One thing I didn't understand until recently was why there was any interest in detonation engines for rockets -- those being engines where the fuel is not burned smoothly but rather mixed and then detonated. I'd heard of the idea before, but it seemed such a bizarre concept that I didn't pay it much attention. Yet there is considerable interest in such engines; they're currently making headlines as a new, promising way to increase the specific impulse of rockets, and well-funded teams have been working on them and reporting successes. Specific impulse ($$I_{sp}$$) is the most important figure of merit there is for a rocket engine, and increases of 15% or even 25% are talked of. The concept is not really new: they've been talked about since the 1950s, and have been promising (and not delivering) for all that time. On looking into the concept, it seems to me that even the theory is in error here, and that scarcely any improvement is in fact possible.

The idea behind those promised increases in $$I_{sp}$$ isn't that combustion is any more efficient in a detonation. It's not; conventional rocket engines achieve nearly complete combustion (in the sense that the burning in the combustion chamber comes close to achieving thermal equilibrium).

The idea also isn't that detonation velocities are greater than rocket exhaust velocities. That is true for many explosives, but irrelevant: detonation velocity is the speed at which a detonation wave sweeps through an explosive; the material in front of and behind the wave moves much more slowly. It's the average speed of the material which matters for rocketry.

The idea also isn't that with detonation you can do without a nozzle. A conventional rocket nozzle is already nearly optimal at converting heat energy into thrust, so detonation engines still have nozzles of the same sort -- at least detonation engines of the sort that are being touted for their $$I_{sp}$$ do. People have built simple detonation engines that omit the nozzle, but their efficiencies could be improved by adding one. On an even more primitive level, if you have a thin layer of explosive on a surface and detonate it, you'll get mainly velocity perpendicular to the surface. But that isn't the idea either: it could perhaps be an improvement on grounds of simplicity (at least until you wonder how to keep spreading a new layer), but not on overall efficiency.

Instead, the promised $$I_{sp}$$ increases come from comparing thermodynamic cycles. Heat engines can be thought of as being instances of abstract themodynamic cycles, and some of those cycles have higher efficiencies than others. Jet engines, for instance, are commonly described as following the Brayton cycle. In that cycle, air is first compressed adiabatically (that is, without heat flowing into it or out from it, though the air does heat up just from being compressed). Then in the next part of the cycle, heat is added at constant pressure. (In jet engines, this is done by injecting fuel and burning it.) In the next part the now-heated air is expanded adiabatically, lowering its temperature while extracting mechanical work from it. In jet engines, this happens first in a turbine (which powers the compressor, and in modern jet engines also powers the ducted fan that produces most of the thrust) and then continues in the exhaust nozzle (in which the work being produced goes into shooting the exhaust out the rear of the engine).

That's the last part of the cycle that actually happens in a jet engine, but a theoretical cycle has to be a literal cycle that returns the engine to its starting state, so has one additional part: the still-somewhat-hot exhaust then gets cooled down to its starting temperature, again at constant pressure. Actual jet engines, of course, take in fresh air instead of recycling it; thermodynamically this would not matter if their exhaust were at ambient pressure, but of course generally it isn't. They also of course differ from the ideal cycle in the usual ways: there are minor air leaks, minor leakages of heat, and so forth; there is also the energy required to inject the jet fuel. But classing them as "Brayton cycle" devices is close enough to be useful.

Now, suppose we alter that cycle to incorporate detonation. Detonation is so fast that it can be regarded as constant-volume combustion. The first part of the cycle is still adiabatic compression, but then instead of the air-fuel mix being burned at a constant pressure it's burned at a constant volume. After that the rest of the cycle is the same. This is called the Humphrey cycle. It saves on compression work: instead of the intake air being compressed mechanically all the way to the engine's maximum pressure in the first stage, it's compressed much less than that and then combustion does the rest of the compressing. This means the cycle is more efficient.

So for jet engines the theory indeed says that there are efficiency gains possible, though one may doubt the practicality of it. With the detonation giving a nearly instant pressure rise, one would presumably need a valve to prevent backflow into the compressor. To prevent air from piling up on the compressor side of that valve when closed, it would perhaps be best to have multiple combustion chambers, with at least one of the valves leading into them open at any one time. And then there's the question of how the pressure in each combustion chamber is to be lowered enough that fresh air can flow in. Putting another valve on the output side of the chamber seems necessary to stop backflow during the times when the pressure is lower than the pressure feeding the turbine. But by itself it won't be enough, unless there is low pressure on the turbine side of that valve -- which there couldn't be if the turbine is to be driven by the higher post-combustion pressure. So perhaps the way to lower the pressure would be to have a piston in each combustion chamber, and lower it mechanically. This would produce work, to be added to the work produced by the turbine. And since we'd have to be returning the piston to its starting point, we might as well use that to do some of the compression.

