Jet efficiency at high altitudes

[Finals19]

Doing some reading here and trying to define why a jet is more efficient than a TP at high altitudes...?

In conclusion, I found that:

A turbo prop is only efficient up to a certain altitude, beyond which you will get no increase in specific fuel consumption and SAR. This is due to thrust decreasing with increasing TAS at a lower altitude than a jet.

Jet engine is more efficient at altitude due to better thermal efficiency and air being less dense. Better SFC and SAR higher up you go, unlike a TP where it starts to flatten off?

Can we also assume that increasing TAS is also limited by propellor tip speed on a TP?

[TruBlu351]

Jet engines make awesome thrust at low level, due to the huge volume of dense air sucked in by the compressor, which needs a huge amount of fuel to balance the air/fuel combustion ratio. You'd get somewhere real quick, but probably only halfway there due to running out of fuel!!.....you'd also overspeed the airframe big time. Thrust at sea level well exceeds drag.

Jet engines are most efficient operating up at their higher rpm, 90'ish % rpm. Thrust decreases with increasing alt due to less dense air as you mentioned, so there's a nice balance found at high altitude where the thrust at ~90% balances drag to give a comfortable airspeed and mach number.

Unlike a prop, with a jet engine the thrust increases with TAS due to ram effect in the intake.....which can become too fast where you have supersonic air hitting the front compressor (which is bad), so it's slowed down just enough with a diverging intake.

The airframe is then designed around this regime using a lower camber swept wing.............allows high altitude speed with delayed onset of mach induced drag.

The turbo prop hit's a brick wall in terms of efficiency as speed increases. At a standstill the blade cuts the air almost perpendicular to it's direction.....lots of thrust. If you can imagine it now doing 300+kts.....the prop, which will now be very coarse, is cutting the air almost parallel to it's direction....the vector diagram on the prop blade results in less thrust being produced.....up to the brickwall, whereas the jet keeps going hard as it doesn't have this physical limitiation.

Again, the airframe is built to optimise this arena....no sweep, higher aspect ratio.

There's also supersonic mach issues with props at high alt....at tips as you mention.

[enicalyth]

The thrust developed by an air-breathing jet engine is derived from two terms commonly, but erroneously known as the momentum and pressure states. Pprune does not support mathematical symbols so let me begin by defining m1 as the mass flow rate of air into the engine and m2 as the mass flow rate out of the engine. The difference is due to the amount of fuel added which we intend to be as small as possible.

Similarly I am now going to define u1 as the aircraft speed in still air on a standard day whereas u2 is the exhaust speed coming out the back.

You’re already with me, p1 is the ambient air pressure and p2 is the pressure across the back nozzle but I’m even further ahead than you because for the present I’ll just assume that they are they same and drop all reference to the pressure state.

I can now reduce Whittle’s equation to one stark line

Thrust T = (m2 x u2) – (m1 x u1)

It is therefore inevitable that to produce any thrust at all and be fuel efficient, the exhaust speed must be higher than the aircraft speed. Don’t worry for the present that the exhaust may be supersonic because the exhaust was very recently very much on fire and its idea of the local speed of sound (which is directly proportional to the square root of absolute temperature) is very high!!

You’ve got it in one. To go extremely fast your exhaust must go extremely faster. Even then when the exhaust velocity does reach its own internal local speed of sound the exit nozzle will choke. All you can do to get more thrust is to exploit the pressure state which I have ignored and increase the size of the nozzle cross-sectional area.

So turbojets can go extremely fast but in doing so they waste all that kinetic energy in the jet efflux and that energy comes from the fuel. Lots of it.

Let’s be happy with subsonic transport and invent the turbofan. Now Whittle’s equation has two stark lines. One for the hot stream of burnt gas and the other for the cold stream propelled by the fan. A very substantial amount of thrust is produced by the fan but unfortunately the hot exhaust stream is still travelling faster than the aircraft but not as fast as it would for a turbojet. So the kinetic energy wasted in the unmixed turbofan efflux is not so large as it otherwise could have been and relatively speaking with regard to the turbojet the fuel consumption is much reduced. We like that but excluding consideration of the frontal area of the engine(s) the schoolboy dream of vast turbofan engines powering Concorde Mark 2 at speeds up to Mach 3 just does not hold, er, air.

Lets go further still and invent the propeller aircraft. Now we don’t give two hoots about the thrust produced by the hot stream emanating from Rubbra’s ejector exhausts (though Stanley Hooker, Harry Reed and Alan Yarker did give several hoots). As far as we are concerned the thrust is all produced in the cold stream which still has to move faster than the aircraft, but only just. Consequently the wasted kinetic energy is not a lot and the fuel bill is quite low. But you can’t fly very fast and yes one limiting factor is going to be the vector sum of aircraft forward speed and propeller tip speed. And oops, if you are not flying very fast then a propeller driven aircraft just cannot ingest enough air in mass flow rate terms so you run out of altitude basically.

When we talk about thrust specific fuel consumption by the way we mean the mass flow rate of fuel divided by the thrust. So in a turbojet which has only one stream to consider, (the hot), and if the fuel-air ratio is defined as f then Whittle’s second equation reduces to

TSFC = f/[(1+f) x (u2-u1)].

I’ll leave you to derive the turbofan and prop versions where both hot and cold streams come into play. If you get stuck buy “Mechanics and Thermodynamics of Propulsion by Hill and Peterson” , ISBN 0201146592 and have them do it for you. Also buy “The Performance of a Supercharged Aero Engine” ISBN 1872922112.

I hope that this sketch of mine has been both simple and helpful. When Whittle and Hooker met the former began to describe his engine by saying “Well its all rather simple really..”. Stanley Hooker stopped him immediately and said “Don’t worry, I’m here to complicate it for you”. See what happens when one giant stands on the shoulders of another. And these days I have trouble climbing out of bed.

Best regards

The “E”

PS: Farnborough is also known as “Buy the Fat Australian a Beer Week”. I drink Toohey’s. Lots of it and to hell with wasted kinetic energy.