The Fragmentation Energy-Fan Model in Quarry Blasts
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Abstract
This paper investigates fragmentation in an aggregate quarry in the light of the fragmentation-energy fan concept. Six blasts were monitored, located one behind the other in the same quarry area. The rock structure was blocky, the water level and the usage of explosives were variable within and between the blasts; however, the distribution of explosive energy in the blocks was relatively uniform and the performance of both explosives per unit mass appeared to be similar. The powder factor above grade was between 0.28 and 0.44 kg/m^{3}. Fragmentation of the plant feed was measured using an online digital image analysis system and three belt scales in the crushing plant. The oversize material that was not directly fed to the plant after the blast corresponds to 3–9.3% of the total. Data from image analysis was used to build the size distribution in the coarse range (fragments above 120 mm), and the passing fractions at 120 and 25 mm obtained from belt scales data were used in the range 120–25 mm. The non-dimensional percentile sizes between 10 and 90% for each blast are described through power functions of the powder factor above grade; the model coefficients (i.e. nine prefactors and nine exponents) are statistically significant. From them, using the principles of the fragmentation-energy fan and the Swebrec distribution properties, the fragmentation can be expressed in terms of the powder factor by means of five parameters: the fan focus coordinates and the three parameters of a function of the exponents versus the percentage passing. The model shows a good capability to describe fragmentation from 10 to 90 percentile sizes, with an expected error below 6% and a maximum likely error less than 15%.
Keywords
Image analysis Belt scales Fragmentation energy-fan Swebrec distribution Rock blasting1 Introduction
Fragmentation is a key factor for optimizing the mining process and minimizing production costs. The size distribution of blasting products can only be measured accurately enough by sieving the muckpile, a complicated, costly and disruptive to production task. Although enhanced 2D technologies (often called 3D though they are not really determining fragmentation of a rock volume) to monitor fragmentation are available, such as LiDAR imaging—laser imaging detection and ranging (McKinnon and Marshall 2014; Oñederra et al. 2015; Thurley 2013; Thurley et al. 2015) or photogrammetry (Noy 2013, 2015; Bamford et al. 2016), 2D image analysis is still the most common tool used. Image analysis systems may show a poor performance at small sizes especially when they have not been calibrated (Sanchidrián et al. 2009) and it is advisable to use additional tools to correct raw data on a blast per blast basis in order to get a good estimation of the actual size distribution (Sanchidrián et al. 2006).
The installation of these systems upstream of the primary crusher allows assessing the influence of blast characteristics on fragmentation. In particular, the influence of powder factor on the fragment sizes (especially the median) was studied since the early 1970s by Soviet researchers (Koshelev et al. 1971; Kuznetsov 1973); these works were years after extended by Cunningham (1983, 1987, 2005) in his Kuz–Ram model, and other models derived from it, such as the crush-zone model (Kanchibotla et al. 1999; Thornton et al. 2001). These models presuppose an underlying size distribution function (the Rosin–Rammler, Rosin and Rammler 1933 for both Kuz–Ram and Crush-zone models), and their goal is then to predict the function parameters (i.e. median size and uniformity index) as a function of rock and blast characteristics. The performance of these models is relatively poor with errors ranging from − 75 to 50% in about half of the cases, when compared with fragmentation obtained by sieving (Sanchidrián and Ouchterlony 2017). One of the reasons for such results is that the Rosin–Rammler function is not a good description of the fragment size distribution from blasting, or it is in a limited range of passing (Ouchterlony 2005, 2009; Sanchidrián et al. 2012a, b, 2014; Sanchidrián 2015).
