Fire triangle: Heat, Fuel, Oxygen. That covers both starting a fire, and keeping one alive. Once a fire starts, usually its own heat covers the heat requirement.
A strong wind will cool down the fire. Strong enough, and it will put it out… Mostly. There might be smouldering in places the wind can’t easily reach. When a candle is “out”, but glowing slightly and producing a ton of smoke, that’s it smouldering.
The heat of the flame causes flammable gasses to be released by whatever is burning, for example vapourised candle wax. This gas is then burned and forms the flame itself. A strong enough wind physically disperses the gas before it can burn, eliminating the flame. A big fire like a campfire will have enough heat to reignite the gasses after a brief gust, whereas a candle does not typically have enough heat.
There is a neat trick where you can relight a candle immediately after blowing it out by touching a match to the stream of smoke; this smoke contains non-ignited flammable gas that is still coming off the wick.
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