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Peroxide is apparently a very good propellant, but I can't seem to find any instance of it being used. Has it ever been used and if not, why hasn't it? What's its downsides?

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Hydrogen peroxide has a (perhaps not fully deserved) reputation of being hazardous to work with; it decomposes spontaneously, releasing heat, and the decomposition rate increases with temperature, so it's tricky to store safely. An old usenet discussion thread found on Yarchive discusses the hazards in more detail.

It is used sometimes as a monopropellant (using a catalyst bed for ignition) for small reaction-control thrusters, notably in the X-15 spaceplane and in the Soyuz spacecraft's descent module's control thrusters.

As Organic Marble notes, it's also been used to drive turbopumps in early pumped bipropellant rockets such as the German V-2 and US Redstone, and is still used for this on the Soyuz booster.

It's also been used as an oxidizer for larger bipropellant rockets, such as the UK's kerosene-peroxide Black Arrow.

Kerosene-peroxide has also been proposed as a propellant combination for reusable SSTO launchers; while it doesn't have as good mass-specific impulse as hydrogen/LOX, peroxide is relatively dense, making for a smaller launcher with correspondingly less mass spent on thermal protection, and the lower mass-specific impulse actually has an advantage in delta-v needed to reach orbit.

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Hybrid smallsat launch vehicle technology is currently being developed 100% in Taiwan1 by the Advanced Rocket Research Center along with all control electronics and algorithms. The video below shows a test with four hybrid peroxide oxidizer plus rubber thermoplastic solid engines being both vectored and throttled to implement stable hovering. I think that the plan is to include a much larger hybrid engine in the middle for higher altitude (> 100 km) tests in 2021 with the eventual goal of reaching orbit (potentially N2O + thermoplastic) with these providing attitude control.

1 I've been a fan for years but no affiliation.

From the first video below (note: this is a non-technical translation of a snippet of an interview):

Reporter: The team’s goal is to create a satellite launch vehicle. They’ll direct it into orbit at least 100 kilometers above the Earth’s surface, to carry out scientific experiments.

Wu Tsung-hsin (Advanced Rocket Research Center) I went to the U.S. with this thing and their eyes lit up. You can make a liquid-propellant rocket, no problem, but this is really the first hybrid-propellant rocket in the world.

Reporter: National Chiao Tung University professor Wu Tsung-hsin is known by students as Uncle Rocket. With a team of 15 professional researchers and over 30 undergrads and graduate students, he hopes to make the first 100% Taiwanese rocket. That would let the nation launch its own satellites into space, without assistance from institutions overseas.

Example of a 1200 N H2O2 + thermoplastic closed-loop throttling and re-ignition engine test

Example of a 10 kN Isp 224 s (sea level) N2O + thermoplastic engine test

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    $\begingroup$ Burt Rutan is sad to hear that his hybrid motor that took humans to space does not count. " this is really the first hybrid-propellant rocket in the world." $\endgroup$ Dec 19 '20 at 4:07
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    $\begingroup$ @OrganicMarble I'll add a note that the translation by the non-technical reporter may not be accurate. Translations between Chinese and English are fraught with errors. That said, it could also be an accurate translation. It's a snippet, perhaps the difference between air-launch from a very expensive airplane and a far simpler and cheaper ground launch was part of a much longer interview that this was excerpted from. $\endgroup$
    – uhoh
    Dec 19 '20 at 4:12
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In addition to Russel Borogove's mention of Black Arrow, I can add that the engines of Black Arrow were Bristol Siddeley Gamma, where 85% hydrogen peroxide was used not only as an oxidizer (with approximately 8:1 mixture oxidizer/fuel ratio), but also as a nozzle coolant.

As mentioned in the Wikipedia article, already catalytically decomposed (and very hot) peroxide was used in combustion chamber (so oxygen was a really oxidizer, although cold peroxide is chemically an oxidizer itself).

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Armadillo Aerospace attempted to use both 90% hydrogen peroxide and later switched to hydrogen peroxide (50% concentration in water) and methanol as a mixed monopropellant for their vehicle.

My internet-fan memories were that, at least with the 90% solution, they were having severe problems getting a consistent burn and tried many catalyst and injector configurations until they stopped being able to get 90%. Their blog and website appears to have disappeared.

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Yet another addition to Russell Borogove's answer: hydrogen peroxide was used for the Mercury capsule's attitude control, and the AMU experiment attempted on Gemini 9.

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According to "Development of Hydrogen Peroxide Monopropellant Rockets" (.pdf, 2006) by Angelo Cervone, Lucio Torre, Luca d’Agostino, Antony J. Musker, Graham T. Roberts, Cristina Bramanti, and Giorgio Saccoccia:

Alta S.p.A. (Italy) and DELTACAT Ltd. (United Kingdom) are conducting a study, funded by the European Space Agency, into the development of hydrogen peroxide monopropellant thrusters using advanced catalytic beds. The present paper focuses on the design of two different demonstration thrusters with nominal ratings of 5 N and 25 N. Design requirements and specifications are presented, followed by the main results of a concept study, which was conducted to define the approximate dimensions needed. Some details about the specific design of the two prototypes and the choice of the main components are provided, with particular regard to the sensors and transducers to be used during the experimentation. Different catalytic bed configurations, including pure silver gauzes and pellets coated with manganese oxide or platinum, are going to be tested in the prototype thrusters, in order to find the optimum one for further industrial development. A dedicated test bench, designed and realized by Alta S.p.A. for tests on the thruster prototypes, is also illustrated.

And:

The most promising high-energy green propellants, like ADN, HAN and HNF (Wucherer et al.1, Schoyer et al.2), are based on complex organic molecules and compensate the large molecular weight of their decomposition products with high operational temperatures of the exhaust gases. As a consequence, it is necessary to use extremely expensive materials and manufacturing processes for the thrust chamber and, at the same time, the operational life of the catalytic beds is drastically reduced.

Hydrogen peroxide, on the other hand, does not suffer from these disadvantages and has therefore been reconsidered as a promising green propellant for low and medium thrust applications.

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