# Shock Initiation and Propagation of Detonation in ANFO

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}= 0.88 g/cm

^{3}) and 4.79 km/s (at ρ

_{0}= 0.8 g/cm

^{3}), respectively.

## 2. Description of Numerical Models

#### 2.1. Shock Initiation Modeling with AUTODYN

_{Igmax}) is reached, after which the growth term is switched on. The initial reaction rate parameters in our model were inspired by [23], who proposed rate constants that have a physical basis (e.g., compression exponent x = 4 approximates the amount of plastic work required for dynamic void collapse; pressure exponent y = 0.9 describes weak, pressure-dependent laminar burning; etc.). Thus, in our study we used values of constants from [22] (I, a, b, c, d, x, and y) and adjusted only the constant G to reproduce our experimental steady-state detonation velocity charge diameter data (Table 1).

_{0}); E is the detonation energy; and A, B, R

_{1}, R

_{2}, and ω are constants.

_{1}and C

_{0}for ANFO were taken from [26] and given by C

_{0}= 0.92 km/s and S

_{1}= 1.4. for a density of 0.8 g/cm

^{3}.

^{3}density) were taken from the built-in material library in AUTODYN [27]. The PVC confinement was modeled with Shock EOS, whereby the EOS parameters for PVC were taken from the work of [28]. Steel confinement was modeled assuming the material properties of 4340 alloy steel taken from the AUTODYN material library [27]. The dynamic behavior of steel was modeled using the Johnson–Cook failure model, the Johnson–Cook strength model, and Shock EOS [27,29].

_{0}= 1.5 g/cm

^{3}), while steel-confined ANFO charges were initiated by PETN boosters of different masses, with an equivalent booster diameter-to-length ratio of 1. The point of initiation was set at the front end of the booster charge. Gauges were placed at regular intervals within the explosive charge (50 mm on the first third of the charge and 100 mm after that) to register the time of arrival of the detonation wave to determine the shock velocity.

#### 2.2. Calculation of Steady-State Detonation Applying Wood–Kirkwood Detonation Model

^{3}were taken from [26]: C

_{0}= 0.92 km/s and S

_{1}= 1.4.

_{r}) of the detonation products, we used Wood and Kirkwood’s [20] radial expansion model, which relates the rate of radial expansion and the shock front curvature radius:

_{c}is the shock front curvature radius, u is the particle velocity in the shock frame, and D is the detonation velocity.

_{0}is the charge radius, a = 0.4256, and b = 1.3835.

^{0.9}).

## 3. Results and Discussion

#### 3.1. Materials and Methods

^{3}.

^{3}.

#### 3.2. Effect of Charge Diameter on Shock Initiation Behavior

^{3}.

_{SDT}) decreased with increase in the charge diameter and ranged between 500 and 700 mm, i.e., 3–6 charge diameters (Table 2).

_{SDT}, was predicted by the calculations.

_{0}= 35.5 mm (Figure 3a), pressure continuously and slowly decreased, approaching about 0.18 GPa (resulting in VoD = 0.96 km/s) at x = 550 mm. This indicates that steady-state detonation velocity was not achieved after 550 mm. Such an effect of charge diameter on the shock initiation behavior of ANFO resulted from larger radial expansion of the detonation products and, consequently, larger pressure weakening in the case of smaller charge diameters. The final shape of the VoD–x curves was determined by an interplay between the rate of expansion of the detonation products and the rate of reaction for a given charge size. To obtain clearer insight into the dependence of initiation behavior on charge diameter, several calculations were carried out for charge diameters ranging from 71 to 1000 mm. The results of the calculations are shown in Figure 4 and Table 2.

_{SDT}= 600 mm for R

_{0}= 52 mm and x

_{SDT}= 280 mm for R

_{0}= 500 mm). For charge radii slightly above the failure radius (31.5 mm according [37]) and 37.5 mm according to [38]), VoD growth to steady-state detonation was slower, and a steady state was attained only beyond distances of 1000 mm (Table 2 and Figure 5).

_{0}results.

_{0}data at radii larger than 50 mm, with AUTODYN predicting slightly larger values of VoD (Table 2). The error in the predicted VoD values was larger in the vicinity of the failure radius for both the EXPLO5 and AUTODYN calculations. The pressure-based model described the experimental VoD–1/R

_{0}data very well, even in the vicinity of the failure radius, and correctly predicted the failure radius (R

_{f}= 39 mm, which was very close to experimental value of 37.5 mm). AUTODYN and EXPLO5 with the I&G model tended to overpredict the failure radius at between 40 and 45 mm.

