# Group Combustion of Dispersed Spherical Core–Shell Nanothermite Particles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Single Isolated Particle Combustion

#### 2.1. Comparison with Other Single-Particle Combustion Models

#### 2.1.1. Spray Combustion

#### 2.1.2. Coal/Char Combustion

#### 2.2. Particle Consumption Time and D-Square Law

## 3. Mass Conservation and Group Combustion

#### Group Mass-Loss Rate and Correction Factor

## 4. Group Classification

#### 4.1. Isolated Particle Combustion (ISOC)

#### 4.2. Internal Group Combustion (IGC)

#### 4.2.1. Individual Particle Combustion (IPC)

#### 4.2.2. Incipient Group Combustion (IGC)

#### 4.2.3. Partial Group Combustion (PGC)

#### 4.3. External Group Combustion (EGC)

#### 4.3.1. Critical Group Combustion (CGC)

#### 4.3.2. External Particle Combustion (EPC)

#### 4.3.3. Sheath Combustion (SC)

#### 4.4. Comparison with Other Group Combustion Models

## 5. Group Combustion of Non-Spherical Particle Distribution

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the core–shell nanothermite and corresponding nanothermite agglomerate, which can disperse in the carrier phase for combustion.

**Figure 2.**Combustion of a single solid nanothermite; surface denotes nanothermite pellet surface. This combustion of a single-particle behaves the same as a highly concentrated group combustion event (sheath combustion as discussed in Section 4.3.3).

**Figure 3.**Illustration of combustion for a group of nanothermites and the different regions of A, B and C.

**Figure 4.**Variation of the quasi-steady non-dimensional temperature $\left(\frac{T-{T}_{\infty}}{{T}_{s}-{T}_{\infty}}\right)$ of the gas and non-dimensional mass fraction $\left(\frac{Y-{Y}_{\infty}}{{Y}_{s}-{Y}_{\infty}}\right)$ of copper vapors with a non-dimensional radius $\left(\frac{r}{{R}_{G}}\right)$ of nanothermite group. ${T}_{s}$ represents the temperature in proximity to isolated particle combustion as defined in Equation (10).

**Figure 5.**Plot of normalized radial mass flow rate $\left({\dot{m}}_{r}/{\dot{m}}_{G}\right)$ against the normalized radius $\left(r/{R}_{G}\right)$ for different group combustion numbers (G).

**Figure 6.**(

**a**) The fraction variation of cloud combustion rate to sheath combustion for a spherical nanothermite group with group combustion number. (

**b**) The fraction variation of cloud combustion rate to isolated individual particle combustion for a spherical nanothermite group.

**Figure 8.**Illustration of minor group combustion modes for a cluster of nanothermites. ${T}_{PS}$ is the particle surface temperature.

**Figure 10.**(

**a**) The variation of the overall number of particles (N) with normalized interparticle distance ($l/{r}_{s}$) for various group combustion numbers (G) corresponding to different group combustion modes. (

**b**) The variation of non-dimensional group radius (${R}_{G}/{r}_{s}$) with normalized interparticle distance ($l/{r}_{s}$) for various group combustion numbers (G) corresponding to different group combustion modes.

**Figure 11.**Schematic of group combustion in nanothermite jet suspension considering convective heating.

Parameter | Nanothermites | Other Models | Details |
---|---|---|---|

B (Mass) | $\frac{Y-{Y}_{\infty}}{1-Y}$ | $\frac{\left(\frac{{Y}_{C{O}_{2}}}{{c}_{C{O}_{2}}}+\frac{{Y}_{{O}_{2}}}{{c}_{{O}_{2}}}\right)-\left(\frac{{Y}_{C{O}_{2},s}}{{c}_{C{O}_{2},s}}+\frac{{Y}_{{O}_{2},s}}{{c}_{{O}_{2},s}}\right)}{1+\left(\frac{{Y}_{C{O}_{2},s}}{{c}_{C{O}_{2},s}}+\frac{{Y}_{{O}_{2},s}}{{c}_{{O}_{2},s}}\right)}$ | Coal/char [13] |

