1. Introduction
It is estimated that from hundreds of billions to a trillion t of solid substance are extracted from the lithosphere each year. A significant share of the processed rock mass is accounted for the extraction of mineral resources using open-pit mining [
1,
2]. Open-pit mining is followed by intensive emission of particulate pollutants, which is a consequence of rock crushing using blasting, drilling, extraction-and-loading works, transportation of rock mass, as well as the transport of dust in the course of wind erosion of made-made massifs of mined-out rocks [
3]. According to the data on emissions of particulate pollutants into the atmosphere, the share of mineral extraction is approximately 28% [
4] of the total emissions depending on the type of business operation. One of the main problems in areas of mineral resource extraction is associated with deterioration in air quality [
5,
6]. At the same time, it is important to note that microparticles of human-made origin are more dangerous than natural materials, due to their chemical composition [
3,
7,
8].
Explosive technologies are widely used both for strip mining and for rock crushing, and are especially intensively applied in pits where hard rocks are mined. Modern studies in the field of development of new technologies for extracting mineral resources and shattering hard rocks shows that this situation will prevail in the coming decades [
2,
9,
10,
11]. A number of studies indicate that blasting operations are one of the main sources of particulate pollutants in pits [
12,
13,
14,
15,
16]. It is estimated that more than 50% of the total amount of particulate pollutants released into the near-surface layer of the atmosphere during mining mineral resources in open pits happens to be through large-scale blasts [
17]. As an example, we will provide data on the number of large-scale explosions at the Lebedinsky Mining and Processing Plant (Lebedinsky GOK). Thus, the number of explosions with a mass of more than 500 t in the period 1997 to 1999 was an average of 25 explosions per year [
14]. Large-scale explosions with a total TNT mass of 3155.2 t were carried out at the Lebedinsky GOK in 2015, and the blasted-out volume of rock mass was 2855 thousand m
3 [
18]. At the same time, large-scale explosions are carried out once a month. A rough estimation of the annual blast out volume of rock mass at the Lebedinsky GOK gives a value of 34 × 10
6 m
3, and an assessment of the mass of explosives used annually is ≈37.9 × 10
6 kg. According to statistical reports, the total emissions of particulate pollutants to the atmosphere from all registered sources at the Lebedinsky GOK were 6.28 × 10
6 kg in 2017, whereas, the total emissions were 7.05 × 10
6 kg in 2018 [
19].
Despite the wide use of large-scale blasting during development of mineral deposits, using open-pit mining, the emission of particulate pollutants into the atmosphere induced by explosions has not been sufficiently studied yet. This is due to a lack of data and unreliability of available tests [
20]. The numerical simulations can help to solve the problem. Important information on the dispersion of particles inside and outside the pit is provided by studies on particle transport using local wind streams, taking into account the disturbances created by the complex topology of the excavation and surrounding terrain [
21,
22,
23]. The Reynolds-averaged Navier–Stokes equations are usually solved numerically in these cases, and various turbulent transfer models are used for the determination of the eddy viscosity. A comprehensive review of numerical dust dispersion models developed for predicting the dispersion of particles during the extraction of mineral resources using open-pit mining is presented in Reference [
24]. The blast-aided dust cloud is simulated in these calculations by an experimentally defined particle flow from the specified injection regions. The formation of a hot cloud filled with detonation products, air and dust in the vicinity of the explosion epicenter, and the subsequent convective rise of hot gas and dust cloud in the stratified atmosphere are ignored in such models. However, these processes become crucial for dust transfer generated by powerful large-scale (>1 t TNT) explosions. The upper boundary of the gas and dust cloud rises to the heights significantly exceeding the pit depth (hundreds of meters), and the subsequent spread of dust particles is determined by the prevailing winds at these heights. The initial spread of particle mass size distribution, which is formed as a result of the interaction of the explosion with the underlying surface, is modified during the formation and rise of a gas and dust cloud. Therefore, a proper description of the particle dispersion after the cloud has reached the upper position requires defining the concentration of particles in the cloud at this point, which in turn, is associated with the entire complex of gas-dynamic processes that are followed by formation and rise of the gas and dust cloud. It is difficult to determine the distribution of particle concentrations in dust cloud experimentally, and numerical modeling can possibly help to solve the problem.
