Drag Curve - Best range

[Bede]

On a drag curve (drag vs airspeed), where is the speed for best range located? Why?

[photofly]

At the bottom.

Alternative formulation of the question: what airspeed will get me from A to be using the minimum quantity of fuel?

Assumption: fuel quantity is proportional to energy.

Elementary physics dictates that energy required to destination is force x distance, therefore drag x distance, regardless of speed. Since the distance from A to B is fixed, the energy required depends only on drag. Minimize the drag to minimize the energy required, and therefore the fuel used.

If you want to account for variations in engine and propellor efficiency (that is, you don't want to assume that fuel quantity is proportional to energy) it's not difficult to generalize.

[RedAndWhiteBaron]

Energy is also wasted as heat. I don't know the general rule for aero engines, but for liquid cooled automotive engines, around ⅔ of the energy produced is turned into heat rather than forward movement. So even if we accept that fuel quantity is proportional to energy, the assumption that the required energy depends only on drag is incorrect. It also depends on engine efficiency and not just propeller efficiency; not all energy is used to produce forward motion.

At some range of speeds for some airframes and some engines, it is entirely possible that best range is therefore not accomplished merely by maintaining the least drag, but also by wasting the least heat. This may be faster or slower than the best L/D speed.

Notwithstanding engine minutiae though, yes, best range is attained at the minimum drag possible in straight and level flight - or with no engine, the speed which will produce the minimum possible sink rate, which is at the bottom of the L/D curve.

This is all assuming stable air though. Returning to soaring has greatly educated me on how this would change in the case of rising and sinking air masses. I suspect though that this is both beyond my ken and the scope of this question.

[Bede]

At the bottom.

Isn't that best endurance? Least drag, requires least power (therefore least fuel). Best range has to figure the marginal faster speed at a marginally higher power setting.

BTW, I've done the math on this, and photofly may be correct, but it's not what the books say and I can't get the math to work. He maybe right because at Vmd, d^2D/dv^2 = 0, but that would mean that best range and best endurance are equal, which they're not.

Here's where I'm going:
Draw a total drag curve
D(v)=Di(v)+Dp(v)
Di=Cl^2/pi*AR*efficiency ratio
Dp=0.5*Cd*WA*rho*v^2

Convert Cl to airspeed (at equilibrium) Cl=2W/rho*WA*v^2

Di(v)=2W^2/(pi*rho*e*span^2) * 1/v^2
Dp(v)=0.5*Cdp*rho*WA*v^2

W-Weight
AR-aspect ratio
WA-wing area
e-efficiency ratio
Cdp-parasite drag coefficient ~0.027

Most books say best range is a tangent line from the origin to total drag curve. I have no idea how this is derived.

P.S. A LaTeX plug in would be fantastic right about now.

[photofly]

Isn't that best endurance?

Best endurance is at the bottom of the power curve (if you’re prepared to consider power proportional to fuel flow). You asked about the drag curve.

On the power curve, best range is the point where the tangent passes through the origin, and best endurance is at the bottom.

On the drag curve, best range is at the bottom.

Least drag, requires least power (therefore least fuel).

Least drag isn't least power. Drag and power are different physical quantities, connected by velocity. Drag is a force, and power is a rate of change of energy.

Least power = lowest fuel flow = least (drag x velocity), not least drag.

and photofly may be correct, but it's not what the books say

Photofly is correct, and it is also what the books say

[Bede]

Isn't that best endurance?

Best endurance is at the bottom of the power curve (if you’re prepared to consider power proportional to fuel flow). You asked about the drag curve.

Sorry, my bad.

On the power curve, best range is the point where the tangent passes through the origin, and best endurance is at the bottom.

Yes I know. But why is best range where the tangent passes through the origin?

Drag and power are different physical quantities, connected by velocity. Drag is a force, and power is a rate of change of energy.

I know, but to go from the drag curve to the power curve, you should be able to multiply drag (N) by velocity (m/s) to get power (Nm/s), correct?

[photofly]

But why is best range where the tangent [to the power curve] passes through the origin?

There are various different graphical ways to show this, all of which are trivial on a whiteboard and hard to write out in text. (My favourite is to transfer lines of constant drag onto the power curve, where they form rays radiating from the origin. Transforming from the drag curve to the power curve is then an easily imaginable affine transformation, and the result is obvious. Hard to describe in text, clearly.)

In text:

best range => bottom of drag curve => dg / dv = 0 (using g for drag)

p = vg therefore g = p/v

best range therefore d (p/v) / dv = 0

using the quotient rule for differentiating quotients (http://amsi.org.au/ESA_Senior_Years/Sen ... tient,%2B1)%E2%88%9212.)
in this case, p/v, and skipping a couple of lines because the differential of the quotient is here zero:

p dv/dv = v dp/dv

p . 1 = v dp/dv

dp/dv = p/v

In other words the gradient is equal to the coordinate of the point. If you draw a tangent with that gradient at that point, it will pass through the origin.

