Endurance not at Best L/D

[redblue]

Was taught that the best glide speed = best range speed and it happens at the best lift/drag ratio (lowest point on the drag vs air speed curve).

Also taught that the best endurance is the cross of the parasite drag and induced drag curves. Okay so I can see how you can get best endurance from the lowest of the summation of the parasite and induced drag at a particular airspeed.

But how can you explain that the best endurance is not at the best lift/drag ratio point on the drag vs airspeed curve. And also the best endurance speed requires more power than the best range speed?

[Old Smokey]

Are you talking of propeller or jet aircraft ?

The appropriate answers depend upon that.

Regards,

Old Smokey

[Intruder]

Since you already understand the drag curve, consider next that in unaccelerated flight lift = weight and thrust = drag. So, at a given weight, lift is constant in level flight, and thrust required is directly proportional to drag.

So, with constant lift, L/D max is where the D is smallest -- that same point on the bottom of the drag vs airspeed curve. Minimum thrust is required at that point, therefore minimum fuel flow and maximum endurance.

What you apparently misunderstand is that max endurance IS at L/D max. However, max range is at an airspeed and power setting above the max endurance airspeed. Max range is a function of power required, not thrust.

[redblue]

I am refering to single engine propeller aircrafts.
Intruder. I think you were meant "What you apparently misunderstand is that max range IS at L/D max" instead of "What you apparently misunderstand is that max endurance IS at L/D max".

The problem is that the Flight Test Notes p.78 says Best Glide is at the best L/D ratio (lowest point on the drag vs speed curve). And then max endurance is at a lower airspeed and higher drag point along the same drag vs speed curve. This I find doesn't quite make much sense.

[Old Smokey]

redblue,

I note from your profile that you fly C152 aircraft, and that Intruder, who just responded flys B747 aircraft. We still need to know whether your enquiry relates to propeller or Jet aircraft to respond appropriately.

What Intruder has said in the second sentence is absolutely correct for a jet aircraft, but certainly not for a propeller aircraft.

Both Max Endurance and Max Range are highly dependant upon Fuel Flow. There is a direct relationship between Fuel Flow and Power for a propeller aircraft, and between Fuel Flow and Thrust for a jet aircraft. Consider the following -

Power = Force X Velocity, and putting it into aircraft terms -

Power = Thrust X TAS

So, for the prop aircraft, Fuel Flow is on one side of the equation, and for the jet, on the other.

For the jet aircraft, thrust (and therefore Fuel Flow) is ALMOST (but not quite) constant over the entire speed range. Therefore, without even knowing the engine installed, I can extract the best holding speed from the conventional drag curve as the speed for best L/D ratio, i.e. VMD, the bottom of the Drag curve, and Intruder is absolutely correct.

For the propeller aircraft, thrust, for a given power setting declines constantly as speed increases, and the optimum holding speed can NOT be directly found from the conventional Drag curves. Instead, you need to refer to a POWER Required curve, a different beast to the Drag (Thrust required) curves. From this you can extract the speed for Minimum Power, which will be the speed for minimum Fuel Flow, that is, best holding sped. This speed will invariably be BELOW the optimum L/D speed. Just how far below will depend upon the characteristics of the propeller installed.

And we haven't talked of Maximum Range yet, that too, depends upon whether you're asking about a propeller or jet aircraft.

Regards,

Old Smokey

[redblue]

OldSmokey, thanks for pointing out the difference in propeller and jet engine aircrafts. I was referring to single engine propeller aircraft. I found a site here that offers a very very good explanation, like yours. It is a very good site:
http://www.eaa1000.av.org/technicl/p...s/perfspds.htm

To quote:

