2.1. Lab-Scale High-Pressure Roll Crusher Test
The lab-test work was performed using two different lithologies, which were both previously used to determine breakage function parameters [
24]: granite from a tin and tantalum mine, in northwest Spain [
16,
25] and calc-silicate rock from a tungsten processing plant in Austria. Both ores contain low grade strategic metals; around 100 ppm of tantalum in the case of the granite [
16], and 2000 ppm for the calc-silicate [
25]. The experiments were performed at the facilities of the Universitat Politècnica de Catalunya, Manresa, Spain.
The sample preparation consisted of quartering and splitting approximately 250 kg of each material. The samples were previously crushed by a KHD Humboldt Wedag double toggle jaw crusher. After that, the material was screened and classified into narrow particle size fractions and multi-size particle distribution. The narrow-size classes of particles ranged from: −19 + 16 mm, −16 + 14 mm, −14 + 12.5 mm, −12.5 + 11.5 mm, −11.5 + 9.5 mm, −9.5 + 8 mm, −8 + 6.7 mm and −6.7 + 5 mm. For both lithologies, two tests with a heterogeneous particle size distribution were performed to validate the model with a realistic material. The laboratory-scale device is a modified roll crusher adapted with a high-pressure hydraulic system (
Figure 1), where the mechanical overload protection springs were replaced by two 60 mm internal diameter pistons, and the original 3.7 kW motor was also changed to a 15 kW Siemens engine (
Table 1). The equipment uses smooth rolls with dimensions of 250 mm in diameter and 50 mm in width. A transparent cover was built to observe the mechanical response of the material, and a slow-motion camera captured the moment of the particles under the roll breakage action. The horizontal displacement and speed of the rolls were also monitored.
The operative parameters were controlled in all tests (
Table 2). The nomenclature of each experiment explains the variable condition in which each one was performed. Thus, the first character indicates the lithology type: P for the granite and M for the calc-silicate. The second character is related to the observed operative condition: V for surface speed of the rolls, F represents the feed size, P the specific pressing force and G indicates the operative gap. Finally, the last character indicates the experiment number. The bulk density
, which includes the porosity of the material, was measured by means of the quotient of mass
m and total volume of a given sample
VB (Equation (1)).
The feed density is denoted as
and the product density as
δ. Both densities were determined, since they are necessary to calculate several model parameters [
8].
The throughput (Th) was calculated using experimental data (
Table 2). The product particle size distribution was determined using standard ASTME sieves with a ro-tap sieve shaker. In order to obtain disaggregated material, the calibration indicated 15 min dry-sieving.
The specific pressing force (Fsp) for all tests was determined based on a compression strength test for both material types [
24] (
Figure 2). The aim was to test the material in the optimal pressure condition. The 95% of confidence interval for the calc-silicate ranges between 0.79 N/mm
2 and 3.36 N/mm
2, and from 1.94 N/mm
2 to 5.17 N/mm
2 for the altered granite.
In this work, the circumferential velocity, static gap, feed size and specific pressing force were studied in order to determine their influence on the particle size distribution product.
2.2. Mathematical Approach
The literature shows several approaches to HPGR models, sometimes as bi-stage sub-processes, with a single particle compression combined with a batch grinding solution [
8,
9] or by means of the calculation of several matrices between the differential feed and the breakage distribution function [
7]. Based on the current models, an improved approach is proposed, taking into account the following hypotheses:
The breakage distribution function parameters should have different values when running under single particle compression compared to running under bed compression. A previous work investigating these materials [
24] showed clear differences when the particles were under single particle compression or when they were under the bed compression condition.
The improved approach should have a selection function that is able to discriminate particles that are large enough to be nipped under the rolls from the remaining particles, which can be subjected to the bed compression effect. Thus, the selection function should be based on the particle size distribution of the feed, the internal products of the model, the geometric dimensions of the device and other characteristics, such as the material density and operational gap.
From the observations of images captured by the camera installed in front of the rolls (
Figure 3), it was possible to observe the existence of a pre-crushed zone, as defined by Austin [
6]. Other smaller particles, representing systems under choke feed condition also interact with each other, not under the single particle compression condition, but under bed compression. The Austin model [
6,
20] was then used as a new method of vertebral structure along with Evertsson [
26] nomenclature, in which single particle compression and bed compression phases are used simultaneously. A block model was developed based on the size distribution of the feed (
Figure 4).