A natural alternative to this is to focus on determining distribution-independent fragmentation prediction equations for the percentile sizes as functions of powder factor; an early example of this approach was the work by Chung and Katsabanis (2000) that re-analyzed the sieved fragmentation data by Otterness et al. (1991) and obtained equations of the median and the 80-percentile sizes as functions of the powder factor. Fragment sizes obtained by sieving versus powder factor data in production blasts have also been published by Olsson et al. (2003), Ouchterlony et al. (2005) and Ouchterlony et al. (2010, 2015), to name a few. These, and many other half- and small-scale blasts data are analyzed by Ouchterlony et al. (2017). A key conclusion of that work is that the uniformity index of the Rosin–Rammler needs to be variable at different percentiles (i.e. a Rosin–Rammler function with piecewise constant uniformity index) to represent fragmentation at different energy levels—or, in blasting terms, powder factors. This model has been called the fragmentation-energy fan because it derives from the plot of several percentile sizes as function of the powder factor, and the observation that this plot has the form of a set of straight lines in log–log space that converge to a common focal point. This new approach follows the principle that fragmentation should be predicted using distribution-free models that provide fragment sizes for the various percentages passing as a function of rock mass and blast characteristics (Ouchterlony et al. 2017; Sanchidrián and Ouchterlony 2017) without resorting to a particular distribution function. Such fragmentation-energy fan principles are followed in this work to describe fragmentation from blasting in an aggregate quarry.
2 The Site
The experimental work was carried out in the quarry El Aljibe. The quarry is located in central Spain near the town of Almonacid de Toledo and mines mylonite to produce track ballast for high-speed and conventional trains (32/56 mm fraction), for high strength concrete and asphalt mixtures (6/12 mm fraction), and for sub-base and base courses (0/25 mm fraction) in road and rail track construction.
The deposit occurs within the Toledo Shear Zone that separates the Toledo Migmatite Complex unit to the North and the Paleozoic metasediments to the South. Mylonite was produced by ductile deformation of migmatites in amphibolites facies and outcrops in the area in which the quarry is located (Enrile 1991). The mineral composition is similar to a granodiorite or to a tonalitic granite. It has a density of 2710 kg/m^{3} and an average uniaxial compressive strength of 131 MPa. It is qualitatively described as tough rock with good resistance to abrasion. The rock mass has three sub-vertical joint families and a subhorizontal one, their intersections with the highwall making up a blocky face.
Drilling and blasting is used to mine about 0.5 Mt/yr of run of mine (ROM), this amount depending on the market demand. Generally only one blast is loaded at a time during one working shift. A hydraulic excavator mucks the fragmented rock into 54 t average payload trucks that haul the material to the bin of the primary crusher. Boulders greater than the largest size that can be fed to the primary crusher (about 900 mm) are further fragmented with a hydraulic hammer and hauled to the plant mixed with the rest of material (undersize) coming from the blast. The number of these fragments are recorded in two categories: small, which could be hauled but would likely block the entrance of the crusher, with average size somewhat in excess of 1 m (we have used 1.1 m), and large, that could not be loaded, with average size between 1.5 and 2 m (we have used 1.7 m). The maximum boulder size was about 2 m.
When a truck dumps, the crusher operator monitors the time in the crusher’s log; the ROM mass fed to the plant per day (m_{ R }) is obtained as the total number of trucks times the average payload.
In the plant, a grizzly feeder is used to scalp the feed of the primary crusher. The grizzly has wedged slots with maximum opening of 109 mm that corresponds to a cut size (i.e. aperture equivalent to a square mesh) of 120 mm (Departamento Técnico de Benito Arnó e Hijos 2005). The aperture (slot) factor of the grizzly (1.10) is in line with data from other works (Colman 1985; Gluck 1965; Gupta and Yan 2006; Ouchterlony et al. 2006, 2015).