The development of gas and dust clouds from commercial blasting can be conditionally divided into three stages. The first one is characterized by fast processes (hundreds of milliseconds) starting from the generation of detonation and shock waves and ejection of rock particles of various sizes from the crater formed. Then, the explosion products expand to atmospheric pressure and mix with the air and dust particles. As a result, buoyancy-driven gas and dust flow forms, which are generally called thermal. Over the course of time, the shock waves travel long distances and attenuate. The second stage, which lasts for several minutes, is characterized by a relatively slow rise of the thermal in the stratified atmosphere under the influence of buoyancy forces, its expansion, and entrainment of the surrounding cold air in the course of turbulent mixing [
25]. The original quasispherical shape of the cloud of explosion products is destroyed, a toroidal vortex is formed, and the gas and dust cloud acquire a mushroom shape. The cloud rise is limited to the height at which the hydrostatic equilibrium with the ambient air is achieved. The third stage lasts for several hours or more, and the cloud moves at this stage by the action of the wind, continues to mix with the ambient air by the influence of turbulent diffusion, and the dust particles contained in it are gradually settled under the action of gravity force. In this paper, we study the first and second stages of the development of the gas and dust cloud, which make it possible to obtain the initial data for the third stage.
In
Section 2, we describe numerical models: Eulerian model to simulate the initial stage of dust cloud formation and rising and Navier–Stokes LES code to simulate thermal rising and mixing with ambient air. In
Section 3, the results of dust cloud formation after 500 t TNT explosion obtained with the Eulerian model are predicted. In
Section 4, thermal rising is investigated using the Navier–Stokes LES code. In
Section 5, we study the evolution of dust particle size distribution during thermal rising. Some conclusions and discussions are presented in the last section.
3. Modeling the Gas and Dust Cloud Initiated by a 500 t TNT Explosion
Simulation of the initial stage of the blast development was considered on the example of 500 t TNT explosions conducted in 1985 and 1987. The initial data were a hemispherical TNT charge with a radius of 6.8 m and a density of 800 kg/m3 placed on a smooth horizontal surface of quartz. The detonation was initiated in the center of the hemisphere at the boundary between explosive material and the rock. The simulations were run using Eulerian model on an axisymmetric, two-dimensional domain. The mesh consisted of 700 cells in both radial and vertical directions. The initial cell size was 10 cm. During the process of cloud extension, the cell size was increased to 3.2 m.
The particle size distribution of the quartz rock ejected during the crater formation was determined in accordance with experimental data [
44]. The rock samples were taken near the heap of the ejection funnel. Distribution of rock particles by size was obtained for particles in the range from 2 µm to 200 mm. The particle distribution by size can be written as:
where
is the mass of the rock. For the considered 500 t explosion and particles sized from 2 µm to 0.5 mm,
= 0.0049,
k = 0.14 ± 0.01; and for particles sized from 0.5 mm to 200 mm
= 0.0164,
k = 1.13 ± 0.03.
Figure 1 shows the density isolines, spatial distributions of detonation products, rock, and ejected particles during the first 300 ms after the explosion. At the moment when the detonation wave reaches the boundary of the charge, a discontinuity of density, pressure, and velocity occurs, the disintegration of which leads to a blast shock wave in the air and the rarefaction wave in the products of detonation. The rarefaction wave is followed by a shock wave (clearly visible at 10 and 30 ms) moving inside the area of explosion products. This effect, resulting from spherical (or cylindrical) divergence, is well known in explosion physics [
45]. The detonation products on the border with air have a higher density than the air’s density, even if it is compressed on the front of the blast wave, and the velocity of the blast wave, and consequently, the explosion product velocity decreases with time. Thus, there is a classic situation of the development of Rayleigh-Taylor instability, i.e., heavy gas slows down in the light one. The instability development, which can be clearly seen in
Figure 1, leads to the mixing of the explosion products and air and intimately determines the initial size of the explosion cloud.
