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Jet engines perform in two basic ways, the combined effect of which determines how much waste they produce as a byproduct of burning fuel to do thrust work on an aircraft.<ref>An engine applies a thrust force to a stationary aircraft and thrust work is done on the aircraft when it moves under the influence.</ref> First is an energy conversion as burning fuel speeds up the air passing through which at the same time produces [[waste heat]] from component losses (thermal efficiency). Second, part of the power which has been given to the air by the engine is transferred to the aircraft as thrust work with the remaining part being kinetic energy waste in the wake (propulsive efficiency). The two efficiencies were first formulated in the 19th century for the [[steam engine]] (thermal efficiency <math>\eta_{th}</math>) and the ship's propeller (propulsive or Froude efficiency <math>\eta_{pr}</math>).
A visual introduction to jet engine performance, from the fuel efficiency point of view, is the Temperature~entropy (T~s) diagram. The diagram originated in the 1890s for evaluating the thermal efficiency of steam engines. At that time entropy was introduced in graphical form in the T~s diagram which gives thermal efficiency as a ratio of areas of the diagram. The diagram also applies to air-breathing jet engines with an area representing kinetic energy<ref name="Propulsion and Power">{{Cite journal |last1=Kurzke |first1=Joachim |last2=Halliwell |first2=Ian |date=2018 |title=Propulsion and Power |url=https://link.springer.com/book/10.1007/978-3-319-75979-1 |journal=SpringerLink |language=en |doi=10.1007/978-3-319-75979-1|isbn=978-3-319-75977-7 |url-access=subscription }}</ref> added to the air flowing through the engine. A propulsion device, a nozzle, has to be added to a gas turbine engine to convert its energy into thrust. The efficiency of this conversion (Froude or propulsive efficiency) reflects work done in the 1800s on ship propellers. The relevance for gas turbine-powered aircraft is the use of a secondary jet of air with a propeller or, for jet engine performance, the introduction of the bypass engine. The overall efficiency of the jet engine is thermal efficiency multiplied by propulsive efficiency ( <math>\eta_o = \eta_{th} \eta_{pr}</math>).
There have been rapid advances in aero-engine technology since jet engines entered service in the 1940s. For example, in the first 20 years of commercial jet transport from the Comet 1 Ghost engine to the 747 [[Pratt & Whitney JT9D|JT9D]] Hawthorne<ref>{{Cite journal |last=Hawthorne |first=William |date= 1978|title=Aircraft propulsion from the back room |url=https://www.cambridge.org/core/journals/aeronautical-journal/article/abs/aircraft-propulsion-from-the-back-room/771675086CDE0E766BE700CD6B3198E7 |journal=The Aeronautical Journal |language=en |volume=82 |issue=807 |pages=93–108 |doi=10.1017/S0001924000090424 |s2cid=117522849 |issn=0001-9240|url-access=subscription }}</ref> scales up the Ghost to give JT9D take-off thrust and it is four and a half times as heavy. Gaffin and Lewis<ref>{{Cite journal |last1=Gaffin |first1=William O. |last2=Lewis |first2=John H. |date= 1968|title=Development of the High Bypass Turbofan |url=https://nyaspubs.onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1968.tb15216.x |journal=Annals of the New York Academy of Sciences |language=en |volume=154 |issue=2 |pages=576–589 |doi=10.1111/j.1749-6632.1968.tb15216.x |bibcode=1968NYASA.154..576G |s2cid=84722218 |issn=0077-8923|url-access=subscription }}</ref> make an assessment using one company's design knowledge. Using [[Pratt & Whitney JT3D|JT3D]]-level technology (1958) to produce a JT9D cycle (1966), with its higher bypass ratio and pressure ratio, an hypothetical engine came out 70% heavier, 90% longer and with a 9% bigger diameter than the JT9D engine.
==Conversion of fuel into thrust==
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Zhemchuzhin et al.<ref>{{Cite book |last1=Zhemchuzhin |first1=N. A. |url=http://archive.org/details/nasa_techdoc_19770023121 |title=Soviet aircraft and rockets |last2=Levin |first2=M. A. |last3=Merkulov |first3=I. A. |last4=Naumov |first4=V. I. |last5=Pozhidayev |first5=O. A. |last6=Frolov |first6=S. P. |last7=Frolov |first7=V. S. |date=1977-01-01 |others=NASA}}</ref> show an energy balance for a turbojet engine in flight in the form of a [[Sankey diagram]]. Component losses leave the engine as waste heat and add to the heat rejected area on a T~s diagram reducing the work area by the same amount.<ref name=":1" />
The engine does work on the air going through it and this work is in the form of an increase in kinetic energy. The increase in kinetic energy comes from burning fuel and the ratio of the two is the thermal efficiency which equals increase in kinetic energy divided by the thermal energy from the fuel (fuel mass flow rate x lower calorific value). The expansion following combustion is used to drive the compressor-turbine and provide the ram work when in flight, both of which cause the initial rise in temperature in the T~s diagram. The remainder of the T~s diagram expansion work is available for propulsion, but not all of which produces thrust work since it includes the residual kinetic energy<ref name="Aircraft">{{Cite journal |last=Lewis |first=John Hiram |date= 1976|title=Propulsive efficiency from an energy utilization standpoint |url=https://arc.aiaa.org/doi/10.2514/3.44525 |journal=Journal of Aircraft |language=en |volume=13 |issue=4 |pages=299–302 |doi=10.2514/3.44525 |issn=0021-8669|url-access=subscription }}</ref> or RVL.
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