Max Endurance, Max Range... [Archive]

rocket66

13th Oct 2010, 21:25

Hey Lads,

After recently sudying ATPL aerodynamics and systems I think I may be able to help out here,

Best L/D ratio is described as the most lift for the least drag (BEST L/D), it is the same speed as BEST GLIDE SPEED and best ANGLE OF CLIMB that yields the MAX EXCESS THRUST. Becasue it is the speed where the least amount of drag occurs it means an aircraft can fly around all day at the lowest power seting to maintain level flight and so leads to MAX ENDURANCE. In other words it will be the longest time an aircraft can stay airbourne.

Best range occurs at 1.32 VIMD (velocity induced minimum drag). It is also the speed at which the aircraft will yields best excess power which means the speed flown is BEST RATE OF CLIMB and MAX RANGE. Some jet aircraft will fly slightly faster (5%) than 1.32VIMD at speeds called 99% max range speeds. They're called this because the 5% increase in speed only results in a 1% decrease in MAX RANGE.

Hope this explains things well enough for you.

Rocket:ok:

EW73

14th Oct 2010, 02:47

That 5% increase in speed is the difference between MRC Max Range Cruise and LRC Long Range Cruise.

ew73

PBL

14th Oct 2010, 04:58

Best L/D ratio is described as the most lift for the least drag (BEST L/D), it is the same speed as BEST GLIDE SPEED and best ANGLE OF CLIMB that yields the MAX EXCESS THRUST. Becasue it is the speed where the least amount of drag occurs it means an aircraft can fly around all day at the lowest power seting to maintain level flight and so leads to MAX ENDURANCE. In other words it will be the longest time an aircraft can stay airbourne.

You apparently think that best L/D (that is, maximum value of C_L/C_D) is where least drag occurs (beginning of the second sentence). Do you have an argument for that?

You then go on to say that this speed, at which max C_L/C_D is achieved, is the speed for maximum endurance.

You are also assuming that having the engine at "lowest power setting" for level flight leads to maximum endurance. Maximum endurance is obviously related to lowest fuel burn per time period, which indeed is minimal power setting. For a propellor-driven airplane, that occurs at a maximal value of ((C_L)^(3/2))/C_D (Anderson, Introduction to Flight, eqn 6.27).

So, which is it? Max endurance at max C_L/C_D, or max endurance at max ((C_L)^(3/2))/C_D? You can't have both. Or can you?

PBL

rocket66

14th Oct 2010, 08:19

EW73 your correct mate, apologies for leaving that part out.

PBL...I'm not too sure what your asking, but I'll see if I can clarify a little.

Best or (MAX L/D) occurs always at a certain angle of attack. The angle of attack it occurs depends on many different factors dependant on aircraft type. If we are talking about Jet aircraft it gets a little more complex because engine efficiency becomes a factor.

A Jet engine works at its most efficient when it operates in whats called 'Design RPM'. Design RPM is found usually in the top 15% or the power range. If it operates anywhere outside this range, be it above or below, it will be less efficient.

So if a Jet aircraft needs to fly for max endurance, it needs to fly at its best/max L/D speed (angle) as high as possible, with the engines operating in the 'Design RPM' band.

If the same Jet aircraft wants to fly for best range it must also fly as high as possible, with the engines operating in the 'Design RPM' band and at a speed that is 1.32 times as fast as its best L/D speed. E.G if the best L/D speed is 100kts the best range speed will be 1.32 x 100 = 132Kts

If we are talking about Light piston aeroplanes then the principle is the same except the aircraft should be flown at its best L/D speed as high as possible (considering the heights they operate in) with what ever power setting can maintain that speed in level flight.

Hope that helps a little

rocket

rudderrudderrat

14th Oct 2010, 09:08

Hi Ted,

Sometimes a couple of graphs are worth a thousand words.

experimentalaircraft.info (http://www.experimentalaircraft.info/flight-planning/aircraft-range-endurance.php)

PBL

14th Oct 2010, 09:21

PBL...I'm not too sure what your asking, but I'll see if I can clarify a little.