The matrices
, for single particle compression, and
, for bed compression, represent the differential matrices extracted from the cumulative form of
B distribution [
27] but using different parameter values when the sample is under single particle compression compared to when it is under the bed compression effect (Equation (2)).
The parameter
dpi in Equation (2) represents the daughter particle and
dpj is the parent particle, while
k,
n1 and
n2 are function parameters.
n3 and
Y0 are the parameters when the breakage distribution behaves as a bimodal function. This is the case for the altered granite (
Table 3) as is explained in previous studies [
24]. All these parameters
k,
n1,
n2,
n3 and
Y0 can also vary when the function is used under single particle compression and bed particle compression. For calculation purposes, the cumulative form
is transformed into the differential form
.
The feed
enters the pre-crushing zone, where the particles are discriminated by the selection function. Over a certain cut-size, some of them go to the single particle compression and the remaining ones go to the bed compression. The product
is the sum of the results of both sub-processes. In algebraic terms, the expression for each comminution step is given by Equation (3).
The selection function used in this case belongs to the formulation reported by Whiten et al. [
27], used mainly for cone and jaw crushers [
26,
28]; here it was applied for HPGR. Equation (4) describes this physical process in the steps at which single particle compression occurs. This function has the particularity of fixing upper and lower edges, which is related to the device geometry and the mineral characteristics.
The parameter
xn in Equation (3) represents the upper limit of the function and is given by the distance between rolls when the nipping action begins and
d1 is the lower limit where the particles cannot be subjected to comminution under the single particle compression condition due to their size. Therefore, they tend to form in the bed compression zone, due to the interaction between coarse and fine particles. The parameter
γ is related to the mineral characteristics [
26] and describes the shape of the curve of the selection function
Sn. The parameter
xn (Equation (5)) represents the relationship among all geometric characteristics of the device, material feed size and other features such as the material density, gaps and the nip angle (Equation (6)) [
8].
The parameter
(t/m
3) in Equation (6) is the bulk density at the feed zone and
δ (t/m
3) is the bulk density at the extrusion zone, which is to say, the product density. Furthermore,
D (m) is the roll diameter, and
S0 (m) is the gap. The values of the vector
Sn depend on
, which is a function of the angle
α (
Figure 5).
Nt denotes the number of single particle compression stages necessary to simulate the breakage of the material until almost all particles reach the gap size and they can be considered as a final product [
8]. The angles used to evaluate the function
are defined as in Equation (7).
In order to evaluate the value of
Nt in the crushing stages, several simulations of single particle compression were performed, using the breakage function parameters found [
24]. Equation (8) is presented to determine that lumps larger than the gap that must undergo repetitive breakage in order to be realized as a final product [
6]:
Equation (8) is a loop of matrices’ product x that stops when no more particles larger than the gap size are generated, and Nt is determined by the number of steps necessary to reach this size. In this work, the way in which the value of Nt is determined is compared with other characteristics of the material, such as the shape of the particles and the distance of the rollers for each rupture stage.
Matrix notations are presented in
Figure 6, where
I = 1 …
m,
j = 1 …
m and
n = 1 …
Nt, with
m being the
pi vector length (
Figure 6).
The model fit was evaluated by measuring the agreement between the experimental data and the model’s predictions. Although the mean absolute error (MAE) is simpler and direct [
29], the best tool to appraise this difference is the Root Mean Square Error (RMSE) (Equation (9)) mainly due to the fact that most of the model errors were produced on key parameters of a typical particle size distribution (PSD) such as the P80 or P50. RMSE gives higher weight to large errors rather than lower errors in the formalism [
30]. In this case, the evaluation of results is more adverse and, therefore, more demanding.
The parameter N in Equation (9) is the length of the input vector, p is the experimental values of the vector and y represents the simulated values. In order to obtain concise evaluations, two phases or local sections of the obtained curves were defined: the fine phases represented by all particles under 850 microns, and the coarse phase when the particles were over this cut-size.
The percentage variation (Equation (10)) was used to evaluate local values.
In Equation (10), Vi is the initial value and Vf the final value.
The model was programmed in a MATLAB® script (R2016a) and different solver functions from the tool-box were used to run the program in order to minimise the RMSE of the selected data.