Max Endurance, Max Range...
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Originally Posted by Anderson, Introduction to Flight, Section 1.16.2
Maximum endurance for a propellor-driven airplane occurs when the airplane is flying at a velocity such that (C_L)^(3/2)/C_D is at its maximum
Maximum endurance for a propellor-driven airplane occurs when the airplane is flying at a velocity such that (C_L)^(3/2)/C_D is at its maximum
Actually only max L/D is defined only by the lift and drag characteristics of the airplane. Both Endurance and Range involve the propulsion system fuel efficiency, and strictly speaking the optimum speed cannot be defined by reference to cL and cD alone. For a jet transport airplane with an engine perfectly matched to the airframe at optimum weight/altitude/speed, the speed at which the engine operates at its minimum sfc point will be close to the airplane max L/D speed, and hence max range speed will be close to max L/D speed.
regards,
HN39
Last edited by HazelNuts39; 15th Oct 2010 at 09:07. Reason: editorial improvement
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Originally Posted by PBL
Originally Posted by Anderson, Introduction to Flight, Section 1.16.2 ....
Originally Posted by HN39
Anderson apparently makes the simplifying assumption that power-specific fuel consumption is constant, which in practice it is not
The issue is not whether and how one makes simplifying assumptions, which are part and practice of doing any engineering whatsoever, but how sensitive your answers are to those assumptions.
Given typical data on engines, it seems to me that the quadratic effect of velocity on the aerodynamic calculations generally overwhelms the rather linear (with low slope) relation of velocity to available thrust. But I am open to being otherwise persuaded if someone can produce data to say so.
PBL
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I'm not quite sure that the OP has been satisfactorily answered. The 'Anderson' quote is about propeller-driven airplanes, for which many introductory textbooks make the simplifying assumption that P-sfc is constant (fuel flow is proportional to power). Similarly, for jet-powered airplanes, as already clearly explained by Rivet Gun, the usual simplifying assumption is that T-sfc is constant (fuel flow is proportional to thrust). In that case minimum fuel flow in level flight (max. endurance) is minimum thrust for level flight, i.e. max L/D. Right?
regards,
HN39
regards,
HN39
Last edited by HazelNuts39; 16th Oct 2010 at 08:31.
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While Max Endurance and Max Range might not be much different on paper...in reality..if you end up over an Island and have to hold for 4 hours because the one runway is blocked...I doubt you will do it FL450...but something more realistic in the the teens....on the other hand, if you wanted max range, you will probably (in a jet) take it up as high as possible to maximize fuel burn(and TAS) and winds to get the best speeds....
Between the two, in the real world..one is about getting somewhere the other is just hanging up there as long as possible..
Between the two, in the real world..one is about getting somewhere the other is just hanging up there as long as possible..
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hi PBL, thanks for your response, but i am still a bit confused, and i guess it comes down to what i've read (or haven't read).
why for a jet engine, point of Min Total Drag is equal to max endurance (i can see this since thrust is also lowest, and assuming fuel varies directly with thrust this would be the minimum required to keep the plane airborne), and on a piston plane, the point of min drag is used for max range since it gives the best L/D ratio?
from what i understand, max endurance on a piston plane comes from min power, but how come this min power doesn't correspond to the min point on the total drag curve? is it because the fact that piston engine doesn't provide the thrust directly?
sorry for the questions, just finding this really confusing, which it shouldn;t!
cheers,
why for a jet engine, point of Min Total Drag is equal to max endurance (i can see this since thrust is also lowest, and assuming fuel varies directly with thrust this would be the minimum required to keep the plane airborne), and on a piston plane, the point of min drag is used for max range since it gives the best L/D ratio?
from what i understand, max endurance on a piston plane comes from min power, but how come this min power doesn't correspond to the min point on the total drag curve? is it because the fact that piston engine doesn't provide the thrust directly?
sorry for the questions, just finding this really confusing, which it shouldn;t!
cheers,
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johns7022;
The A330 FCOM gives information for holding at 'green dot' speed in clean configuration, which is a speed between minimum fuel speed and minimum drag speed.
