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A Challenging Endurance Problem

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A Challenging Endurance Problem

Old 19th Oct 2014, 18:31
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Hawk,
my reasoning was that we have to chose a single speed and altitude. Given that we are probably above OEI ceiling but below AEO ceiling we are already too low for optimum 2 eng endurance so we either climb now or start throttling back both. If we chose the single eng ALT and speed we get the bonus of height in hand. That gets us ahead of the game as we drift down with one at MCT. Then we only misuse on engine as we throttle back.
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Old 19th Oct 2014, 18:44
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Yes FE Hoppy, to get max endurance as the OP requested, it does seem best to be at single engine and almost a high as possible.
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Old 19th Oct 2014, 19:13
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It is a known fact that max range (and I know this is difference from max endurance, but not by much) is achieved independent of altitude, as per Carson. Because L/D ratio increases with altitude. There is no benefit by going high, especially when you also have to account for the fuel to get up there.

Therefore, and this is an educated guess, I'm pretty certain that a jet will have no endurance benefit by going to higher altitudes. Take off, level off at lowest obstacle clearance altitude and pull back to max endurance (which will be lower than Vbg).
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Old 19th Oct 2014, 20:37
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OK465: But the OP also said:
my one and only goal is to stay in the air as long as possible
so although your neat solution meets the constraints, it doesn't meet the goal. I think the goal has to be met within the constraints!

Hawk: I'm struggling a bit with the idea that min drag may not be the optimum speed for maximum endurance. Min Drag = Min Thrust. Assuming constant altitude, are there situations where a jet engine can produce more thrust whilst burning fuel at a lower rate? It's only the absolute rate at which fuel is consumed that is important here, not the efficiency with which it is consumed.

Last edited by Andrewgr2; 19th Oct 2014 at 20:38. Reason: Spotted error in my statement
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Old 19th Oct 2014, 20:38
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@AdamFrisch
It is a known fact that max range (and I know this is difference from max endurance, but not by much) is achieved independent of altitude,
what?

http://www.pprune.org/tech-log/39997...ml#post5416515
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Old 19th Oct 2014, 21:12
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AdamF,
Like with any aircraft - the max endurance is had at Vbg - best glide.
Why do you think that all aircraft are the same AdamF?
The airframes obey the same laws but the engines are different animals. The minimum rate of fuel flow for a 777 will occur at a speed slower than min drag.
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Old 19th Oct 2014, 21:23
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The missing bit of information in this equation is the nature of the prize.
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Old 19th Oct 2014, 21:31
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Andrewgr2.

When I said max endurance will not be at minimum drag speed, I was primarily meaning that the minimum drag would be at low altitude, at the speed for L/D max. If one believes that thrust vs fuel flow is flat in at this speed, then minimum drag will be max endurance speed. However, the thrust to fuel flow curve depends on the engine, and for those I've seen, at low altitude, at higher speeds the fuel flow for the same thrust is higher. Hence, in this case, the max endurance speed will be slightly lower than the minimum drag speed.
Now let's add some altitude to the flying aircraft. With that comes extra drag due to compressibility. Hence, the minimum drag at that higher altitude is more than the minimum drag at the lower altitudes. A natural conclusion could be that the fuel flow would thus be lower at low altitude than at high altitude, for max endurance. This conclusion is normally wrong. If you look at holding charts for aircraft with jet engines, you will see that the fuel flow usually decreases with altitude. Now I know that holding speed is not necessarily max endurance speed, but it is normally close enough to demonstrate that higher is slightly better for max endurance.
What is the reason for that you ask? The reason is that when holding at low altitudes the jet engine is at a relatively low power setting, at low fuel efficiency, and this means less pounds of thrust produced for a given fuel flow than at the higher power settings, and higher fuel efficiency available when at the higher altitudes for a high level hold. So, although minimum drag is higher at higher altitudes, the lower specific fuel consumption of a jet engine at higher rotational speeds has a more profound effect on the fuel flow, and hence the fuel flow is slightly lower at higher altitudes.

Now shut down an engine and compare the fuel efficiencies and endurance.....that's would need further analysis...