3 The Blasts
Summary of blast characteristics
B1 | B2 | B3 | B4 | B5 | B6 | |
---|---|---|---|---|---|---|
Burden (m) | 2.6sd0.3 | 2.6sd0.2 | 2.6sd0.3 | 2.8sd0.4 | 2.5sd0.3 | 2.5sd0.3 |
Spacing (m) | 2.6sd0.2 | 2.7sd0.1 | 2.7sd0.1 | 2.7sd0.2 | 2.7sd0.2 | 2.6sd0.1 |
Bench height (m) | 16.5sd0.3 | 16.3sd0.6 | 16.2sd0.5 | 17.5sd0.5 | 14.7sd0.5 | 14.6sd0.6 |
Planned hole length (m) | 19.5 | 19.5 | 19.5 | 19.5 | 19.5 | 18.5 |
Blast volume (m^{3}) | 10,037 | 9593 | 10,066 | 11,431 | 7603 | 6852 |
Water amount in the holes | High | Low | Low | Medium | Medium | High |
Explosive mass, bulk (kg/hole) | 33.3sd31.1 | 68.9sd16.7 | 62.4sd22.9 | 49.7sd24.3 | 43.8sd31.6 | 7.1sd13.9 |
Explosive mass, cartridges (d_{ c }= 50 mm) (kg/hole) | 28.3sd17.9 | 7.1sd9.0 | 8.1sd10.3 | 14.6sd13.0 | 16.6sd16.6 | 34.5sd11.1 |
Explosive mass, cartridges (d_{ c }= 60 mm) (kg/hole) | 4.7sd0.8 | 5.3sd2.4 | 5.0sd0.6 | 5.0sd0.4 | 5.0sd1.3 | 4.8sd0.7 |
Total mass powder factor (kg/m^{3}) | 0.51 | 0.61 | 0.60 | 0.47 | 0.60 | 0.49 |
Mass powder factor above grade (kg/m^{3}) | 0.36 | 0.44 | 0.43 | 0.37 | 0.31 | 0.28 |
Energy powder factor above grade (MJ/m^{3}) | 1.434 | 1.709 | 1.66 | 1.476 | 1.223 | 1.156 |
Fragmentation data monitored in mine and plant
B1 | B2 | B3 | B4 | B5 | B6 | |
---|---|---|---|---|---|---|
Number of days monitored | 4 | 5 | 8 | 7 | 5 | 6 |
Mass m_{1} from belt scale S1, t | 4140 | 6092 | 9943 | 6557 | 5449 | 3834 |
Mass m_{2} from belt scale S2, t | 3085 | 4759 | 7222 | 5028 | 4169 | 3073 |
Mass m_{3} from belt scale S3, t | 12,702 | 15,022 | 25,003 | 16,760 | 14,882 | 12,764 |
Passing P_{120} (%) | 30.09 | 37.25 | 35.86 | 35.85 | 33.71 | 28.35 |
Passing P_{25} (%) | 7.67 | 8.15 | 9.81 | 8.36 | 7.92 | 5.63 |
Mass m_{ B } of boulders, t | 416.4 | 854.9 | 1396.3 | 1705.2 | 951.6 | 909.2 |
Passing of plant undersizes, P_{ m } (%) | 96.97 | 94.77 | 94.96 | 90.68 | 94.11 | 93.28 |
Number of images | 3161 | 3190 | 5368 | 4275 | 3883 | 1626 |
The long subdrilling results in a total mass powder factor from 0.47 to 0.61 kg/m^{3} (see Table 1). These values seem to be excessive for the drilling pattern used, and suggests considering the mass powder factor above grade q (i.e. ratio of explosive mass to rock volume discarding the explosive in the subdrill part of the hole) as a better indicator of the specific charge that contributes to fragmentation. The q powder factors of our blasts are in line with values recommended (e.g. AECI 1986) for rocks with good (0.30 kg/m^{3}) and fair (0.45 kg/m^{3}) blast behavior, respectively.
The low powder factor blasts correspond to more water (i.e. more cartridged product) and the high powder factor to less water (hence more bulk). The reason for this is that cartridges loading (not filling the hole section) puts less explosives than bulk, that fills the holes section, even if the density of cartridged product (gelatin dynamite) is higher than that of bulk (ANFO). Energy per unit mass of both products are not too different (the difference is as small as 5%) so, mass-wise, they are nearly interchangeable.