Fuel consumption varies only slightly with altitude, for example at a typical weight it is 2038 kg/h/eng at FL15, 1980 at FL150 and 1960 at FL250.
regards,
HN39
So you fly an Airbus at FL250? If the Island up ahead is weathered in, chances are I will loiter at FL410 rather then go down and hold at FL150, but I would have to look at the actual numbers to determine that....
The A330 FCOM gives information for holding at 'green dot' speed in clean configuration, which is a speed between minimum fuel speed and minimum drag speed.
Fuel consumption varies only slightly with altitude, for example at a typical weight it is 2038 kg/h/eng at FL15, 1980 at FL150 and 1960 at FL250.
regards,
HN39
So you fly an Airbus at FL250? If the Island up ahead is weathered in, chances are I will loiter at FL410 rather then go down and hold at FL150, but I would have to look at the actual numbers to determine that....
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When studied at ATPL level the conventional explanations for best endurance and best range speeds are as follows.
Endurance is the amount of time that we can fly using the quantity of fuel that we have on board the aircraft. So to maximise endurance we must minimise the rate at which we use the fuel. So maximum endurance in any aircraft type is achieved by flying at the speed at which the rate of fuel flow is minimum.
Fuel flow in a jet aircraft is proportional to thrust, so maximum jet endurance is achieved when the thrust required for a given TAS is at a minimum. In straight and level flight, thrust = drag, so we could say that the shape of the drag / TAS curve is representative of fuel flow / TAS curve. Best endurance occurs at the speed at which fuel flow is minimum, so in straight and level flight, the best jet aircraft endurance will be achieved at minimum drag speed, Vmd. Vmd occurs at the lowest point on the drag /TAS curve.
The relationship between fuel flow and thrust is less direct for a propeller aircraft. This is because the engine does not produce thrust directly. The engine produces Power through its output shaft. This power is fed to the propeller where it is used to generate thrust. The efficiency with which the propeller uses the power to produce thrust is not constant, but varies with such things as blade angle, RPM and aircraft TAS.
Fuel flow in a propeller aircraft is proportional to the amount of power that is being produced by the engine. So we can say that the shape of the power required / TAS curve is representative of the Fuel Flow / TAS curve. So maximum propeller aircraft endurance is achieved by flying at the speed at which power required is a minimum, Vmp. Vmp occurs at the lowest point on the Power required. TAS curve.
Note that in both cases maximum endurance occurs at the lowest point on the curve.
To achieve maximum range it is not sufficient to burn the fuel slowly. We must fly as far as possible for each unit of fuel consumed. This means that we will achieve maximum range by flying at the speed at which the ratio of Ground speed to TAS is a maximum. If we assume still air conditions this means that the ratio of TAS to Fuel Flow must be maximised.
For a propeller aircraft we look at the power required (or fuel flow) / TAS curve. To find the speed at which the ratio of TAS to Fuel Flow is maximum we draw a tangent from the origin to touch the curves. The maximum range speed is the point at which this tangent touches the curve. For all aircraft types, A tangent from the origin to the Power Required / TAS curve touches the curve at Vmd. So maximum range speed for a propeller aircraft is Vmd.
For a jet aircraft we draw a tangent from the origin to the Drag (or fuel flow) / TAS curve. This tangent touches the curve at about 1.32 Vmd. So maximum range speed for a jet aircraft is about 1.32 Vmd.
Note that in both cases (props and jets) max endurance is at the bottom of the curve and max range is where a tangent from the origin touches the curve.
BUT
The above explanations are based on the assumptions that the Power Specific Fuel Consumption (for propeller aircraft) and the Thrust Specific Fuel Consumption (for jet aircraft) are constant at all speeds. Because of these assumptions, the predicted speeds for best endurance and best range (Vmp and Vmd for props and Vmd and 1.32 Vmd for jets) are not entirely accurate.
If we wish to study the subject at a higher lever, such as for an Aerospace Engineering Degree, we would need to look into the aerodynamics as PBL has done in his posts. The conclusions would then be in a different format, and may look more impressive, but would still produce essentially the same results. If we limited our examination to aerodynamic factors and ignored the variable nature of SFC, we would again produce results that were not entirely accurate.