${B}_{T}$ (Thermal) | $\frac{\dot{{m}_{s}}{C}_{p}({T}_{\infty}-T)}{{Q}_{s}}$ | $\frac{{C}_{p}(T-{T}_{s,b})}{{L}_{h}}+\frac{{Y}_{{O}_{2,\infty}}}{{c}_{{O}_{2}}}\frac{{h}_{c}}{{L}_{h}}$ | Spray [11] |

**Table 2.**Summary of classifications of nanothermite groups based on range of group combustion number (G). ${G}_{L}$ represents the lower limit and ${G}_{U}$ represents the upper limit of the group combustion regimes, respectively.

Regime | Zone | Class | Range of G | Details |
---|---|---|---|---|

I | Internal group combustion | $G<({G}_{U}\sim 10$) | Dilute, low G | |

i | Isolated combustion | $G<({G}_{U}\sim 0.5)$ | Ultradilute, very low G | |

ii | Individual to partial group combustion | $(0.5\sim {G}_{L})<G<{G}_{U}\sim 10$ | Moderately dilute, marginally low G | |

II | External group combustion | $G>({G}_{L}\sim 10)$ | Dense, high G | |

iii | Critical to external particle combustion | $(10\sim {G}_{L})<G<({G}_{U}\sim 100)$ | Moderately dense, marginally high G | |

iv | Sheath Combustion | $(100\sim {G}_{L})<G$ | Ultradense, very high G |

Combustion | Nanothermites | Analogous Coal | Analogous Spray | |||||
---|---|---|---|---|---|---|---|---|

Present Study | [43] | [39] | ||||||

Parameter | $\frac{{\mathit{M}}_{\mathit{G}}}{{\mathit{M}}_{\mathit{SC}}}$ | Group No. (∼$\mathbf{G}$) | Group No. (G) | Group No. (G) | ||||

Regime | Min. | Max. | Min. | Max. | Min. | Max. | Min. | Max. |

Isolated Group Combustion (ISOG) | 0 | 0.14 | 0 | 0.5 | 0 | 0.3 | 0 | 0.01 |

Individual Particle Combustion (IPC) | $0.14$ | 0.46 | $0.5$ | 3 | 0.3 | 2 | 0.01 | 0.006 |

Incipient Group Combustion (IGC) | $0.46$ | 0.6 | 3 | 6 | 2 | 3 | 0.06 | 0.1 |

Partial Group Combustion (PGC) | $0.6$ | 0.7 | 6 | 10 | 3 | 4 | 0.1 | 1 |

Critical Particle Combustion (CPC) | $0.7$ | 0.87 | 10 | 60 | 4 | 40 | 1 | 10 |

External Particle Combustion (EPC) | $0.87$ | 0.9 | 60 | 100 | 40 | 100 | 10 | 100 |

Sheath combustion (SC) | $0.9$ | 1 | 100 | ∞ | 100 | ∞ | 100 | ∞ |

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**MDPI and ACS Style**

Rahman, M.M.; Saieed, A.; Khan, M.F.; Hickey, J.-P.
Group Combustion of Dispersed Spherical Core–Shell Nanothermite Particles. *Thermo* **2022**, *2*, 209-231.
https://doi.org/10.3390/thermo2030016

**AMA Style**

Rahman MM, Saieed A, Khan MF, Hickey J-P.
Group Combustion of Dispersed Spherical Core–Shell Nanothermite Particles. *Thermo*. 2022; 2(3):209-231.
https://doi.org/10.3390/thermo2030016

**Chicago/Turabian Style**

Rahman, Mustafa Mutiur, Ahmed Saieed, Muhammad Fasahat Khan, and Jean-Pierre Hickey.
2022. "Group Combustion of Dispersed Spherical Core–Shell Nanothermite Particles" *Thermo* 2, no. 3: 209-231.
https://doi.org/10.3390/thermo2030016