# Simplified Interfacial Area Modeling in Polydisperse Two-Phase Flows under Explosion Situations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flow Model

## 3. Viscous Drag

## 4. Polydisperse Particles

#### 4.1. General Relations

#### 4.1.1. Liquid Volume Fraction and Number of Particles

#### 4.1.2. Interfacial Area

#### 4.1.3. Moments of the Probability Distribution

#### 4.2. Distribution Law

#### 4.2.1. Determination of the Initial Conditions

#### 4.2.2. Illustration of the Initial Gamma and Inverse Gamma PDFs

#### 4.3. Impact of the Probability Density Functions on the Interfacial Area of the Two-Phase Flow and Mean Radius of the Polydisperse Droplets

## 5. Numerical Results

^{−5}Pa . s. The various equation-of-state parameters are provided in Table 2.

#### 5.1. Monodisperse

#### 5.2. Polydisperse

#### 5.2.1. Common Initial Interfacial Area ${A}_{I}^{t=0}$ between the Polydisperse and Monodisperse Computations

#### 5.2.2. Common Initial Mean Radius ${\overline{{R}_{2}}}^{t=0}$ between the Polydisperse and Monodisperse Computations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Log-Normal and Rosin–Rammler Probability Density Functions

**Figure A1.**Initial Gamma, Inverse Gamma, Log-Normal and Rosin–Rammler probability density functions (Equations (32), (33), (A1) and (A2)) versus the particle radius, for various $\kappa $, $\sigma $ and $\delta $ parameters (Table A1). The ${\beta}_{t=0}$, ${\nu}_{t=0}$ and ${\eta}_{t=0}$ coefficients are computed via Equations (40), (A3) and (A4), such that the initial mean radius ${\overline{{R}_{2}}}^{t=0}$ of the polydisperse particles is ${\overline{{R}_{2}}}^{t=0}=10$ $\mathsf{\mu}$m.

**Table A1.**Various parameters of the Gamma, Inverse Gamma, Log-Normal and Rosin–Rammler PDFs displayed in Figure A1.

Plot | Shape Parameter: $\mathit{\kappa}/\mathit{\sigma}/\mathit{\delta}$ | |
---|---|---|

(a) | Gamma | 30 |

Inverse Gamma | 30 | |

Log-Normal | $0.18$ | |

Rosin–Rammler | 6 | |

(b) | Gamma | 5 |

Inverse Gamma | 5 | |

Log-Normal | $0.5$ | |

Rosin–Rammler | $2.5$ |

## Appendix B. Monodisperse Limit of the Interfacial Area

#### Appendix B.1. Gamma

#### Appendix B.2. Inverse Gamma

## Appendix C. Common Initial Specific Interfacial Area

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**Figure 1.**A cylindrical explosive charge is initially surrounded by a liquid layer. When the charge explodes the liquid layer transforms to a cloud of droplets forming highly dynamical particle jets. Experimental results are presented on the left. Same jetting effects appear when the liquid is replaced by a granular layer. These jets are present in cylindrical and spherical dispersal explosions. On the right, a schematic representation of the initial cylindrical gas-liquid explosive system is depicted. The internal cylinder is initially filled with a dense gas at high pressure. The external cylinder is initially filled with liquid water at atmospheric pressure. Atmospheric air surrounds both cylinders. Material interfaces are initially present.

**Figure 2.**Initial Gamma and Inverse Gamma probability density functions (Equations (32) and (33)) versus the particle radius, for various $\kappa $ parameters. The ${\beta}_{t=0}$ coefficient is computed via Equation (40) such that the initial mean radius ${\overline{{R}_{2}}}^{t=0}$ of the polydisperse particles is ${\overline{{R}_{2}}}^{t=0}=10$ $\mathsf{\mu}$m.

**Figure 3.**Comparison of the interfacial area ${A}_{I}$ and mean radius $\overline{{R}_{2}}$, in the monodisperse and polydisperse situations, for various liquid volume fractions ${\alpha}_{2}$. The specific number of droplets ${N}_{2}$ is constant and is set to ${N}_{2}={10}^{12}$ droplets per unit volume. For the monodisperse situation, ${A}_{I}$ and ${R}_{2}$ are computed via Equation (42). For the polydisperse situation, ${A}_{I}$ and $\overline{{R}_{2}}$ are computed via Equation (43).

**Figure 4.**Comparison of the interfacial area ${A}_{I}$ and mean radius $\overline{{R}_{2}}$, in the monodisperse and polydisperse situations, for various $\kappa $ parameters. The specific number of droplets ${N}_{2}$ and liquid volume fraction ${\alpha}_{2}$ are constant and are set to ${N}_{2}={10}^{12}$ droplets per unit volume and ${\alpha}_{2}=0.2$. For the monodisperse situation, ${A}_{I}$ and ${R}_{2}$ are computed via Equation (42). For the polydisperse situation, ${A}_{I}$ and $\overline{{R}_{2}}$ are computed via Equation (43).

