Numerical simulation of dust dispersion using molecular-kinetic model for description of particle-to-particle collisions

doi:10.1016/j.jlp.2015.02.006

Journal of Loss Prevention in the Process Industries

Volume 36, July 2015, Pages 223-229

https://doi.org/10.1016/j.jlp.2015.02.006 Get rights and content

Highlights

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Mathematical description of collisional dynamics in gas particle suspensions.
•
Numerical modeling of shock wave and detonation processes.
•
Analysis of collisional effects in detonations.
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Analysis of dust dispersion from layers under shock wave actions.

Abstract

A mathematical model of two-phase medium for the description of shock wave processes in gas particle suspensions with regard for particle-to-particle collisions is presented. The model is based on the molecular-kinetic approaches of theory of granular materials. A numerical technology for 2-D calculations is based on the Harten TVD scheme for gas and the Gentry-Martin-Daly scheme for particles. The effects of collisional particle dynamics are analyzed in shock-wave and detonation processes. In the flows of heterogeneous detonation in gas suspensions of reactive and inert particles the collisions of inert particles do not affect the detonation velocity, cell size, and cellular structures but provide spreading the inert phase layer-type structures in the far zone. The problems of a shock wave passage over a dense layer and of an explosive shock wave interaction with a layer are considered. It is confirmed a weak influence of the Saffman force and a significant effect of the Magnus force in the process of particle lifting from the layer directly behind the shock wave. It is shown that the development of chaotic motion and collisions is also one of the important mechanisms providing the dispersion of particles and formation of the dust clouds in the shock-wave processes.

Introduction

Formation of explosive dust clouds is one of the major risk factors of dust explosions at industrial production related to yielding and use of powders and in coal mines. Processes of particle dispersion under the influence of shock waves have been extensively studied experimentally and theoretically. A fairly detailed review on the problem of mixing under shock wave and detonation processes is presented by Fedorov (2004). The major factors of layer expansions in diluted suspensions are the interactions of shock-wave structures: refractions and reflections, as well as the development of instability in the surface layer, see Borisov et al., 1967, Fedorov et al., 2002. Studies on the motion of single particles in the flow field behind the shock waves showed an influence of the Saffman force on the dust lifting height (Gosteev and Fedorov, 2002). The Saffman force is taken into account in modelling of dust suspensions (Wang et al., 2005, Fan et al., 2007). A determining role of the Magnus force in processes of dust lifting due to development of the rotational particle motion in the shear flow behind the shock wave front was proposed by Boiko and Papyrin, 1987, Klemens et al., 2001, Kiselev and Kiselev, 2001. In the study by Kiselev and Kiselev (2001) the effect of Magnus forces was associated with the particle rotations in the development of their chaotic motion and collisions. Some attempts have been made to estimate the input of interaction between particles in processes of shock wave dust dispersion on the base of different models. Sakakita et al. (1992) studying the behavior of layers used a bulk density model, taking into account the pressure associated with contact interactions of particles. Zydak and Klemens (2007) proposed an empirical model for estimation of particle collisions in the particle phase dynamics and analyzed the influence of this factor. They obtained that the collisional effects may be sufficiently greater than an influence of the Saffman and Magnus forces. Ilea et al. (2008) analyzed group effects of particle-to-particle collisions and collisions with rough solid wall in the problem of a dust layer interaction with a shock wave. In the framework of the Lagrangian approach for description of the dynamics of particles Ilea et al. (2008) obtained pictures of the particle layer dispersion but do not reveal the shock-wave structures inside the layer. Semenov et al. (2013) considered problems of lifting and dispersing of a dust layer behind the propagating shock wave as well as ignition, combustion and dust-layered detonation formation. The model based on the Magnus force provides reliable results at the initial stage but the authors note that the usage of additional models such as intergranular pressure is necessary. Utkilen et al. (2014) performed 3-D calculations of the processes of dust lifting under an action of shock waves in the frame of the turbulent model for gas – particle media. To prevent unrealistic growth of particle volume concentration some kind of “intergranular pressure” as additional force in the particle phase was introduced.

Thus, the role of the random motions of particles and their collisions in the dispersion process and dust cloud formation was not determined and is of interest. Besides, the contact interactions of particles (collisions) can influence the property of dustiness (Klippel et al., 2012), the processes of particle agglomeration (Eckhoff, 2012) as well as on the ignition conditions of dusts.

Khmel and Fedorov (2014) presented a detailed description and analysis of a theoretical model of dense gas particle suspension in which particle-to-particle collisions are described using the molecular-kinetic approaches of theory of granular materials (Goldshtein and Shapiro, 1995, Goldshtein et al., 1996).

In the present paper numerical simulations of some shock wave and detonation flows are performed in the frame of the model by Khmel and Fedorov (2014) which takes into account collisional effects. The purpose of this paper is to estimate the contribution of particle random motion and particle-to-particle collisions as one of the mechanisms resulting dispersion processes in dust suspensions. The numerical technique used earlier by Fedorov and Khmel, 2002, Fedorov and Khmel, 2005 is modified here for an extended range of particle volume concentrations. The problem of propagation of a cellular detonation in a mixture of reactive and inert particles with regard for collisional dynamics of the inert component, as well as the problem of a shock wave interaction with an inert particle layer will be considered.

