Propulsive efficiency
In aircraft and rocket design, overall propulsion system efficiency formula_1 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 formula_2, where formula_3 is the cycle efficiency and formula_4 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. 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: 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., formula_11), for no change in entropy. Thus: where formula_13 is the absolute temperature of  the hot source and formula_14 that of the cold sink, usually measured in kelvins. Note that formula_15 is positive while formula_16 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. Propulsive efficiency. Propulsive efficiency is defined as the ratio of work done on the fluid to the propulsive power, i.e. thrust times velocity of the vehicle. In generic terms, the propulsive power can be calculated as  follows: The thrust can be computed from intake and exhaust massflows (formula_20 and formula_21) and velocities (formula_22 and formula_23): The work done by the engine to the flow, on the other hand, is the change in kinetic energy per time. This does not take into account the efficiency of the engine used to generate the power, nor of the propeller, fan or other mechanism used to accelerate air. It merely refers to the work done to the flow, by any means, and can be expressed as the difference between incoming kinetic energy flux and exhausted kinetic energy: The propulsive efficiency  can therefore be computed as: Depending on the type of propulsion used, this equation can be simplified in different ways, demonstrating some of the peculiarities of different engine types. The general equation already shows, however, that propulsive efficiency improves when using large massflows and small velocities compared to small mass-flows and large velocities, since the squared terms in the denominator grow faster than the non-squared terms. The losses modelled by propulsive efficiency are explained by the fact that any mode of aero propulsion leaves behind a jet moving into the opposite direction of the vehicle. The kinetic energy  flux in this jet is formula_29 for the case that formula_30. Jet engines. The propulsive efficiency formula for air-breathing engines is. It can be derived by setting formula_31 in the general equation. , and assuming that formula_32. This cancels out the mass-flow and leads to: For pure jet engines, particularly with afterburner, a small amount of accuracy can be gained by not assuming the intake and exhaust massflow to be equal, since the exhaust gas also contains the added mass of the fuel injected. For turbofan engines, the exhaust massflow may be marginally smaller than the intake massflow because  the engine supplies "bleed air" from the compressor to the aircraft. In most circumstances, this is not taken into account, as it makes no significant difference to the computed propulsive efficiency. By computing the exhaust velocity from the equation for thrust (while still assuming formula_36), we can also obtain the propulsive efficiency as a function of specific thrust (formula_37): 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. Rocket engines. A rocket engine's formula_3 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 (formula_4) 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. 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 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 formula_43 is engine output in horsepower, converted to foot-pounds/second by multiplication by 550. Given that specific fuel consumption is  C = "h"/"P" and H = 20 052 BTU/lb for gasoline, the equation is simplified to: expressed as a percentage. Assuming a typical propulsive efficiency formula_4 of 86% (for the optimal airspeed and air density conditions for the given propeller design), maximum overall propulsion efficiency is estimated as: