# Why do jet engines get better fuel efficiency at high altitudes?

I'm told that this is true, but I can't imagine why. It seems like the fact that there is less air would make the engines less efficient... But that probably just shows how little I know about jet engines.

• On a slightly different note, the Concorde was super-efficient at its cruise altitudes when flying at Mach 2.00+. The same can't be said about its low altitude subsonic efficiency :) – shortstheory Feb 8 '14 at 3:25
• Less air makes the engines less powerful. But efficiency is ratio of power generated to fuel consumed . – Jan Hudec Mar 26 '14 at 12:45
• Probably because designs require trade-offs and the engineers chose to optimize the engine for high altitude operation since that's where it spends most of the time. – Roman Oct 13 '14 at 23:31

For a quick explanation, you need to know that

• Thrust is the difference between the entry impulse of the air entering the engine and the exit impulse of the heated fuel-air mixture leaving the engine. Impulse is mass times velocity, and expressed with a mass flow $$\dot m$$, thrust T is $$T = \dot m \cdot (v_{exit} - v_{entry})$$
• The exit impulse is increased by accelerating the airflow through the engine, and the acceleration is achieved by heating the air.
• Each gram of fuel heats up a given mass of air by a certain number of centigrades. The definition of the energy content of fuels is given as the capacity to heat one pound of water by one degree Fahrenheit. The definition of one Calorie is similar but in metric units. Since the heat capacity of both water and air are almost constant at moderate temperatures, the starting temperature makes little difference to the absolute temperature increase when a given amount of energy is added.

## Thermal efficiency

Thermal efficiency is the ratio between the mechanical work extracted as thrust and the heat energy spent on heating the air, and it is affected indirectly by the flight altitude. Please see the Wikipedia article on the Carnot cycle. This and similar cycles describe the workings of all combustion engines in thermodynamic terms. Basically, it says that the efficiency of a combustion engine cannot be greater than the temperature ratio between the temperature increase from ambient ($$t_{amb}$$) to the maximum temperature $$t_{max}$$ of the process, divided by the maximum temperature. All temperatures must be expressed as total temperatures, where 0° means 0 K or -273.15°C. Operating in colder air makes the ratio bigger and improves efficiency. $$\eta_t = \frac{t_{max} - t_{amb}}{t_{max}}$$

If $$t_{amb}$$ is 290 K (16.85°C or 62°F) and the fuel heats up the air to 1400 K (2060°F), the thermal efficiency according to the formula above is 79.3%.

At cruise altitude $$t_{amb}$$ is only 220 K (-53.15°C or -63.7°F), and the same fuel flow relative to air flow will lift the maximum temperature only to 1320 K (in reality even less; for more precise reasoning see below). Now the thermal efficiency is 83.33%! If the maximum temperature is maintained, both thrust and thermal efficiency will go up; the latter to 84.3%.

In reality, the total efficiency will be lower because we have not included propulsive efficiency, friction effects or power offtake by bleed air, pumps and generators. Propulsive efficiency describes how well the acceleration of air is performed.

## Heating the fuel-air mixture

Burning a fuel-air mixture will add thermal energy to it, about 43 MJ for every kilogram of kerosene (if we assume complete combustion). The isobaric heat capacity or specific heat of air (close enough, the mixture has very little fuel but lots of air in it) is 29 J per mol and per K, so those 43 MJ will heat 1000 mol of air by 1483 K. The heat capacity changes slightly with humidity and temperature, but little enough that we can consider it constant for this purpose. If the air starts at 220 K, precompression in the intake will heat it to approx. 232 K, further compression in the engine will heat it up to approx. 600 K if we assume a compression ratio of 25, and this is the temperature at the entry of the combustion chamber.

Those 1000 mol of air weigh about 29 kg, and adding a full kilo of fuel and burning the mixture will heat it to 2083 K. If you want more details about the parameters in a typical jet engine, please see the diagram in this answer. Since the mixture picks up speed as it burns, the fuel mass is also heated and combustion is never complete, the maximum temperature given here will not be reached in reality.