To maximize the endurance, we want to maximize the amount of time that we can stay in the air. In order to do this, we must minimize the fuel flow. Since the fuel flow is proportional to the power required, the fuel flow will be minimized at the point where the power required is a minimum. The speed corresponding to the bottom of the power required curve is the speed for maximum endurance

and,

To minimize the pounds of fuel per nautical mile, we can minimize the ratio of power over velocity. Looking at the power required chart, a line from the origin to any point on the curve has the slope of power over velocity (P/V). As you trace a line from the origin to each point on the curve, the slope will be a minimum when the line is tangent to the power required curve. Therefore, the maximum range airspeed occurs where a line from the origin is tangent to the power required curve. This also corresponds to the minimum point on the thrust required curve (drag polar).

and for Carson's Speed,

Unfortunately the maximum range airspeed is generally a lot slower than most people wish to fly. After all, you built an airplane to get places fast. Since we are also interested in getting places fast, we must consider speed. So consider a parameter of fuel flow per knot (Fuel Flow/knot). This would tell us how much fuel per hour we are burning for each knot of velocity. The optimum speed would then be the speed where this parameter is a minimum. Mathematically, the derivative with respect to velocity would equal zero

[Old Smokey]

Hi redblue,

It seems that we were simultaneously earlier, don't you guys ever sleep in CYVR?

It seems that you've done a good job of finding the answers anyway. Knowledge is Power, and that will add more Thrust to your arguments.

Good Luck,

Old Smokey

[Intruder]

What Intruder has said in the second sentence is absolutely correct for a jet aircraft, but certainly not for a propeller aircraft.

So, at a given weight, lift is constant in level flight, and thrust required is directly proportional to drag.

AFAIK, the L/D max / max endurance equation works for any powered aircraft. I don't know how max endurance at a constant altitude can be at a lower airspeed.

OTOH, I am also aware of the minimum sink speed for a glider, which is at the maximum lift speed ( just above stall), and lower than the max endurance speed for a powered aircraft.

Maybe there are some technical nuances to the definintions used in the various publications...

[Capt Pit Bull]

OTOH, I am also aware of the minimum sink speed for a glider, which is at the maximum lift speed ( just above stall), and lower than the max endurance speed for a powered aircraft.

I don't think thats correct.

Obviously with a glider you remove any powerplant issues regarding thrust or power output versus speed, and it purely becomes a lift and drag versus airspeed problem.

I agree that the minimum sink speed is very low, but I don't think that is the point of maximum lift.

Given that Lift = W Cos (Angle of Descent) you will be at the point of maximum lift when you minimise the angle of descent, i.e. at best L/D ratio.

Where as the minimum sink speed is the point of minimum power required. as mentioned above, the point of minimum (Drag x TAS).

So in effect, by waffling along at low speed you put up with more drag (being below Vmd) for the bonus of a proportionally lower TAS. The downside is that this is not the optimum L/D ratio and therefore your descent angle is actually steeper, and hence lift is smaller.

So, in summary, minimum sink speed is not equal to maximum lift.

On the other hand, its been a while since I taught this so I could be talking out of my fundamental orifice. It wouldn't be the first time!

BTW since power=drag x TAS, and given that at any particular altitude TAS is proportional to RAS, you can find the minimum power required speed from a drag graph. Graphically, drag x TAS is proportional to the area of a rectangle formed by drawing lines from your chosen point on the graph to both axes. It is reasonably obvious (pictorially) that this point is way below Vmd.

CPB

[bookworm]

Both Max Endurance and Max Range are highly dependant upon Fuel Flow. There is a direct relationship between Fuel Flow and Power for a propeller aircraft, and between Fuel Flow and Thrust for a jet aircraft.

I don't know how much complexity it is appropriate to add, but the idea that power is dependent only on fuel consumption is an approximation, at least for a piston engine. Such engines are usually more efficient for high power delivery. Add in the efficiency of the propeller, and I believe there can be small but significant variations in brake specific fuel consumption with speed.

I say significant because we're finding the minimum of a curve, where the curve is by definition not very steep, so fairly large speed changes cause only small changes in drag. In the same way that the best rate of climb speed is usually substantially above the min sink speed because power available increases with speed, so the best range speed may be considerably higher than min drag, and best endurance may be higher than min power required.