Endurance is the amount of time that we can fly using the quantity of fuel that we have on board the aircraft. So to maximise endurance we must minimise the rate at which we use the fuel. So maximum endurance in any aircraft type is achieved by flying at the speed at which the rate of fuel flow is minimum.
Fuel flow in a jet aircraft is proportional to thrust, so maximum jet endurance is achieved when the thrust required for a given TAS is at a minimum. In straight and level flight, thrust = drag, so we could say that the shape of the drag / TAS curve is representative of fuel flow / TAS curve. Best endurance occurs at the speed at which fuel flow is minimum, so in straight and level flight, the best jet aircraft endurance will be achieved at minimum drag speed, Vmd. Vmd occurs at the lowest point on the drag /TAS curve.
The relationship between fuel flow and thrust is less direct for a propeller aircraft. This is because the engine does not produce thrust directly. The engine produces Power through its output shaft. This power is fed to the propeller where it is used to generate thrust. The efficiency with which the propeller uses the power to produce thrust is not constant, but varies with such things as blade angle, RPM and aircraft TAS.
Fuel flow in a propeller aircraft is proportional to the amount of power that is being produced by the engine. So we can say that the shape of the power required / TAS curve is representative of the Fuel Flow / TAS curve. So maximum propeller aircraft endurance is achieved by flying at the speed at which power required is a minimum, Vmp. Vmp occurs at the lowest point on the Power required. TAS curve.
Note that in both cases maximum endurance occurs at the lowest point on the curve.
To achieve maximum range it is not sufficient to burn the fuel slowly. We must fly as far as possible for each unit of fuel consumed. This means that we will achieve maximum range by flying at the speed at which the ratio of Ground speed to TAS is a maximum. If we assume still air conditions this means that the ratio of TAS to Fuel Flow must be maximised.
For a propeller aircraft we look at the power required (or fuel flow) / TAS curve. To find the speed at which the ratio of TAS to Fuel Flow is maximum we draw a tangent from the origin to touch the curves. The maximum range speed is the point at which this tangent touches the curve. For all aircraft types, A tangent from the origin to the Power Required / TAS curve touches the curve at Vmd. So maximum range speed for a propeller aircraft is Vmd.
For a jet aircraft we draw a tangent from the origin to the Drag (or fuel flow) / TAS curve. This tangent touches the curve at about 1.32 Vmd. So maximum range speed for a jet aircraft is about 1.32 Vmd.
Note that in both cases (props and jets) max endurance is at the bottom of the curve and max range is where a tangent from the origin touches the curve.
BUT
The above explanations are based on the assumptions that the Power Specific Fuel Consumption (for propeller aircraft) and the Thrust Specific Fuel Consumption (for jet aircraft) are constant at all speeds. Because of these assumptions, the predicted speeds for best endurance and best range (Vmp and Vmd for props and Vmd and 1.32 Vmd for jets) are not entirely accurate.
If we wish to study the subject at a higher lever, such as for an Aerospace Engineering Degree, we would need to look into the aerodynamics as PBL has done in his posts. The conclusions would then be in a different format, and may look more impressive, but would still produce essentially the same results. If we limited our examination to aerodynamic factors and ignored the variable nature of SFC, we would again produce results that were not entirely accurate.
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Keith,
So the following statement, with regard to jet aircraft, is incorrect...
"At this point, the least amount of power is required for both the maximum lift and minimum drag. This will provide Max endurance, Max range and Best Glide Speed."
Thanks all for the in-depth discussion.
So the following statement, with regard to jet aircraft, is incorrect...
"At this point, the least amount of power is required for both the maximum lift and minimum drag. This will provide Max endurance, Max range and Best Glide Speed."
Thanks all for the in-depth discussion.
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Whether we are talking about propeller aircraft or jets, there is no single speed that will provide Max endurance, Max range and Best Glide Speed."
Max endurance occurs at the speed where fuel flow in minimum. Max range occurs where the tangent touches the appropriate (power or drag) curve.
When gliding (with engines shut down) it doesn't matter what kind of engines we have, they are not contributing anything useful.