If we start on the ground with an air temperature of 290 K, the temperature in the intake would drop slightly because we will not be flying fast enough for any precompression to happen in the intake. Now the compressor will heat the air to 730 K, and again adding and burning that kilo of kerosene will heat 1000 mol of air to 2213 K. Ideally.

In reality, the engine control will see that the limit temperatures are not exceeded, but here we can play with the numbers as we like. The exact values will certainly be slightly different (more frictional heating in the compressor, heat loss to the outside, slight drift in specific heat with temperature), but the gist of the explanation is correct.

## Explanation in layman's terms

Burning the fuel-air mixture heats it and makes the gas expand. This happens at nearly constant pressure and in a restricted volume, so the only way to make room for this expansion is for the gas to flow faster. Nearly constant pressure means that the density of the gas must decrease. The density ratio between the heated and the unburnt gas is proportional to its temperature ratio, measured in absolute temperature.

However, the amount of fuel burnt determines the absolute temperature increase, the difference in degrees between the burnt gas inside the combustion chamber and the unburnt gas at the intake. For a given amount of fuel, the temperature ratio which can be achieved with an absolute temperature increase becomes smaller the higher the temperature of the unburnt gas is. Thus, efficiency decreases with a higher temperature of the intake air.

• Wow, great expansion! I know this is asking a bit, but could you include a section before "thermal efficiency" that talks a bit about how fuel is transferred into heat? Perhaps a formula that shows it? I'm a little confused... Why is it that the same amount of fuel will raise the temperature of the exhaust gas the same amount even if the tempurature is lower? It feels like if I was starting at -50c and going to 50c that would take a lot more energy than 25c to 125c. Or am I thinking wrong? Either way, adding a section about that would be fantastic, then I maymark your answer correct... – Jay Carr Jan 11 '15 at 21:08
• @JayCarr: Thanks for your encouraging comments, I am working on it ... – Peter Kämpf Jan 11 '15 at 21:16
• I like the addition, but I'm still a little confused on why the same amount of energy is needed from the fuel to heat the air, no matter what temperature you start at. – Jay Carr Jan 11 '15 at 22:18
• @JayCarr: The definition of the energy content of fuels is given as the capacity to heat one pound of water by one degree Fahrenheit. The definition of one Calorie is similar, but in metric units. Since the heat capacity of both water and air are almost constant at moderate temperatures, the starting temperature makes little difference to the absolute temperature increase when a given amount of energy is added. – Peter Kämpf Jan 11 '15 at 22:52
• @JayCarr: I suck at chatting. I need time to research, type and refine a response. – Peter Kämpf Oct 7 '15 at 20:23

What matters for a jet engine is the pressure and temperature differentials between the exhaust gas and the ambient atmosphere. It is the expansion and high kinetic energy of the exhaust gas as it exits the engine that provides the thrust (and noise) of a jet (note this does not take into account the bypass portion of a turbofan).

The ambient pressure is atmospheric pressure, which for example at the surface is roughly 1000 hPa and at cruise might be 200 hPa or roughly a fifth of the pressure at the surface. The temperature at that altitude is also typically around -50 C.

The exhaust gas pressure and temperature is controlled by a few things:

• The compression by the N2 compressor stages -- Increases temperature and pressure
• The hot section -- Greatly increases temperature and pressure
• The N1/N2 turbine stages -- slight decrease in temp/pressure (work done on moving the turbines).

As the outside pressure is dropping as we climb, to maintain the same pressure differential in the engine, we need less temperature and pressure in the engine, and one way to do this is to reduce airflow into the engine and fuel added to that air. The atmosphere takes care of reducing airflow (there is just less of it up at cruise, though this also depends on airspeed) and the FADEC takes care of adjusting the fuel flow. The net result is less fuel needed to produce the same pressure differential when the air outside has a lower pressure, e.g. cruise flight.

