To start off, you need to know the difference between a detonation and a deflagration. Deflagration is basically when the energy released from combustion doesn’t form a shockwave because it’s not burning faster than the speed of sound. A detonation on the other hand creates a shockwave because it’s moving faster than the speed of sound.
A real world example would be dumping gasoline on the ground and lighting it on fire. It doesn’t form a shockwave, and you can see the fire spread over the gasoline. A detonation on the other hand would be like when the Mythbusters exploded a cement truck. You can see a shockwave form.
So what’s the practical difference for rocket engines? Well, a rocket engine’s efficiency is partially determined by how fast it can throw the exhaust out the back. It’s why ion engines are so much more efficient than chemical engines. All chemical engines today use deflagration, and have a slower exhaust speed. Ion engine “exhaust” (the ions being thrown out the back) move insanely fast. A detonation engine by definition would have much faster exhaust, and therefore much more efficiency.
It’s physically impossible to continuously feed a combustion chamber with fuel/oxidizer fast enough to keep up with a detonation. Not to mention that constantly pounding your rocket with repeated high acceleration/no acceleration cycles would likely be detrimental to most payloads, especially humans.
So how do they get it to work? If you look at detonation engines today, you will basically see a ring of nozzles. Typically, 2 opposite nozzles are detonating and the others are recharging fuel/oxidizer or waiting to detonate. They will basically detonate one after the other, circling the ring. This creates basically a constant “wave” of detonations and solves the issue of shaking the rocket and its payload to pieces.
Here’s a brilliant video that dives into it:
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