# Aerodynamics and fuel consumption of a powered glider

I want to optimize the flight distance of a powered glider model airplane, without the use of soaring. I've studied some aerodynamics books, but still feel unsure about my conclusions. Please tell me if I'm wrong in the following:

1) The best fuel economy (liters per km) is for planes with the highest lift/drag ratio, typically so called gliders. The reason a 747 doesn't look like a glider to save fuel, is that it would be too difficult to build strong enough wings of that dimension.

2) Gliders are typically designed so that the best lift/drag ratio is when the airplane is in cruising mode with zero attack angle (horizontal). This is typically also when all control surfaces are neutral.

3) Optimum long distance cruising speed is at max lift/drag ratio, and when the lift is equal to the aircraft weight, given that the engine and propeller are also optimized for it. In other words, the speed where it sits stable in the air without any control. I can find this speed for my model glider by flying it at zero attack angle and with neutral control surfaces, and then increase the throttle gradually until the plane stops sinking.

4) Although the lift/drag ratio varies with speed, it varies very little around the speed that is optimum for long distance cruising. This is because lift-induced drag is a very small component of form drag (overall drag) at that speed. In other words, form drag is very close to proportional to the square of the speed.

5) If I want to increase the weight of the aircraft (for example add more batteries), all I have to do is fly proportionally faster to stay in the same zone of optimization for distance. Lets say the normal take-off weight of the glider is 1.4 kg. I quadruple the take-off weight to 5.6 kg. The lift equation says that my speed must now be doubled to make the airfoil provide four times the lift. This new speed will be very close to optimum cruise speed for distance, because of 4) above.

7) The speed has doubled, which means the drag has approximately quadrupled, so a new engine and propeller with four times the power and optimized for twice the speed must be installed.

• Point #1 is wrong. The reason the 747 (and other airliners) doesn't have wings like a glider is that it travels at high speed. You could probably get much more fuel efficiency with glider wings (which wouldn't be that hard to build) and a 50 kt airspeed, but I suspect few people would be willing to fly with your airline :-) Dec 26, 2018 at 18:41
• Point #4 is also wrong. At best L/D induced drag is half of overall drag. What varies little around best L/D speed is overall drag - lift equals weight and does not vary, so near the flat optimum, drag will vary little as well. Dec 27, 2018 at 3:45
• What is point #6? Dec 27, 2018 at 3:45
• Thanks jamesqf, that makes a lot of sense! Dec 27, 2018 at 19:26
• Thanks Peter Kämpf. There is no #6, I made an error. But I don't understand your point. Looking at a drag/speed diagram, I can see that lift induced drag is very strong at low speeds and almost doesn't exist at high speeds, where parasitic drag is the major (only) component. Around best L/D, the drag equation must still be valid, so overall drag must vary a lot with speed (however the lift induced portion of it is probably weak). Or am I missing something here? Dec 27, 2018 at 19:31

## 1 Answer

I can talk about a couple of these...

3) Optimum long distance cruising speed is at max lift/drag ratio, and when the lift is equal to the aircraft weight, given that the engine and propeller are also optimized for it. In other words, the speed where it sits stable in the air without any control.

Well, there are two speeds you're talking about:

• The most efficient cruising speed, which is also the speed that maximizes L/D, which is also the best glide speed.
• The "trim speed", which is the speed where the aircraft tends to maintain a constant pitch ("sit stable in the air") when the pilot is not applying any forces to the control.

For full-scale aircraft, the trim speed can generally be altered by the pilot in flight, using the trim wheel (or whatever trim control that aircraft has). So, the trim speed is usually not equal to the best glide speed.

That said, you certainly can (and it's probably a good idea to) design your powered glider so that its trim speed is constant, and equal to the best glide speed.

I can find this speed for my model glider by flying it at zero attack angle and with neutral control surfaces, and then increase the throttle gradually until the plane stops sinking.

Yes, you can find the trim speed (but not the best glide speed) of your model glider by controlling the throttle so that the height remains constant. If the controls are neutral and your aircraft is not pitching, then it is flying at trim speed.

(I'm not sure what you mean by "flying it at zero attack angle", since you can't explicitly control the angle of attack without moving the control surfaces. For steady flight, the angle of attack is determined by the speed and the bank angle.)

7) The speed has doubled, which means the drag has approximately quadrupled, so a new engine and propeller with four times the power and optimized for twice the speed must be installed.

That's almost right. If the speed doubles, then the drag quadruples. But power equals force times speed, so if the speed doubles, and the drag quadruples, then the needed power increases eightfold, not fourfold.

Responses to your comments:

However, as I understand it the "trim" is just a minor adjustment to the elevator, correct? So in effect we are nudging at the angle of attack so the lift perfectly matches the weight and the plane can sit stable, but we aren't really flying at the optimum L/D?

I wouldn't call it "just a minor adjustment". In a Cessna 172, trim is what determines whether the aircraft flies at 55 knots or 155 knots. Other airplanes and gliders work the same way: there is one particular speed at which the aircraft will fly steadily without any force on the control stick, and trim is used to determine what that speed is.

Trim isn't used for making the lift match the weight. If the lift is too high, then the aircraft will begin climbing (or stop descending), and if the lift is too low, then the aircraft will begin descending (or stop climbing). You control this by using the stick and the throttle.

The aircraft can "sit stable" with the trim in any position, but for each trim position, there is only one corresponding speed at which the aircraft will fly steadily without force on the stick.