This is less intuitive than a whole bunch of graphical demonstrations, but I don't have the facilities to sketch graphs easily.

Curiously today I was discussing with someone if there's an experimental method to determine a best range airspeed. This would be the method - plot a power vs speed curve, and find that tangent point.

[Bede]

Thanks!

That's the answer I was looking for. For fun, I plotted a theoretical drag curve for a C150 and it turns out pretty close to the same as real life (Vmd, Vs, etc.) The only surprising thing is that the parabola for Dp is much, much wider than what is typically shown in texts.

[photofly]

This might interest you:
https://www.av8n.com/how/htm/power.html ... ower-curve

[Bede]

Denker is fantastic. I have no time for other "pilot" texts on aerodynamics and performance.

[digits_]

Curiously today I was discussing with someone if there's an experimental method to determine a best range airspeed. This would be the method - plot a power vs speed curve, and find that tangent point.

Wouldn't this work for fixed pitch props:
- fly at lowest rpm that keeps you airborne (gives you your best endurance speed as well)
- add 50 rpm, note airspeed increase.
- repeat, keep track of airspeed increase (eg 2 kts)
- once the airspeed increase decreases, you found your max range setting

I believe this is described in the purple FTM as well, but it's been a while.

[photofly]

The biggest airspeed increase occurs at that best endurance airspeed. You already know what that is.

[RedAndWhiteBaron]

Wouldn't this work for fixed pitch props:
- fly at lowest rpm that keeps you airborne (gives you your best endurance speed as well)
- add 50 rpm, note airspeed increase.
- repeat, keep track of airspeed increase (eg 2 kts)
- once the airspeed increase decreases, you found your max range setting

I believe this is described in the purple FTM as well, but it's been a while.

I have the 4th edition purple FTM. It does describe a similar procedure to determine best endurance (that being the lowest power setting that enables level flight) when not specified by the manufacturer, but not one for best range. For best range, it simply refers the reader to the range charts in the PoH, and remarks that in almost all cases, nobody flies for best range anyway.

Still an interesting academic exercise, though.

[Bede]

For best range, it simply refers the reader to the range charts in the PoH, and remarks that in almost all cases, nobody flies for best range anyway.

I find that the charts don't go low enough for most aircraft.

[digits_]

The biggest airspeed increase occurs at that best endurance airspeed. You already know what that is.

Have you tried the technique I described?

I tried it a few times in a 172 with a fixed pitch prop, and while it takes a long time to stabilize at the different RPMs, it does give a value that is close to the book one. And you do get a different value for best endurance airspeed and max range speed (assuming zero wind, although you could likely use the same technique but look at the relationship RPM - ground speed).


in my older notes, I found this description of the technique, which is what I used to try it out:

o Start from speed for maximum endurance
o Add power 100 RPM at a time, let airspeed stabilize and trim
 look at the airspeed increase
o Continue until the largest increase in airspeed for a 100 RPM increase is noted, set the RPM at the setting for just above the largest increase
o lean the mixture

[photofly]

I understand that’s what you’ve been taught, and what you wrote down, and it appears to give about the right answer, but it’s still wrong.

If you want to find a still-air best range airspeed, carefully plot the power curve and find the tangent that passes through the origin.

Trivially, that is not the flattest (or steepest) part of the curve, so any recipe that involves finding a maximum (or minimum) change in airspeed for some fixed change in RPM will not identify this point.

[digits_]

I understand that’s what you’ve been taught, and what you wrote down, and it appears to give about the right answer, but it’s still wrong.

Okay, but what you wrote earlier "The biggest airspeed increase occurs at that best endurance airspeed" is not what happened in that 172 when I tried it out.

It might be wrong, but then I'm interested in learning why there is such a dfiference between the theory and practice.

[photofly]

I understand that’s what you’ve been taught, and what you wrote down, and it appears to give about the right answer, but it’s still wrong.

Okay, but what you wrote earlier "The biggest airspeed increase occurs at that best endurance airspeed" is not what happened in that 172 when I tried it out.

It might be wrong, but then I'm interested in learning why there is such a dfiference between the theory and practice.

I don’t find there’s a difference between theory and practice. When I plotted a power curve, all the way from slow flight to cruise, it was, within the limits of what you might expect, u-shaped, and it was possible to discern where best range airspeed might be.

It’s very difficult to get an accurate measurement for a best endurance airspeed because the curve is flat at the bottom, so I wouldn’t put a lot of reliance on any data there.

Best range power is typically only 100 or 150 rpm more than best endurance power and changing RPM by 100 each time is too coarse a change for any real measurement purpose. On the other hand I doubt most pilots in a 172 can manually stabilize airspeed at 25 or 10 RPM increments accurately enough, and it would take a lot of repeated experiments to gather enough data to see the results through the measurement noise.

The link I posted to on Denker’s site on how to plot a power curve for your airplane is probably a better way to approach things than trying to plot airspeed vs power while trying to maintain level flight.

[RedAndWhiteBaron]

In my older notes, I found this description of the technique, which is what I used to try it out:

o Start from speed for maximum endurance
o Add power 100 RPM at a time, let airspeed stabilize and trim
 look at the airspeed increase
o Continue until the largest increase in airspeed for a 100 RPM increase is noted, set the RPM at the setting for just above the largest increase
o lean the mixture

I understand that’s what you’ve been taught, and what you wrote down, and it appears to give about the right answer, but it’s still wrong.

If you want to find a still-air best range airspeed, carefully plot the power curve and find the tangent that passes through the origin.

As I understand, the method Digits describes will in fact approximate the power setting at the point where the tangent passes through the origin. Further increases in power will yield progressively less increase in airspeed, and further decreases in power will yield progressively more decrease in speed. Or to put it another way, you will find diminishing returns in both directions. You're simply finding best range experimentally rather than mathematically. I think that while Digit's answer is "correct", PF's answer is "more correct".

It's also correct to say that my lawn is grass - but it's more correct to say it's Kentucky Bluegrass (well, and clover and dandelion, but that's not the point)

[photofly]

As I understand, the method Digits describes will in fact approximate the power setting at the point where the tangent passes through the origin.

No, it doesn't.

The point his method claims to isolate is an approximation to the point where Δv/ΔRPM reaches a maximum value. The biggest change in airspeed occurs for a small change in RPM. That actually happens at best endurance, where a range of airspeeds is available all at the same power setting - with no change in RPM - therefore Δv/ΔRPM (max) is in fact unboundedly large.

The tangent that passes through the origin is actually where Δv/ΔRPM = v/RPM. That is, ratio of the marginal change in airspeed per 100rpm is equal to the total airspeed divided by the total rpm. This is not the maximum or minimum or any extremal value of this quantity.

To give a numeric example: if the best range airspeed for an aircraft was at 75 KIAS which was achieved at 1900 rpm, then the marginal rate of change of airspeed per 100rpm around that airspeed would be 3.94 knots per 100rpm. That would make the tangent at that point pass through the origin. At higher airspeeds, an extra 100rpm would gain you less than 3.94 knots. But that same airplane achieves best endurance at 1780RPM and 62.5 knots. At 1800RPM the airplane will maintain level flight at 60 or 65 knots - because the curve is so flat. At 65 knots the marginal change in airspeed per 100rpm is approximately 12.5 knots, much bigger than at best range airspeed. At exactly 62.5 knots the marginal change in airspeed per 100RPM is infinitely large. This is the extremal value.

If you are unhappy with infinite quantities then it's equally ok to look at change in rpm required per change in airspeed, the reciprocal quantity. Digits' described method attempts to isolate where ΔRPM/Δv is smallest. This actually occurs at best endurance airspeed where zero change in rpm is required to maintain level flight across a range of airspeeds, because the power curve is flat at the bottom.



If the method he posts said that max range is approximately where Δv/ΔRPM = v/RPM, that would be correct. Very difficult to determine with any accuracy in practice, but at least correct. But it doesn't say that. It says max range is where Δv/ΔRPM is maximum, which is actually at best endurance.

[RedAndWhiteBaron]

As I understand, the method Digits describes will in fact approximate the power setting at the point where the tangent passes through the origin.

No, it doesn't.

Okay, how about a theoretical exercise then? Where would you think that the exercise would land you on the power curve? I think it would land you pretty close to best range, and I can see that visually on a power curve, because increases in power after that point produce progressively less increase in speed, however I haven't done the math. It appears to me to be the best power to airspeed ratio, and I think we're assuming that power is linearly or at least geometrically proportional to fuel flow for this argument.

And apologies if I missed the math earlier in the thread. And I posted while you edited your comment, I'll need to read that now.

[photofly]

Okay, how about a theoretical exercise then? Where would you think that the exercise would land you on the power curve? I think it would land you pretty close to best range, and I can see that visually on a power curve, because increases in power after that point produce progressively less increase in speed, however I haven't done the math. It appears to me to be the best power to airspeed ratio, and I think we're assuming that power is linearly or at least geometrically proportional to fuel flow for this argument.

And apologies if I missed the math earlier in the thread. And I posted while you edited your comment, I'll need to read that now.

Best range power is probably achieved about 100 to 150 rpm above the power required to achieve best endurance.

Looking in "Aerodynamics for Naval Aviators" the authors write results that indicate best range is achieved at 13.6% more power than best endurance power and at a speed 31.6% faster than best endurance speed. For a C172 which (we might guess) has best endurance at 63 knots and 1750rpm, best range would in theory be at 1988rpm and 83knots. This is based on the simple model of parasite and induced drag.

You are correct to say that range is indeed the biggest airspeed/power ratio, or the smallest power/airspeed ratio. That would mean you could experimentally find it merely by finding the max value of v/rpm.

[RedAndWhiteBaron]

Yep, I see it now. Took a bit of head scratching but now it seems obvious. Digit's method will find best endurance.