I was trying gently to point out to you that your post contains a number of mistakes. Sorry that my message was not so clear. Let me make it clearer.

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,

not C_L/C_D as you proposed. (Unless C_L is 1!!!!)

That means that your reasoning to your conclusion is mistaken (for you gave that reasoning for a generic airplane, so for example a propellor-driven one). It might be worth while for you to inquire where the mistake lies. My questions gave some hints.

PBL

Wizofoz

14th Oct 2010, 10:29

PBL is in fact correct here Rocket, Max L/D will not necessarily giove yu Max endurence, though it will be in the same ballpark.

The simplest explanation I can recall is Max Endurence will occur at Max excess Thrust (and full throttle height for that condition), max range occurs at max excess power.

Is that more or less where the formula lead PBL?

FE Hoppy

14th Oct 2010, 10:38

rudderrudderrat Hi Ted,

Sometimes a couple of graphs are worth a thousand words.

experimentalaircraft.info

Those swine have plagiarised my post on here from a few years ago!

have a look! (http://www.pprune.org/tech-log/399975-some-doubts-ace-technical-pilot-interview.html#post5416515)

rudderrudderrat

14th Oct 2010, 11:13

Hi FE Hoppy,

I always thought you were a genius, but I didn't realise you were a Time Lord also.

Those swine have plagiarised my post on here from a few years ago!
The post is dated Jan 2010

FE Hoppy

14th Oct 2010, 13:12

lol

not so much time lord as senile!!

PBL

14th Oct 2010, 15:51

The simplest explanation I can recall is Max Endurence will occur at Max excess Thrust (and full throttle height for that condition), max range occurs at max excess power.

Is that more or less where the formula lead PBL?

As I understand it, it goes as follows.

You have a given amount of energy in your fuel tank. Energy is force x distance, and power is energy per unit time.

Let's apply a constant thrust to attain a constant velocity. Then

Energy = Force x distance = Force x Velocity x Time = constant

because that energy represents a fixed amount of fuel sitting in the tank.

If you want to maximise Time for this given energy, you need to minimise Force x Velocity, which is power. So maximising endurance is minimising power (P) required to hold level flight.

If you want to maximise distance for this given energy, the distance is Velocity x Time, so you need to minimise Force. So maximising range is minimising Force required to hold level flight.

Force is required thrust, T, and since you are constant velocity, this must equal drag, D. Since you are level, lift L must equal weight W. So T/W = D/L, trivially, which means T = W / (L/D). W is given constant (except for fuel burn, which we can ignore here), so minimising T means maximising L/D. So maximum range is achieved at maximum L/D.

Minimising P is a little trickier, although it is still easy algebra:

P = Thrust x Velocity = (W / (C_L/C_D) ) x V,

using the expression for T above and the relation L/D = C_L/C_D. But now

L = 1/2 x rho x V^2 x S x C_L = W

where rho and V are the freestream density and velocity, and S the wetted area. This is direct from the definition of C_L. So, rearranging,

V = sqrt( (2 x W) / (rho x S x C_L) )

and substituting into the above expression for P gives

P = (W / (C_L/C_D) ) x sqrt( (2 x W) / (rho x S x C_L)
= sqrt( (2 x W^2 x (C_D)^2) / (rho x S x (C_L)^3)

of which the variable part is C_D / (C_L)^(3/2). So minimising this is maximising (C_L)^(3/2) / C_D.

I think the reason this result is proposed for "propellor-driven" aircraft is that, obviously, incompressible dynamics is being used here; no compressibility corrections.

PBL

rocket66

14th Oct 2010, 21:11

Gday PBL

The way I have explained it is the way it had been explained in the text I used to sit the ATPL exam. I'll go back and review it at some stage and see if I can see where I may have made a mistake.

I must put in I have not been lucky enough to fly jets yet but tech logs like this one will certainly help in the interviews getting there.