(There's an exception to the above: with some aircraft in some circumstances, if the trim is too far back, then constant forward pressure on the stick or yoke is required in order to prevent the aircraft from stalling.)

If you're sitting in an aircraft and you want to fly at optimum L/D, you'll usually do that by knowing what airspeed corresponds to optimum L/D, and setting the trim for this airspeed. (The process for setting the trim appropriately is: use the stick to fly at that airspeed, hold the stick in place, and then use the trim control to relieve the force the stick is exerting on your hand.)

I'm just assuming that airfoils are designed to be at their best L/D at zero angle of attack, is that really the case?

I doubt it. In particular, some aircraft have symmetric airfoils, which have an L/D of zero at zero angle of attack.

When I said "flying at zero attack angle", I mean that the plane sits completely horizontal in the air, and it does so without any compensation from control surfaces.

If the plane is completely horizontal, that's zero pitch, not zero angle of attack. Zero angle of attack is when the nose is pointed exactly into the oncoming air, rather than higher or lower.

The pitch that an airplane settles at depends on both the trim and the power setting.

Last part doesn't seem right to me. If you look at it from the perspective of the engine, it creates a forward force. This force is constant, no matter the aircraft speed (taking aside the fact that a propeller cant be equally effective at all speeds). However the drag is related to the speed, and the drag is four times stronger. So it feels to me like the needed power is only four times more, not eight.

Well, the thing there is, an engine doesn't really create a constant force. In particular, you need to produce enough force to counteract drag. If your aircraft experiences four times as much drag, then your engine also needs to produce four times as much thrust in order to fly the same way.

So in order to fly twice as fast, you need to make the engine produce four times as much force (either by opening the throttle more, or by using a more powerful engine). The engine also needs to operate at twice the airspeed. Power is force times speed, so together, these two facts mean that the engine needs to produce eight times as much power.

Further responses to your comments:

Re angle of attack and pitch, I might have just confused the terms. As I understand the terms, "angle of attack" refers to the angle of incoming air relative to the chord of the airfoil, so it relates to the wings only. Pitch is the angle of incoming air relative to the aircraft itself. So if the wing chord is mounted exactly horizontal, angle of attack and pitch are identical.

Pitch doesn't have anything to do with the angle of the oncoming air; it's exclusively defined by the way that the aircraft is oriented in space, relative to the horizon. Specifically, pitch is the angle by which the nose is pointing above or below the horizon. If it's pointing 10 degrees above the horizon, the pitch is 10 degrees; if it's pointing 20 degrees below the horizon, the pitch is -20 degrees. This is regardless of how the aircraft and the air are moving.

It makes more sense to me when I imagine putting a piston engine with a propeller in a wind tunnel. Would the thrust force of that engine/propeller vary greatly with incoming airspeed when running the engine at constant power? I can't imagine why it would. As the air speed increases, the engine would increase its RPM to exert the same power on the air, and the propeller efficiency would vary slightly, but the engine power would still be the same. And overall the thrust force would be approx the same. Or am I thinking the wrong way here?

Well, the best way I know how to put it is "that's physics". If you want to push something in the same direction that it's already going (as an airplane propeller does—it pushes the air aft, even though the air is already moving aft), and you want to push it with a constant force, then the faster that thing is going, the more power is required in order to push it. I don't think it's easy to see why this is the case, but this is, in fact, the case.

Maybe it would be helpful to imagine an airplane flying straight up. Imagine that this airplane is heavy but sleek, so that the drag is very small compared to the weight. Then, in order for this airplane to fly straight up at any speed, its engine will simply have to generate thrust equal to the airplane's weight. (The only time the engine would need to generate more thrust than the airplane's weight, is when it's making the airplane accelerate.) And yet, if this airplane were to fly twice as fast, surely the engine would need to generate more power, right?

The moral of the story is that power depends on both force and speed. That's physics; there's no way around it. If you had an engine that generated a constant amount of thrust, using a constant amount of power, regardless of speed, then you'd be able to make a perpetual motion machine out of it.

• minor nitpick: Zero angle of attack is when the direction of movement is parallel to the chord line, the line connecting the nose and the trailing edge of the airfoil. The other angle is the zero-lift angle of attack. Dec 26, 2018 at 20:02
• @PeterKämpf Thanks, I removed my mistaken statement there. Dec 26, 2018 at 21:08
• Thanks Tanner, this was enlightening, I stand corrected. However, as I understand it the "trim" is just a minor adjustment to the elevator, correct? So in effect we are nudging at the angle of attack so the lift perfectly matches the weight and the plane can sit stable, but we aren't really flying at the optimum L/D? I'm just assuming that airfoils are designed to be at their best L/D at zero angle of attack, is that really the case? I hope I'm making sense here. Dec 27, 2018 at 19:42
• Tanner, when I said "flying at zero attack angle", I mean that the plane sits completely horizontal in the air, and it does so without any compensation from control surfaces. Perhaps that is impossible because center of lift isn't exactly over center of gravity. Looks like Im simplifying here. Just trying to understand how to get to the point of most economical flight. Dec 27, 2018 at 19:46
• Tanner, the last part doesn't seem right to me. If you look at it from the perspective of the engine, it creates a forward force. This force is constant, no matter the aircraft speed (taking aside the fact that a propeller cant be equally effective at all speeds). However the drag is related to the speed, and the drag is four times stronger. So it feels to me like the needed power is only four times more, not eight. Dec 27, 2018 at 19:56