# Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden

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## Abstract

**:**

## 1. Introduction

## 2. Construction of Theoretical Calculation Model of Blast Throwing Energy

#### 2.1. Casting Velocity Dimension

_{r}, and tensile strength σ

_{t}, and the remaining six are collated using dimensionalization. The control parameters affecting the initial casting velocity are represented by the equation of dimensionality consisting of six π

_{i}terms, as shown in Table 1.

_{4}using π-theorem, we obtain:

#### 2.2. Calculation Model of Spherical Cartridge Throwing Energy

_{1}are factors related to the rock and explosive properties, and k, α

_{1}can be obtained by a linear fit.

^{2}(when α < 5°, sinα = α, and rα = L). A function of time is denoted by r(t). m is the mass of the unit, m = πρ

_{r}W

_{i}(αW

_{i}/2)

^{2}/3. a is the acceleration, which is denoted by dv/dt. Equation (6) can be expressed as:

_{0}; at time point t, the initial casting velocity is reached. The initial casting velocity equation of the unitary body is obtained after the integral variation of Equation (8):

#### 2.3. Column Charge Blast Throw Energy Calculation Model

_{ij}. The model for calculating the magnitude of the casting velocity at point j can be expressed as:

_{ij}of the i-th cartridge at unit j point can be expressed as:

_{r}, as shown in Figure 3. Given differences in the initial casting velocity of a micro-element at different locations in step blasting, its functional relationship is supposed as follows:

## 3. The Model Test

#### 3.1. Test Program

#### 3.2. Making Models and Testing the Mechanical Properties of Materials

#### 3.3. Blasting Parameters

#### 3.4. Survey Program

## 4. Discuss Test Results

#### 4.1. Throwing Process

#### 4.2. Casting Velocity Analysis

#### 4.2.1. Casting Velocity Distribution Mechanism

_{jt}is the velocity of the mass point j at the time point t, m/s; (x

_{t}

_{−1},y

_{t}

_{−1}) and (x

_{t},y

_{t}) are the coordinate positions at time t − 1 and time t, respectively. Aiming at the fluctuation of broken rock in the process of throwing, the maximum velocity is chosen as the casting velocity in the calculation model. The specific calculation equation is:

- (1)
- In addition, a comparative analysis of the bulge morphology of each model reveals that the timing of the initial bulge movement varies among models; the bulging time of Model 1 is about 1 ms, while Models 2 and 3 are about 2 ms, and Models 4 and 5 are about 3 ms. This indicates that as the minimum burden increases, the penetration time of the blast crack will be delayed and the free surface bulge start time is lagging. During the same movement time, the displacement of the broken rock on the free surface decreases with the increase of the burden. This indicates that the casting velocity shows a decreasing trend with the increasing burden;
- (2)
- The time from the beginning of the bulge to the completion of the initial acceleration is between 2 ms to 6 ms. Afterward, the casting velocity fluctuates due to the explosion of overflowing gas and collisions between broken bodies, etc. The peak value of the initial casting velocity shows a clear downward trend with the increasing burden.

#### 4.2.2. Throwing Kinetic Energy Distribution Law

_{max}and y

_{min}, respectively. Then, the equation for calculating the kinetic energy of throwing can be transformed by Equation (20), shown as follows.

## 5. Conclusions

- (1)
- Based on dimensional theory, the calculation equations for the casting velocity and throwing energy of the rock fragments were constructed during spherical cartridge and step columnar cartridge blasting. Model tests verified the feasibility of the calculated equation;
- (2)
- Through high-speed photography technology, it was found that the casting velocity of rock in the broken zone of step blasting has different burdens. This shows a normal distribution law along the vertical direction of the steps, and the fit correlation is high;
- (3)
- The model experiment successfully obtained the energy consumption of throwing broken rocks under explosive load. This shows a decreasing trend with an exponential relationship with the increasing burden. The trend of the energy proportion is similar.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Acceleration |

a_{1} | Hole distance |

A | Fit coefficient |

D | Explosive burst velocity |

E | Modulus of elasticity |

Ev | Throwing kinetics |

Et | Explosive energy |

k | Rock coefficient |

L | Arc length, pressure area diameter |

m | Unit mass |

m_{1} | Fit coefficient |

M | Blasting muckpile statistical quality |

P | Pressure |

Q | Explosive quantity |

r | Unit radius |

R^{2} | Correlation coefficient |

S | Pressure area |

v | Initial casting velocity |

v(y) | Variation formula of casting velocity with ordinate |

W | Minimum burden |

y_{0} | Fit coefficient |

α | Radian angle |

α_{1} | Explosive coefficient |

$\mathsf{\mu}$ | Fit coefficient |

π | Number π |

π_{i} | Similarity criterion |

ρ, ρ_{r} | Rock density |

ρ_{b} | Explosive density |

σc | Compressive strength |

σt | Tensile strength |

λ | Number of micro-elements |

Hv | Throwing kinetics ratio |

y_{max}, y_{min} | Mass point coordinate |

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**Figure 5.**Division diagram of explosion cavity: (