Will let you know

rocket

Rivet gun

14th Oct 2010, 22:08

The problem with "theoretical" approaches to this question is that they have to make simplifying assumptions about engine and propulsive efficiency.

For example, the "power out" of a propulsion unit is thrust*TAS. Fuel flow corresponds to "power in". If the efficiency of converting fuel flow to power out (i.e combination of engine efficiency and propulsive efficiency) was constant over a range of speeds then you could say that minimum fuel flow would correspond to minimum (drag*TAS). But can you be sure that efficiency is constant even for a piston engine - prop?

For jets the common assumption is that they follow actuator disc theory and therefore propulsive efficiency increases with speed. This leads to the assumption that fuel flow is proportional to thrust and therefore min fuel flow corresponds to min drag. But again, is this an accurate assumption for a modern turbofan?

The only accurate solution is to use the graphs supplied by the manufacturer.

johns7022

15th Oct 2010, 00:01

Maybe some simplification is in order here....max endurance is about staying up in the air the longest amount of time...max range is about going the farthest distance..

Both require max L/D but the dif is in the engines....if you can hang over an island, or weather, flying at say130kt going around in circles burning the least amount of fuel...that's going to be your max endurance......on the other hand, max range will be some power/speed ratio gives you most forward speed for the least amount of fuel burned...

orangeboy

15th Oct 2010, 02:25

i know i am getting something seriously wrong here, so hoping you guys can help me clarify something.

does fuel flow increase with thrust? i would assume it increases right?

then if thats the case, if SFC is the ratio of the amount of fuel used to the amount of thrust produced, then as thrust increases, fuel flow would increase also, but thrust will increase more in proportion hence the SFC ratio falls? is that right? so there would be a point where the increase in fuel used is offset by the biggest gain in thrust, hence the best SFC?.

then if thats the case, if flying for max endurance, if flying at the lowest possible thrust setting to maintain SL flight will give relatively less fuel flow, then why fly higher? isn't the engine having to work harder to produce the same amount of thrust than at low altitudes - so fuel flow will increase as the engine is trying to produce the same amount of thrust, but has to work harder due to the less dense air? isn't the optimum operating range just where SFC is achieved, but not necessarily where the least fuel flow is achieved as a given quantity?

sorry if the answer is really obvious to you guys, but i am hoping to learn more, and this is something that is getting me a bit confused!:confused:

thanks!

PBL

15th Oct 2010, 06:18

TedUnderwood, rocket66, johns7022,

it is well to keep in mind that there appear to be texts out there with significant elementary mistakes in them.

It is usual for texts, even academic engineering texts, to have mistakes in them, but successful texts such as Anderson go through many editions (Anderson is on his sixth) and there is much more chance that the mistakes will have been pointed out by users and corrected. What I understand is that there are study texts out there for practical pilot training which have significant mistakes in them, which are not corrected over time. It is probably wise to be aware of this!

The math as I gave it is elementary, and thus acts as a good guide to reality.

As Rivet gun points out, it is not a perfect guide to the question posed. What it relates, accurately enough, is the amount of required power and thrust to the conditions which minimise them for a given quantity of energy.

What it doesn't deal with is the amount of fuel a given engine will need to burn to achieve that required thrust, respectively power. An engine can be considered as a converter, which converts latent energy (in the fuel) into, say, force. We all know that an engine will need to use more energy than comes out, because a certain amount is lost in the conversion process. And if that relation between potential energy in and force out is not relatively simple, then using force out, respectively power out, as a proxy for energy in, as my math did, could be somewhat misleading.