I agree with FE Hoppy, of course maximum range increases with altitude.

Hope that helps.

Last edited by hawk37; 19th Oct 2014 at 21:40. Reason: changed "better" to "lower", an oops on my part....
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Old 19th Oct 2014, 22:51
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I don't know enough about turbines, so I will happily defer that there are added benefits by going higher for these propulsive systems.

But.

Here's a specific reference saying that endurance is independent of altitude even for jets. Only max range benefits from higher altitudes as TAS goes up. So, I was half right since we were talking about endurance initially. My mistake was that I though they both were independent of altitude.

Aircraft Performance - Maido Saarlas - Google Books
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Old 19th Oct 2014, 23:49
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Here's a specific reference saying that endurance is independent of altitude even for jets
Adam, I'm afraid he is incorrect as he has not taken into account the efficiency of turbine engines with respect to altitude. The higher the altitude the more efficient they are in terms of fuel burnt for the thrust produced. Higher means colder air for consumption, and the rotor/s are operating at higher RPM, both elements increasing efficiency. Theoretically a jet will have its greatest endurance at or near the tropopause. (As taught by the USN and USAF)

What is taught by the RAF and RAAF
Principles

1. Broadly, since fuel flow is proportional to thrust, fuel flow is least when thrust is least; therefore maximum level flight endurance is obtained when the aircraft is flying at the IAS for minimum drag (VIMD), because in level flight thrust is equal to drag.

2. Maximum endurance is obtained at an altitude which is governed by engine considerations. Although for a given set of conditions the IAS for minimum drag remains virtually constant at all altitudes, the engine efficiency varies with altitude and is lowest at the lowest altitudes where rpm must be severely reduced to provide the low thrust required.

3. To obtain the required amount of thrust most economically, the engine must be run at maximum continuous rpm. Therefore maximum endurance is obtained by flying at such an altitude that, with the engine(s) running at or near optimum cruising rpm, just enough thrust is provided to realize the speed for minimum drag, or MCRIT, whichever is the lower. Above the optimum altitude little, if any, additional benefit is obtained, and in some cases there may be a slight reduction because burner efficiency decreases at or about the highest altitude at which level flight is possible at VIMD. In general, optimum endurance is obtained by remaining between 20,000 ft and the tropopause at the recommended IAS and appropriate rpm. The greater the power/weight ratio of the aircraft, the greater will be the optimum height. With aircraft having high power/weight ratios, maximum endurance is obtained at the tropopause.

4. Altitude should only be changed to that for maximum endurance if the aircraft is above or near the endurance ceiling, otherwise if the aircraft is climbed from a much lower altitude, a considerably higher fuel flow will be required on the climb thereby reducing overall endurance.

5. On engines having variable swirl vanes, the consumption increases markedly if the rpm are so low that the swirl vanes are closed. If the altitude is low enough to cause the swirl vanes to close at the rpm required for VIMD, the aircraft should either be climbed to the lowest altitude at which VIMD can be obtained with the swirl vanes open, or the rpm increased to the point at which the vanes open, accepting the higher IAS.
Effect of Weight

6. Drag and thrust at the optimum IAS are functions of the all-up weight; the lower the weight the lower the thrust and fuel flow. Endurance varies inversely as the weight and not as the square root of the weight as in range flying because in pure endurance flying the TAS has no importance.
Effect of Temperature

7. In general, the lower the ambient air temperature, the higher the endurance, due to increased thermal efficiency, and vice versa. However, the effect is not marked unless the temperature differs considerably from the standard temperature for the altitude. In any case, the captain can do nothing but accept the difference, since any set of circumstances requiring flying for endurance usually ties the aircraft to a particular area and height band.

Twin and Multi-Engine Aircraft

8. When flying for endurance in twin or multi-engine aircraft at medium and low altitude, endurance can be improved by shutting down one or more engines. In this way the live engine(s) can be run at rpm closer to the optimum for the thrust required to fly at VIMD, thus improving GFC. Provided that the correct number of engines are used for the height, altitude has virtually no effect on the endurance achieved.
Use of the Fuel Flowmeter

9. The fuel flowmeter is a useful aid when flying for endurance. If the endurance speed is unknown, the throttle(s) should be set at the rpm which give the lowest indicated rate of fuel flow in level flight for the particular altitude.