**Figure 5.**Simplified two-phase explosion test. A layer of liquid water is initially present in a 1D domain. The liquid volume fraction is ${\alpha}_{2}=0.9999$ in this zone, and atmospheric conditions are considered $p={10}^{5}$ Pa, ${\rho}_{2}=1050$ kg/m

^{−3}. The liquid layer is surrounded by air on both sides. Material interfaces are then initially present. On the left side, the air is initially dense and at elevated pressure: $p={10}^{7}$ Pa, ${\rho}_{1}=12$ kg/m

^{−3}, and represents initial explosion conditions. On the right side, the air is at atmospheric conditions: $p={10}^{5}$ Pa, ${\rho}_{1}=1.2$ kg/m

^{−3}. The air volume fraction is initially ${\alpha}_{1}=0.9999$ on both sides. The 1D domain is $2.5$ m long. The liquid layer is placed at abscissa $x=1.4$ m and is $0.025$ m wide. The mesh consists on $2,500$ regular elements, yielding a space step of $\Delta x=1$ mm. Boundary conditions are non-reflective.

**Figure 6.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse results with various initial radii: ${R}_{2}^{t=0}=3$ $\mathsf{\mu}$m, ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m and ${R}_{2}^{t=0}=300$ $\mathsf{\mu}$m. Computation in the absence of viscous drag force is also considered. All results are given at time $t=1.2$ ms. The air volume fraction ${\alpha}_{1}$, the pressure ${p}_{1}={p}_{2}=p$, the air and liquid densities, respectively ${\rho}_{1}$ and ${\rho}_{2}$, as well as the air and liquid speeds, respectively ${u}_{1}$ and ${u}_{2}$, are presented. A close-up view of the liquid density ${\rho}_{2}$ is also shown.

**Figure 7.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse results with various initial radii: ${R}_{2}^{t=0}=3$ $\mathsf{\mu}$m (top plots), ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m (middle plots) and ${R}_{2}^{t=0}=300$ $\mathsf{\mu}$m (bottom plots). All results are given at time $t=1.2$ ms. The specific number ${N}_{2}$ and the radius ${R}_{2}$ of the liquid particles are presented.

**Figure 8.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse results with various initial radii: ${R}_{2}^{t=0}=3$ $\mathsf{\mu}$m, ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m and ${R}_{2}^{t=0}=300$ $\mathsf{\mu}$m. All results are given at time $t=1.2$ ms. The specific interfacial area ${A}_{I}$ is presented for all radii on the left. On the right, a close-up view is shown.

**Figure 9.**Initial droplet distributions provided by the Inverse Gamma density function with various $\kappa $ parameters. On the left, a common initial interfacial area ${A}_{I,\mathrm{mono}.}^{t=0}={A}_{I,\mathrm{poly}.}^{t=0}$ is considered between the monodisperse and polydisperse computations through an appropriate determination of the initial mean radius ${\overline{{R}_{2}}}^{t=0}$ of the polydisperse computation. The single radius ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m of the monodisperse computation is considered as an input data. The initial mean radius ${\overline{{R}_{2}}}^{t=0}$ of the polydisperse computation is determined with the help of Equation (39) (see Appendix C). The initial mean radius ${\overline{{R}_{2}}}^{t=0}$ is used to find the ${\beta}_{t=0}$ coefficient (Equation (40) in Section 4.2.1). On the right, the initial mean radius ${\overline{{R}_{2}}}^{t=0}$ is directly used as an input datum. The initial monodisperse and polydisperse mean radii are considered equal: ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m.

**Figure 10.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial specific interfacial area is the same for both computations ${A}_{I,\mathrm{mono}.}^{t=0}={A}_{I,\mathrm{poly}.}^{t=0}$ and is computed via an initial monodisperse radius of ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =5$. All results are given at time $t=1.2$ ms and are presented in terms of ${\alpha}_{1}$, p, ${\rho}_{1}$, ${\rho}_{2}$, ${u}_{1}$ and ${u}_{2}$ for both computations. The two computed solutions are very close, only slight differences appear in the various profiles.

**Figure 11.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial specific interfacial area is the same for both computations ${A}_{I,\mathrm{mono}.}^{t=0}={A}_{I,\mathrm{poly}.}^{t=0}$ and is computed via an initial monodisperse radius of ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =5$. All results are given at time $t=1.2$ ms and are presented in terms of ${A}_{I}$, u, ${N}_{2}$ and ${R}_{2}$ for both computations. The two computed solutions are very close, only slight differences appear in the various profiles.

**Figure 12.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial specific interfacial area is the same for both computations ${A}_{I,\mathrm{mono}.}^{t=0}={A}_{I,\mathrm{poly}.}^{t=0}$ and is computed via an initial monodisperse radius of ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =3.1$. All results are given at time $t=1.2$ ms and are presented in terms of ${\alpha}_{1}$, p, ${\rho}_{1}$, ${\rho}_{2}$, ${u}_{1}$ and ${u}_{2}$ for both computations. As the $\kappa $ parameter tends towards its lower limit, differences between the two solutions are visible.

