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Here is my assumption:

In straight and level flight of a jet turbine engine:

One IAS = One drag = One thrust = One fuel flow

Therefore, the fuel flow will be approximately the same for a given IAS, regardless of altitude, temperature, or RPM necessary to maintain this IAS.

Is that correct? If not, where is the fallacy?

EDIT : Here is a screenshot from ATPL course material which supports this idea and led me to ask the question as it seems doubtful to me. Propulsive efficiency vs RPM at high altitude at the end is not clear as well. enter image description here

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Arnaud BUBBLE
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    This is very much dependant on altitude, air pressure, temperature... Thinner air creates less drag, has less oxygen, etc. Not too mention that "IAS" changes with altitude all by itself... – Ron Beyer Aug 30 '19 at 12:35
  • Related: Why do jet engines get better fuel efficiency at high altitudes? – Bianfable Aug 30 '19 at 12:48
  • @Ron Beyer. It's true that IAS increases (in TAS terms) with altitude, but the OP presumes a fixed IAS... – xxavier Aug 30 '19 at 13:26
  • @Ron Beyer as IAS is fixed, thinner air will not create less drag. IAS reflects dynamic pressure, so if maintained constant, dynamic pressure will be the same, and so will drag. It's true though that air gets thinner with altitude, but it is compensated by the increased TAS (more air, so overall dynamic pressure is constant). This part by itself is counter intuitive – Arnaud BUBBLE Aug 30 '19 at 13:34
  • @Bianfable Thanks for the link, I'm familiar with this topic. Engines get better fuel efficiency at high altitudes because for a same IAS, so a same fuel flow from my point of view, the TAS is much greater: therefore the consumption per unit of distance is much less – Arnaud BUBBLE Aug 30 '19 at 13:38
  • @ArnaudPROST But the higher TAS for same IAS is due to the lower air density at higher altitudes. The better fuel efficiency (at least in part) is due to the lower temperatures, so you should get lower fuel flow for the same IAS at higher altitudes. – Bianfable Aug 30 '19 at 13:53
  • @Bianfable this is the point, but I'm not sure. If the temperature is lower (density higher), the TAS necessary to get this IAS will be lower. But the drag will be the same, and so will, in level flight, the thrust. And for me in a jet engine, if you set a thrust, the fuel flow is set as well. – Arnaud BUBBLE Aug 30 '19 at 14:01

2 Answers2

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The most obvious difference is due to the temperature of the air.

Both turbine and piston engines are heat engines. They work by converting thermal power into mechanical power. The theoretical absolute maximum efficiency you can achieve is called the Carnot efficiency, $$\eta = 1-\dfrac{T_C}{T_H}$$

This is the efficiency of an ideal engine using the Carnot cycle, that works by transferring heat from a hot reservoir with temperature $T_H$ to a cold reservoir with temperature $T_C$. A typical jet engine is approximated by the Brayton cycle, and a piston engine the Otto or Diesel cycle, but neither can ever by more efficient than the efficiency noted above. The efficiency of a Brayton cycle is $$\eta = 1-\dfrac{T_C}{T_E}$$ with $T_E$ the EGT.

When flying higher, the temperature of the cold reservoir (the atmosphere) drops lower. You can see that the maximum efficiency of the engine will also increase (even if $T_H$ or $T_E$ respectively drop simultaneously with $T_C$). This means that, even if the required power would stay constant for constant thrust, the fuel flow changes, because a single unit of energy from a drop of fuel can be converted into more mechanical power.

