In aircraft and rocket design, overall propulsive efficiency
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Mathematically, it is represented as
Cycle efficiency
Most aerospace vehicles are propelled by heat engines of some kind, usually an internal combustion engine. The efficiency of a heat engine relates how much useful work is output for a given amount of heat energy input.
From the laws of thermodynamics:
whereIn other words, a heat engine absorbs heat energy from the high temperature heat source, converting part of it to useful work and delivering the rest to the cold temperature heat sink.
In general, the efficiency of a given heat transfer process (whether it be a refrigerator, a heat pump or an engine) is defined informally by the ratio of "what you get out" to "what you put in".
In the case of an engine, one desires to extract work and puts in a heat transfer.
The theoretical maximum efficiency of any heat engine depends only on the temperatures it operates between. This efficiency is usually derived using an ideal imaginary heat engine such as the Carnot heat engine, although other engines using different cycles can also attain maximum efficiency. Mathematically, this is because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir (i.e.,
where
Mechanical efficiency
From momentum considerations, propulsion requires material to be pushed backwards to push the vehicle forwards. In general, energy efficiency is highest when the air or exhaust gas used to propel the vehicle end up travelling as slow as possible for the required thrust, in the frame of reference of the Earth.
Jet engines
For all airbreathing jet engines the propulsive efficiency (essentially energy efficiency) is highest when the engine emits an exhaust jet at a speed that is as close as possible to the vehicle velocity. The exact formula for air-breathing engines as given in the literature, is
where c is the exhaust speed, and v is the speed of the aircraft.
A corollary of this is that, particularly in air breathing engines, it is more energy efficient to accelerate a large amount of air by a small amount, than it is to accelerate a small amount of air by a large amount, even though the thrust is the same.
Rocket engines
A rocket engine's
Rocket engines have a slightly different propulsive efficiency (
As with ducted jet engines, matching the exhaust speed and the vehicle speed gives optimum efficiency in principle. Although in practice rocket exhausts are high and typically fixed, this can be a useful theoretical consideration. Unlike ducted engines, rockets give thrust even when the two speeds are equal.
In 1903, Konstantin Tsiolkovsky discussed the average propulsive efficiency of a rocket, which he called the utilization (utilizatsiya), the "portion of the total work of the explosive material transferred to the rocket" as opposed to the exhaust gas.
Propeller engines
The calculation is somewhat different for reciprocating and turboprop engines which rely on a propeller for propulsion since their output is typically expressed in terms of power rather than thrust. The equation for heat added per unit time, Q, can be adopted as follows:
where
expressed as a percentage.
Assuming a typical propulsive efficiency