In calculations, the initial perturbations are caused by the calculation errors, in particular, by the inability to correctly describe the hemisphere by using the square cells. However, in real-life, very strong perturbations occur at large-scale explosions immediately, which is due to the heterogeneity of the charge. As a rule, the charge surface shape is far from the perfect hemisphere, as it includes ventilation ducts, a chamber for placing the blast unit, etc. All these can result in very strong perturbations occurring immediately when the detonation wave reaches the charge boundary. According to calculations, the initial heterogeneity can increase the size of the formed gas-dust cloud by 10–15%.
The crater begins to form almost immediately after the detonation wave reaches the boundary of the explosive charge and the initial size of the crater coincides with the TNT hemisphere radius. During the first 10–20 ms, the rock particles move from the explosion center at speeds up to 1 km/s. At a later stage, the dust spreads almost diffusely, gradually filling the entire cloud. However, large particles, i.e., stones with a radius of more than one centimeter, move at angles to the horizon not exceeding 45°. At the boundary with the rock, the detonation products are slowed down by friction and because they are forced to bend around the crater’s elevated edges and expanding cloud of particles. As a result, the envelope shock wave specific for the supersonic flow occurs, which is clearly visible at 10 ms after the explosion. This leads to the take-off of the current and cloud from the surface, which is visible after 30 ms from the explosion start.
Figure 2 shows the temperature distributions at the same time, as in
Figure 1. It is seen that the explosive products are very quickly cooled by adiabatic expansion, while the temperature of the air inside the shock wave reaches several thousand degrees. The explosive product temperature remains lower than the temperature of the heated air till the mixing is complete. The mixing in a narrow hot layer starts almost immediately, which explains the spotted structure of the glowing area described in the experiments [
46]. The temperature of the heated gas and dust cloud starting to rise is 1000–1500 K when the shock wave separates from the cloud.
The explosive products have completely mixed with air by 300 ms after the explosion, i.e., the scale of individual particles of products becomes comparable with the size of the counting cell. From this point on, the mixed mixture is considered as a single gas with thermodynamic properties of air. To estimate further propagation of the explosion products, passive markers moving at the velocity of gas are introduced. The markers are assigned to coordinates of cells containing explosive products. Besides, these cells are assigned masses of products that are not involved in calculations, but estimates the product concentrations at any time and point.
We have compared the calculated data with the results of processing images of 500 t experiments.
Figure 3 shows the experimental and calculated time dependencies of the explosion cloud radial size during the first second after the detonation. These dependencies coincide well both qualitatively and quantitatively. However, the good coincidence of the cloud’s geometric size cannot be achieved for the subsequent period of thermal rise. The rise and radius values of the toroidal cloud obtained from the calculations are 1.5–2 times less than those observed. This discrepancy is not related to the volume of dust captured. If the condensed particles are completely removed from the thermal at any timepoint, which can be easily done at calculations, the result practically does not change. The results do not depend on cell size as well. Probably, a reason for such discrepancy is using Eulerian equations with numerical viscosity modeling the real turbulence.
It should be said that a notable difference, for example, of the cloud rise values, was registered for the two different explosions with the same energy 500 t (see
Figure 4).
Figure 4 shows the time dependencies of the values of clouds upper edge rise. However, the difference between experimental data is still much smaller than that between experimental and calculated data (
Figure 4, yellow rectangles). It is possible to get closer to the experimental data if additional heating of the cloud is provided at the timepoint of about 1 s. The calculations have shown that additional 30% of energy release in the cloud’s upper part at this timepoint allows obtaining a picture of thermal rise comparable to the experimental (
Figure 4, yellow dots, and
Figure 5). Nevertheless, an increase of the initial energy by the same value does not provide the same result.
Figure 4 (curves 3 and 4) also demonstrates the thermal rise calculated with an account of turbulent eddy viscosity at the ad-hoc Smagorinsky coefficients in the Navier–Stokes LES code.
Section 4.1 contains the discussion of these curves.