(I presume that there is a similar effect with jets in that their TSFC is better at lower thrusts. For example this page claims to derive from Shevell a TSFC that increases substantially with Mach no. Thus best endurance might actually be below minimum drag speed.)

[Intruder]

Given that Lift = W Cos (Angle of Descent) you will be at the point of maximum lift when you minimise the angle of descent, i.e. at best L/D ratio.

Actually, not. Maximum lift will give you the minimum RATE of descent, without regard to angle of descent or distance covered in the glide.

In a glider there are several speeds normally considered (these are only quick descriptions, not technical diatribes):

Minimum sink: Just above stall speed, it gives you the least rate of descent, which translates to the greatest rate of climb when in a column of lift. It also gives you the smallest turn radius, helpful for staying inside narrow columns of lift.

Max glide: At L/D max, gives you the greatest still-air distance in the glide, without regard to the time it takes to get to the ground.

"Speed to fly": Calculated dynamically, based on the specific aircraft, winds, available lift/sink, distance to go, and time constraints. Used when racing, to get the farthest the fastest without hitting the ground.

[Capt Pit Bull]

Intruder,

hmm - I don't disagree with anything you say in your discriptions regarding minimum sink and max glide, where you find them in speed terms or the applications thereof.

I just disagree with your assertion that minimum sink = maximum lift.

1.) Best Glide Speed = Best L/D ratio.
2.) Best L/D ratio = smallest descent angle.
3.) Smallest descent angle yields biggest possible value for lift in the equation L=W Cos (Angle Of Descent.)

Could you clarify which of points 1, 2 or 3 you feel is invalid and maybe we can thrash it out from there.

Cheers,

CPB.

[enicalyth]

HI redblue!!!

The trick is to remember thrust is proportional to the square of speed whereas power is proportional to the cube of speed. Your best endurance speed conforms to the bottom of the bathtub curve of required power versus speed. Clearly you stay in the air longer if you use as little power as possible. (Provided that that speed is above the stall!). Looking at the same bathtub curve of required power versus speed or indeed of drag versus speed if you draw a line passing through the origin and tangent to the curve then you have the speed for maximum range which is also best glide slope speed. For aircraft where compressibility drag is negligible this occurs when parasite and induced drag are equal as you have discovered. But the speed for minimum power is 76% of the speed for maximum range because the two are in the ratio of 1/(3^0.25). I think your ground school can show you "Fuel Efficiency of Small Aircraft," which is AIAA Paper AIAA-80-1847 and was written by B H Carson.

Vancouver eh? How goes the Gastown Steam Clock and its steam whistle "chimes"?

[Intruder]

I just disagree with your assertion that minimum sink = maximum lift.

3.) Smallest descent angle yields biggest possible value for lift in the equation L=W Cos (Angle Of Descent.)

Could you clarify which of points 1, 2 or 3 you feel is invalid

Not invalid, just misinterpreted... You have to consider descent RATE as well as ANGLE.

Using your same equation, note that weight is constant, while lift and descent angle are varied with airspeed.

When airspeed is decreased from L/D max to max lift (stall), L increases. So, for a constant weight the ANGLE of descent will indeed increase. However, the sine of the descent angle is defined as the vertical speed divided by the airspeed. If airspeed is decreased sufficiently, even a slight increase in descent angle will result in a lesser vertical speed.

For example, assume a glider has an L/D max speed of 45 knots (4557 fpm), and a min sink speed of 300 fpm at 40 knots (4051 fpm). [Note: These are not "real" numbers, but are in the ballpark for a low-performance glider like an SGS 2-33.] At min sink speed the descent angle is 4.24 degrees -- arcsin 300/4051. Now assume that at 45 knots (L/D max) the sink speed is 325 fpm. The descent ANGLE will be only 4.09 degrees -- arcsin 325/4557 -- even though the descent RATE is greater!

