# Numerical Modeling of Formation and Rise of Gas and Dust Cloud from Large Scale Commercial Blasting

^{*}

## Abstract

**:**

## 1. Introduction

^{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].

## 2. Numerical Models for Describing the Gas and Dust Cloud

#### 2.1. A numerical Model of Fireball Formation (Eulerian Model)

_{cr}, the energy value increases by an amount $Q{\rho}_{dm}^{0}/{\rho}_{exp}$, where ${\rho}_{dm}^{0}$ is initial density of the detonating mixture, ${\rho}_{exp}$ is current explosive density, which exceeds ${\rho}_{dm}^{0}$, due to compression in the shock wave, and Q is the heat of the reaction. Such operation ensures total energy conservation, correct detonation velocity, and parameters after the detonation wavefront.

^{5}–10

^{10}with proximate trajectories, and accordingly, having proximate velocities, dimensions, and other characteristics. The reference number of such markers in actual calculations is 10

^{4}–10

^{5}. Dust diffusion is calculated using the Monte Carlo method. At each time step τ, the particle movement δ

**r**is defined as $\mathsf{\delta}r=u\mathsf{\tau}+j\sqrt{D\mathsf{\tau}}$, where D is the diffusion coefficient,

**u**is the particle velocity, and

**j**is a unit vector, whose direction is selected randomly.

**g**is gravitational acceleration, ${C}_{d}$ is drag coefficient. The first and second term of the right-hand side of Equation (1) combine the Stokes drag force, the predominant effect of which is experienced by the mass at low Reynolds numbers, and the drag force at high Reynolds numbers [35]. A specially designed implicit algorithm for calculating the momentum transfer [26] ensures a proper solution for both large particles which follow ballistic trajectories largely unaffected by the ambient air, and for the case of microparticles that are frozen in the gas stream. The scheme also ensures a proper particle settling limiting velocity (depending on their size) under the influence of gravity force.

#### 2.2. Numerical Model of the Rising Thermal (Navier–Stokes LES Code)

## 3. Modeling the Gas and Dust Cloud Initiated by a 500 t TNT Explosion

^{3}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.

## 4. The Gas and Dust Cloud Rise Initiated by a TNT Explosion

#### 4.1. The Rise of the Thermal Initiated by a 500 t TNT Explosion

#### 4.2. The Gas and Dust Cloud Rise Initiated by 1–1000 t TNT Explosion

## 5. The Size Distribution of Dust Particles

_{p}(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 R

_{p}(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).

^{*}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 ${r}_{p}$ = 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.

## 6. Discussion of Results and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Density isolines. The black-filled areas correspond to rock. The grey-colored areas mark the products of detonation. The dots show the distribution of dust particles. Distance in kilometers is plotted on the axes. Time after the detonation of explosive: (

**a**) 10 ms, (

**b**) 30 ms, (

**c**) 100 ms, (

**d**) 300 ms.

**Figure 2.**Temperature distribution at the initial stage of the explosion. The lighter shade corresponds to a higher temperature; white corresponds to values: (

**a**) 5300 K, (

**b**) 3300 K, (

**c**) 1900 K, (

**d**) 1200 K Distance in kilometers is plotted on both axes. Time periods corresponding to each image are the same as in Figure 1.

**Figure 3.**Dependence of the gas and dust cloud radius on time. The grey color line depicts the experimental data, black color line—the calculated data.

**Figure 4.**Time dependency of the position of the upper edge of the gas-dust cloud: 1—experimental dependence for the explosion in 1985; 2—experimental dependence of explosion in 1987; yellow rectangles show a numerical simulation of the Euler equations; yellow circles show a numerical simulation of the Euler equations in the case of additional 30% of energy release; curves 3 and 4 describe the numerical simulation by the Navier–Stokes LES code with different values of the Smagorinsky coefficient (see Section 4.1).

**Figure 5.**(

**a**) The photo of the gas-dust cloud at 15 s after detonation. (

**b**) The calculated data and the isolines of the density of the explosion products (gray lines on the left part) and the dust (dots on the right part) at the same timepoint. The distances in kilometers are plotted along the axes.

**Figure 6.**The predicted time dependency of the relative rock particles mass (i.e., divided by the explosive mass) remaining in the atmosphere.

