Performance

The question

Lesson 1 promised that a detonation adds heat with less entropy, so the exhaust should leave at a higher total pressure than the gas came in with. Eleven lessons later we have met injectors that throw pressure away, fuel that burns outside the wave, and waves that run slow. Does a rotating detonation engine actually gain pressure?

Measuring something that won't sit still

In a conventional combustor, you measure the total pressure at the inlet and at the outlet, and the ratio is the pressure loss — typically 3–6%. In an RDE the exhaust pulses at kilohertz frequencies, swirls, and varies across the annulus. Averaging the pressure the usual way gives a misleading answer: different averages give different numbers.

Researchers at NASA, notably Daniel Paxson and Tom Kaemming, proposed a cleaner yardstick: the . It is the total pressure of a steady, uniform flow that would give the same ideal thrust through a nozzle as the RDE's real exhaust. It can be inferred from a measured thrust and mass flow. If the EAP exceeds the plenum pressure, the engine has truly gained pressure.

Where the gain leaks away

ExploreThe pressure-gain ledger

Compressor exit temperature T₃700 K

Turbine entry temperature T₄1700 K

Injector stiffness1.60 · 38% lost

Fuel burned in the wave75%

Wave speed85% of D_CJ

Unsteady-exhaust efficiency96.0%

Ideal gain at these temperatures× 1.56biggest when the compressor does little

Net, plenum to exit0.76pressure loss

vs conventional-19%

Worse than a conventional combustor: the losses have eaten the gain.

Challenges0/3

Beat a conventional combustor (net above 0.95).
Achieve true pressure gain with an injector stiffness of at least 1.3.
Find the compressor temperature where the ideal gain is largest.

Follow the total pressure from the plenum to the exit of an RDE used as a gas-turbine combustor. The ideal gain comes from the cycle analysis of Lesson 1 (the same heat added with less entropy); the losses are the ones this course has met along the way. The dashed line is a conventional constant-pressure combustor, which loses about 5%. This is a simple model: it treats the injector loss as the whole cost of getting the mixture to the wave. In the course's solver, the fresh layer expands well below the chamber's average pressure before the wave arrives, so real engines fall shorter than this ledger suggests.

The ledger makes the challenge clear. An ideal detonation multiplies the total pressure by a factor of 1.5 to 3, depending on how much compression came before it and how much heat is added. Then reality takes its share:

Injector stiffness (Lesson 10) costs 20 to 50% before anything burns.
Fuel burned outside the wave — at the contact surface (Lesson 9), or after the wave because it mixed late (Lesson 10) — releases its heat at constant pressure and gains nothing.
Slow waves gain less: a wave at 85% of the CJ speed has a pressure ratio only about 70% as large.
The unsteady, swirling exhaust loses a few percent more as it mixes out.

Most laboratory RDEs measured so far have shown a net total-pressure loss from plenum to exit, though often a smaller one than a conventional combustor with the same injectors would have, and the best designs have approached break-even. Getting across that line — true pressure gain — is the field's most important open problem.

The course's own solver tells the same story. Measure the equivalent available pressure of its exhaust for six engines fed the same supply, differing only in their injectors: the standard engine delivers 48% of its supply pressure, the stiffest 27%, and only the softest injector (stiffness about 1.1, blocked for a fifth of every lap) just breaks even. The detonation itself does its job — it multiplies the pressure of the gas it burns by 1.5 to 2 — but in the standard engine that fresh layer sits at only about a quarter of the supply pressure by the time the wave reaches it, having expanded through the injectors and into the chamber. The pressure is lost before the wave, not in it.

Why waves run slow

The is one of the most robust observations about RDEs: measured waves run at 70–95% of $D_{CJ}$ , sometimes less. The culprits, in rough order:

Imperfect mixing: part of the layer is too rich or too lean to detonate properly (Lesson 10).
Contact burning: part of the fuel has already burned before the wave arrives, and the products dilute the layer (Lesson 9).
Losses through the top of the fill layer: a detonation with no confinement above it loses energy sideways, like one in a narrow tube.
Heat and friction losses to the walls of a channel only a few millimetres wide.
Wave interactions in multi-wave and counter-rotating modes (Lesson 11).

What a detonation is worth to a rocket

Rockets are attracting at least as much RDE work as gas turbines: a rotating detonation rocket engine (RDRE) burns propellants in an annulus at very high pressure and expands the products through an aerospike or a conventional nozzle. NASA, AFRL, Purdue, JAXA and others have fired RDREs on hydrogen, methane and kerosene with oxygen.

ExploreA detonation rocket against a conventional one

Manifold pressure40 bar

RDRE injector stiffness1.50

RDRE effective pressure gain× 1.30

Conventional288.3 sp_c 32 bar

RDRE291.0 seffective p_c 35 bar

Difference+0.9%

Challenges0/2

Find settings where the RDRE beats the conventional engine by more than 3% in Isp.
In vacuum, give the RDRE a 1.9× pressure gain and see how little its Isp rises.

Specific impulse of an ideal rocket against chamber pressure, from the rocket course's propellant model — at sea level with the nozzle expanded to 1 atm, or in vacuum with a fixed nozzle. Both engines are fed from the same manifold pressure. A conventional thrust chamber loses about 20% across its injector; an RDRE loses more across its injector but may gain some of it back in the detonation. In vacuum the curve is almost flat; at sea level, pressure buys expansion, and expansion buys Isp.

The plot holds a sobering lesson. A rocket's specific impulse depends only weakly on chamber pressure — especially in vacuum, where a fixed nozzle extracts nearly the same velocity from the gas whatever its pressure. So even a sizeable effective pressure gain buys only a few percent of Isp at sea level, and fractions of a percent in vacuum. The more compelling case for RDREs is often elsewhere: the chamber can be very short — a few centimetres instead of tens — because a detonation releases the heat in microseconds rather than milliseconds, which saves mass and cooling area.

Estimate first

A vacuum rocket engine with a fixed nozzle area ratio gains 30% in effective chamber pressure. By roughly how many percent does its specific impulse rise?

0.0100.10110100

Drag along the line to place your guess — each tick is ten times the last.

Check your intuition

Which change would most directly raise the fraction of fuel that releases its energy inside the detonation wave?

Recap

What to carry forward

Pressure gain is measured with the equivalent available pressure: the steady total pressure giving the same ideal thrust.
The ideal gain (×1.5–3) is eaten by injector stiffness, fuel burned outside the wave, slow waves and the unsteady exhaust; most measured RDEs show a small net loss.
Waves run at 70–95% of $D_{CJ}$ because of mixing, contact burning, lateral relief, wall losses and wave interactions.
For rockets, Isp barely depends on pressure; the RDRE's bigger promise is a shorter, lighter chamber.

Next: where rotating detonation engines will fly first — and what still stands in the way.