A Challenging Endurance Problem
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A Challenging Endurance Problem
Suppose I'm flying a Boeing 777-200ER (Trent 892), and am already at 500KTAS and 35,000ft. I weigh 217,800kg, of which 43,800kg is fuel.
Suppose my one and only goal is to stay in the air as long as possible. My only constraint: I must select a single speed (as fast or slow as I please) and altitude (as high or low as I please), head to it ASAP, and stay with it until flameout.
At what speed (in KTAS) and altitude (in ft) should I fly, and why?
Prize goes to whoever not only provides the answer, but PROVES how all OTHER combinations of speed & altitude MUST burn more fuel per hour.
Huge thanks to whoever tries to tackle this.
Suppose my one and only goal is to stay in the air as long as possible. My only constraint: I must select a single speed (as fast or slow as I please) and altitude (as high or low as I please), head to it ASAP, and stay with it until flameout.
At what speed (in KTAS) and altitude (in ft) should I fly, and why?
Prize goes to whoever not only provides the answer, but PROVES how all OTHER combinations of speed & altitude MUST burn more fuel per hour.
Huge thanks to whoever tries to tackle this.
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Engineer, not jet pilot response
The objective is to stay airborne for as long as possible at a constant ALT and KTAS.
This will be achieved by picking the speed and altitude that results in the lowest average fuel flow.
As fuel is burnt off, an ever lower fuel flow will maintain the set ALT and KTAS.
So the problem is surely to find the ALT and KTAS with the lowest fuel flow at which the plane can just remain airborne at its initial weight?
Problem is complicated slightly by fuel used to get to the initial setting and glide time after fuel exhaustion, but I would have thought these considerations wouldn't significantly change the answer.
This will be achieved by picking the speed and altitude that results in the lowest average fuel flow.
As fuel is burnt off, an ever lower fuel flow will maintain the set ALT and KTAS.
So the problem is surely to find the ALT and KTAS with the lowest fuel flow at which the plane can just remain airborne at its initial weight?
Problem is complicated slightly by fuel used to get to the initial setting and glide time after fuel exhaustion, but I would have thought these considerations wouldn't significantly change the answer.
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zlin
That will give max range, not endurance.
wind tunnel
Given your constraint of a single speed... Too difficult, because optimum endurance speed reduces as the a/c weight reduces due to fuel burn. Simple pilots don't have the tables to work it out.
That will give max range, not endurance.
wind tunnel
Given your constraint of a single speed... Too difficult, because optimum endurance speed reduces as the a/c weight reduces due to fuel burn. Simple pilots don't have the tables to work it out.
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So, to get back to the question -
What alts / speeds would give minimum fuel flow at:
starting weight / 50% of starting fuel / nearly 0 fuel? If they are similar - not much problem. If very different, we have to think more deeply.
What alts / speeds would give minimum fuel flow at:
starting weight / 50% of starting fuel / nearly 0 fuel? If they are similar - not much problem. If very different, we have to think more deeply.
Last edited by Andrewgr2; 19th Oct 2014 at 12:08. Reason: Punctuation & wording improved to clarify post
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Clearly, you would need detailed performance tables for the aircraft in order to answer WindTunnel's question. For starters, is there a holding table giving fuel flow at various altitudes based on weight of aircraft?
Secondly, is this table at the minimum fuel flow necessary to sustain flight at that altitude? If the answer is yes, then you have the solution to WindTunnel's question.
The general solution is for the aircraft to be nearly at its maximum altitude. This is because the engines would be required to operate at high power settings where the thrust produced per lb of fuel consumed will be greater than that at low altitudes and low power settings. Holding tables often demonstrate this, with lower fuel flow's at the higher altitudes
Secondly, is this table at the minimum fuel flow necessary to sustain flight at that altitude? If the answer is yes, then you have the solution to WindTunnel's question.
The general solution is for the aircraft to be nearly at its maximum altitude. This is because the engines would be required to operate at high power settings where the thrust produced per lb of fuel consumed will be greater than that at low altitudes and low power settings. Holding tables often demonstrate this, with lower fuel flow's at the higher altitudes
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General thoughts:
Max endurance is a holding problem, so fly at optimum holding altitude and speed. To satisfy the constraint of of a single speed and altitude for the rest of the flight, you could use an average weight for the rest of the flight. However, fuel flow will be higher at the initial weight, so I would bias toward that. So optimum holding at 2/3 fuel of the initial condition is probably good.
