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The SABRE engine is designed to power the Skylon SSTO vehicle. It reduces the amount of oxygen that needs to be carried by cooling and compressing air in the early part of the flight. More hydrogen is needed for cooling than is used in the engine, so it is sent to ramjets surrounding the main nozzle. Presumably the ramjets would generate no thrust at engine start.

Why have ramjets been used? Wouldn't it be more efficient to use slush hydrogen and condense the hydrogen for use later in the flight?

Alternatively, wouldn't it be more efficient to design the SABRE as an air-augmented rocket? Wouldn't that mean that the spare hydrogen would generate thrust from engine start? Could it also improve specific impulse after transition to rocket mode?

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At engine start the ramjets would provide no thrust - but at this stage of flight there are not excessive cooling needs, so there's no "excess hydrogen" to burn in them anyway.

In the middle of the flight regime the air intake is decelerating more air than is needed, by reheating this extra air with the excess hydrogen in the ramjet burners, this extra ram drag is mitigated.

At the end of the flight, when the incoming air is hottest almost no air is available for the bypass burners so they again produce little thrust.

There is a happy coincidence that the conditions under which a ramjet are most effective are also the conditions that the SABRE nacelles have spare hydrogen and air.

Using slush hydrogen (directly) would reduce the amount of fuel needed to cool the incoming air, and so reduce the wastage at the final part of the flight, but potentially removing the advantage given by the ramjets in the middle part of the flight.

If slush hydrogen was used as you suggest it would increase the system complexity - hot hydrogen would have to be brought back into the tank area, and held in tanks which would need to be able to cope with increasing pressure throughout the flight.

Using slush hydrogen would increase the complexity of the fuelling - One of the Skylon design goals is to have simple, fast and automated fuelling procedures (40 minutes to connect, fill and disconnect). It would add additional untested technologies into the mix another Skylon design goal is to minimise development risk by using well understood techniques and materials - at least for the first batch.

For solid fuel, an air augmented rocket has about double the ISP. Apparently a study into an air augmented LH2/LOx engine expected an ISP of around 830 - but I haven't found a source I trust for that. SABRE is expected to deliver an ISP peaking at 3600 in it's air-breathing phase

A further reason for there being little benefit in using slush hydrogen is the latent heat of fusion is low - about 58 j/g and the specific heat capacity of liquid hydrogen below the planned temperature 18K is also low - about 8j/gK, so using slush hydrogen would gain around 80j/g. When compared to 447j/g for the latent heat of fission and the 10-15 j/gK SHC over the 18-500K operation range of the precooler giving around 6400j/g using slush hydrogen would represent a saving of around 1.25% of the wasted fuel at the high altitude & Mach portion of the flight.

OrangePeel52 rightly points out hydrogen embrittlement is an issue; to minimise it in the SABRE design helium is used as a working fluid in a "loop" to cool the incoming air but is itself in turn cooled (and effectively compressed in a Brayton cycle) by the hydrogen, so it is the hydrogen the heat is dumped into.

Ultimately, they've designed the engine this way because they think it's the most achievable solution to the problem they have in mind. Other engines may be better for other scenarios.

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