#### 3.3. Effects of Booster Mass and Confinement on Shock Initiation Behavior

_{SDT}became shorter for larger boosters, supports the aforementioned hypothesis.

_{SDT}on booster mass in a log–log scale, and it is similar to the so-called “Pop Plot”, which represents the dependence of run-to-detonation-distance on initial impact pressure [36].

_{SDT}and booster mass (m

_{B}) data was linear. For all the charge diameters, x

_{SDT}decreased with increase in the booster mass and, for m

_{B}= 1000 g, x

_{SDT}was approximately 250 mm, regardless of the charge radius and the existence of confinement. From the slopes of the lines, it follows that the effect of booster mass was more pronounced for smaller charge radii (slope is larger). The slope for steel-confined charges was the smallest, which indicates the smaller effect of booster mass compared to lightly confined (unconfined) charges. For illustration, the x

_{SDT}for m

_{B}= 1000 g equaled approximately 250 mm for all the charges. However, for m

_{B}= 1 g of booster charge, x

_{SDT}= 450 mm for the steel-confined charge of a 25.4 mm of radius, x

_{SDT}= 380 mm for the lightly confined charge of a 77 mm radius, and x

_{SDT}= 600 mm for the lightly confined charge of a 52 mm radius.

_{SDT}is much shorter than for unconfined charges, even at much lower charge radii. Based on this, it can be expected that, with stronger confinement where radial expansion is highly supressed, the effect of booster mass on x

_{SDT}may be less pronounced compared to weak confinement. It should be added that other factors (e.g., shape and size of the booster charge, position of booster charge relative to the ANFO charge, etc.) can affect, to a certain extent, the shock initiation and growth of detonation to attain a steady state.

## 4. Conclusions

- (1)
- It was demonstrated that AUTODYN with an I&G model could qualitatively describe the initiation process and predict steady-state detonation velocity as a function of charge diameter with an error of up to 6%. The WK detonation model incorporated into EXPLO5 thermochemical code could predict accurately the steady-state detonation velocity of unconfined (i.e., lightly confined) charges with an error of up to 3.5% for R
_{0}> 50 mm. - (2)
- For lightly confined charges initiated by a constant mass of booster, steady-state detonation was established faster (at shorter distances) for charges of larger diameters. The initial drop in VoD and the minimum VoD reached were related to booster mass, i.e., initiating pressure impulse, and thus did not change with charge diameter. However, the shock velocity (and pressure) growth rate increased with charge diameter. This was associated with a greater expansion of products at smaller charge diameters, which resulted in stronger pressure weakening and a slower reaction rate at smaller charge diameters. Faster reaction, in turn, resulted in faster generation of additional pressure and energy that strengthened the initial shock wave, hence leading to faster growth in pressure and shock velocity and, ultimately, to a shorter distance to steady-state detonation.
- (3)
- Both the experiments and the calculations confirmed that booster mass strongly affected the resulting VoD–x profiles: the minimum VoD increased with booster mass and shifted to larger distances, the shock velocity growth rate increased, and the run-to-steady-state detonation decreased. On a log–log scale, x
_{SDT}vs. booster mass showed linear dependence analogous to a “Pop Plot”, which represents the dependence of run-to-detonation distance on initial impact pressure. - (4)
- The log(x
_{SDT})–log(m_{B}) graphs for lightly confined and steel-confined charges showed that the x_{SDT}was smaller for steel-confined charges, which supports the hypothesis that the expansion of detonation products plays an important role in the shock initiation of ANFO charges. The effect of booster mass was less pronounced for steel-confined charges. - (5)
- Considering the wide range of parameters that could affect the shock initiation and propagation of detonation in non-ideal explosives such as ANFO, this study aimed to contribute a better understanding of the impacts of some of these factors, which is important for better tailoring the effects of ANFO for use in the mining industry.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Experimental and calculated detonation velocities of slightly confined ANFO charges having different radii as a function of distance from the booster.