Aircraft in flight are constantly dissipating energy as they move forward against the drag force. The purpose of the engines is to provide energy to the aircraft, to replace the energy that is being lost. When the engines fail the aircraft has a limited store of kinetic and potential energy, which can no longer be replenished. This energy will be used up during the subsequent glide. To maximise glide endurance we must fly at the speed at which the stored energy is dissipated as slowly as possible. Energy dissipation rate is power, so for maximum glide endurance we must glide at Vmp.
If you sketch the forces acting on an aircraft in a steady glide you will find that the tangent of the glide angle = D/L. For maximum glide range we need minimum glide angle. Tangents increase with the angle, so for maximum glide range we need minimum D/L. Minimum D/L means maximum L/D, which occurs at Vmd. So best glide range speed is Vmd.
So best glide endurance occurs at Vmp and best glide range occurs at Vmd. These points on the curve are the same as for propeller aircraft. So for gliding use the Power Curve just as we did for propeller aircraft.
Looking at all tree situations (props, jets and gliding) we have.
1. Vmp, where (C_L)^(3/2)/C_D) is a maximum, gives best Glide endurance and best prop endurance.
2. Vmd, where (C_L/C_D is a maximum, gives best prop range, best glide range and best jet endurance.
3. 1.32 Vmd, Where TAS / Drag) is a maximum, gives best jet range.
BUT REMEMBER
All of the above ignores the fact that SFC is not constant. So in the real world the required speeds are likely to be slightly different.
Max endurance occurs at the speed where fuel flow in minimum. Max range occurs where the tangent touches the appropriate (power or drag) curve.
When gliding (with engines shut down) it doesn't matter what kind of engines we have, they are not contributing anything useful.
Aircraft in flight are constantly dissipating energy as they move forward against the drag force. The purpose of the engines is to provide energy to the aircraft, to replace the energy that is being lost. When the engines fail the aircraft has a limited store of kinetic and potential energy, which can no longer be replenished. This energy will be used up during the subsequent glide. To maximise glide endurance we must fly at the speed at which the stored energy is dissipated as slowly as possible. Energy dissipation rate is power, so for maximum glide endurance we must glide at Vmp.
If you sketch the forces acting on an aircraft in a steady glide you will find that the tangent of the glide angle = D/L. For maximum glide range we need minimum glide angle. Tangents increase with the angle, so for maximum glide range we need minimum D/L. Minimum D/L means maximum L/D, which occurs at Vmd. So best glide range speed is Vmd.
So best glide endurance occurs at Vmp and best glide range occurs at Vmd. These points on the curve are the same as for propeller aircraft. So for gliding use the Power Curve just as we did for propeller aircraft.
Looking at all tree situations (props, jets and gliding) we have.
1. Vmp, where (C_L)^(3/2)/C_D) is a maximum, gives best Glide endurance and best prop endurance.
2. Vmd, where (C_L/C_D is a maximum, gives best prop range, best glide range and best jet endurance.
3. 1.32 Vmd, Where TAS / Drag) is a maximum, gives best jet range.
BUT REMEMBER
All of the above ignores the fact that SFC is not constant. So in the real world the required speeds are likely to be slightly different.
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So maximum range speed for a jet aircraft is about 1.32 Vmd.
Excellent summary!
I'm wondering, would it be appropriate to add, just for symmetry, to your item (1): Vmp is about Vmd/1.32 ?
regards,
HN39
P.S. I just came to remember/reconstruct the derivation of the factor 1.32 (3^(1/4) rounded to two decimal places). The reciprocal value applies to Vmp, doesn't it? And in your item (3): ... where cL/cD^2 is maximum?
Last edited by HazelNuts39; 18th Oct 2010 at 09:19. Reason: P.S.