**Figure 7.**The distributions of the relative density of pollution compared to the velocity field (

**a**,

**c**) and buoyancy (

**b**,

**d**) shown in the radial section of the gas and dust cloud at 2 and 4 min (panels (

**a**,

**b**) and (

**c**,

**d**), respectively). Brown shades (panels (

**a**,

**c**)) indicate changes in the relative dust concentration (divided by the initial concentration). Dust concentration values are given in color bars. Shades of color (panels (

**b**,

**d**)) indicate changes in buoyancy values. The buoyancy values are given in color bars. The maximum values of velocities are given in rectangles on panels (

**a**,

**c**). Distance in kilometers is plotted on both axes.

**Figure 8.**The distributions of the relative density of particles with a size of 0.4 mm (panels (

**a**,

**b**)) and 0.8 mm (panels (

**c**,

**d**)), at 2 min (panels (

**a**,

**c**)) and 4 min (panels (

**b**,

**d**)), respectively. Shades of color are the same as in Figure 7a. Distance in kilometers is plotted on both axes.

**Figure 9.**The distributions of the relative density of pollution particles are similar to those shown in Figure 8 (panels (

**a**,

**b**)) and 0.8 mm (panels (

**c**,

**d**)), at 2 min (panels (

**a**,

**c**)) and 4 min (panels (

**b**,

**d**)), but these do not account for diffusion. Shades of color are the same as in Figure 7a. Distance in kilometers is plotted on both axes.

**Figure 10.**(

**a**) The dependence of the Smagorinsky parameter ${\mathrm{C}}_{\mathrm{sm}}$ on the charge mass W, the charge-mass dependent maximum height ${\mathrm{H}}_{\mathrm{m}}/{\mathrm{r}}_{\mathrm{T}}$ of the cloud upper edge, and charge-mass dependent maximum radius ${\mathrm{R}}_{\mathrm{m}}/{\mathrm{r}}_{\mathrm{T}}$ of the gas and dust cloud cap. The two latter are obtained for this dependence ${\mathrm{C}}_{\mathrm{sm}}$ (W). (

**b**,

**c**) The time dependences of the upper-edge height $\mathrm{H}/{\mathrm{r}}_{\mathrm{T}}$ and the cloud cap radius $\mathrm{R}/{\mathrm{r}}_{\mathrm{T}}$ for 3-, 30-, and 500-t TNT explosions (curves 1, 2, 3, respectively).

**Figure 11.**A 500 t TNT explosion: (

**a**) The time dependence of the mass fraction of particles with radius ${r}_{p}$ in CDP (related to the mass of particles with this radius in CDP and SDP); (

**b**) the upper edge height of the cloud of particles with radius ${r}_{p}$; (

**c**) the radius of CDP of particles with radius ${r}_{p}$. Curves 1–6 correspond to ${r}_{p}$ = 1.5 × 10

^{−4}, 9.5 × 10

^{−3}, 0.019, 0.075, 0.15, 0.3 cm.

**Figure 12.**The curves for the one-tonne TNT explosion: (

**a**) The time dependence of the mass fraction of particles with radius ${r}_{p}$ in CDP (related to the mass of particles with this radius in CDP and SDP); (

**b**) the upper edge height of the cloud of particles with radius ${r}_{p}$; (

**c**) the radius of CDP of particles with radius ${r}_{p}$. Curves 1–4 correspond to ${r}_{p}=$ 1.5 × 10

^{−4}, 4.75 × 10

^{−3}, 9.5 × 10

^{−3}, 0.019 cm.

**Figure 13.**A 500 t TNT explosion: The distribution of dust particles mass by size at the initial moment of time (black squares), at 30 s (blue dots), and at 5 min (red triangles) after the detonation.

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**MDPI and ACS Style**

Khazins, V.M.; Shuvalov, V.V.; Soloviev, S.P.
Numerical Modeling of Formation and Rise of Gas and Dust Cloud from Large Scale Commercial Blasting. *Atmosphere* **2020**, *11*, 1112.
https://doi.org/10.3390/atmos11101112

**AMA Style**

Khazins VM, Shuvalov VV, Soloviev SP.
Numerical Modeling of Formation and Rise of Gas and Dust Cloud from Large Scale Commercial Blasting. *Atmosphere*. 2020; 11(10):1112.
https://doi.org/10.3390/atmos11101112

**Chicago/Turabian Style**

Khazins, Valery M., Valery V. Shuvalov, and Sergey P. Soloviev.
2020. "Numerical Modeling of Formation and Rise of Gas and Dust Cloud from Large Scale Commercial Blasting" *Atmosphere* 11, no. 10: 1112.
https://doi.org/10.3390/atmos11101112