Finally, I suspect for a B777 single-engine holding is a lower fuel flow than holding with both engines operating (one engine provides sufficient thrust while operating at a more efficient RPM).
I would look at optimum SE holding at 200 or 205T. Probably the altitude is in the high teens/low 20s. If you need to take the descent into account, use a SE descent at the hold speed.
Max endurance is a holding problem, so fly at optimum holding altitude and speed. To satisfy the constraint of of a single speed and altitude for the rest of the flight, you could use an average weight for the rest of the flight. However, fuel flow will be higher at the initial weight, so I would bias toward that. So optimum holding at 2/3 fuel of the initial condition is probably good.
Finally, I suspect for a B777 single-engine holding is a lower fuel flow than holding with both engines operating (one engine provides sufficient thrust while operating at a more efficient RPM).
I would look at optimum SE holding at 200 or 205T. Probably the altitude is in the high teens/low 20s. If you need to take the descent into account, use a SE descent at the hold speed.
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It seems likely that any published figures would be for holding speeds safely above a minimum fuel flow which, I would expect to correspond to a speed not too far above the stall...
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I like AtoBsafely's thinking.
Without any data to reference my instinct would be to shut down an engine and drift down on one at current OEI dftftdown speed to current OEI ceiling and stick with that speed and alt. reducing thrust as weight reduced.
To be honest the problem isn't worth any more thought than that as the constraints of single alt and speed make it nothing more than a theoretical Q with no practical application.
Without any data to reference my instinct would be to shut down an engine and drift down on one at current OEI dftftdown speed to current OEI ceiling and stick with that speed and alt. reducing thrust as weight reduced.
To be honest the problem isn't worth any more thought than that as the constraints of single alt and speed make it nothing more than a theoretical Q with no practical application.
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Nothing to do with MH370? 
The question, as posed, has some mutually incompatible parameters. As fuel burns off, the aircraft will tend to want to climb higher, reducing drag and increasing ground speed. Constraining the aircraft to a single speed and altitude will not result in maximum endurance in either time or distance.
The question, as posed, has some mutually incompatible parameters. As fuel burns off, the aircraft will tend to want to climb higher, reducing drag and increasing ground speed. Constraining the aircraft to a single speed and altitude will not result in maximum endurance in either time or distance.
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Like with any aircraft - the max endurance is had at Vbg - best glide. If such is not published, it's pretty close to Vy, normally. I don't know how this pertains to jets, but the fundamentals are the same. And what else is that max endurance is independent of altitude - you get no benefit for flying high, at least not with pistons. I would assume the same would be true for turbines as they're both aspirated.
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I think most of the points necessary to solve the Ops question have now been made:
Gouli - I was assuming that the intention was to set KTAS and ALT and then (for whatever reason) leave the setting alone whilst the aircraft flew to flameout. The fact that this would not achieve max endurance was not relevant to the OP's specification and, so far as I can see, does not make the original question invalid.
AdamFrisch - Has raised the important point that what we need to do is minimise drag in order to get max endurance and that this is achieved at Vbg or Vy. This will vary with weight but if,as AtoBsafely has suggested, we assume 2/3 of the original fuel, we won't be far off.
AtoBSafely has raised the point that we would be more efficient operating on one engine at a higher power than two engines at lower power. Would the additional drag caused by asymmetric flight outweigh the advantage and would the on-board systems feed all the fuel to the one running engine without further intervention?
AdamFrisch has also suggested that endurance is independent of altitude. Instinctively this sounds right - maybe someone can comment on whether fuel flow varies for a given KTAS (approx Vy) at different altitudes.
So the question may boil down to - what is the fuel burn, independent of altitude, on a single engine, to fly at Vbg (or Vy)at approx 2/3 of the original fuel load? Given that we start with 43,800 kg of fuel, how long will this last?
Gouli - I was assuming that the intention was to set KTAS and ALT and then (for whatever reason
) leave the setting alone whilst the aircraft flew to flameout. The fact that this would not achieve max endurance was not relevant to the OP's specification and, so far as I can see, does not make the original question invalid.AdamFrisch - Has raised the important point that what we need to do is minimise drag in order to get max endurance and that this is achieved at Vbg or Vy. This will vary with weight but if,as AtoBsafely has suggested, we assume 2/3 of the original fuel, we won't be far off.
