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I have a chart where the thrust ($F$) and the thrust-specific fuel consumption (TSFC) are plotted against the aircraft flying speed, for several altitudes (i.e. sea level, 3000 meters and 11000 meters). This is for a generic turbojet.

chart01

I don't know the source of the image, I apologize for that.

At the left, we see that the thrust decreases with altitude. At the right, we observe that the TSFC actually decreases with altitude as well.

However, I think this is counter-intuitive. I understand that $F$ decreases with altitude, since density (and therefore mass flow rate) decreases when the airplane goes up.

What I don't understand is that the TSFC, defined as: $$\text{TSFC}\equiv\dfrac{\text{fuel mass flow rate}}{F}$$ decreases with altitude. In other words, the higher we fly, the more thermodynamically-efficient the airplane is. How does that happen?

From a pure mathematical point of view, it doesn't make sense that TSFC decreases with altitude since, following the aforementioned formula, $F$ is decreasing, which leads me to think that the fuel mass flow rate diminishes with altitude faster than thrust.

In a nutshell: why does the TSFC decreases with the flying altitude, having the last equation into consideration?

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    $\begingroup$ This answer should answer most of it. $\endgroup$ – Jordy Jun 1 '16 at 20:45
  • $\begingroup$ That's a great answer (thanks for linking it), but I was most likely looking a mathematical answer, considering the equation I posted before. I'd like @PeterKampf to chime in and shed some light here! $\endgroup$ – Jose Lopez Garcia Jun 1 '16 at 21:03
  • $\begingroup$ Hi Jose, this is applied thermodynamics. As usual, Wikipedia has plenty of formulas. I linked to the Carnot cycle; the Brayton cycle is just a special case and its Wikipedia article has fewer formulas. $\endgroup$ – Peter Kämpf Feb 8 '17 at 20:15
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I can't provide the math, but yes, this is correct. Fuel flow decreases as air density decreases. The engine becomes more efficient because of the greater temperature differential between inlet and exhaust gases combined with the lower fuel flow.

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    $\begingroup$ Hey Nate. In order to accept this answer, could you please elaborate more on it at least a bit? You don't need to use maths, a pure "textual" answer will work. Maybe you can provide a chart along with a bit more information? $\endgroup$ – Jose Lopez Garcia Jun 2 '16 at 18:21
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    $\begingroup$ I'll have to dig through some reference books and see if I can't find something more useful. I'll get back with you on it $\endgroup$ – Nate Graham Jun 6 '16 at 15:57
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    $\begingroup$ Jose, here are some good references for more information on turbine engine efficiency at altitude: This video for the maths: youtube.com/watch?v=E8VfieYhsjg $\endgroup$ – Nate Graham Jun 6 '16 at 19:03
  • $\begingroup$ This wikipedia article may help you to briefly understand the brayton cycle efficiency of gas turbines: en.wikipedia.org/wiki/Thermal_efficiency And lastly, check out this guys post on the airline pilot's forum about Jet engine efficiency. It's not deep but you may find his notes helpful: theairlinepilots.com/forum/viewtopic.php?t=476 Hope that helps $\endgroup$ – Nate Graham Jun 6 '16 at 19:10
  • $\begingroup$ Alright @Nate, thanks for the help and your time. $\endgroup$ – Jose Lopez Garcia Jun 6 '16 at 19:13
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Gas turbine engine works more efficiently at higher altitudes

Altitude increases - Air density reduces - Mass flosw reduces - Maximum thrust reduces.

To maintain thrust as altitude increases - Compressors must rotate faster.

Hight altitude - Less air density - Lesser resistance - Less fuel required to spin the compressor faster.

There is an optimum altitude in reference to speed and thrust which increases as weight reduces.

Altitude increase - Maintaining a constant TAS - Reduction in fuel flow and SFC from sea level up to the optimum altitude. for more information visit http://www.theairlinepilots.com/forum/viewtopic.php?t=476

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Multiple parameters have a role in this, but high altitudes are not the best operating conditions in general for an engine: see hot and high tests, but are the best for an aircraft !

High altitude means - less thrust because of lower air density - less power in the gas generator due to less oxygen density - far less losses from air resistance to the fuselage

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Less oxygen is present at higher altitudes, so not decreasing the fuel flow during climb will eventually throw the air to fuel mixture so far out of balance that the engine will run rough or even shut down or flame out. By definition you will get better fuel economy at higher altitudes but I would not say it is "more efficient" due to the significant reduction in power that can actually be developed due to the lack of oxygen. Any engine will produce more power as a direct result of fuel burn and maximum available oxygen at sea level. However, in jet engines one must take into account EGT temperature differentials and the resulting expansion of exhaust gases that do produce additional thrust dependent on greater temperature differences.

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