**Figure 13.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial specific interfacial area is the same for both computations ${A}_{I,\mathrm{mono}.}^{t=0}={A}_{I,\mathrm{poly}.}^{t=0}$ and is computed via an initial monodisperse radius of ${R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =3.1$. All results are given at time $t=1.2$ ms and are presented in terms of ${A}_{I}$, u, ${N}_{2}$ and ${R}_{2}$ for both computations. As the $\kappa $ parameter tends towards its lower limit, differences between the two solutions are visible.

**Figure 14.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =50$. All results are given at time $t=1.2$ ms and are presented in terms of ${\alpha}_{1}$, p, ${\rho}_{1}$, ${\rho}_{2}$, ${u}_{1}$ and ${u}_{2}$ for both computations. The two solutions are very close.

**Figure 15.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =50$. All results are given at time $t=1.2$ ms and are presented in terms of ${A}_{I}$, u, ${N}_{2}$ and ${R}_{2}$ for both computations. The two solutions are very close.

**Figure 16.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =7$. All results are given at time $t=1.2$ ms and are presented in terms of ${\alpha}_{1}$, p, ${\rho}_{1}$, ${\rho}_{2}$, ${u}_{1}$ and ${u}_{2}$ for both computations. Differences between the two solutions are visible.

**Figure 17.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =7$. All results are given at time $t=1.2$ ms and are presented in terms of ${A}_{I}$, u, ${N}_{2}$ and ${R}_{2}$ for both computations. Differences between the two solutions are visible.

**Figure 18.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =3.5$. All results are given at time $t=1.2$ ms and are presented in terms of ${\alpha}_{1}$, p, ${\rho}_{1}$, ${\rho}_{2}$, ${u}_{1}$ and ${u}_{2}$ for both computations. Differences between the two solutions are visible.

**Figure 19.**Simplified two-phase explosion test depicted in Figure 5. Comparison of the monodisperse and polydisperse results. The initial mean radius is the same for both computations ${\overline{{R}_{2}}}^{t=0}={R}_{2}^{t=0}=30$ $\mathsf{\mu}$m. For the polydisperse computation, the Inverse Gamma distribution is used with $\kappa =3.5$. All results are given at time $t=1.2$ ms and are presented in terms of ${A}_{I}$, u, ${N}_{2}$ and ${R}_{2}$ for both computations. Differences between the two solutions are visible.

**Table 1.**Minimum and maximum radii for which the initial Gamma (32) and Inverse Gamma (33) PDFs are numerically zero. Outside from the radius interval $[{R}_{\mathrm{min}},{R}_{\mathrm{max}}]$, the PDFs are less than $\u03f5=0.001\times \mathrm{max}\left(f\left({R}_{2}\right)\right)$.

${\mathit{\kappa}}_{\mathbf{Gamma}}$ | ${\mathit{R}}_{\mathbf{min}}$ | ${\mathit{R}}_{\mathbf{max}}$ | ${\mathit{\kappa}}_{\mathbf{Gamma}}^{\mathbf{Inverse}}$ | ${\mathit{R}}_{\mathbf{min}}$ | ${\mathit{R}}_{\mathbf{max}}$ |
---|---|---|---|---|---|

30 | 4.5 $\mathsf{\mu}$m | 18 $\mathsf{\mu}$m | 30 | 5 $\mathsf{\mu}$m | 20 $\mathsf{\mu}$m |

5 | 0.7 $\mathsf{\mu}$m | 33 $\mathsf{\mu}$m | 5 | 2 $\mathsf{\mu}$m | 50 $\mathsf{\mu}$m |

1 | 0 $\mathsf{\mu}$m | 70 $\mathsf{\mu}$m | 3.1 | 1.2 $\mathsf{\mu}$m | 70 $\mathsf{\mu}$m |

0.5 | 0 $\mathsf{\mu}$m | 70 $\mathsf{\mu}$m | - | - | - |

**Table 2.**Stiffened-Gas coefficients for air and liquid water [32].

Coefficients | $\mathit{\gamma}$ | ${\mathit{P}}_{\mathit{\infty}}\left(\mathbf{Pa}\right)$ |
---|---|---|

Gas Phase | $1.4$ | 0 |

Liquid Phase | $4.4$ | $6\times {10}^{8}$ |

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**MDPI and ACS Style**

Feroukas, K.; Chiapolino, A.; Saurel, R.
Simplified Interfacial Area Modeling in Polydisperse Two-Phase Flows under Explosion Situations. *Fire* **2023**, *6*, 21.
https://doi.org/10.3390/fire6010021

**AMA Style**

Feroukas K, Chiapolino A, Saurel R.
Simplified Interfacial Area Modeling in Polydisperse Two-Phase Flows under Explosion Situations. *Fire*. 2023; 6(1):21.
https://doi.org/10.3390/fire6010021

**Chicago/Turabian Style**

Feroukas, Konstantinos, Alexandre Chiapolino, and Richard Saurel.
2023. "Simplified Interfacial Area Modeling in Polydisperse Two-Phase Flows under Explosion Situations" *Fire* 6, no. 1: 21.
https://doi.org/10.3390/fire6010021