In aerospace engineering, concerning aircraft, rocket and spacecraft design, overall propulsion system efficiency η {\displaystyle \eta } 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 η = η c η p {\displaystyle \eta =\eta _{\mathrm {c} }\eta _{\mathrm {p} }} , where η c {\displaystyle \eta _{\mathrm {c} }} is the cycle efficiency and η p {\displaystyle \eta _{\mathrm {p} }} 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:
d W = d Q c − ( − d Q h ) {\displaystyle dW\ =\ dQ_{\mathrm {c} }\ -\ (-dQ_{\mathrm {h} })}
where
d W = − P d V {\displaystyle dW=-PdV} is the work extracted from the engine. (It is negative because work is done by the engine.)
d Q h = T h d S h {\displaystyle dQ_{\mathrm {h} }=T_{\mathrm {h} }dS_{\mathrm {h} }} is the heat energy taken from the high-temperature system (heat source). (It is negative because heat is extracted from the source, hence ( − d Q h ) {\displaystyle (-dQ_{\mathrm {h} })} is positive.)
d Q c = T c d S c {\displaystyle dQ_{\mathrm {c} }=T_{\mathrm {c} }dS_{\mathrm {c} }} is the heat energy delivered to the low-temperature system (heat sink). (It is positive because heat is added to the sink.) 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.
η c = − d W − d Q h = − d Q h − d Q c − d Q h = 1 − d Q c − d Q h {\displaystyle \eta _{\mathrm {c} }={\frac {-dW}{-dQ_{\mathrm {h} }}}={\frac {-dQ_{\mathrm {h} }-dQ_{\mathrm {c} }}{-dQ_{\mathrm {h} }}}=1-{\frac {dQ_{\mathrm {c} }}{-dQ_{\mathrm {h} }}}}
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