Note that the descent rate at 45 knots will have to be 337 fpm to equal the descent angle at min sink speed. The situation satisfies your equation.

[Capt Pit Bull]

Hmm.

I agree with the idea that a steeper descent angle can be consistent with a lower descent rate (along with 99% of the rest of your post). You just have to fly at the minimum power required speed to make that happen, which is well below best range / best L/D ratio speed.

What I don't see is how, given that you accept L=w cos (descent angle), you can maintain that maximum lift happens anytime other than minimum descent angle.

(Take it to extremes. If you really dirtied up, (say by opening a huge parachute) you'd decrease your descent rate by virtue of massive drag, even though your descent angle would be 90 degrees and lift had disappeared.)

The only thing I can think is that perhaps when you say maximum lift, you mean maximum lift coefficient.

Hmm. Its late. I need more beer.

Regards,

CPB

[Intruder]

You just have to fly at the minimum power required speed to make that happen, which is well below best range / best L/D ratio speed.

...which is also referred to as stall speed! Continuing remembering your basic aerodynamics... An airplane will continue flying at some point below the power-off stall speed, but power must be increased to do so.

Since a glider has no internal power available, glider pilots make use of this principal when they fly at "minimum sink" speed while thermalling.

What I don't see is how, given that you accept L=w cos (descent angle), you can maintain that maximum lift happens anytime other than minimum descent angle.

The key is here:

However, the sine of the descent angle is defined as the vertical speed divided by the airspeed. If airspeed is decreased sufficiently, even a slight increase in descent angle will result in a lesser vertical speed.

Maximum lift represents minimum descent RATE, not ANGLE. The descent angle in the equation you cite is actually a result, not an input parameter. Maybe the equation would be better stated as

Descent angle = arccos (L/W)

The only thing I can think is that perhaps when you say maximum lift, you mean maximum lift coefficient.

That may well be... Aerodynamics equations always make my head hurt. I'm thinking of the top of the hump in the curve that represents stall speed, so if that's a Cl vs airspeed (or AOA) instead of a Lift vs AOA curve, that's what I mean.

[Wizofoz]

Intruder,

Continuing remembering your basic aerodynamics

OK, in an aircraft flying straight and level, LIFT = WEIGHT. If lift is any other value than weight, the aircraft will accelerate vertically either up or down as LOAD FACTOR = LIFT/WEIGHT.

When talking about "Maximum lift" you obviously mean "Maximum CO-EFFICIENT OF LIFT" which is indeed achievied at the stalling angle of attack. This has nothing to do with either max endurence in a powered aircraft (achieved at Max excess thrust) or minimum sink in a glider (achieved at minimum drag).

Both these speeds are well above stall,as Total Drag rapidly increases as stall is approached because of a massive increase in Induced Drag.

The term "Maximum lift" is meaningless. L= CL*1/2rho*Vsquared. As V can have any value, Lift can increase exponentially until the wings fall off!

[bookworm]

Look at it via the basic equations for flight performance for unaccelerated flight. T= thrust, D = drag, L = lift, W = weight.

T = D + W * sin(flight_path_angle)
L = W * cos(flight_path_angle)

Thus

(T - D)/L = tan(flight_path_angle)

The maximum flight_path_angle is at maximum (T - D)/L, i.e. at maximum excess thrust. For T = 0, the maximum flight_path_angle is best glide angle and reduces to maximum -D/L, which is equivalent to maximum L/D.

Multiply through by airspeed v to make the right hand side vertical velocity and get:

(T - D)*v/L = rate_of_climb

Thus best rate of climb occurs at maximum excess power and for the T = 0 case, best rate of climb is minimum rate of descent and reduces to maximum -D*v/L, which is equivalent minimum power required. This is unrelated to stall speed, though it's possible in some cases that it might occur very close to stall.