10. Whenever reporting aircraft endurance, the time for which the aircraft can remain airborne should be given. It is confusing and dangerous to report endurance in terms of amount of fuel remaining because of the possibility of mis-interpretation.
Conclusions

11. Maximum endurance is achieved by flying at an altitude where optimum rpm give the minimum drag speed. It will rarely pay to climb to a higher altitude unless the commencing altitude is very low; in any event, the instruction or need to fly for endurance may preclude this. At the lower altitudes, maximum endurance may be obtained either by flaming-out engines to use optimum rpm on the remainder, or by using near-optimum rpm to give VIMD. It should be remembered that:

a. The importance of flying at VIMD outweighs engine considerations, always assuming that an engine, or engines, will be stopped in the low level case.

b. At lower altitudes there will be a slight decrease in endurance due to the higher ambient temperature reducing engine thermal efficiency.
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Old 20th Oct 2014, 00:47
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Adam;

"endurance is independent of altitude even for jets" only applies if you consider two things:

1. there is no additional drag on the aircraft due to compressibility, and

2. the fuel required to produce a certain amount of thrust is independent of the aircraft's speed.

For simplicity, many aerodynamics texts make these assumptions.
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Old 20th Oct 2014, 00:51
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Linktrained,

Just to be clear, shutting down an engine is being suggested in this hypothetical problem only. Not recommended for normal operations.

Megan,

Thanks for the reference. I was thinking of an older generation engine, and having reviewed data for a new engine I see that best holding is up high. So I revise my guess to fly at SE cruise ceiling and descend in a drift down.
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Old 20th Oct 2014, 01:03
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Just to be clear, shutting down an engine is being suggested in this hypothetical problem only. Not recommended for normal operations.
Not hypothetical in the military, regularly used in the P-3 community, and sob, the Nimrod when it was with us. Not an airline though I would hope, though it might give a airline beancounter somewhere an idea if s/he hears of the procedure.
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Old 20th Oct 2014, 02:03
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Yes airlines too. Shutting down an engine in cruise was standard operating procedure with Aeroflot in the early 90's - at least that was my experience as a SLF back then. Personally I see nothing wrong with it.

AdamFrisch is correct with his reference that endurance (not range) is independent from altitude. These curves are normally not published in AFMs unless you operate a military patrol aircraft. When you look at these curves, best endurance is surprisingly flat between some low altitude like 10000ft right up to tropopause.
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Old 20th Oct 2014, 07:18
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From 'megan' and RAF
the engine must be run at maximum continuous rpm
- my understanding is that that related to 'standard' axial flow engines and now-a-days it should read 'at design rpm'.

Shutting down a donk was SOP on the Lightning if short of fuel (weren't we always......) for that very reason, plus there was very little trim drag associated with the event unlike a 'normal' twin.
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Old 20th Oct 2014, 13:42
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GOING BACK TO BASICS
The aircraft initially has a certain quantity of stored usable energy made up of:

a. Chemical energy by virtue of its fuel load.
b. Potential energy by virtue of its height.
c. Kinetic energy by virtue of its speed.

To achieve maximum endurable we must expend the stored energy as slowly as possible.

MINIMIZING CHEMICAL ENERGY EXPENDITURE RATE
The rate of expenditure of chemical energy is essentially:

Fuel consumption rate = Thrust required x Specific Fuel Consumption.

(SFC is the mass of fuel that is required per hour to generate each unit of thrust).

Looking at the above equation we can see that to minimize the fuel consumption rate a good starting point would be to minimize the thrust required and to minimize the SFC.

Thrust required is equal to drag, so our ideal speed should be VMD. The drag at any given EAS does not vary significantly with altitude until compressibility effects kick in, so provided we do not go too high, the altitude should not affect the drag, thrust required, or fuel consumption.

