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.

中文翻译：

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离散颗粒云中火焰传播的尺寸标度

分析和数值研究了火焰在氧化剂中的燃料颗粒悬浮液中传播所需的临界尺寸。考虑了两种类型的模型：首先，通过传统的有限差分技术求解连续模型，其中单个颗粒源没有被解析并且热释放被假定为空间均匀的。其次，离散源模型，其中来自单个源的热扩散通过每个源的格林函数的叠加建模，用于检查介质的随机、离散性质的影响。热传递到冷的等温壁和反应介质周围的惰性气体层被认为是损失机制。考虑了反应介质的圆柱形和矩形（平板）几何形状，火焰速度分别作为区域直径和厚度的函数进行测量。在具有惰性气体限制的连续介质模型中，发现临界直径与临界厚度的比例接近 2:1。在离散源模型中，由于与粒子间扩散时间相比，源放热的时间尺度变小，圆柱和平板之间的几何尺度显示值大于 2:1。火焰在离散状态下在比连续标度预测的更薄的板中传播的能力归因于火焰能够利用整个板中浓度的局部波动来维持传播。随着源的热释放时间增加，离散源模型恢复到与连续模型一致的结果。