I want to make a burnt edge on the jet engine. How do you do that? What colors should I use? Any other alternatives other than charcoal?

I'm not exactly sure what you're aiming for, but powdered charcoal applied with a cotton swab or small brush works well. Some folks airbrush flat black, but I find that treatment too hard to control. The important thing to remember about paint effects is that when you reach the point where one more application will do, stop.

The engine afterburner cans do not smoke to my knowledge.
Get some reference material before attempting to do this.
The jet exhaust is too hot for smoke to build up.

If you really wanted to , you could get some soot from a candle and apply with a brush or q-tip

Were you going for the black chalk effect, because I'd lean towards the burnt-iron look on jet engines. If you've ever seen a soldering-iron before and after it's first use, that's kinda how an engine looks.