In aircraft and rocket design, overall propulsion system efficiency is the efficiency with which the energy contained in a vehicle's fuel is converted into kinetic energy of the vehicle, to accelerate it, or to replace losses due to aerodynamic drag or gravity. Mathematically, it is represented as , where is the cycle efficiency and is the propulsive efficiency.
The cycle efficiency is expressed as the percentage of the heat energy in the fuel that is converted to mechanical energy in the engine, and the propulsive efficiency is expressed as the proportion of the mechanical energy actually used to propel the aircraft. The propulsive efficiency is always less than one, because conservation of momentum requires that the exhaust have some of the kinetic energy, and the propulsive mechanism (whether propeller, jet exhaust, or ducted fan) is never perfectly efficient. It is greatly dependent on exhaust expulsion velocity and airspeed.
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:
In other words, a heat engine absorbs heat from some heat source, converting part of it to useful work, and delivering the rest to a heat sink at lower temperature. In an engine, efficiency is defined as the ratio of useful work done to energy expended.
The theoretical maximum efficiency of a heat engine, the Carnot efficiency, depends only on its operating temperatures. Mathematically, this is because in reversible processes, the cold reservoir would gain the same amount of entropy as that lost by the hot reservoir (i.e., ), for no change in entropy. Thus:
where is the absolute temperature of the hot source and that of the cold sink, usually measured in kelvins. Note that is positive while is negative; in any reversible work-extracting process, entropy is overall not increased, but rather is moved from a hot (high-entropy) system to a cold (low-entropy one), decreasing the entropy of the heat source and increasing that of the heat sink.
Conservation of momentum requires acceleration of propellant material in the opposite direction to accelerate a vehicle. In general, energy efficiency is highest when the exhaust velocity is low, in the frame of reference of the Earth, as this reduces loss of kinetic energy to propellant.
where is the exhaust expulsion velocity and is the airspeed at the inlet.
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. This is why turbofan engines are more efficient than simple jet engines at subsonic speeds.
A rocket engine's is usually high due to the high combustion temperatures and pressures, and the long converging-diverging nozzle used. It varies slightly with altitude due to changing atmospheric pressure, but can be up to 70%. Most of the remainder is lost as heat in the exhaust.
Rocket engines have a slightly different propulsive efficiency () than air-breathing jet engines, as the lack of intake air changes the form of the equation. This also allows rockets to exceed their exhaust's velocity.
Similarly to jet engines, matching the exhaust speed and the vehicle speed gives optimum efficiency, in theory. However, in practice, this results in a very low specific impulse, causing much greater losses due to the need for exponentially larger masses of propellant. 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.
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 H = calorific value of the fuel in BTU/lb, h = fuel consumption rate in lb/hr and J = mechanical equivalent of heat = 778.24 ft.lb/BTU, where is engine output in horsepower, converted to foot-pounds/second by multiplication by 550. Given that specific fuel consumption is Cp = h/Pe and H = 20 052 BTU/lb for gasoline, the equation is simplified to:
expressed as a percentage.