Impact of the injector size on the transfer functions of premixed laminar conical flames

The transfer function (FTF) of premixed laminar conical flames submitted to flowrate modulations is investigated over a wide variety of injection conditions. The response of methane/air and propane/air flames for cylindrical injectors of radii R=11, 7, 1.5 and 1.0 mm is examined. The steady flames investigated all feature the same flame tip half-angle α=14.5o, i.e., the same aspect ratio h/R=4, where h is the flame height. When the injector radius R is large compared to the flame thickness δ, the FTF measurements are shown to collapse on the same response curve when they are plotted as a function of the reduced frequency ω*=ωR/(SLcosα), where ω is the forcing angular frequency and SL the laminar burning velocity. When the injector size is reduced and δ/R becomes sizable, additional parameters are needed to fully describe the FTF. The Lewis number and flame temperature are shown to alter the low frequency behavior of the FTF of flames stabilized above small injectors. One of the main features of these flames is FTF gain values exceeding unity, called gain overshoots, at low reduced frequencies. Larger FTF gain overshoots are found as the injector size is reduced or as the flame temperature is reduced. A model accounting for mutual flame interactions and unsteady heat and mass transfer at the flame base is derived for the reduced frequency ω*0 corresponding to the peak FTF gain values. This expression is shown to better match measurements than previous models based on planar flames that only consider unsteady heat and mass transfer between the flame stand-off position at ψ0 and the burner. The main finding is that mutual flame interactions due to interpenetrating diffusion layers and unsteady heat transfer at the flame base both lead to FTF gain values exceeding unity but the former mechanism is largely dominant for the configurations investigated in this study. It is finally suggested that the FTF of flames stabilized over small injectors may be fully described by four dimensionless parameters ω*, α or h/R, ψ0/R and δcos α/R.