A spray flame propagating in a nonadiabatic duct with varying cross-sectional area

The influence of flame stretch, preferential diffusion, internal heat transfer, and external heat loss on the extinction of dilute spray flames propagating in a nonadiabatic duct with varying cross-sectional area is analyzed using activation energy asymptotics. A completely prevaporized mode and a partially prevaporized mode of flame propagation are identified. Internal heat transfer, resulting from droplets gasifying, varies with the liquid-fuel loading and the initial droplet size in the spray and also provides internal heat loss for rich sprays but heat gain for lean sprays. A spray flame propagating in a divergent (convergent) duct experiences positive (negative) stretch. The results show that the burning intensity of a lean (or rich) spray is enhanced (or reduced) with an increased liquid-fuel loading or smaller initial droplets. The positive stretch coupled with the effects of the Lewis number (Le) weakens a lean methanol-spray flame (Le>1Le>1), but intensifies a rich methanol-spray flame (Le<1Le<1). For a positively stretched flame with Le<1Le<1 or a negatively stretched flame with Le>1Le>1, without external heat loss, no extinction occurs by increasing the stretch. However, irrespective of heat loss, a flame with Le>1Le>1 experiencing positive stretch or a flame with Le<1Le<1 enduring negative stretch can be extinguished by increasing the stretch. Flame extinction characterized by a C-shaped curve is dominated by stretch or external heat loss. Note that for a methanol-rich spray flame (Le<1Le<1) experiencing positive stretch and enduring a partially prevaporized spray composed of a large enough liquid loading and sufficiently large droplets, an S-shaped extinction curve can be obtained. The S-shaped curve, which differs from the C-shaped one, indicates that flame extinction is governed by internal heat loss.