There are many types of rocket engines in use today for various specific tasks. However, there are only two standard types of engines currently being used for space program launch vehicles: solid-fuel and liquid-fuel engines. Hybrid-fuel engines are being considered as a third choice for launch vehicles. In order to appropriately compare hybrid engines against those currently in use, it is necessary to have a basic understanding of the operation of each of these three engines.

All rocket engines used for launch vehicles consist of a fuel and an oxidizer which are mixed together and burned in a combustion chamber. As the fuel and oxidizer react, large amounts of high-energy gasses are released. These gasses expand, and as they escape through a nozzle at the end of the engine, they create the tremendous forces that launch the rocket into space.

Solid-Fuel Engines
Solid-fuel rocket engines consist of a hollow tube of solid fuel with a nozzle at the end. This solid core is actually a fuel and an oxidizer mixed together with a binder which causes the mixture to solidify. The hollow center of the tube of fuel acts as the combustion chamber. The fuel is usually made from a powdered metal such as aluminum or magnesium mixed with an oxidizer, usually ammonium perchlorate.
More information on solid rocket engines can be found here.

Liquid-Fuel Engines
The design of a liquid-fuel engine consists of two tanks, one for the fuel and one for the oxidizer, and a separate combustion chamber. The fuel and oxidizer each have a pump system which pump the them into the combustion chamber, where they mix and are burned. Currently, highly refined kerosene (RP-1) and cryogenic liquid hydrogen (LH2) are the most commonly used fuels in liquid-fuel engines. Most current liquid-fuel engines used in the space program use cryogenic liquid oxygen (LOX) as the oxidizer. Most military liquid systems use Inhibited Red Fuming Nitric Acid (IRFNA). (Thanks, Jurph, for that info.)
More information on solid rocket engines can be found here.

Hybrid-Fuel Engines
The overall design of a hybrid-fuel engine involves a hollow core of solid fuel similar to those used in solid-fuel engines as well as a tank of liquid oxidizer and a pump system like those used in liquid-fuel engines. Possible fuels for hybrid engines could include aluminum, magnesium, polyethylene, hydroxyl-terminated polybutadiene (HTPB), or beryllium. Possible oxidizers could include cryogenic LOX or hydrogen peroxide.

Comparisons of hybrid-fuel engines to existing engines are difficult because hybrids are currently only experimental. Full-scale hybrid-fuel engines have never been built and tested. However, enough experimental results have been produced to extrapolate and compare the performances of full-scale hybrid-fuel engines to solid and liquid-fuel engines.
Hybrid engines cannot produce the extremely high efficiency of liquid-fuel engines, but compare quite favorably with solid-fuel engines. In terms of thrust to weight ratio, hybrids can approximately equal solids. Data suggests that the achievable specific impulse of a hybrid-fuel engine may even exceed that of solid-fuel boosters currently in use.

Hybrid engines combine the simplicity of the solid-fuel core with the complexity of pump systems for the liquid oxidizer. However, because pumps are only needed for the oxidizer, only about half of the pump systems used in liquid-fuel engines are required in hybrid-fuel engines. In addition, because the solid-fuel core is smaller than that used in solid-fuel rockets, hybrid-fuel rockets have less of a chance for burn-through problems that sometimes affect solid-fuel rockets. Hybrid-fuel engines must deal with the same cryogenic handling difficulties and costs as liquid-fuel engines. These costs can be avoided by using hydrogen-peroxide as he oxidizer instead of LOX.
Additional savings can be achieved because, like liquid-fuel engines, hybrid-fuel engines have the capability to throttle. This allows cheaper testing, and also reduces the cost and difficulty of an emergency launch abort.
Hybrid-fuel engines should be able to offer a significant savings over liquid-fuel engines, because they only experience half of the difficulties associated with liquid pump systems, and can avoid difficulties with cryogenics entirely. In relation to solid-fuel engines, they can reduce the problems regarding the solid-fuel core because it is much smaller. In addition, the extra expense of the liquid system may be offset by the savings gained in safety precautions and throttleability.
Even with the advancements being made in the field, liquid-fuel engines remain much more expensive than solid-fuel engines. Though a quantifiable comparison is difficult to make, it appears that hybrid-fuel engines represent similar costs to solid-fuel engines. Because of the additional advantages of increased safety and throttling capability, hybrid-fuel engines may even represent savings over the cost of current solid-fuel engines.

Environmental Impact
Liquid-fuel engines are the cleanest burning of the three types of rocket engines. The primary components of their exhaust are water and carbon-dioxide. Solid-fuel engines produce several pollutants, including hydrochloric acid (HCl). Though recent developments will significantly reduce the amount of HCl in engine exhaust, some acid will still be produced. Hybrid-fuel engines produce some of the same pollutants as solids, but because they use a liquid oxidizer rather than ammonium perchlorate, hybrids produce none of the hydrochloric acid that solid-fuel engines produce.

Hybrid-fuel engines are very comparable to solid-fuel engines. Hybrids can achieve higher performances than solids. Hybrid-fuel engines also represent an overall savings over solid-fuel engines. Finally, hybrid-fuel engines have a similar environmental impact to solid-fuel engines, with the additional advantage that no hydrochloric acid is produced at all. In the future, hybrid-fuel engines could be a viable alternative to solid-fuel engines in launch vehicle applications. Hybrids would increase performance, decrease cost, and eliminate a major environmental concern of current solid-fuel engines.

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