EDIT:

Some of the other answers/comments make reference to mass flow through the jet, and particularly the mass flow through the exhaust nozzle. I agree with that, but I didn't mention it directly because that mass flow is setup by the pressure gradient within the engine. I should also clarify that pressure at the nozzle will be at or very close to ambient atmospheric pressure and it is the pressure gradient between that ambient pressure and that in the hot section that establishes the mass flow rate out of the engine.

Lastly, to address the bypass ratio comment, see Lnafziger's comment. The turbofan engines on the EMB-145 are similar in that the bypass provides more thrust at sea level than cruise. This perhaps relates to increased fuel efficiency at cruise in that the N1 fan is doing less work and thus the N1 turbine is extracting less energy from the engine.

• the pressure differential between the exhaust gas an the ambient atmosphere should be a low a possible. It is not the gas exiting the engine that provides the thrust. It are the compressor and the combustion chamber that produce most of the thrust. – DeltaLima Feb 7 '14 at 21:06
• How then do the compressor and combustion sections produce the thrust if not through the modification of the air passing through the jet engine? The compressor and combustion work on the air, and the air in turn works on the turbines and the atmosphere. – casey Feb 7 '14 at 21:13
• Does this mean that this answer, if modified, is correct? ..some sort of reference might be wise at this point too. – Jay Carr Feb 7 '14 at 21:35
• en.m.wikipedia.org/wiki/Bypass_ratio – yankeekilo Feb 8 '14 at 10:39

They work better at high altitude firstly because the air is cooler. Cool air expands more when heated than warm air. It is the expansion of the air that drives combustion engines.

The second reason is the low density of the air. Low density causes low drag and therefore the aircraft flies much faster at high altitude than on low altitude when it is given the same thrust. At this high speed, the mass flow through the engine is comparable to the mass flow at low speed in high density air (low altitude). The amount of energy needed is to heat the air to exhaust temperature is comparable between high and low altitudes. But since the aircraft at high altitude flies much faster, the amount of power generated is higher $(Power=Thrust\times{Speed})$ at altitude.

The difference with propeller aircraft is that at high speeds the propeller loses efficiency, and therefore the available power decreases with altitude.

• You could also add the concept of ram air compressing to make this answer a bit more complete. – shortstheory Feb 8 '14 at 3:27

For a non-math approach:

Let's think how a jet engine works and compare low altitude with high altitude flights. The engine takes air from the intake situated at the front. As you are climbing, the air becomes less dense (there is less air mass in a volume) so you need to go a little faster just so that the mass of the air coming in through the intake is the same in a given second. You will actually get the same mass flow of air at high altitudes as you will get at low altitudes, but you are actually travelling faster.

Then you compress that air, remembering that as you are now travelling faster higher up, the ram effect will help you out and compress some of that air for you, just by 'ramming' your engines into it at high speed. As you compress it you pass it to the combustion chamber where it burns. This combustion stage is the same for both high and low altitudes, although the fact that at higher altitudes air is colder actually helps a little bit, as we can burn more fuel without reaching dangerous temperatures, so that's nice.

After burning it up, the air is passed through a turbine then expelled out the back. Now here it gets a bit complicated: You see, it is more efficient to accelerate a lot of air (mass) a little bit(small dv), than to accelerate a little bit of air(small mass) to a very fast speed(dv). This means in turn that the faster the airplane moves, the better the jet propulsive efficiency gets. So as you climb, you go faster and the flow gets better efficiency, plus the lower air pressure behind you means there is less of a force pushing against your outflow.

So what do we have in low vs high flight:

Same amount of air intake, same amount of combustion, same amount of fuel used, better jet propulsion at higher altitudes and better speed at higher altitudes. You just get more bangs for your bucks at higher altitude.