**Figure 3.**Calculated pressure–time profiles along the charge axis for (

**a**) 35.5 mm and (

**b**) 52 mm charge radii.

**Figure 6.**Comparison of experimental and calculated detonation velocity and inverse charge radii data [7].

**Figure 7.**Experimental and calculated VoD–x profiles as a function of mass of booster for confined ANFO charges [13].

**Figure 8.**Calculated effect of booster mass on VoD–x curve (note: charge size is the same in all cases).

**Figure 9.**Summarized presentation of effects of booster mass, charge diameter, and confinement on initiation behavior of ANFO.

Reacted ANFO ^{(a)} | Unreacted ANFO ^{(b)} | Lee–Tarver I&G Reaction Rate Parameters ^{(c)} |
---|---|---|

A = 81.6492 GPa | A = 1454.25 GPa | I = 10 1/μs |

B = 1.7537 GPa | B = −0.347 GPa | a = 0.2 |

R_{1} = 4.588863 | R_{1} = 21.8866 | b = 0.222 |

R_{2} = 1.021101 | R_{2} = 0.7874 | x = 4 |

ω = 0.32021 | ω = 3.4613 | G = 0.086 1/(μs GPa ^{^y}) |

D = 4.78 km/s | E_{0} = −0.1549 kJ/cm^{3} | c = 0.222 |

p_{CJ} = 4.61 GPa | d = 0.666 | |

E_{0} = 3.4481 kJ/cm^{3} | y = 0.9 | |

F_{Igmax} = 0.3 |

^{(a)}calculated by EXPLO5;

^{(b)}derived from Murnaghan EOS data;

^{(c)}all parameters except G are taken from [23], and G is adjusted to reproduce our experimental VoD data.

**Table 2.**Experimental and calculated detonation velocities and run-to-steady-state detonation velocity distances at different charge radii.

d_{in}(mm) | R_{0}(mm) | Detonation Velocity (km/s) | x_{SDT} (mm) | x_{SDT}/d_{in} | ||||
---|---|---|---|---|---|---|---|---|

Expt. | AUTODYN (at x > x _{SDT}) | EXPLO5 (PB Model) | EXPLO5 (I&G Model) | Expt. | AUTODYN | |||

71 | 35.5 | 1.23 | 0.96 | - | - | - | - | |

90 | 45 | 1.63 | 2.48 | 2.34 | 1000 * | 11.11 | ||

98 | 49 | 2.69 | 2.78 | 2.61 | 720 | 7.35 | ||

104 | 52 | 2.85 | 2.88 | 2.94 | 2.77 | 600 | 600 | 5.77 |

119 | 59.5 | 3.15 | 3.31 | 3.22 | 3.10 | 650 | 420 | 3.53 |

154 | 77 | 3.68 | 3.89 | 3.58 | 3.57 | 500 | 380 | 2.47 |

250 | 125 | 4.59 | 4.01 | 4.10 | 300 | 1.20 | ||

1000 | 500 | 4.85 | 4.56 | 4.57 | 280 | 0.28 |

Booster (g) | Experiment | AUTODYN | ||
---|---|---|---|---|

VoD * (km/s) | VoD ** (km/s) | VoD_{min} (km/s) | x_{SDT} (mm) | |

1 | 2.35 | 3.34 | 0.81 | 450 |

5 | 3.34 | 1.10 | 400 | |

20 | 3.04 | 3.28 | 1.72 | 330 |

50 | 3.19 | 3.31 | 2.10 | 320 |

100 | 3.46 | 3.33 | 2.25 | 290 |

200 | 3.32 | 2.45 | 260 | |

500 | 3.32 | 2.55 | 250 |

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**MDPI and ACS Style**

Bohanek, V.; Štimac Tumara, B.; Serene, C.H.Y.; Sućeska, M.
Shock Initiation and Propagation of Detonation in ANFO. *Energies* **2023**, *16*, 1744.
https://doi.org/10.3390/en16041744

**AMA Style**

Bohanek V, Štimac Tumara B, Serene CHY, Sućeska M.
Shock Initiation and Propagation of Detonation in ANFO. *Energies*. 2023; 16(4):1744.
https://doi.org/10.3390/en16041744

**Chicago/Turabian Style**

Bohanek, Vječislav, Barbara Štimac Tumara, Chan Hay Yee Serene, and Muhamed Sućeska.
2023. "Shock Initiation and Propagation of Detonation in ANFO" *Energies* 16, no. 4: 1744.
https://doi.org/10.3390/en16041744