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interesting theme , i just looked at the FOM of our turbo-seneca, one ac type i ride in our company .
maximum range is reached due to the book at 45% power. this setting aims for roughly 125kt indicated airspeed ( TAS depends of course on altitude)
the best glide speed at this plane is just 90 kt IAS , so max range occours far above this speed.
when we come to ( especially turbocharged) pistons you have to consider one more thing that was not mentioned and is unique here.
at low power settinngs you can lean out the engine aggressively, at higher settings you have to enrich.
since at high settings you need exessive fuel to cool the cylinders and keep the EGT in a range what is healthy for the turbo SFC and fuel flow is not nearly linear vs power output-at a turbine this feature does not occour.
looking at the book you see it clearly: the range difference between 45 and 55% power is nearly nothing, at 65% percent noticable lower, at 75% significant lower and at 85% ( the highest recommended cruise at the continental TSIO-360) dramatically lower.
the reason is that at up to 55 percent you can lean hard, above this not anymore, and at any power setting above 75% continental calls for full rich setting. ( especially with the altitude corrected fuel injection system on this engines )
best regards!
maximum range is reached due to the book at 45% power. this setting aims for roughly 125kt indicated airspeed ( TAS depends of course on altitude)
the best glide speed at this plane is just 90 kt IAS , so max range occours far above this speed.
when we come to ( especially turbocharged) pistons you have to consider one more thing that was not mentioned and is unique here.
at low power settinngs you can lean out the engine aggressively, at higher settings you have to enrich.
since at high settings you need exessive fuel to cool the cylinders and keep the EGT in a range what is healthy for the turbo SFC and fuel flow is not nearly linear vs power output-at a turbine this feature does not occour.
looking at the book you see it clearly: the range difference between 45 and 55% power is nearly nothing, at 65% percent noticable lower, at 75% significant lower and at 85% ( the highest recommended cruise at the continental TSIO-360) dramatically lower.
the reason is that at up to 55 percent you can lean hard, above this not anymore, and at any power setting above 75% continental calls for full rich setting. ( especially with the altitude corrected fuel injection system on this engines )
best regards!
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originally posted by orangeboy
why for a jet engine, point of Min Total Drag is equal to max endurance (i can see this since thrust is also lowest, and assuming fuel varies directly with thrust this would be the minimum required to keep the plane airborne), and on a piston plane, the point of min drag is used for max range since it gives the best L/D ratio?
from what i understand, max endurance on a piston plane comes from min power, but how come this min power doesn't correspond to the min point on the total drag curve? is it because the fact that piston engine doesn't provide the thrust directly?
sorry for the questions, just finding this really confusing, which it shouldn;t!
why for a jet engine, point of Min Total Drag is equal to max endurance (i can see this since thrust is also lowest, and assuming fuel varies directly with thrust this would be the minimum required to keep the plane airborne), and on a piston plane, the point of min drag is used for max range since it gives the best L/D ratio?
from what i understand, max endurance on a piston plane comes from min power, but how come this min power doesn't correspond to the min point on the total drag curve? is it because the fact that piston engine doesn't provide the thrust directly?
sorry for the questions, just finding this really confusing, which it shouldn;t!
We know that for props we can make the simplifying assumption that fuel flow is proportional to power required, whereas for jets we assume fuel flow is proportional to thrust required.
But props and jets are subject to the same laws of physics: they're both devices which create thrust by causing a mass of air to accelarate. There is really no fundamental difference between them. So the question becomes: why do the ATPL textbooks make different (and apparantly contradictory) simplifying assumptions for props and jets?
The answer to this question lies in propulsive efficiency. We can define propulsive efficiency as power_out / power_in.
For level flight, Power_out = power required = drag*TAS = thrust*TAS.
Power_in = the rate at which kinetic energy is added to the air to create thrust. Since the engine converts chemical energy in the fuel to kinetic energy in the air let's assume that fuel flow is proportional to the rate of adding kinetic energy to the air (another simplifying assumption!). So far no difference between props and jets.
How does propulsive efficiency vary with speed (TAS). This is where I would really like to draw a graph, but I'll try to describe it in words (if you have D P Davies' Handling The Big Jets, see p50).
Imagine you are at the beginning of the runway, running up the engine just prior to brake release. At this point you are creating thrust and burning fuel but going nowhere. Propulsive efficiency is therefore zero. So our graph of propulsive efficiency against TAS will begin at the origin (zero TAS, zero efficiency)for both props and jets.
