Predicting extinction limits of concurrent smoldering spread by a reduced analytical model

Smoldering is a flameless combustion mode occurring on the surface of charring fuels, such as wood and cigarettes. Although the smoldering process is slow and has a low temperature compared to flaming, it is easy to be initiated by a weak heat source and persists under poor oxygen conditions. Extensive work has been done for flame extinction to develop scaling models to predict the limiting oxygen concentration (LOC), but limited work is available for the smoldering extinction. This study develops a reduced analytical model to predict the extinction limits of smoldering. The model simultaneously solves smoldering propagation rate, surface temperature, and surface oxygen mass fraction as part of the solutions. The extinction limit is determined as the critical condition where solutions satisfying all governing equations cease to exist. The model provides a qualitative description and captures the essential characteristics of a previous experiment. The smoldering rate decreases with increasing fuel diameter, and a larger-diameter fuel is easier to extinguish. The mechanisms of the extinction process are investigated, showing the dominant role of radiative heat loss in the smothering limit at low airflow velocities and convective heat loss near the blowoff limit at high airflow velocities. Further analysis of the effect of oxygen concentration shows an increasing trend of LOC with fuel diameter, and the smothering branch cannot be predicted without considering the heat loss through radiation from the solid surface. Novelty and Significance Statement: The novelty of this research is the prediction of smoldering propagation rates and extinction limits across various fuel diameters using a reduced analytical model. The model is modified to accommodate a thin fuel configuration and high airflow velocity condition, where convective heat and mass transfer play a more important role. The model's significance is that it not only provides essential insights into the mechanisms but also has the capability to simultaneously determine key parameters such as spread rate, reaction temperatures, and surface oxygen concentration as part of the solutions. Additionally, this study demonstrates the model's ability to reproduce the limit conditions, including smothering/blowoff limits and limiting oxygen concentration (LOC).