A simple theoretical model for the onset of flame front inversion for flames propagating in closed or half-open ducts

The present work aims to develop a simple theoretical model to study the propagation of flames in closed or halfopen ducts in the early stages of the process including the onset of flame front inversion. It is known from previous studies that flame propagation in these configurations is characterized by four stages, namely the spherical flame stage, the finger flame stage, the flat flame stage, and the flame front inversion stage. The fourth stage is also called the tulip flame stage. The instant at which the flame skirt region touches the side walls of the duct (or tube) is here considered the onset of the flame front inversion phenomenon. In order to accurately represent the early stages of flame propagation and the onset of flame front inversion, a theoretical model was developed that takes into account the effects of compressibility. It was also possible to obtain an analytical solution for this model. The effect of compressibility was incorporated into the model through a parameter that depends on the initial Mach number of the mixture and on the expansion ratio of the flame. The results obtained with the model were compared with experimental results from the literature concerning mixtures of hydrogen, carbon monoxide and air in different compositions and equivalence ratios, in closed and half-open ducts. The model was shown to be adequate to represent the early staged for flames propagating in closed or half-open ducts. The model for determination of the time at which the flame skirt touches the side walls of the duct showed an average value of the relative errors of 1.84% when compared to available experimental data for H2/ CO/air.