We've just reinvented the turbocompounded piston engine, such as was in general use just prior to jet engines. The airliners were glad to get rid of those monsters, since they were maintenance nightmares, but they were the mainstay of the business for years. Perhaps with today's computer controls and today's emphasis on efficiency their time will come again, this time with the engine's power used to drive a ducted fan rather than an open propeller. Detonation, though, likely wouldn't make it in: once you have valves there's no need for detonation, and detonation is well known for wrecking piston engines.

Or maybe there's some alternate valveless scheme via which a detonation wave is cleverly reflected so that it mostly fails to make it back into the compressor, and is followed by a rarefaction wave which lowers pressure enough that fresh air from the compressor can enter... and maybe this can be done without creating so much noise that passengers are buzzed to death. In any case, with jet engines there are definite possibilities for superior thermodynamic cycles, even if it means reverting to old-fashioned designs.

Rocket engines, though, are a different matter. They also are spoken of as using the Brayton cycle, but whereas jet engines really incorporate only three of the four parts of the cycle, rocket engines incorporate only two. Instead of compressing intake air they inject liquid fuel. Liquid fuel is much denser than air, so the amount of power used to pump it is much less. (Neglecting inefficiencies, the power needed to drive a pump is the pressure the pump produces multiplied by the volume per second that is pumped.) The heat engine aspect of compression, where a highly compressible gas soaks up energy and heats up as it's compressed, and later needs to be persuaded to disgorge that energy in a productive manner, is also nearly absent, since liquids are much less compressible.

When comparing jet engine cycles it's common to neglect the work needed to inject the jet fuel, since it's so much less than the energy needed to compress the intake air; in rocketry, not only the fuel but the oxidizer injection is in that "negligible" category. The reason the Humphrey cycle would be more efficient for jet engines is that it saves work on compression by having the combustion do part of the compression; when work on compression is negligible to begin with, so are the potential savings.

There is a fine point here. With high pressure rocket engines, the amount of fuel that is burned to drive the pumps can be significant. (Rocket engines often operate at much higher pressures than jet engines do: for instance the Space Shuttle Main Engine operates at 3000 psi chamber pressure, about 200 times atmospheric pressure.) This is why high pressure rocket engines tend to be of the "staged combustion" variety, where the exhaust from the preburners which power the turbopumps gets fed into the main combustion chamber rather than dumped overboard. But this is because the process of pumping is quite inefficient: to power the pumps you normally use a far off-stoichiometric mix so that your turbines don't melt, and then both turbines and pumps have major inefficiencies, and then downstream of the pump some energy gets wasted in fluid friction. Those inefficiencies multiply together, so the amount of propellant needed to power the pumps is substantial -- and if you dump it overboard after the preburner, you lose all its reaction mass and much of its energy.

But if you use staged combustion (or the expander cycle, which also passes all the propellant mass through the main combustion chamber), none of these pumping inefficiencies matter: the exhaust from the far off-stoichometric mix goes into the chamber and finishes burning, and almost all the waste heat from the other inefficiencies ends up in the chamber where it makes the chamber hotter just as it would if the combustion had occurred there. So the only energy you really lose is the actual pressurization energy (P*V). Forcing a liter of kerosene into a chamber at a pressure of 300 bar takes 30 kilojoules, and burning it provides about as many megajoules. Even though you also have to inject about twice the volume of liquid oxygen, that's not a big energy loss. Maybe it gets up to half a percent, if you also allow for the combustion energy being lower than the usual figure due to the fact that even main chamber combustion is somewhat fuel-rich. For a half percent energy loss, the $$I_{sp}$$ loss is a quarter of a percent, because exhaust velocity scales as the square root of energy.

That quarter of a percent is what a detonation-engine scheme for rockets is trying to eke savings out of. If you want the number to be higher, run it for liquid hydrogen, the least dense liquid propellant; but even then it's not going to be much higher. That, and not the advertised 15% or 25%, is the real maximum savings.

