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
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]:
where
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. The product of the values (
A × b), which represents the derivative of Equation (16) with respect to the specific impact energy (
Esp) 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
tn [
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
where
and
are fitting parameters. The optimal parameters from Equation (17) are listed in
Table 2, while
Figure 3 demonstrates the reasonable agreement between experimental and fitted values for each
tn analyzed.
The Whiten crusher model uses tabulated values relating selected
t10 values to the various
tn values, and the data are listed in
Table 3. 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.
4.3. 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, 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
,
and
, whereas parameter
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. 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.