The artificial procedure (additional 30% of energy release) applied to calculate thermal rise allows using the solution for one-minute periods and determining the particle dynamics at the initial stage of rise.
Figure 6 shows the time dependence of the total mass of rock particles, including large fragments moving out at the excavation stage. The maximum value of the dust mass and the mass of rock excavated from the crater have the same order of magnitudes. At about 20 s after the explosion, only dust captured by the rising cloud (a little under half of the charge mass) remains in the atmosphere. The size distribution of the mass of particles remaining in the cloud will be given in
Section 5 below.
The Eulerian model described in this section allows us to describe the initial stage of dust cloud formation and rising and to estimate its initial size and dust concentration.
5. The Size Distribution of Dust Particles
The distribution of dust particles within the cloud is very interesting from the viewpoint of air pollution. Let us consider the dust transport within the cloud in more detail. For each particle size, we will define a cap of the cloud of particles of this size CDP (Cap of Dust Particles) as a spherical volume. The CDP radius Rp(t) is defined as the maximum radius of the cloud of particles of considered size, and its center is on the axis at a distance Rp(t) from the top cloud edge. Part of the cloud below the CDP will be called a stipe of dust particles (SDP). CDP and SDP are different for particles of different sizes. As follows from the calculations, the larger the particle size, the smaller the CDP radius. Let the CDP* be the CDP with the maximum radius. The size of CDP* is determined by the transfer of passive pollution (the smallest particles) at the time the cloud rise is complete. The upper-edge height and radius of the CDP* are determined by ratios (14) and (15).
Figure 11a demonstrates the redistribution of particles of different sizes between CDP and SDP in the case of the 500 t TNT explosion. Here
is the mass of particles of a particular size in CDP,
is the mass of such particles in CDP and SDP. At the time, the cloud rise is complete (5 min),
of larger particles is significantly lower than the initial mass
. The CDP
* has a radius of ~700 m. The lower boundary of this volume is marked with a dotted line in
Figure 11b. The cloud of particles with
= 0.3 cm (
Figure 11b, curve 6) is partially included into the CDP
* during the rise; it rises to its maximum height for 3 min after the detonation, then it starts to sink, and by the time the cloud rise is complete, the particles of the said size have fallen out of the CDP
*. Thus, according to our estimation, the 500 t explosion cloud cap includes particles with dimensions not exceeding 0.1–0.3 cm at the time when the cloud rises to its maximum height.
When the charge mass is reduced, the characteristic flow velocities in the gas and dust cloud decrease, and the redistribution of dust particles between CDP and SDP is more influenced by the particle precipitation.
Figure 12, similar to
Figure 11, shows the redistribution of particles of different sizes between CDP and SDP for a 1 t TNT explosion. In this case, only particles less than 0.01–0.02 cm are left at the time when the gas and dust cloud rise is complete.
In conclusion, we shall describe the dynamics of the dust particles for a 500 t explosion (
Figure 13).
The black curve in
Figure 13 shows the initial particle mass distribution by size (9). This distribution is characterized by two branches of the power function. After 30 s, the cloud contains only a part of the particles which mass is about half of the charge mass (
Figure 6). The particle mass distribution by size (the blue curve in
Figure 13) at this time is characterized by a maximum in the area of particles with a radius of 0.1–1 cm. This is most likely due to the superposition of the two power laws (9) and (1), which can lead to the appearance of a local maximum. All particles captured in the gas and dust cloud for the period of 30 s after the detonation remains in the cloud during its further rise, but there occurs the redistribution of the particles between the CDP and SDP. When the cloud stops to rise, the mass of particles of each size in the CDP is less than the total mass of particles of these sizes in the CDP and SDP. The larger the size of the particles, the smaller the mass of these particles remains in the CDP. Large particles with sizes exceeding 0.1–0.3 cm fall out of the CDP completely. The red curve in
Figure 13 demonstrates the final particle distribution by size at the time the cloud stops to rise.