And to get those tiny savings is a task of huge difficulty. There's a reason that the military fills artillery shells with compounds that detonate: they do an excellent job of destroying things. A detonation engine would have to be very heavily built; yet in rocketry light weight is of prime importance. Combustion instability is normally regarded as an engine-wrecking event to be avoided; here extreme combustion instability would be normal operation which the engine would have to be beefed up to withstand. The simpler sort of detonation engines also suffer from a low duty cycle: the chamber fills with propellant, it is detonated, and then there's a wait while the chamber empties to where propellants can be injected again -- because in order to get the thermodynamic savings, the injection pressure needs to be considerably lower than the pressure after the detonation. So the engine waits until the chamber pressure drops before injecting the next batch of propellants. This low duty cycle further lowers the thrust-to-weight ratio.

The sort of detonation engine that is currently making headlines is the rotating detonation engine. It attempts to solve the problem of low duty cycle. In it, the combustion chamber is ring-shaped; a detonation wave repeatedly and continuously circles the top of the ring, with fresh fuel being injected behind the detonation front so that before it has returned there is enough fresh fuel to detonate again. From the top of the ring, exhaust gases proceed downwards until they join together at the nozzle entrance. (For those who'd like help picturing this, Scott Manley has a good video). Yet this raises the question: since the combustion chamber pressure isn't to be lowered by waiting, how is it to be lowered?

Getting a detonation wave to reliably propagate in an engine like this is difficult. People have achieved such propagation, but it's something that they feel entitled to boast about. They have had to surmount several difficulties, the first of which is starting it in the first place. Just igniting it with a spark, for instance, can result in two waves propagating in opposite directions until they collide on the other side of the ring and extinguish each other. Another issue is that after a detonation has passed, the combustion products can be hot and chemically active enough to ignite the fresh propellants that are being injected, resulting in them burning right away rather than accumulating to be detonated. The detonation wave may also self-extinguish due to random turbulence; or multiple detonation waves may become active at the same time, chasing each other around the ring.

But the problem isn't just getting a reliable detonation wave; it's getting a reliable detonation wave while achieving thermodynamic advantage. That can't be done by injecting fuel in the way a normal rocket engine does, where the pressure inside the injector manifold is higher than the combustion chamber pressure. Instead, the pressure inside the injector manifold would need to be significantly lower than the combustion chamber pressure as seen by the nozzle.

That last qualifier deserves explanation. In a normal rocket engine the combustion chamber pressure is a well-defined quantity: the chamber is barrel-shaped and the pressure inside it is relatively uniform, the main non-uniformity being that as gas accelerates towards the nozzle its pressure decreases as its energy is turned into kinetic energy. Rotating detonation engines are different: the pressure fluctuates tremendously at the top of the combustion chamber, where the detonation is flashing around and around the ring. But as the combustion gases proceed down toward the nozzle and converge there, the output of the whole ring gets averaged together, so that the nozzle sees a pressure that doesn't fluctuate much. It's that average pressure that governs the nozzle efficiency and thus the $$I_{sp}$$.

And that's the pressure which the pressure in the injector manifold must be less than, to get the promised thermodynamic savings. Then, and only then, is the engine making good use of the increase in pressure from constant-volume combustion. Increase in pressure due to constant-volume combustion can vary considerably depending on starting temperature, which propellants are used, and so forth; but it might be a factor of three here. So to get the full benefit, you might be trying for injector manifold pressure of about a third of the chamber pressure as seen by the nozzle -- not "an excess of a third", as might be used in a normal rocket engine, but simply a third.

This is not necessarily impossible. Where there are pressure waves bouncing around, there are generally rarefaction waves too. Where there are extreme pressure waves (as in detonations), there can be strong rarefaction waves (though rarefaction is limited: pressure can't go below zero). Putting a tiny check valve inside each injector orifice might work, by preventing backflow the majority of the time and then letting propellant through when a strong rarefaction wave passed by. But getting such a scheme to work, and work reliably, would at the very least add considerable difficulty to what is already a difficult problem. When the upside would be a quarter of a percent performance increase, there's no way it's worth it, especially when any resulting engine would be much heavier than a normal one.

Now from a practical point of view, the people who are trying to build rotary detonation engines have probably done the computations for how much energy it takes to inject fuel as rockets usually do (that is, with injector manifold pressure higher than average combustion chamber pressure), found that it's insignificant, and thus have not even started to consider this sort of crazy scheme of saving power by lowering injector manifold pressure. Which is a correct approach, as far as it goes; but an even more correct approach would be to realize that these meager savings are where the purported thermodynamic gains come from, and thus that the whole enterprise of detonation engines for rockets just amounts to playing with loud noises.

(Thanks to participants on the arocket mailing list, particularly Jim Davis and Henry Spencer, for first telling me what the idea is as to why detonation engines would be more efficient, and then defending that idea against my questioning attack, helping me make the attack more thorough and understandable. But any errors in the above are all my own.)