AtoBSafely has raised the point that we would be more efficient operating on one engine at a higher power than two engines at lower power. Would the additional drag caused by asymmetric flight outweigh the advantage and would the on-board systems feed all the fuel to the one running engine without further intervention?
AdamFrisch has also suggested that endurance is independent of altitude. Instinctively this sounds right - maybe someone can comment on whether fuel flow varies for a given KTAS (approx Vy) at different altitudes.
So the question may boil down to - what is the fuel burn, independent of altitude, on a single engine, to fly at Vbg (or Vy)at approx 2/3 of the original fuel load? Given that we start with 43,800 kg of fuel, how long will this last?
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AtoBsafely FEHoppy
Look up
Linktrained PPrune "Fuel not used"
Sorry not able to give a better reference.
The aircraft was a twin engined biplane ( ! ) and with own my lack of experience, I was reluctant to shut down my Port engine. I was required to fly at 3000 ft for the task, which I did.
Before it got dark, I had not been able to see that the Port exhaust was glowing (or not !). So the first hour was flown using both engines.
After returning to Liverpool The G/Es wanted to know why I had only refuelled one side. ( I hadn't !)
No performance information was available to me, anyway !
Look up
Linktrained PPrune "Fuel not used"
Sorry not able to give a better reference.
The aircraft was a twin engined biplane ( ! ) and with own my lack of experience, I was reluctant to shut down my Port engine. I was required to fly at 3000 ft for the task, which I did.
Before it got dark, I had not been able to see that the Port exhaust was glowing (or not !). So the first hour was flown using both engines.
After returning to Liverpool The G/Es wanted to know why I had only refuelled one side. ( I hadn't !)
No performance information was available to me, anyway !
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Taking this as an academic question, I wouldn't know where to begin without knowing such specifics such as the drag values and fuel vs thrust.
From a practical perspective, one simply reduces thrust until minimum fuel burn is achieved. Given that the drag curve on a transport jet with a clean wing tends to be very flat at the bottom, there probably is negligible difference for a relatively wide range of speeds, but flaps up min maneouvering speed would probably be the best start point. If the situation is particularly dire, reduce below min manoeuvring speed if the fuel burn is lower.
Also, probably best to disconnect the A/T too, to prevent unnecessary thus lever movement.
From a practical perspective, one simply reduces thrust until minimum fuel burn is achieved. Given that the drag curve on a transport jet with a clean wing tends to be very flat at the bottom, there probably is negligible difference for a relatively wide range of speeds, but flaps up min maneouvering speed would probably be the best start point. If the situation is particularly dire, reduce below min manoeuvring speed if the fuel burn is lower.
Also, probably best to disconnect the A/T too, to prevent unnecessary thus lever movement.
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AdamFrisch - Has raised the important point that what we need to do is minimise drag in order to get max endurance and that this is achieved at Vbg or Vy. This will vary with weight but if,as AtoBsafely has suggested, we assume 2/3 of the original fuel, we won't be far off.
AtoBSafely has raised the point that we would be more efficient operating on one engine at a higher power than two engines at lower power. Would the additional drag caused by asymmetric flight outweigh the advantage and would the on-board systems feed all the fuel to the one running engine without further intervention?
AdamFrisch has also suggested that endurance is independent of altitude. Instinctively this sounds right - maybe someone can comment on whether fuel flow varies for a given KTAS (approx Vy) at different altitudes.
So the question may boil down to - what is the fuel burn, independent of altitude, on a single engine, to fly at Vbg (or Vy)at approx 2/3 of the original fuel load? Given that we start with 43,800 kg of fuel, how long will this last?
1. Re "AdamFrisch". Max endurance will not necessarily be at least drag for a 777. Max endurance will be at minimum fuel flow, and minimum fuel flow will be a function of not only drag, but engine thrust per pound of fuel consumed.
2. Re operating on one engine vice 2; it may be possible at low altitudes for a jet to have more endurance (less fuel flow on one than two), but I have not seen any numbers to suggest that this can be true at higher altitudes
3. max endurance (time) is certainly not independent of altitude. As I said earlier, the thrust specific fuel consumption is much higher at lower power settings than higher ones, hence higher power settings will give more thrust per pound of fuel burned. Caveat, I use the term "high power settings" to indicate those not far below maximum N1.