For a mathematical approach:

• I assume, though, that every engine must have an altitude of maximum efficiency then? At some point the air is going to just be to thin to allow for maximized efficiency on the input, right? – Jay Carr Feb 7 '14 at 23:17
• no, not really: at some point the air becomes so thin you need to go very, very fast to keep the mass of air going in. And as you approach transonic speeds you get into trouble with a conventional engine. If you plan on a ramjet/scramjet engine you could go higher/faster, but you fuselage is now going to melt. Bottom line is: the top ceiling is never an engine-performance issue, it's always some other limitation – Radu094 Feb 7 '14 at 23:28

This is because the air is cooler and less dense which means that there is less fuel to air mixture at higher altitudes giving it a better fuel efficiency

The higher the altitude the thinner the atmoshere means less air resistance or drag on the airplane so it needs less engine thrust to push it. Thats lucky because the engine loses thrust with altitude at almost the same rate because as less air is available to the engine, the fuel system must reduce fuel to maintain the correct air/fuel ratio to support combustion and keep the engine alight. Its a win win situation.

The engine of an airliner is designed to be as efficient as possible over a journey containing a take-off, a climb, and most of its time at 35000 to 40000 feet where air pressure is about 1/4 to 1/5 of ground level. The engine has a few extra compression stages to work efficiently at normal cruise at the expense of overheating if flown for long at full power near ground level due to too much compression at the intake. Look up water injection for an interesting way to get take-off boost in a medium altitude engine in the 707.

I think most are simply thinking too much. The easiest and likely the most complete answer is resistance (or friction). High altitude air is less dense, making it easier to pass through. Oxygen content in high altitude is exactly the same as sea level. While the air up there is the same air we breath, there is less of that air in the same volume container. Space vehicles do not use jet engines. To turn or do any movement actually they have "jets" in various locations around the shuttle. "Jets" in this case are not jet engines, they are simply small nozzles that pressurized gases gets released through. Keep in mind that with zero air, there is no resistance to movement, remember newton's laws of motion: every action has an equal an opposite reaction.

• This answer would greatly benefit from a re-wording. Please consider editing your answer and explain in more detail the various arguments you bring forward. Consider also that formatting helps making an answer more readable – Federico May 16 '14 at 12:48

As you know as altitude increasing pressure and temp both are reducing up to stratosphere after that temp remain constant pressure drop continues so density of air reducing so its create less drag aircraft travel at high speed this pressure loss is overcome by ram pressure rise at the inlet of engine and aircraft required less power to move faster at 36,000 ft to 40,000 ft above more power full engine required to run faster so the blade tip does not get stall.

• Welcome to Stack Exchange! Please consider to rework your post; at its current state it is marred by spelling errors which make it hard to understand. – Peter Kämpf Sep 20 '14 at 13:08
• I fixed some of your worst spelling errors, but I cannot figure out where your sentences are. Sorry, but -1. – CGCampbell Sep 20 '14 at 16:08

I'm not even close to an expert. But here it goes.

Air like water is thick. Submarines are slower than boats. Jets are faster than boats. Automobiles are faster than boats.

Space has no friction because there is no matter. But I think jets still work in space. Of course they need oxygen. Just like Superman doesn't need ground friction to run fast, while some other super heroes do. Which is why I find it unrealistic how a super heroes that need ground friction are able to run so fast and make sharp turns without doing serious damage to floors.

So I'm guessing that being that the air is less thick up higher, that the easier it is to travel through. The jets don't depend on friction like propellers do. Superman doesn't need friction like Flash Gordon or Wonder Woman. So in space, Wonder Woman would be helpless because her propellers don't work while Superman's jets will work just fine.

Of course jet's need oxygen. So I'm not sure how all that works.

And something I didn't think of was what was mentioned in another post. Sound needs air. So yeah. Sound can increase friction.

• I don't know if I should vote you down for being ridiculous or vote you up for being creative. I guess they cancel each other out, so I'll just leave it alone ;). – Jay Carr Feb 13 '14 at 18:46

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