A quantitative theory for heterogeneous combustion of nonvolatile metal particles in the diffusion-limited regime

TL;DR

This work develops a quantitative analytical theory for the diffusion-limited heterogeneous combustion of nonvolatile metal particles, incorporating Stefan flow and surface chemisorption on a spherical particle. Closed-form expressions for burn time and transient particle temperature are derived from gas-phase conservation and particle enthalpy balance, with coupling achieved through film-layer transport properties and an iterative procedure. Application to micron-sized iron particles shows strong agreement with experiments for burn times and reasonable agreement for peak temperatures at low oxygen, while deviations at higher oxygen are attributed to neglected evaporation and potential regime transitions, indicating the model’s validity within its stated assumptions. The framework provides clear scaling laws (e.g., $t_b \propto d_0^2$) and a robust basis for extending analytical treatment of diffusion-limited, nonvolatile particle combustion to broader metals and operating conditions.

Abstract

The paper presents an analytical theory quantitatively describing the heterogeneous combustion of nonvolatile (metal) particles in the diffusion-limited regime. It is assumed that the particle is suspended in an unconfined, isobaric, quiescent gaseous mixture and the chemisorption of the oxygen takes place evenly on the particle surface. The exact solution of the particle burn time is derived from the conservation equations of the gas-phase described in a spherical coordinate system with the utilization of constant thermophysical properties, evaluated at a reference film layer. This solution inherently takes the Stefan flow into account. The approximate expression of the time-dependent particle temperature is solved from the conservation of the particle enthalpy by neglecting the higher order terms in the Taylor expansion of the product of the transient particle density and diameter squared. Coupling the solutions for the burn time and time-dependent particle temperature provides quantitative results when initial and boundary conditions are specified. The theory is employed to predict the burn time and temperature of micro-sized iron particles, which are then compared with measurements, as the first validation case. The theoretical burn time agrees with the experiments almost perfectly at both low and high oxygen levels. The calculated particle temperature matches the measurements fairly well at relatively low oxygen mole fractions, whereas the theory overpredict the particle peak temperature due to the neglect of evaporation and the possible transition of the combustion regime.

Figures (9)

• Figure 1: Schematic illustration of the combustion process for a single, non-volatile metal particle in the diffusion-limited regime.
• Figure 2: Calculation procedure for obtaining the quantitative solution of the burn time and temperature history .
• Figure 3: Theoretical temperature history and fraction of unburned iron during the combustion of a $40\,$µ m and a $60\,$µ m iron particles burning in the $21\%\text{O}_2/79\%\text{N}_2$ mixture at $T_\mathrm{g,\infty}=300\,$K and $P_{\infty}=1\,$ atm, (a) without radiation and (b) with radiation ($\varepsilon = 0.7$).
• Figure 4: Scaled temperature history and fraction of unburned iron during the combustion of a $40\,$µ m and a $60\,$µ m iron particles burning at the room condition: $21\%\text{O}_2\slash79\%\text{N}_2$, $T_\mathrm{g,\infty}=300\,$K, and $P_\mathrm{\infty}=1\,$atm, (a) without radiation and (b) with radiation ($\varepsilon = 0.7$).
• Figure 5: Time histories of temperature for a $50\,$µ m particle when burning at different oxygen mole fractions and ambient pressures. The oxygen mole fraction of 14% is used to limit the maximum particle temperature below 2500K at the ambient temperature of 900K.