Sanchises
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  • thanks, I'm almost convinced. Here are my doubts: 1/ even though the Carnot cycle efficiency is very much dependent on temperature difference between hot and cold reservoir, is it the case for the Brayton cycle ? The maximum (Carnot efficiency) could vary with this temperature while the Brayton, staying below, would not be affected. 2/ Why wouldn't the hot reservoir just become colder (EGT lower) ? 3/ If the temperature is colder, density higher therefore TAS will be lower to fly at same IAS. Doesn't this compensate the extra energy from cold air ? – Arnaud BUBBLE Aug 31 '19 at 12:21
  • 1/ I quoted Carnot as to avoid the inevitable question "what if we used another cycle". I'll add it to the answer. 2/ If the temperature rise is approximately constant (which makes sense because $c_p$ is approximately constant as the atmosphere is approximately an ideal gas) you still get a rise in efficiency $1-\frac{T_C}{T_h}<1-\frac{T_C-\Delta T}{T_H-\Delta T}$. 3/ That effect is already accounted for if we only look at the IAS. – Sanchises Sep 01 '19 at 06:15
  • Could you edit your answer to elaborate on 3/ ? It's not clear to me, and the whole difficulty of the problem. If we had fixed TAS, the efficiency difference would be sufficient to explain different fuel flows. But we fixed IAS, so even though more energy is produced out of a unit of fuel in colder air, in this colder air TAS will be lower to maintain the IAS. So less work will be needed, and less fuel. – Arnaud BUBBLE Sep 01 '19 at 08:24
  • @ArnaudPROST You're thinking the wrong way around. IAS is dynamic pressure, so what the aircraft actually 'feels' - it doesn't matter if that's due to temperature or altitude that the density drops. But after establishing that constant IAS leads to roughly constant thrust, you can then wonder how efficient thrust is generated. Density doesn't really come into the whole thermodynamic efficiency picture. – Sanchises Sep 01 '19 at 22:07
  • I agree, but thermodynamic is not the only parameter influencing the efficiency of thrust generation, TAS is one as well. Thrust is approximately: $T=\dot{m_{air}}(V_{jet}-TAS)$ Thermo tells you how hard it is to accelerate the airflow to $V_{jet}$, but for a same thrust, you will need more or less of it depending on TAS – Arnaud BUBBLE Sep 01 '19 at 22:31
  • Your answer focus on thermal efficiency, which is only part of the overall propulsive efficiency which determines fuel flow $\eta=\eta_{cycle}\eta_{prop}$ with $\eta_{prop}=\frac{2}{1+\frac{V_{jet}}{TAS}}$. From "Gas turbine theory" by Cohen and Rogers, overall propulsive efficiency is proportional to $\frac{TAS}{SFC}$. My assumption is that on iso-IAS lines, this value is constant – Arnaud BUBBLE Sep 02 '19 at 08:20
  • @ArnaudPROST Perhaps you are right along adiabatic lines, but it is possible to have the same IAS and TAS with a different temperature. So while your statement may be true by approximation, there's no fundamental law that says it should be. – Sanchises Sep 02 '19 at 11:04
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The fallacy is

One thrust = One fuel flow

The thrust-specific fuel consumption (TSFC) of a jet engine (the mass of fuel consumed per unit of thrust) isn't constant, so that equality is violated. TSFC will vary with all the usual factors: ambient temperature and pressure, airspeed, etc.

Erin Anne
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    Could you perhaps elaborate why this is the case? – Sanchises Aug 31 '19 at 07:53
  • Honestly, come to think of it, you can have multiple level flights at the same IAS with different AoAs or configurations, and thus different drags, too, so that part's also trivially fallacious, but the question also seemed to be about engines instead. Where it's also trivially fallacious because there's a whole concept devoted to the idea that one thrust doesn't equal one fuel flow. – Erin Anne Aug 31 '19 at 09:49
  • @Erinn Anne this is true about configurations, but the question of course considers a constant configuration. Could you elaborate on different drags and AoAs for the same IAS on level flight ? It sounds wrong to me, from ATPL at least. For your answer, I also disagree: the difficulty is that ambient temperature and pressure changes, as well as airspeed, are included in IAS parameter (from density). IAS being fixed, it does not matter if one is changed, the others will compensate to maintain the same value of IAS and it could therefore be the same from the engine point of view. – Arnaud BUBBLE Aug 31 '19 at 12:11
  • On configuration: you could deploy some drag, like flaps or spoilers, add an equal amount of thrust, and now you're at the same IAS (because thrust - drag is still 0 and haven't changed altitude). So one IAS absolutely doesn't correspond to one drag (hence what we call the "drag coefficient", which changes with configuration, or with airframe...)

    On thrust: TSFC is an empirically observed fact. Look at the tables in the wikipedia article I linked. You can thought experiment at it all you like, but the reality is what it is.

     – Erin Anne Aug 31 '19 at 21:55
  • @Erin Anne: configuration is constant. When you said "you can have multiple level flights at the same IAS with different AoAs OR configuration" it's false. For a level flight and an IAS you have only one AoA. Your reference to TSFC is not clear enough, and not linked to the statements of the question. Please elaborate your reasoning: "reality is what it is" does sound like an authority argument which does not help to understand underlying phenomena – Arnaud BUBBLE Sep 01 '19 at 08:10
  • @Erin Anne: in addition, TSFC is not equal to fuel flow, so the logic of the argument in your answer is violated. – Arnaud BUBBLE Sep 01 '19 at 16:54
  • Per my answer above, TSFC is equal to fuel flow per unit thrust, (not just simply fuel flow) and it empirically changes in response to factors like altitude (one of the factors you specifically name "regardless of" in the question), so it invalidates "one thrust = one fuel flow." It is literally the number that tells you how much fuel you will burn at a given thrust. – Erin Anne Sep 01 '19 at 20:13
  • I understand your point, but I do think it is incomplete, because it does not take into account the iso-IAS hypothesis. You look at it on first order only. TSFC of course changes with temperature, but also with airspeed, RPM, pressure, as you said. IAS links all those parameters together (you act on the engine so RPM to maintain the IAS, airspeed changes with temperature for the same IAS, etc...). Since then, it is absolutely possible than on iso-IAS lines, TSFC is constant. At least, you answer does not prove the opposite – Arnaud BUBBLE Sep 01 '19 at 22:56