SFC is more problematic because it varies with a number of factors including air temperature, air pressure and RPM. The most significant of these factors is RPM.
At very low RPM the majority of the fuel energy is used simply in overcoming friction and keeping the engine running, so little thrust is produced for each unit of fuel burned. This means that the SFC is very high at low RPM. As RPM increases, the basic running costs (in terms of energy) represent a decreasing proportion of the total fuel flow. The aerodynamic and thermal processes also become more efficient. This means that the SFC gradually decrease as RPM increases. Most text books quote a between 90% to 95% RPM as the RPM range within which SFC is lowest. The precise value of the optimum RPM will of course vary from engine to engine, but the figure of 90% to 95% is a reasonable starting point for the purposes of this thread.

Combining the above factors shows that for our maximum endurance we need to fly at VMD with our engines (or engine) running at 90% to 95% RPM.

At low altitude the thrust of both engines running at 90% to 95% RPM will be too great to balance the drag at VMD. Increasing drag using spoilers or flaps would simply waste fuel so this is not an option. But the thrust at any given RPM decreases as altitude increases, so if we climb to suitably high altitude we will achieve the required balance between drag at VMD and thrust at 90% to 95% RPM. But even at this altitude, the fuel flow with both engines running at 90% to 95% RPM will be far greater than that with one engine shut down and the other running at 90% to 95%. So a better solution would be to fly at the altitude where single engine thrust at 90% to 95% RPM is balances the drag at VMD.


MINIMIZING POTENTIAL ENERGY AND KINETIC ENERGY EXPENDITURE RATE
The power required is the rate at which an aircraft expends energy. So to minimize the expenditure rate of stored potential end kinetic energy during the descent to cruise altitude, the best option would be to shut down both engines and fly at VMP. So ideally we would descend with all engine out at VMP then star one engine and run it up to 90%-95% RPM as we level off at cruise altitude at VMD.

Unfortunately this contravenes the requirements to “fly at a single speed” and to “reach the cruise altitude asap”. Reaching the cruise altitude in minimum time would require a maximum speed descent, but then maintaining max sped in the cruise would not achieve maximum endurance.

ADJUSTING FOR REAL-WORLD FACTORS
All of the above does not of course take into account the actual performance characteristics of the specified aircraft and engines. To do this we would need to examine the appropriate data and adjust our airspeed, RPM and altitude accordingly.
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Old 20th Oct 2014, 14:03
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Well I don't have 777 documents but for the E190 the figures look something like this:

Speed - 210kt chosen because: a) I have numbers for this speed at all weights. b) Its just behind the drag curve at heavy weights and plenty ahead at light weights (probably the best compromise for fixed speed)

Driftdown not calculated.

Start weight 50T End weight 30T (only the lineage version can carry this amount of fuel)

All Engines operating
FL 300 - 869 minutes
FL 250 - 872 Minutes
FL 150 - 850 minutes

One Engine shutdown
FL 150 - 868 Minutes

These values assume Air conditioning AND Anti-Ice ON. (only tables with fixed speed) So the engine bleed will have them running at higher RPM which probably skews the results a little.

Not sure why I spent an hour on this but there you go.
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Old 21st Oct 2014, 00:25
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AtoBsafely


I had explained in that quote:
"I throttled back the Port engine until the exhaust was dark enough although still giving some power..."
Of course this would be at an inferior specific consumption of petrol on the Port engine. The Starboard remained at its normal cruising power, whatever that may have been ! (My Type Rating has lapsed, now.)
I had remained close to Blackpool aerodrome, which I see has just closed after over a century of use.


I spent most of my time on 4 engines with a F/E, so operating with one shut down was a part of life, sometimes.
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Old 21st Oct 2014, 02:28
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I'd go to current Max Altitude and speed at the top of the yellow arc (min safe speed). Overall, for the rest of the flight, that would probably be closest to max endurance altitude and speed.
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Old 21st Oct 2014, 04:15
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my understanding is that that related to 'standard' axial flow engines and now-a-days it should read 'at design rpm'
I don't know BOAC, not having the necessary experience, but as a 737 driver, how close to max continuous was your cruise RPM? The heavy drivers have an answer for their types? Do you in fact ever get close to max continuous (besides take off)?
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