6. Discussion of Results and Conclusions
To estimate the impact of powerful industrial explosions on the environment outside mining operations, one should have data on the pollution source, which in this case, is a mushroom-shaped dust cloud cap. The main parameters of these data include the dust cloud height, horizontal dimension of the dust cap, the dust mass contained in the cloud cap, and distribution of the dust mass by the particle size. The initial gas and dust cloud are formed near the ground surface after the expansion of detonation wave, dust ejection from the crater, and mixing the dust and detonation products with the ambient air. For a 500 t TNT explosion, the duration of the initial gas and dust cloud formation is about one second and proportional to the cube root of the charge mass. The Eulerian numerical model of this stage of the explosion, we have built, allows one to determine the initial size and temperature of the gas and dust cloud, estimate the mass and size distribution of the dust particles contained in the cloud. The resulting gas and dust cloud is lighter than air, due to its high temperature. The cloud starts to rise first, due to buoyancy, then by inertia. In the process of this rise, the size and mass of the cloud is increased by mixing with the ambient air, and at the same time, the mass is reduced by settling of the heaviest particles by gravity. The Navier–Stokes numerical model of this second stage, built by us, allows one to determine the size and maximum rising height of the buoyancy-driven gas and dust cloud, to study the variations of the dust mass and particle size distribution in the cloud during its rise. The performed calculations have shown that variations of the dust mass and particle-size distribution in the cloud during its rise significantly depend on the explosion scale. As the mass of TNT is reduced, the maximum particle size in the cloud cap reduces. Hence, the maximum particle size is linked to explosive mass.
The results show that numerical simulations make it possible to investigate details of explosive processes that are difficult to register and measure during the experiment (e.g., the evolution of particle size distribution). However, the results of the calculations again strongly depend on input parameters that can be reliably determined experimentally directly or indirectly. Here we, first of all, mean constants in Equation (9), which determines the particle mass distribution by size at the initial stage of explosion, and the Smagorinsky coefficient for the model of thermal rise. Tests of the large-eddy simulation turbulence model for the study of thermal rise have shown that the turbulent eddy viscosity coefficient in the used gas-dynamic model depends on the spatial resolution (grid cell size). It is not excluded that this dependence is determined by the numerical viscosity, which is summed up with the turbulent eddy viscosity. The coarser the computational grid, the greater the numerical viscosity becomes. This increase in numerical viscosity is compensated by reducing the turbulent eddy viscosity. Indeed, the increase in cell size requires a decrease in the Smagorinsky coefficient to obtain the same height as the cloud rise. However, one can successfully simulate the gas and dust cloud rise if the numerical solution is configured by the Smagorinsky constant required to match numerical simulations and experimental data. No less important is the question of the value of the turbulent Schmidt number. However, no universally-accepted values of this parameter have been established [
52]. Some available values of the turbulent Schmidt number from atmospheric systems literature are given in Reference [
52].
values vary from 0.1 to 2.5,
value of 1 is considered as an acceptable choice. We used this value. However, this question requires further study.
Numerical calculations are significantly less time-consuming than a large-scale experiment with full registration of all related processes. Therefore, a reasonable approach is to carry out separate expensive experiments to build and configure numerical models for using them in mass calculations aimed at solving environmental problems. We have elaborated on the Eulerian numerical model of the dust cloud formation and initial rising, which allows one to determine the initial size and temperature of the gas and dust cloud, to estimate the mass and size distribution of the dust particles contained in the cloud. The specific feature of the model is that an extension of dust is described not as the movement of a solid continuous medium, but as a motion of discrete particles taking into account their interaction with the gas flow. The Navier–Stokes LES code has been elaborated, which allows one to predict the size and maximum rising height of the buoyancy-driven dusty cloud and to study the variations of the dust mass and particle size distribution in the cloud during its rise. However, we cannot now provide a simple relationship for the amount of material in the cloud at it is at the highest point and particle size distributions for different charge masses. It is a problem for future investigations. It was found that the value of the Smagorinsky coefficient depends on both the charge mass and the grid cell size. The values of the Smagorinsky coefficient were found for charges with a mass of 1–1000 t using a specific grid.