Section snippets

Physical and mathematical model

The Euler equations for two-dimensional flow of two-phase mixture of gas and two fractions of particles follow from conservation laws for mass, momentum, and energy (the subscripts 1, 2, and 3 stand for gas, inert particles, and reactive particles, respectively): $\frac{\partial W}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} = Γ$ $W = (\begin{array}{l} W_{1} \\ W_{2} \\ W_{3} \end{array}), F = (\begin{array}{l} F_{1} \\ F_{2} \\ F_{3} \end{array}), G = (\begin{array}{l} G_{1} \\ G_{2} \\ G_{3} \end{array}), Г = (\begin{array}{l} - {\tilde{Γ}}_{2} - Г_{3} \\ Г_{2} \\ Г_{3} \end{array})$ $W_{1} = (\begin{array}{l} ρ_{1} \\ ρ_{1} u_{1} \\ ρ_{1} v_{1} \\ ρ_{1} E_{1} \end{array}), F_{1} = (\begin{array}{l} ρ_{1} u_{1} \\ m_{1} p_{1} + ρ_{1} u_{1}^{2} \\ ρ_{1} u_{1} v_{1} \\ ρ_{1} u_{1} E_{1} + m_{1} p_{1} u_{1} \end{array}), G_{1} = (\begin{array}{l} ρ_{1} v_{1} \\ ρ_{1} u_{1} v_{1} \\ m_{1} p_{1} + ρ_{1} v_{1}^{2} \\ ρ_{1} v_{1} E_{1} + m_{1} p_{1} v_{1} \end{array}) .$ $W_{2} = (\begin{array}{l} ρ_{2} \\ ρ_{2} u_{2} \\ ρ_{2} v_{2} \\ ρ_{2} E_{2} \\ ρ_{2} E_{c} \end{array}), F_{2} = (\begin{array}{l} ρ_{2} u_{2} \\ ρ_{2} u_{2}^{2} + m_{2} p_{2} \\ ρ_{2} u_{2} v_{2} \\ ρ_{2} u_{2} E_{2} + m_{2} u_{2} p_{2} \\ ρ_{2} u_{2} E_{c} + η m_{2} u_{2} p_{2} \end{array}$

Collisional effects in detonation waves

In heterogeneous detonations the volume concentration of the reacting particles is small excepting the ρ-layers. To estimate the collisional effects in detonations we consider a detonation wave propagation in a mixture of aluminum particles (d₃ = 2 μm) and oxygen with addition of inert aluminum oxide particles (d₂ = 10 μm). The mechanisms of the cellular heterogeneous detonation development and the properties of its propagation in monodisperse aluminum–oxygen suspensions were discussed by

Shock wave passage over a dust layer surface

The model and numerical method was tested on the problem of interaction of a planar shock wave with a loose dusty bulk layer corresponding to the experimental conditions of Fan et al. (2007). Experiments were carried out for starch particles with additive PbO particles of similar size (data on ratio of their concentrations not shown); the mean particle diameter is about 10 μm. Fig. 5a schematically shows the shock wave shape and the surface of starch particle dispersion in accordance with the

Conclusions

The paper presents a mathematical model of two-phase medium for the description of shock wave processes in gas suspensions of solid particles considering the chaotic particle motion and collisions. Description of the collisional dynamics is based on the molecular-kinetic theory applications to granular media approaches.

On the base of numerical simulation of shock wave and detonation processes taking into account the particle collisional dynamics established the following:

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The effect of

Acknowledgments

The work was supported by the Russian Foundation for Basic Research (grant 13-08-00110).

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View full text

Numerical simulation of dust dispersion using molecular-kinetic model for description of particle-to-particle collisions

Highlights

Abstract

Introduction

Section snippets

Physical and mathematical model

Collisional effects in detonation waves

Shock wave passage over a dust layer surface

Conclusions

Acknowledgments

J. Comput. Phys.

Chem. Eng. Sci.

J. Loss Prev. Process Industries

J. Loss Prev. Process Industries

J. Loss Prev. Process Industries

Dynamics of the formation of a gas suspension behind a shock wave sliding over the surface of a loose material

Combust. Explos. Shock Waves

Instability of the surface of a granular medium behind sliding shock and detonation waves

Combust. Explos. Shock Waves

Influence of dispersibility and coagulation on the dust explosion risk presented by powders consisting of nm-particles

Interaction of a shock wave with a loose dusty bulk layer

Shock Waves

Mixing in wave processes propagating in gas mixtures (review)

Combust. Explos. Shock Waves

Numerical simulation of detonation initiation with a shock wave entering a cloud of aluminum particles

Combust. Explos. Shock Waves

Numerical simulation of formation of cellular heterogeneous detonation of aluminum particles in oxygen

Combust. Explos. Shock Waves

Formation and degeneration of cellular detonation in bi-disperse gas suspension of aluminum particles

Combust. Explos. Shock Waves

Calculation of detonation wave propagation in a gas suspension of aluminum and inert particles

Combust. Explos. Shock Waves