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| In [[aircraft]] and rocket design, '''overall propulsive efficiency''' <math>\eta</math> is the efficiency, in percent, with which the energy contained in a vehicle's propellant is converted into useful energy, to replace losses due to [[air drag]], gravity, and acceleration. It can also be stated as the proportion of the mechanical energy actually used to propel the aircraft. It is always less than 100% because of kinetic energy loss to the exhaust, and less-than-ideal efficiency of the propulsive mechanism, whether a [[Propeller (aircraft)|propeller]], a jet exhaust, or a fan. In addition, propulsive efficiency is greatly dependent on air density and airspeed.
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| Mathematically, it is represented as <math>\eta = \eta_c \eta_p</math><ref>,[http://www.hq.nasa.gov/pao/History/SP-468/ch10-3.htm ch10-3<!-- Bot generated title -->]</ref> where <math>\eta_c</math> is the [[cycle efficiency]] and <math>\eta_p</math> is the propulsive efficiency. The cycle efficiency, in percent, is the proportion of energy that can be derived from the energy source that is converted to mechanical energy by the [[engine]].
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| ==Estimation==
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| ===Jet engines===
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| [[Image:JetWirkungsgrad.png|thumb|right|Dependence of the energy efficiency (η) from the exhaust speed/airplane speed ratio (c/v) for airbreathing jets]]
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| For all [[airbreathing jet engine]]s the ''propulsive efficiency'' (essentially [[Energy conservation|energy efficiency]]) is highest when the engine emits an exhaust jet at a speed that is the same as, or nearly the same as, the vehicle velocity. The exact formula for air-breathing engines as given in the literature,<ref>K.Honicke, R.Lindner, P.Anders, M.Krahl, H.Hadrich, K.Rohricht. Beschreibung der Konstruktion der Triebwerksanlagen. Interflug, Berlin, 1968</ref><ref name=spt>Spittle, Peter. [http://users.encs.concordia.ca/~kadem/Rolls%20Royce.pdf "Gas turbine technology"] p507, ''[[Rolls-Royce plc]]'', 2003. Retrieved: 21 July 2012.</ref> is
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| :<math>\eta_p = \frac{2}{1 + \frac{c}{v}}</math>
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| where ''c'' is the exhaust speed, and ''v'' is the speed of the aircraft.
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| 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 little bit than a small amount by a large amount, even though the thrust is the same.
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| [[File:Propulsive efficiency.png|thumb|right|Dependence of the propulsive efficiency (<math>\eta_p</math>) upon the vehicle speed/exhaust speed ratio (v/c) for rocket and jet engines]]
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| ===Rocket engines===
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| A rocket engine's <math>\eta_c</math> is usually high due to the high combustion temperatures and pressures, and long nozzle employed. The value varies slightly with altitude due to atmospheric pressure on the outside of the nozzle/engine, but can be up to 70%. Most of the remainder is lost as heat energy in the exhaust.
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| Rocket engines have a slightly different propulsive efficiency (<math>\eta_p</math>) than airbreathing jet engines as the lack of intake air changes the form of the equation. This also means that rockets are able to exceed their exhaust velocity. See diagram.
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| :<math>\eta_p= \frac {2 \frac {v} {c}} {1 + ( \frac {v} {c} )^2 }</math><ref name="RPE">George P. Sutton & Oscar Biblarz, ''Rocket Propulsion Elements'', pg 37-38 (seventh edition)</ref>
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| 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.
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| Rockets often also have an additional significant source of energy, because they are able to leverage the kinetic energy of their propellant with the [[Oberth effect]]. This is also a factor for air-breathing aircraft, but much smaller one due to their lower speed.
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| ===Propeller engines===
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| 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:
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| : <math>550 P_e = \frac{\eta_c H h J}{3600},</math>
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| where <math>P_e</math> is engine output in [[horsepower]], converted to foot-pounds/second by multiplication by 550. Given that [[specific fuel consumption (shaft engine)|specific fuel consumption]] is ''C''<sub>''p''</sub> = ''h''/''P''<sub>''e''</sub> and using the aforementioned substitutions for ''H'' and ''J'', the equation is simplified to:
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| : <math>\eta_c = \frac{14}{C_p}.</math>
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| expressed as a percentage.
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| Assuming a typical propulsive efficiency <math>\eta_p</math> of 86% (for the optimal airspeed and air density conditions for the given propeller design), maximum overall propulsive efficiency is estimated as:
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| : <math>\eta = \frac{12}{C_p}.</math>
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| ==See also==
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| *[[Specific fuel consumption (thrust)]]
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| ==References==
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| *{{cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468|url=http://www.hq.nasa.gov/pao/History/SP-468/cover.htm|accessdate=2006-04-22}}
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| *{{cite web|author=Loftin, LK, Jr.|title=Quest for performance: The evolution of modern aircraft. NASA SP-468 Appendix E|url=http://www.hq.nasa.gov/pao/History/SP-468/app-e.htm|accessdate=2006-04-22}}
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| ==Notes==
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| <references/>
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| [[Category:Aerodynamics]]
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