In our prop aircraft, as TAS increases the propulsive efficiency rises steeply at first, then the graph bends over and the efficiency approaches a constant value. Eventually if the prop tips go supersonic the efficiency decreases. However, for speeds around Vmd we assume that we are on the bit of the graph where propulsive efficiency is roughly constant. This leads to the conclusion that fuel flow is proportional to power required and hence maximum endurance corresponds to minimum power required.
For the turbojet, the propulsive efficiency increases with speed at a much shallower gradient. Eventually, at very high speeds, the graph bends over towards a constant, but at speeds around Vmd we assume that we are on a bit of the graph where the propulsive efficiency is increasing roughly linearly with TAS. If propulsive efficiency is proportional to TAS and power out = thrust*TAS the speed term cancels and we are left with power_in proportional to thrust, hence fuel flow proportional to thrust. From this it follows that maximum endurance correspends to minimum thrust required or minimum drag.
But what about a modern turbofan at high altitude (High TAS, Low EAS). Which bit of the graph are we on now? Perhaps we are on the curved bit where neither simplifying assumption applies?
Simplifying assumptions are great provided you understand their origin and limitations.
Last edited by Rivet gun; 19th Oct 2010 at 08:30.
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Drag and speed
Last edited by HazelNuts39; 18th Oct 2010 at 16:11.
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Now, working it backwards... Old Smokey made a reference in a different thread related to Max Endurance/Max Range and in that he talked about FMC users and Cost Index. Which Cost Index values will relate to Max Endurance and Max Range? In a round about way, what I am getting to is; can I manually plot different values (altitudes, temperature, cost index etc) to create my own performance charts when I don't have access to the data that the FMS uses to calculate the values? I know I'm reinventing the wheel, but if I can plot this as I'm flying, it will help me to understand the relationship between these values, ie altitude, temperature, weight, etc. Am I making this more difficult than it needs to be???
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Cost index of zero will give max range speed.
I doubt the exact algorithms used to progam the FMCs are available, they may be proprietary information.
I doubt the exact algorithms used to progam the FMCs are available, they may be proprietary information.
Last edited by Rivet gun; 19th Oct 2010 at 21:03.
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Thats actually an interesting point rivetgun,
johns7022 cost index is a number entered into the FMS at the flight planning stage to determine the efficiency or speed of the flight. Best endurance and LRC will always occur at the same angle of attack but at different speeds for different weights.
Typically if 0 is entered as the cost index number the FMS will calculate speeds etc to fly for minimum fuel burn (and usually speed) and generally this will be as slow as possible. If you entered 999 (the cost index usually goes from 0-999) the FMS will calculate the flight to be flown at max speed and not worry at all about fuel flow.
Can anyone with answer rivetguns Q?? would be interested to know also.
rocket
johns7022 cost index is a number entered into the FMS at the flight planning stage to determine the efficiency or speed of the flight. Best endurance and LRC will always occur at the same angle of attack but at different speeds for different weights.
Typically if 0 is entered as the cost index number the FMS will calculate speeds etc to fly for minimum fuel burn (and usually speed) and generally this will be as slow as possible. If you entered 999 (the cost index usually goes from 0-999) the FMS will calculate the flight to be flown at max speed and not worry at all about fuel flow.
Can anyone with answer rivetguns Q?? would be interested to know also.
rocket
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CI 100 changed to 999
Originally Posted by TedUnderwood
Which Cost Index values will relate to Max Endurance and Max Range?
regards,
HN39
Last edited by HazelNuts39; 20th Oct 2010 at 09:53.
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That's interesting, thanks...never heard of planning a flight using cost index...
Sounds like cost index is tied to fuel burn.......
As a general rule, when the fuel prices were low, going fast made sense, as engine overhaul costs outweighed the fuel burn savings...when fuel prices were stupid, some peeps were pulling the levers back...
I found that as a general rule, if I could eliminate some of the stops and fuel purchase, and cycles, that really saved some dough...
Sounds like cost index is tied to fuel burn.......
As a general rule, when the fuel prices were low, going fast made sense, as engine overhaul costs outweighed the fuel burn savings...when fuel prices were stupid, some peeps were pulling the levers back...
I found that as a general rule, if I could eliminate some of the stops and fuel purchase, and cycles, that really saved some dough...