Maximum lift as an absolute quantity for unaccelerated (1G) flight will indeed occur at minimum flight path angle, i.e. level flight if you have the power to stay level, but it's not a very useful concept. Maximum lift coefficient does indeed occur at stall, but by the time you've reduced the speed to that point, it's not the same as maximum lift.

[Dick Whittingham]

I hope that this, concerned only with staying level or going down, may be of value.

There are three basic speeds defined in level flight.

The lowest is Vmp, where the minimum power is needed to keep the aircraft flying. This is not Vs. It is a little bit above Vs, where it is difficult to fly, as you are on the wrong side of the drag curve.

The next highest is Vmd, where the drag is minimum. Remember, this is in level flight, so the thrust required to keep the aircraft flying is also minimum

The third is the speed that gives the best Speed/drag ratio. This would be 1.32 times Vmd if the theory is working right, but in real life it could be rather more or less.

If you are gliding, and there is no power or thrust delivery to worry about, then these speeds, based only on the aircraft aerodynamics, apply to pistons, jets, and gliders. To get the minimum sink rate in the glide, fly at Vmp. This will allow you to stay up longer – gliding for endurance. If you are heavy you will sink faster and vice versa. If you want to glide as far as you can then fly at Vmd. This gives you the angle of attack for best lift/drag ratio, and your best glide angle is, in fact, exactly your lift/drag ratio and as long as you fly the correct speed is not affected by aircraft weight. This is gliding for range.

These speeds are defined in level flight, and, as lift required equals weight, speeds (not best glide angle) vary as the weight of the aircraft changes. They are faster for a heavy aircraft, slower for a light aircraft. When you are gliding lift no longer equals weight. It is now the lower value, W x Cosine of the glide angle. This makes no practical difference up to glide angles of about 15deg, but in theory, you should fly in the glide at slightly lower speeds than the level flight quoted values.

Now, if you are under power, holding level flight, it makes a difference what type of power delivery you have. In piston engines, approximately, fuel flow is proportional to the Engine Horsepower (EHP) delivered. So, to use minimum fuel you would fly at the minimum power required speed, Vmp. In practice it is so difficult to do this that your Flight Manual will probably suggest a higher speed. In jet engines, fuel flow is approximately proportional to thrust delivered. In this case, to stay up longest you would fly at the minimum thrust required speed, minimum drag speed, Vmd. Again, to make things easier for you your Flight Manual may recommend a higher value

A similar difference applies if you want maximum range in level flight. Best speed/power required works out as Vmd, so Vmd is range speed for piston aircraft. Best speed/thrust required works out at 1.32Vmd and is jet range speed. For various practical reasons, including increasing engine efficiency at higher speeds, your Flight Manual may quote higher speeds, and at altitude, transonic drag problems will define range speed at a different value.

Glider heads will recognise that at the beginning of the day, with lift all over the place, they will take off with full ballast tanks. Vmd will be high, but best glide angle is unchanged, so they transit rapidly from Cu to Cu. As the lift dies, they will dump the ballast to lighten ship and fly for endurance to stay up that last minute longer.


Dick W

[Intruder]

Wizofoz said:
for the T = 0 case, best rate of climb is minimum rate of descent and reduces to maximum -D*v/L, which is equivalent minimum power required. This is unrelated to stall speed, though it's possible in some cases that it might occur very close to stall.

Dick W said:
The lowest is Vmp, where the minimum power is needed to keep the aircraft flying. This is not Vs. It is a little bit above Vs, where it is difficult to fly, as you are on the wrong side of the drag curve.

Hmmm... I think that's what I said at first, before confusing myself with all the terms: mininimum sink speed in a glider is just above stall speed, and less than the speed for best glide angle.

I also found the graph I was looking for, and it is indeed the Cl vs AOA graph, which is often used to describe the stall AOA (or airspeed in unaccelerated flight), that I had in mind. So where I said "max lift" earlier, I did really mean max Cl.