The different charging schemes resulted in different explosive-to-rock coupling conditions: the bulk explosive (ANFO) was coupled to the borehole wall whereas the gelatin cartridges were coupled through the water annulus. Works by Ash (1973),^{1} Smith (1976), Brinkmann (1982) and Bleakney (1984) show that the water coupling is able to transmit a high pressure shock to the rock so that the fragmentation effect does not differ much from when the explosive is fully coupled. Based on that, the action of the water-coupled gelatin on the rock is assumed to be equivalent (only different due to the slightly different specific energy and different mass) to the ANFO directly coupled, filling the hole section. Some shock attenuation must take place in the water annulus for the decoupled charges and, to some degree, the higher detonation pressure of the gelatin (given its higher density and VOD) than the ANFO compensates for that so that the performance of both is not apparently different.
Water saturation of the rock mass may enhance fragmentation by: (1) reducing compressive/tensile strengths; (2) reducing shock wave attenuation; (3) decreasing the cohesion and the frictional properties of joints (Singh and Narendrula 2007; Zhang 2016). The water absorption coefficient of the mylonite was measured at 0.10 ± 0.03% hence the effect of water on shock wave transmission, joints opening and crack propagation was probably limited.
The explosive was bottom initiated with 500 ms non-electric detonators. The blasts were initiated from the center or from the end of the row using an open (B2, B3 and B4 blasts) or closed (B1, B5 and B6 blasts) chevron initiation pattern; a delay of 17 ms was employed within the rows and of 42 ms between rows.
4 Fragmentation Measurements
Fragmentation monitoring was completed with the online digital image analysis system, Split-Online^{®}. To get images of the ROM before the crusher, a video camera was installed perpendicularly above the discharge of the crusher feeder at the end part of the grizzly (see Fig. 1). The camera was triggered 20 s after the crusher feeder started working and the mass flow in S1 was greater than 20 t/h. While both conditions were fulfilled, the camera took a photo every 8 s. Each image was sent via wireless to a computer where it was analyzed to get a fragment size distribution curve.
The bottom (black and white) images in Fig. 4 show the segmented (binary) images corresponding to the upper photographs; areas in white are delineated particles and areas in black correspond to non-delineated material with sizes below a threshold size of 32 mm, below which no reliable delineation is obtained. They show the quality of the automatic analysis; the large fragments splitting and small fragments merging (disintegration and fusion errors) are observed in some of the binary segmented images. Data from delineated particles are considered to form the passing fractions from 32 mm upwards. At this size the passing is calculated from the surface of non-delineated particles corrected by a factor. This factor defines the percentage of non-delineated material that is actually made of fines (Sanchidrián et al. 2009), and was set during the system commissioning to a low value as no significant amount of fines is apparent in the images (since they bypass the crusher feed). Below 32 mm, the system extrapolates fragmentation through an analytical size distribution function which parameters are calculated from two points in the range of sizes in which delineation is reliable (Split Engineering 2010).
5 Processing of Fragmentation Measurements
The global size distribution of blasted material was determined from the belt scale-based points, i.e. (25 mm, P_{25}) and (120 mm, P_{120}), and the image analysis-based points for sizes greater than 120 mm.
The size distributions as measured by image analysis on the grizzly are combined into a single distribution by averaging the passing at each size of all the individual distributions of a blast. The maximum size of each blast is the greatest of all measured maxima. However, since boulders had been previously broken this is not the maximum size of the blast; in fact, percentile sizes higher than P_{ m } (90.7–97.0% depending on the blast, see Table 2) cannot be determined. The resulting size distributions for each blast are given in Table 5 in the “Appendix”. The maximum sizes measured are given in the footnote of Table 5; they are in line with the sizes of the largest fragments that are fed into the crusher (about 900 mm).
- 1.
The size range 25 mm ≤ x_{ j } < 120 mm is interpolated linearly in log–log space from the belt scales data (25 mm, P_{25}) and (120 mm, P_{120}) on the same size points used by Split. The size distributions are not extended beyond 25 mm as the fines tail (< 25 mm) of the blasts under analysis was not measured.
- 2.