(Unfortunately, Rivetg phrases this using the old trope of "theoretical" versus - what? - "practical", I presume. Let me warn against taking this to have much real meaning. For most people, "theoretical" is "what I don't understand" (often math) and "practical" is "what I do understand". That is OK as long as it is regarded as a personal attribute, and not something objective. But for people such as myself who actually use these intellectual tools on a daily basis to get answers to real questions, it is a more or less meaningless distinction. When you have a nut to loosen, you go find the right spanner, and when you use it, you have to keep in mind that if the fit is too loose, you might round the edges of the nut and make it difficult to loosen even with a tighter-fit spanner. No one explains this caveat by saying "the problem with theoretical approaches is that you might round the corners of the nut". They say "watch out that you don't round the nut because of a loose fit." So, in this case, watch out that the efficiency characteristics of the engine doesn't affect the result too much. As Rivetg says, in a practical case it is best to use manufacturer's data.)

orangeboy,

it seems to me your considerations about SFC and burn are appropriate. But they do not lead you by themselves to an answer to the question originally posed. You are missing out the aerodynamic component, which I concentrated on. As Rivetg points out, you need both engine efficiency (relation between fuel-energy in and force, resp power output) and aerodynamics (relation between force, resp. power required and range, resp. endurance) to get an answer as to how far, resp. long you can go with a given amount of fuel.

PBL

B777Heavy

15th Oct 2010, 06:25

That 5% increase in speed is the difference between MRC Max Range Cruise and LRC Long Range Cruise.

ew73

Boeing FCTM says LRC provides 1% less fuel mileage than MRC. And read this on the Boeing Aero Mag, that on DESC, Speed on FMC is given as M.XX/250 which equates to L/D Max approximately.

HazelNuts39

15th Oct 2010, 08:11

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 maximumAnderson apparently makes the simplifying assumption that power-specific fuel consumption is constant, which in practice it is not. Also thrust-specific fuel consumption for a jet engine is not constant. It typically has a minimum, and increases from there towards idle and towards max. thrust.

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

PBL

15th Oct 2010, 11:06

Originally Posted by Anderson, Introduction to Flight, Section 1.16.2 ....

I wrote that? So I did. It's section 6.12.1, p448.

Anderson apparently makes the simplifying assumption that power-specific fuel consumption is constant, which in practice it is not

He discusses these issues in depth.

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

HazelNuts39

15th Oct 2010, 18:55

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

johns7022

16th Oct 2010, 02:17

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..

orangeboy

16th Oct 2010, 08:07

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,

johns7022

16th Oct 2010, 13:26

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....

Keith.Williams.

16th Oct 2010, 15:53

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.

TedUnderwood

17th Oct 2010, 02:22

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.

Keith.Williams.

17th Oct 2010, 09:52

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.

orangeboy

17th Oct 2010, 10:15

thanks for the great explanantion Keith, really appreciate the time you and others in this thread have put in to explain things :ok:

HazelNuts39

17th Oct 2010, 10:55

So maximum range speed for a jet aircraft is about 1.32 Vmd.Keith;

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?

aerobat77

17th Oct 2010, 11:08

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!

Rivet gun

18th Oct 2010, 11:32

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!

Well this whole thing certainly confused me when I was doing my ATPL.

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.

HazelNuts39

18th Oct 2010, 13:20

Not related to the preceding post - Just for fun ... (https://docs.google.com/leaf?id=0B0CqniOzW0rjNzdiMWE3ZGYtMjgyZS00NWNjLWIxYjktNjZlNmI wNzgyYTVk&hl=en_GB&authkey=CNyHuO4M)

regards,
HN39

TedUnderwood

19th Oct 2010, 16:21

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???

johns7022

19th Oct 2010, 17:44

Can you define cost index?

Rivet gun

19th Oct 2010, 17:51

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.

rocket66

19th Oct 2010, 21:46

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

HazelNuts39

19th Oct 2010, 22:26

Which Cost Index values will relate to Max Endurance and Max Range?Cost index is a feature of FMS which enables the user to specify a ratio between the cost he assigns to the Time necessary to cover a certain distance, and the cost of Fuel to cover that distance. Depending on that ratio, the FMS selects a speed between that for max. range (cost index 0) and the maximum speed attainable (cost index 999). If you plot a diagram of fuel used versus time for a certain distance flown with various speeds, the cost index corresponds to the slope of a local tangent to that line. It will not give you the speed for max. endurance, because that is slower than max. range (i.e it would correspond to a negative cost index, can you enter that?).

regards,
HN39

johns7022

20th Oct 2010, 00:11

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...

