Dimensional scaling of flame propagation in discrete particulate clouds

Abstract

The critical dimension necessary for a flame to propagate in suspensions of fuel particles in oxidiser is studied analytically and numerically. Two types of models are considered: First, a continuum model, wherein the individual particulate sources are not resolved and the heat release is assumed spatially uniform, is solved via conventional finite difference techniques. Second, a discrete source model, wherein the heat diffusion from individual sources is modelled via superposition of the Green's function of each source, is employed to examine the influence of the random, discrete nature of the media. Heat transfer to cold, isothermal walls and to a layer of inert gas surrounding the reactive medium are considered as the loss mechanisms. Both cylindrical and rectangular (slab) geometries of the reactive medium are considered, and the flame speed is measured as a function of the diameter and thickness of the domains, respectively. In the continuum model with inert gas confinement, a universal scaling of critical diameter to critical thickness near 2:1 is found. In the discrete source model, as the time scale of heat release of the sources is made small compared to the interparticle diffusion time, the geometric scaling between cylinders and slabs exhibits values greater than 2:1. The ability of the flame in the discrete regime to propagate in thinner slabs than predicted by continuum scaling is attributed to the flame being able to exploit local fluctuations in concentration across the slab to sustain propagation. As the heat release time of the sources is increased, the discrete source model reverts back to results consistent with the continuum model. Implications of these results for experiments are discussed.

Keywords:

• metal particle combustion
• flame propagation limit
• percolation
• heterogeneous combustion
• critical phenomena
• statistical phenomena
• geometric scaling

Acknowledgements

Computing resources used in this work were provided by Compute Canada. The authors are grateful to F.D. Tang and C. Wagner for their contributions to the early stages of this study, and thank S. Goroshin, J.M. Bergthorson, and J. Palečka for valuable discussions and feedback.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project was supported under the Canadian Space Agency – Flights and Fieldwork for the Advancement of Science and Technology (FAST) funded project, “Sounding Rocket Flight to Explore Percolating Reactive Waves.”