A population balance approach for predicting the size distribution of oxide smoke near a burning aluminum particle

In this article, we present a fully Eulerian framework for modelling the envelope flame around a burning, spatially resolved aluminum particle, including detailed descriptions of the particle's reactive surface, the gas phase composition and the polysized oxide droplets condensing from gaseous precursors. The size distribution of this oxide smoke influences deposition and emission mechanisms and plays a decisive role in the design of oxide particle extraction techniques, for example, as part of the exhaust fume treatment for metal dust flames. In order to describe the reacting gas-droplet dispersion, we combine the population balance equation governing the size distribution of oxide smoke fines with the balance equations for gas phase species and the dispersion's mass, momentum and enthalpy. In this regard, the droplets' bulk velocity and temperature are assumed to equilibrate instantaneously with those of the carrier gas. Additionally, a detailed kinetic framework governing the rates at which gas phase and surface reactions as well as droplet nucleation, growth/dissociation and coagulation occur is augmented by the transport parameters governing species differential diffusion, droplet diffusion and thermophoresis. A major novelty of our physical model lies with the prediction of the oxide smoke size distribution at every location in the flow domain, the accommodation of size-sensitive kinetic and transport processes as well as the prediction of possible NOx pollutants. Considering the steady combustion of a single aluminum particle in an O2/Ar mixture, the droplet formation and interaction kinetics are adjusted in such a way that fair agreement with existing experimental measurements is achieved. This calibration shows not only that the interplay of nucleation and dissociation affects the envelope flame through the chemical equilibrium between aluminum suboxides and oxide smoke, but also that the shape of the smoke halo is mainly determined by the droplets' sizes and their diffusional mobility. Concomitantly, we are led to the conclusion that coagulation does not occur in the flame zone, possibly due to electrostatic droplet repulsion induced by thermionic emissions and ionization reactions. Following the consolidation of the droplet formation kinetics, the combustion of an aluminum particle in air is investigated as validation case. For the consolidated droplet formation kinetics, the predicted radial AlO and oxide smoke profiles compare qualitatively very well with the available experimental data, while deviations in the temperature profile remain.