HazelNuts39

20th Oct 2010, 11:43

Sounds like cost index is tied to fuel burn.......
To explain in different words, the FMS evaluates fuel burn and flight time as a function of speed, and outputs a speed schedule that minimizes Cost = a * fuel burn + b * flight time, where a and b are constants and the ratio between them is defined by the cost index.

Rivet gun

21st Oct 2010, 09:40

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

Boeing publishes a manual which you can download here:

Jet Transport Performance Methods (http://www.flightwork.com/resources/document-download.html?no_cache=1&tx_mmdamfilelist_pi1%5Bviewmode%5D=category%3A11&tx_mmdamfilelist_pi1%5Bmode%5D=category%3A6&tx_mmdamfilelist_pi1%5Boldmode%5D=category%3A6&tx_mmdamfilelist_pi1%5Bpointer%5D=0&tx_mmdamfilelist_pi1%5BshowUid%5D=722)

This gives the principles for performance calculations and also the reference wing area for Boeing types, from which you can calculate coeficient of lift. However in order to replicate the FMC I think you would also need the airframe specific drag polars and engine fuel flow data which are not given in detail.

If you look at page 32-20 figure 32-21 there is an example of how to determine best endurance (holding) speed for a B757 at FL350. We previously discussed the theory about speeds for range and endurance of "jet" aircraft which is usually taught to ATPL students. According to this theory, best endurance occurs at the speed for minimum drag. The theory implies constant TSFC, i.e the blue fuel flow lines in figure 32-21 would be horizontal . However in this Boeing example the fuel flow lines slope downwards (TSFC decreases with speed) such that the speed for best endurance is about 23 knots TAS (13 knots EAS) slower than minimum drag speed.

I think this is a significant difference. In this example best endurance speed may perhaps be closer to Vmp than Vmd. Should we be teaching student pilots theories about range and endurance of "jet" aircraft which are quite innacurate for modern turbofan transport types?

CabinMaster

21st Oct 2010, 21:29

TSFC also increases with lower throttle settings.
TSFC is actually quite complex. And we have to keep in mind: TSFC is no physical value, it is simply fuel flow divided by thrust. Sometimes it can be misleading. For example: TurboProps have low TSFC, but the lower cruise speed might result over higher fuel burn for a given distance (depends heavily on some other variables, too).

But Flight Performance is always fun.
But Flight Performance requires good data, for aircraft drag and engine performance. Both are hard to obtain (but are available to some extent). With an Excel-Spreadsheet however you will probably run into problems.

HazelNuts39

21st Oct 2010, 23:16

With an Excel-Spreadsheet however you will probably run into problems.I agree with what you wrote, but don't understand the last sentence. Please explain. (I'm just fond of Excel, as you may have noticed).

regards,
HN39

TedUnderwood

21st Oct 2010, 23:20

That Boeing document is solid gold. Thanks.

CabinMaster

23rd Oct 2010, 10:28

Flight performance means different weights, altitudes and some other parameters. Excel is limited to 2D-matrices.
For example: I want to calculate Rate of Climb for a given weight over the entire Mach/Speed-Altitude region.
First I need local speed of sound and air density for each altitude-Mach pair.
Then the lift coefficient.
Then using an analytical drag model the drag (if I have a look-up table or a hybrid with analytical polar but look-up zero lift drag it gets tricky).
Thrust is usually a look-up table, so again a big pain when using Excel.
The resulting calculations from this point on are rather simple.

One can use Excel with some Visual Basic scripts, but then a real programming language might quickly become more handy. I am using Matlab, a similar but free software is Scilab. Easy to program, easier than Excel VB.