A tripropellant rocket is a rocket that uses three propellants, as opposed to the more common bipropellant rocket or monopropellant rocket designs, which use two or one fuels, respectively. Tripropellant rockets appear to offer fairly impressive gains for single stage to orbit designs, although to date no tripropellant rocket design has been developed to the point of testing that would prove the concept.
There are two principally different kinds of tripropellant rockets. One is a rocket engine which mixes three separate streams of propellants. For example, a mixture of lithium, hydrogen, and fluorine produced a specific impulse of 546 seconds; the highest ever of any chemical rocket motor. The other kind of tripropellant rocket is one that uses one oxidizer but two fuels, switching between the two in mid-flight. In this way the motor can combine the high thrust-to-mass of a dense fuel like kerosene early in flight with the high specific impulse of a lighter fuel like liquid hydrogen (LH2) later in flight. The result is a single engine providing some of the benefits of staging.
Although liquid hydrogen delivers the largest specific impulse of the plausible rocket fuels, it also requires huge structures to hold it due to its low density. These structures can weigh a lot, offsetting the light weight of the fuel itself to some degree, and also result in higher drag while in the atmosphere. While kerosene has lower specific impulse, its higher density results in smaller structures, which implies less loss to atmospheric drag. In addition, kerosene-based engines generally provide higher thrust, which is important for takeoff, reducing gravity drag. So in general terms there is a "sweet spot" in altitude where one type of fuel becomes more practical than the other.
Traditional rocket designs use this sweet spot to their advantage via staging. For instance the Saturn Vs used a lower stage powered by RP-1 (kerosene) and upper stages powered by LH2. Some of the early Space Shuttle design efforts used similar designs, with one stage using kerosene into the upper atmosphere, where an LH2 powered upper stage would light and go on from there. The existing Shuttle design is somewhat similar, although it uses solid rockets for its lower stages.
Almost all of the cost of operating the Shuttle is for the payroll for the army of workers needed to refurbish the Shuttle after it has landed. The fuel used is orders of magnitude cheaper, and, if a single stage to orbit design SSTO avoided some of this refurbishment, costs would drop, although this could require more repairs. But in this case the staging solution is not available, by definition, so it becomes harder to use both fuels.
SSTO rockets could simply carry two sets of engines, but this would mean the spacecraft would be carrying one or the other set "turned off" for most of the flight. With light enough engines this might be reasonable, but an SSTO design requires a very high mass fraction and so has razor-thin margins for extra weight.
And thus the tripropellant engine. The engine is basically two engines in one, with a common engine core with the engine bell, combustion chamber and oxidizer pump, but two fuel pumps and feed lines. The engine is somewhat heavier and more complex than a single-fuel engine, but the complexity is generally a little less than 50% more than a single engine, hence less than two engines would be. Of course there are numerous practical reasons why this would be more complex.
At liftoff the engine typically burns both fuels, gradually changing the mixture over altitude in order to keep the exhaust plume "tuned" (a strategy similar in concept to the plug nozzle but using a normal bell), eventually switching entirely to LH2 once the kerosene is burned off. At that point the engine is largely a straight LH2/LOX engine, with an extra fuel pump hanging onto it.
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