In the range x_{ j }≥ 120 mm, starting from (x_{0}, P_{0}) = (120 mm, P_{120}), the successive points (x_{ j }, P_{j}) are calculated upwards from the image analysis points (x_{ j }, P_{ j,S }) as follows:
Summary of Swebrec fits to final fragmentation data
Blast # | x_{50} (mm) | b | RMSE^{a} | R ^{2} |
---|---|---|---|---|
B1 | 199 | 3.915 | 0.0396 | 0.9926 |
B2 | 168 | 4.214 | 0.0158 | 0.9991 |
B3 | 176 | 3.771 | 0.0244 | 0.9965 |
B4 | 178 | 3.982 | 0.0179 | 0.9991 |
B5 | 185 | 4.002 | 0.0198 | 0.9980 |
B6 | 207 | 4.255 | 0.0233 | 0.9977 |
Mean | 186 | 4.023 | 0.0235 | 0.9972 |
Standard dev. | 15 | 0.183 | 0.0085 | 0.0024 |
6 Analysis of Fragmentation
Output of linear least squares fit of x_{ P } in m and x_{ P }/L_{ c } through Eq. 8
P | x _{ P } | x_{ P }/L_{ c } | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Coefficient | p-value^{a} | R ^{2} | Coefficient | p-value^{a} | R ^{2} | |||||
A _{ P } | α _{ P } | A _{ P } | α _{ P } | A_{ P }′ | α_{ P }′ | A_{ P }′ | α_{ P }′ | |||
90 | 0.449 | 0.054 | < 0.0001 | 0.456 | 0.145 | 0.058 | 0.224 | < 0.0001 | 0.004 | 0.895 |
80 | 0.337 | 0.144 | < 0.0001 | 0.047 | 0.667 | 0.044 | 0.314 | < 0.0001 | 0.001 | 0.939 |
70 | 0.244 | 0.246 | < 0.0001 | 0.015 | 0.808 | 0.031 | 0.415 | < 0.0001 | 0.004 | 0.901 |
60 | 0.181 | 0.299 | < 0.0001 | 0.018 | 0.791 | 0.023 | 0.469 | < 0.0001 | 0.011 | 0.835 |
50 | 0.136 | 0.359 | < 0.0001 | 0.031 | 0.728 | 0.018 | 0.529 | < 0.0001 | 0.018 | 0.787 |
40 | 0.092 | 0.472 | < 0.0001 | 0.048 | 0.664 | 0.012 | 0.642 | < 0.0001 | 0.031 | 0.728 |
30 | 0.062 | 0.537 | < 0.0001 | 0.059 | 0.631 | 0.008 | 0.707 | < 0.0001 | 0.038 | 0.698 |
20 | 0.036 | 0.629 | < 0.0001 | 0.041 | 0.690 | 0.005 | 0.799 | < 0.0001 | 0.028 | 0.738 |
10 | 0.014 | 0.807 | < 0.0001 | 0.047 | 0.669 | 0.002 | 0.977 | < 0.0001 | 0.033 | 0.718 |
A comparison of the fits in Table 4 shows that the fan lines are steeper and have been shifted downwards when x_{ P } is normalized by L_{ c }; these changes are constant for every percentile: the exponent increases by 0.17 (α_{ P }– \(\alpha_{P}^{'}\) ≈ − 0.17), and the prefactor is divided by 7.75 (i.e. A_{ P }/\(A_{P}^{'}\) ≈ 7.75).
Significant fits to a 0.05 level for all passing fractions from 10 to 90% are also obtained with the energy powder factor above grade (see Table 1) when an Eq. 9-type formula is used. The variability of the data explained by this model is, however, smaller, and the determination coefficients for all passing below 40% are lower than 0.7. Thus, mass powder factor above grade seems to be the best explosive concentration descriptor in this case.
Equation 14 can be viewed as a function \(A_{P}^{'} = f\left( P \right)\), since \(\alpha_{P}^{'}\) is itself a function of P. The focus coordinates (q_{0}, x_{0}′) can be determined as parameters of the fit of Eq. 14 to the nine pairs (\(A_{P}^{'}\),\(\alpha_{P}^{'} \left( P \right)\)). This way, the set of 18 parameters that define the fan (nine slopes \(\alpha_{P}^{'}\) and nine prefactors \(A_{P}^{'}\), if nine fan lines are selected) is reduced to five parameters, q_{0}, x_{0}′, a_{1}, a_{2} and a_{3}. Figure 8 (bottom graph) shows the \(A_{P}^{'}\) values (blue circles) as function of the passing using a logarithmic scale for the ordinates; the regression line from fitting Eq. 14 using the calibrated expression for \(\alpha_{P}^{'}\) in Eq. 10 is also plotted (green line). The determination coefficient of the fit is high, 0.998 and the values of the focal point (q_{0}, \(x_{0}^{'}\)) of the fan lines are significant to a 0.05 level; these and their 95% confidence intervals are given in Fig. 8 (bottom graph).
