured using Bond’s procedure [33,34], are plotted in Figure 4, while average values are listed in Table 1. The distributions in the figure presented a high coefficient of variation (around 50%), with values varying from 1.9 to 15.5 kWh/t and 3.6 to 26.7 kWh/t for the Compact and Supercompact Itabirites, respectively.

4.2. Industrial Surveys

Figure 5 presents the feed size distributions of all surveys performed with the cone crusher. The significant variation in feed top size is evident. In addition, large proportions of fines were observed in the feed in selected surveys, suggesting poor screening efficiency in the respective operations depicted in the surveys. Experimental campaigns with operations close to ideal classification exhibited steep size distributions with particles mostly contained in the size range between 100 and 25 mm.

Figure 6 summarizes the measured product size distributions from samples of the various surveys. They show clear non-normalizable breakage behavior with segregation of particles in the range between approximately 500 and 100 µm, whereas data from breakage tests identified the inflection at 240 µm (Figure 2). Minor differences could be observed in the slope of the non-normalizable region in Figure 6. This same non-normalizable breakage behavior has been observed elsewhere for some other iron ores [37], quarry rocks [38] and platinum ores [39], and its mathematical description does not follow the original normalized breakage function [40] or normalized appearance function [26].

A comparison of the values of 80% passing sizes in the feed and the product from the individual surveys demonstrates that the reduction ratio varied from 1.6 to 4.4, which is within the lower range of typical operation of cone crushers [41]. Such variations evidence the significant effect of the CSS and the feed size distribution in the different surveys (Table A1).

4.3. Model Fitting

Values of parameters in the sub-models describing the effects of impact work index, feed size, moisture and CSS on crusher throughput have been fitted to data, and a comparison is presented in Appendix B. A comparison between measured the fitted values in Figure 7 shows that Equation (10) was able to fit the data appropriately, with a mean sum of squares of the absolute relative deviations of 0.23, besides highlighting that both model predictions and experiments are within the limits defined by Bruno^© for the coarser (green line) and finer (blue line) feed size distributions. Such deviations may be explained by variations in liner condition and feed blends. A summary of model parameters is presented in

Table 4. Values of parameters of the throughput model (Equation (10)) for the HP500 crusher in size reduction of Itabirite iron ore.

Sub-Model	Parameter	Value
Work index	η	0.15
Feed coarseness	$\ni$	0.26
Feed moisture	$\xi$	0.09
Closed-side setting	$\upsilon$	1.27
Throughput model	$\lambda$	0.84

, where the optimal value of parameter λ was obtained based on industrial survey #10 (Table A1).

Based on the appearance function in Table 3, the present work then fitted the classification function parameters (Equation (2)) and the value of t₁₀ for each industrial survey performed in the present work (Table A1). The model was fitted to describe the parameters $K_1$, $K_2$ and $t_{10}$, whereas parameter $K_3$ was kept constant at 2.3 as suggested elsewhere [24].

A comparison between the model and the experiments is first presented in Figure 8 for industrial survey #8 (Table A1), comparing the model fit considering the standard normalizable and the non-normalizable breakage function for particles below 240 µm (Mod. Whiten’s model in the figure). It shows the importance of incorporating the modified breakage function for the particular iron ore in order to reach a good fit to data at fine sizes. The optimal values of parameters defining the non-normalizable breakage function (Equation (9)) were n₃ = 0.47 and y₀ = 0.24 mm.

A comparison of data from selected surveys (#9 and #10) is presented in Figure 9, showing the good fit of the model to data for the crusher operating with the same feed, but different CSS values. The good agreement between the model and experiments highlights the robustness of the model and its ability to extrapolate its predictions to different operating conditions.

A general comparison between experimental and fitted values for the percent passing 12.5 mm is presented in Figure 10 for all industrial surveys performed in the present work and highlights the good agreement between the model and experiments in the coarse part of the product size distribution. However, it is worth noting that different sets of parameters were used in fitting data from most industrial surveys.

To obtain a set of parameters that describes the breakage behavior across all surveys, multiple linear regression was performed for each value of K₁, K₂ and t₁₀ as a function of CSS and f₈₀ to fit parameters in Equations (3)–(5). Satisfactory correlations were obtained for the t₁₀, K₁ and K₂ parameters, and optimal values are presented in

Table 5. Optimal parameters for the Andersen–Whiten crusher model to describe results from industrial surveys performed in the present work.

Function	Parameter *	Value
Classification function K1	$A_1$	0.23
	$A_3$	0.30
Classification function K2	$B_0$	12.0
	$B_1$	0.55
	$B_3$	0.40
Appearance function t10	$D_0$	64.0
	$D_1$	0.12
	$D_3$	−0.23
Power	Pn	110 kW
	$\zeta$	1.30

* Absent parameters are equal to zero.

. The results show that K₁ and K₂ increase with CSS, which is indeed the parameter that predominantly characterizes this effect in the classification for breakage.

