The idea we had was actually a coincident, JLMC's idea was to combust the still-combustible hydrogen to get more thrust out of it, while my original idea was to attempt to mitigate the issue of leaking hydrogen propellant during deep space missions, by storing the hydrogen in the form of water, which spiraled down into reintroducing the oxygen into the hydrogen exhaust to combust and generate more thrust.
Since JLMC have given you guys a rough idea of such an engine's operation, I shall give you my take.
First I am going to assume that the spacecraft is using nuclear as a power source and has no issue with energy shortage.
Hydrogen that is stored as water will be electrolyzed into Oxygen and Hydrogen, cooled down into their liquid form and stored into their separate propellant tanks. The amount of water electrolyzed depends on the planned burn time.
The engine concept that I've proposed above has two modes; Normal and High-Thrust.
Normal mode (above image, left) has the external bell attached to the main exhaust nozzle, where the superheated hydrogen will flow through that and generate normal thrust. LOX will still run along the internal combustion chamber to prevent long term heat damage. The hot oxygen will be sent back to the heat exchanger to be cooled back down.
High-Thrust mode (bottom) has the external bell pushed down to attach to the nozzle of the inner combustion chamber via the use of actuators, the exhaust hydrogen is diverted into the inner combustion chamber and the LOX injection plug valve is opened. This combustion chamber does not require an ignition system, as the exhaust hydrogen's temperature far surpasses its flash point, simply introducing it to the oxygen will allow for immediate self-ignition.
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