Influence of inert sprays on extinction of premixed flames propagating in a duct with varying cross-sectional area

The extinction of a premixed flame propagating in a duct with varying cross-sectional area under the influences of flame stretch, preferential diffusion, and water sprays is analyzed using activation energy asymptotics. A completely prevaporized mode and a partially prevaporized mode are identified on the basis of a critical value of the initial droplet size (r′ _c) for completing the vaporization process at the flame. The liquid loading and droplet size of the water spray induces internal heat loss through liquid vaporization. It is found that the burning intensity of a flame with water sprays is reduced with increasing liquid loading or decreasing initial droplet size, independent of the value of Lewis number (Le). Small droplets can absorb heat from the flame more effectively than large ones. However, it is not beneficial to reduce droplet size below the critical initial droplet size. The negative (or positive) stretch weakens (or strengthens) a Le < 1 flame but intensifies (or reduces) a Le > 1 flame. A positively stretched flame with Le > 1 or a negatively stretched flame with Le < 1 can be extinguished by increasing the effect of stretch, and this behavior is characterized by a C-shaped extinction curve. For a Le < 1 flame enduring positive stretch or a Le > 1 flame experiencing negative stretch, extinction does not occur for the completely prevaporized mode. However, for the partially prevaporized mode, the S-shaped extinction curve occurs if the liquid-loading water is large enough and the droplet size is also sufficiently large. Note that extinction characterized by a C-shaped curve is dominated by flame stretch; whereas extinction, characterized by an S-shaped curve, is governed by internal heat loss.