Since all tests were run under choke-feeding conditions, no dependence on throughput appeared (A₂ = B₂ = 0). Parameters K₁ and K₂ also increased with 80% passing size in the feed. The t₁₀ value decreased with CSS and 80% passing size in the feed. The optimal value for the parameter A₁ (Equation (3)) is similar to the optimal values of 0.80 found by Karra [9], 0.65 by King [10] and 0.60 by Neves and Tavares [11]. This comparison shows that the final set of parameters proposed in the present work is in general agreement with previous applications of the model, in which K₁ (Equation (2)) is often 20% to 70% lower than the CSS. For parameter B₁ (Equation (4)), the optimal value was slightly lower than the usual range observed in some of the previous studies: 1.72 for Karra [9], 1.70 for King [10] and 1.45 for Neves and Tavares [11]. These differences in magnitude are explained by the strong dependence of parameter K₂ (Equation (2)) with respect to the 80% passing size in the feed, which was taken into account in the present work and not in the cited studies [9,10,11].

To demonstrate the validity of the set of values proposed in Table 5 describing the parameters fitted separately for each industrial survey campaign (Table A1), a comparison is presented in Figure 11. The results show good agreement between fitted and predicted values for the classification function parameters (K₁ and K₂ in Figure 11a) and for the breakage function parameter (t₁₀ in Figure 11b). The results in Figure 11a also show the clear difference in magnitude between parameters K₁ and K₂. The values of parameter t₁₀ in Figure 11b varied from 20% to 50%, which is higher than the usual range observed for secondary crushing stages [25]. Nevertheless, the results presented in Figure 1 and Figure 3 highlight the high values observed for the t₁₀ when processing Compact Itabirite, which partially explains the large values observed in Figure 11b.

A comparison between the experimental and predicted product size distributions for selected industrial surveys is presented in Figure 12. The red line in Figure 12 shows the fitted values for each industrial survey, highlighting the good agreement between the model and experiments, as already shown in Figure 10. Simulated values using the set of parameters presented in Table 5 show good agreement between the model and experiments, with the model prediction slightly overestimating the coarse part of the product size distribution for industrial survey #3 (Figure 12b) and slightly overestimating data from industrial survey #8 in the fine part of the product size distribution (Figure 12d).

The validity of fitting parameters in Table 5 is again demonstrated in Figure 13 for a comparison of experimental and predicted percent passing 12.5 mm, as previously shown in Figure 10 for fitting each industrial survey separately. The agreement between the experimental and simulated results was reasonably good, although not as good as that in Figure 10. This is not surprising, given that the variations in liner conditions, moisture contents and crushability could not be accounted for explicitly in the modeling approach. The absolute relative deviation from measurements was up to 15%.

Finally, the parameters of the power model are also listed in Table 5, and a comparison between measurements and simulations is presented in Appendix C, which showed only fair agreement, likely associated with issues during the measurement of power from the different crushers operating in parallel.

5. Conclusions

Ten industrial surveys performed over the period of several years of operation of the secondary crushing stages of the Minas Rio Plant (Anglo American), fed with a blend of Compact and Supercompact Itabirites, were used to assess the crusher performance under different feed characteristics and crushing conditions. Significant variations were observed in the performance of the crushers, with reduction ratios varying from 1.6 to 4.4. Nevertheless, a distinct non-normalizable breakage behavior was evident with an inflection point at around 240 µm. Details of the crusher power and throughput for the various industrial surveys also demonstrated important variabilities, associated with crushing conditions and feed size.

The application of the Andersen–Whiten crusher model, implemented in the commercial software Integrated Extraction Simulator (IES) from Orica Inc., first relied on the description of the normalized appearance (breakage) function based on drop-weight tests with particle sizes varying from 13.2–16.0 mm to 53–63 mm. The results showed a higher breakage intensity for the Compact Itabirite when compared to the Supercompact Itabirite, with A × b varying from 677.3 to 126.2, respectively. The t₁₀-t_n relationship was well grouped, and the incomplete beta function allowed the description of the data.

Classification function parameters were then fitted for each industrial survey separately by estimating parameters K₁ and K₂ on the basis of feed coarseness and closed-side setting. A modification to the model was implemented to account for the non-normalizable nature of the size reduction of the Itabirite iron ore, demonstrating the improved model prediction in the fine end of the product size distribution for the different industrial surveys. In order to obtain a single set of parameters that described the crushing response of the ore across all experimental campaigns, multiple linear regression was performed, and the optimal values of parameters that could describe the variation in K₁, K₂ and t₁₀ as a function of operating conditions were fitted. A good agreement between the model and experiments for the different industrial survey campaigns was reached.

A model for predicting the maximum throughput of the crusher when operating in choke-feeding conditions was proposed from simulations in the software Bruno^© (Version 4.2.2.0, exe v4.1.2.9, DB v9.50, Metso Corporation, Helsinki, Finland), from Metso, and information from crusher manuals. The model describes the throughput based on closed-side setting, ore bulk density, crushability, moisture content and feed coarseness. In spite of the empirical nature of the model, its robustness was demonstrated by predicting the HP500 cone crusher throughput with good confidence when compared to data from all industrial surveys.

Crusher power prediction demonstrated a limitation in the model by capturing variabilities in the different industrial surveys and of crushers operating in parallel that were surveyed, besides unaccounted fluctuations in the proportion of Compact and Supercompact Itabirites in the feed.

The application of this modified version of the Andersen–Whiten model is key for describing industrial-scale cone crushing of Itabirite iron ores by capturing the intrinsic non-normalizable breakage behavior of the material. The use of a throughput model is key for allowing a proper design and analysis of cone crushers operating in integrated circuits, besides allowing one to understand how to optimize crushing conditions to better capture the process dynamics of the crusher.