**a**) standard specimens, (

**b**) compression test, (

**c**) acoustic testing.

Variables | Dimension | π-Term | After Sorting |
---|---|---|---|

Initial casting velocity v | LT^{−1} | π_{1} | $v/\sqrt{{\sigma}_{\mathrm{c}}/{\rho}_{\mathrm{r}}}$ |

Minimum burden W | L | π_{2} | |

Explosive density ρ_{b} | L^{−3} M | π_{3} | ρ_{b}/ρ_{r} |

Explosive quantity Q | M | π_{4} | Q/ρ_{r}W^{3} |

Explosive burst velocity D | LT^{−1} | π_{5} | $D/\sqrt{{\sigma}_{\mathrm{c}}/{\rho}_{\mathrm{r}}}$ |

Tensile strength σ_{t} | L^{−1} MT^{−2} | π_{6} | σ_{t}/σ_{c} |

Compressive strength σ_{c} | L^{−1} MT^{−2} | π_{7} | |

Rock density ρ_{r} | L^{−}^{3} M | π_{8} | |

Modulus of elasticity E | L^{−1} MT^{−2} | π_{9} | E/σ_{c} |

Density kg∙m ^{−3} | Longitudinal Wave Velocity/m∙s ^{−1} | Poisson’s Ratio | Compressive Strength/MPa | Modulus of Elasticity/GPa |
---|---|---|---|---|

1850 | 2326 | 0.235 | 8.38 | 10.02 |

Explosive Category | Line Density ρ _{l}/(kg∙m^{−1}) | Explosion Heat S/(kJ∙kg ^{−1}) | Explosive Power/mL | Detonation Velocity /(m∙s^{−1}) |
---|---|---|---|---|

Hexogen | 0.025 | 5600 | 480 | 8300 |

Hole Depthd_{h}/m | Cartridge Diameter D/m | Explosive Quantity Q/g | Charge length l_{0}/m | Powder Factor q_{m}/kg·m^{−3} | Minimum Burden W/m | |
---|---|---|---|---|---|---|

1 | 0.225 | 0.006 | 1.58 | 0.04 | 0.49 | 0.12 |

2 | 0.245 | 0.33 | 0.14 | |||

3 | 0.265 | 0.24 | 0.16 | |||

4 | 0.285 | 0.17 | 0.18 | |||

5 | 0.305 | 0.13 | 0.20 |

**Table 5.**The function relationship between the initial casting velocity of the particle and the y-coordinate value.

Number | Minimum Burden W/m | Functional Relationship | Correlation Parameters R ^{2} | |||
---|---|---|---|---|---|---|

y_{0} | A | m_{1} | $\mathit{\mu}$ | |||

1 | 0.12 | 4.60 | 0.33 | 0.03 | 0.03 | 0.91 |

2 | 0.14 | 0.93 | 0.59 | 0.03 | 0.02 | 0.99 |

3 | 0.16 | 3.08 | 0.20 | 0.03 | 0.02 | 0.98 |

4 | 0.18 | 2.19 | 0.14 | 0.02 | 0.02 | 0.98 |

5 | 0.20 | 2.64 | 0.11 | 0.02 | 0.02 | 0.96 |

Number | Minimum Burden W/m | Explosive Energy E _{t}/kJ | Throwing Kinetics E _{v}/J | Throwing Kinetics Ratio η _{v}/% |
---|---|---|---|---|

1 | 0.12 | 8.848 | 1306.88 | 14.77 |

2 | 0.14 | 8.848 | 1024.29 | 11.58 |

3 | 0.16 | 8.848 | 985.11 | 11.13 |

4 | 0.18 | 8.848 | 786.06 | 8.88 |

5 | 0.20 | 8.848 | 747.49 | 8.45 |

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## Share and Cite

**MDPI and ACS Style**

Huang, Y.; Zhao, Z.; Zhang, Z.; Zhou, J.; Li, H.; Li, Y.
Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden. *Appl. Sci.* **2022**, *12*, 6524.
https://doi.org/10.3390/app12136524

**AMA Style**

Huang Y, Zhao Z, Zhang Z, Zhou J, Li H, Li Y.
Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden. *Applied Sciences*. 2022; 12(13):6524.
https://doi.org/10.3390/app12136524

**Chicago/Turabian Style**

Huang, Yonghui, Zixiang Zhao, Zhiyu Zhang, Jiguo Zhou, Hongchao Li, and Yanlin Li.
2022. "Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden" *Applied Sciences* 12, no. 13: 6524.
https://doi.org/10.3390/app12136524