7 Discussion
The fragmentation energy fan concept stems from the observation that the percentile sizes, if represented as function of the powder factor (or, in general, energy concentration) tend to form straight lines when plotted in log–log space. Such lines have decreasing slopes at growing percentage passing so that they form a curve bundle with a relatively well defined common intersection point, or focus; the curve set can be described by the coordinates of that point and the slopes. It further turns out that, if the underlying size distribution is of Swebrec-type (Ouchterlony 2005, 2009), the slopes must follow (see Ouchterlony et al. 2017 for the derivations) a certain functional form of the percentage passing, that simplifies the description of the slopes variation to only three parameters a_{1}, a_{2} and a_{3}. These, plus the focus coordinates (q_{0}, \(x_{0}^{'}\)) are the only parameter set required to define the whole fragmentation spectrum as function of the powder factor, by means of Eqs. 9, 10 and 13. Dimensionless sizes, scaled with a characteristic size, x_{ P }/L_{ c }, also follow the fan pattern as proved by Ouchterlony et al. (2017); they have been used here since they provide better fits to the fan lines.
The procedure followed is (primes are used for the parameters with dimensionless sizes): (1) α_{ P }′ and A_{ P }′ are obtained for the P-percentiles from Eq. 9-type fits; (2) Eq. 10 is fitted to the \(\alpha_{P}^{'}\), so that a_{1}, a_{2} and a_{3} are obtained, and (3) Eq. 14 is fitted to the \(A_{P}^{'}\), \(\alpha_{P}^{'}\) pairs from (1), from which the focus coordinates q_{0}, x_{0}′ are obtained.
The first and third quartiles of e_{ LP } are − 0.052–0.043, respectively (equivalent to relative errors of − 5.1 and 4.4%). These values show a relatively centered distribution of errors, and are estimates of bounds for errors expected 50% of the times. The maximum likely errors estimated from the 95% percentile of the absolute e_{ LP } values is 0.136, which is roughly equivalent to a relative error of 14.6%. These figures are in the lower bound of the uncertainty range of fragmentation measurements by sieving estimated by Sanchidrián (2015), 8–22%. One of the reasons for such good performance of a simple model with only two P-dependent coefficients, \(A_{P}^{'}\) (P) and \(\alpha_{P}^{'}\)(P)—while nine P-dependent parameters are required in the model developed by Sanchidrián and Ouchterlony (2017)—is that most of the variables with a significant influence on fragmentation (rock strength and rock mass description, bench geometry and delay) have limited variability within our blasts and their effect is well lumped by the factor \(A_{P}^{'}\).
The analysis presented assumes that water coupling is able to transmit a high pressure shock to the rock so that the fragmentation effect does not differ much from when the explosive is fully coupled (in line with results by Ash 1973, Smith 1976, Brinkman (1982) and Bleakney 1984). Based on that, the action of the water-coupled gelatin on the rock is assumed to be equivalent to the ANFO directly coupled. The mass of gelatin and ANFO are lumped together to calculate the powder factor, either directly for the mass factor or combined with their specific energies for the energy factor. The fragmentation data are sensitive to the powder-factor so calculated; not surprisingly, they are also, almost equally, sensitive to the energy powder factor.
Water in rock may influence blast fragmentation due to higher wave velocities and less attenuation of the shock wave in water saturated rock, possibly resulting in a better (finer) fragmentation. In this work, blasts with high powder factor correspond to drier rock than blasts with low powder factor. The fragment sizes obtained show an inverse relation with powder factor (hence a direct relation with the water content), so the powder factor influence has probably overridden the water aspect. The low water absorption of the rock, measured at 0.10%, probably makes the water effect on the rock mass negligible.
