# Dimensional scaling of flame propagation in discrete particulate clouds

## TL;DR

This paper investigates how the critical size for flame propagation in particulate clouds depends on geometry and particle discreteness, revealing different scaling laws in continuum versus discrete models.

## Contribution

It introduces a discrete source model for flame propagation in particulate media and compares it with a continuum model, highlighting the influence of particle discreteness on critical dimensions.

## Key findings

- Continuum model shows a universal 2:1 scaling of critical diameter to thickness.
- Discrete model exhibits greater than 2:1 scaling when heat release is rapid.
- Flame propagation in discrete media can occur in thinner slabs than continuum predictions.

## Abstract

The critical dimension necessary for a flame to propagate in suspensions of fuel particles in oxidizer 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 modeled 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.