Steady Smoldering of Fuel Rods: Relationship Between Propagation Velocity and Fume Thickness on Schlieren Photographs

The steady smoldering of rod-shaped fuels, a traditional Chinese disinfection and pest control technique, presents unique challenges in theoretical modeling. Conventional analytical approaches based on energy and mass conservation equations form an underdetermined system, failing to uniquely resolve three critical parameters: temperature field, char morphology, and propagation velocity. This study establishes a quantitative relationship between smoldering propagation velocity and smoke schlieren thickness through integrated experimental and theoretical methodologies. Systematic experiments were conducted on vertically oriented fuel rods (upward and downward configurations), measuring propagation velocity, char cone geometries, and schlieren photographs. By incorporating surface oxidation kinetics and oxygen transport mechanisms into a theoretical model, we revealed an inverse proportionality between propagation velocity and schlieren thickness, thereby introducing a third constraint to resolve the system. Comparative analysis demonstrated excellent agreement between calculated and measured velocities for downward smoldering, with deviations below 20% for biomass rods and 60% for commercial incense rods. Significant discrepancies in upward smoldering were attributed to smoke plume entrainment effects. This work enhances the mechanistic understanding of smoldering propagation dynamics in anisotropic fuel systems.