The variability in the charging pattern (usual in the quarry sector in Spain, where bulk trucks are little used) in blasts where the water content is variable across the block must inevitably bring variability to the data. Still, the results obtained prove, if anything, that the powder factor is the primary controlling variable of fragmentation by blasting.
8 Conclusions
A fragmentation by blasting model is developed for a mylonite quarry with blocky rock structure from the fragmentation-energy fan principles. The data set comprised six production blasts located one behind the other in the same block, so the variability in rock strength and rock mass characteristics was limited. The water level and the usage of explosives (cartridged gelatin water-coupled to rock, and bulk ANFO) were variable within and between the blasts, but the distribution of explosive energy within each block was relatively uniform and the performance of both explosives per unit mass appeared to be similar. Mass powder factor above grade, considered the best descriptor of the specific charge that contributes to fragmentation, ranged from to 0.28–0.44 kg/m^{3}.
Fragmentation has been measured with three belt scales located in the processing plant, from which the fraction passing at two sizes, 120 (cut size of the grizzly) and 25 mm can be calculated, and with an online digital image system installed above the discharge of the grizzly crusher feeder. This prevents to assess the actual size of the boulders, the amount of which was estimated at 3–9.3% of the material. The size distributions of the blasted material are obtained from the image analysis data in the coarse range, where image analysis is more reliable, and from the belt scales data between 25 and 120 mm. The resulting size distributions are well described with the Swebrec function. The size distributions have been interpolated to get the desired percentile sizes in the range where the data are considered to be representative of the muckpile fragmentation, about 10–90% passing.
The percentile fragment sizes of each blast, nondimensionalized by a characteristic length (the geometric mean of the bench height and the hole spacing), form a set of convergent lines when represented as a function of the powder factor above grade in log–log space. The pre-factor and the exponent of the size-to-powder factor power laws has been obtained for nine percentiles, 10–90; the fits are all significant to a 95% level, providing a good description of the data variability with determination coefficients above 0.70. Analytical formulae to calculate fragmentation as function of the mass powder factor above grade have been obtained by deriving expressions for both parameters of the model (pre-factors and exponents) as functions of the fraction passing, based on the principles of Ouchterlony et al.’s (2017) fragmentation energy-fan. The 18 parameters of the nine fan lines (nine exponents and nine prefactors) are this way reduced to five: the fan focus coordinates and three parameters of the function of the slopes with the percentage passing.
The model explains 99% of the variance in the non-dimensional sizes at percentiles between 10 and 90% and it is statistically significant. The expected and maximum errors for sizes in that percentile range are less than 6 and 15%, respectively.
Footnotes
- 1.
Ash’s work is one of the seminal references in the history of modern rock blasting, where some of the basic ratios of blast design come from (e.g. burden ≈ 30 hole diam., stemming ≈ 2/3 burden, etc.). Ash performed his tests in boreholes with the explosive charges (ammonia dynamite in that case) coupled to the rock by means of water added in the annulus between the explosive and the rock (he also included a thin polyethylene tube as cartridge). He used ½ inch diameter charges in 7/8 inch holes. Burden was 15 inch (i.e. 30 times de charge diameter). He states (Ash 1973, p 150): “From the general appearance of the fragmented material […] satisfactory coupling of the energy from the polyethylene-packaged charge encased in water into the surrounding rock seemed to have resulted”.
Notes
Acknowledgements
The experimental work has been partially funded by MAXAM Civil Explosives program “Cátedra MAXAM” (Project UPM/FGP 007033). The support of Benito Arnó e Hijos during the whole project is also acknowledged; especial recognition is due to the technical staff of El Aljibe quarry for their enthusiastic involvement. The authors would also like to thank Split Engineering for their cooperation and support for the project success. The help and contribution of the students involved in the work is also recognized. We would also like to thank the anonymous reviewers for their valuable suggestions all across the paper.
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