This discussion thread is a follow-on to several conversations I've had with people in the forums who are particularly interested in the aerodynamics of vertical take off and landing (VTOL) aircraft.  Much of the dialog in these forums appropriately surrounds the mechanisms for robotic automation of VTOL aircraft, and in those contexts, I am much more a listener than a contributor.  There are some brilliant engineers, code smiths, and experimenters who frequent these hallowed pages.  The group effort to yield such a marvel as the APM platform is nothing short of astounding.

However, I think we can all agree that the primary functionality of anything that flies is related to how it generates forces to oppose gravity.  Much of the focus here has been on the control system, for a myriad of reasons.  Seemingly ignored is the aerodynamics of propeller thrust, but fairly speaking, it is unromantic as having been largely figured out 90 years ago.  In fact, here's a link to the NACA (forerunner to NASA) original paper entitled "The Problem of the Helicopter", dated 1920.  It is of interest to note that we widely applaud Sikorsky for inventing the modern helicopter, but his contribution was one of a control scheme; he gave us cyclic pitch variation for thrust vectoring coupled with a variable pitch tail rotor to counterbalance torque.

If technical papers like that make your eyes glaze over, perhaps an essential basic treatise is in order.

We go back to Newton's basic laws here, and one in particular: Force=Mass X Acceleration, or F=MA.  In order for our craft to fly, we need it to generate a force equal to and directly opposing the force of gravity.  To produce this force, we normally take the air around our craft as our readily available mass, (except in the case of the rocket and to some degree, the jet engine, where the mass is a product of combustion), and accelerate it (add to its velocity) toward the ground.  Yes, rotors, wings, and propellers all do this, and they all rely on the same principles.

However, there is another factor to consider.  While this particular law is not attributable to Newton, it is still a primary expression: energy is equal to half the mass times the velocity squared, or E= 1/2M X V^2.  So while the lifting force is linearly proportional to mass and acceleration, the energy required to perform the acceleration increases exponentially with the change in velocity.  It naturally follows, then, that taking a lot of air and accelerating it a little takes a lot less energy than taking a little air and accelerating it a lot.  This is why heavy-lift helicopters have such large rotor spans, and their technically analogous cousins, sailplanes, have long wings.  (I drive some people in the pseudo religion of ducted fan technology crazy by pointing out that all their purported efficiency gains can be had by merely making the propeller blade longer...ah, but I digress...) 

In the final analysis we must be concerned about lifting efficiency.  The basic expression for us in comparing efficiencies of different designs can be simplified to merely the number of watts (power) it takes to produce a pound of thrust (mass).  Of course, we cannot simply make our rotors infinitely large and fly with no power expended at all.  There are therefore some engineering compromises which must be made in a VTOL aircraft design.  I hope you can see now why aerodynamic designers first examine the ratio of lifting surface area to the weight lifted as an indicator of potential efficiency.  In the rotary wing world, this ratio is called disk loading, and it is expressed as so many pounds per square foot of total rotor swept area.

Disk loading is a basic predictor of hovering efficiency, but it is by no means the only one.  In my next message, I'll get into evaluating basic rotor (or propeller) blade design criteria.

I hope you've enjoyed this little introduction, and yes, I do plan to eventually show that electric multicopters can be a very viable solution for large payloads compared with conventional helicopters.  However, we need to "level set" on the concepts.  Let the discussions begin.

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Very interesting perspective and I'm looking forward to reading more on this..


Brad, very interesting thoughts indeed - especially in the case of ducted fans!

These days I was considering that if amperage is our "fuel capacity", than efficiency could be calculated based upon it. The common power/mass is a good expression, but how much air time can we achieve with the same amount of amperage? In a petrol world, a more km/l (mpg) way of thought, instead of displacement(cc)/power.

I would not go into equations with you, but I have faith in yours. :)

In my previous posting I tried to convey how important it is to have a large lifting area for a given VTOL aircraft weight.  The next step is to take a look at some practical examples to show how much this can affect the amount of power required to produce lift.  Just as a definition of terms, I tend to use "rotor" and "propeller" interchangeably, just as lift and thrust are the same if you're hovering in a VTOL aircraft.

A basic Newtonian physics analysis of a rotor blade thrust is known as "disk actuator" theory.  Such a disk is modeled as merely a round, infinitely thin plane producing some change in velocity of an air column equivalent to its diameter.  Using simple math we can easily calculate exactly how much power a "perfect" rotor disk of a given diameter would need to produce a certain amount of thrust in average density air.  This "ideal power" is the theoretical measure upon which the actual aerodynamic performance of a rotor disk can be compared using a standard ratio called a Figure of Merit.

FM = ideal power/actual power

Remember that FM is really only applicable to efficiency if comparisons are made to similar disk loadings.  In other words, FM is a relative expression of the quality of the aerodynamic design of the rotor blade system, and is not an indicator of efficiency of the entire aircraft.  In the end, you have to look at both disk loading and FM to derive actual efficiency, although it's much easier to look directly at total watts of input divided by measured thrust output.  Watts-per-pound is where the proverbial "rubber meets the road", so to speak. 

@Marcos: You're absolutely on the right track, but don't forget to multiply your amps by the voltage under load to get watts.  Battery manufacturers rely on amperage as a rating because in a certain family of cell chemistry, the voltage is considered a constant.

I know everybody hates math story problems, so I'll try to keep this short.  How about an example close "to home"?

We have two quadcopters, each with a 4 pound total mass.  One has 9" props and the other, 11" props.  Because it's a quad, each prop has to produce 1 pound of thrust.

9" prop copter: square root of (1 pound/2 * 0.00238 * 0.442sqft.) = 21.8 feet/second ^-1 and times our thrust is 21.8 pound feet/sec ^-1.  Dividing by 550 yields an ideal power of 0.0396 horsepower or * 746 = 29.54 watts.

11" prop copter: square root of (1 pound/2 * 0.00238 * 0.660sqft.) = 17.8 feet/second ^-1 and times our thrust is 17.8 pound feet/sec ^-1.  Dividing by 550 yields an ideal power of 0.0324 horsepower or * 746 = 24.2 watts.

So, moving to an 11" prop from a 9" prop should cut your power consumption by 18%.  Disk loading matters.  It's the difference between 24 watts per pound and nearly 30 watts per pound in this example.

I know what you're thinking.  I have to be making this stuff up  Nobody, but nobody here has a quad that comes even close to consuming only 100 watts in hover (that's a 4C LiPoly battery delivering 133 watts to 75% efficient BLDC motors or a paltry 9.5 amps total current to all four motors combined).

I encourage you to check my math.  I refer you to page 47 (2002 edition) of my favorite technical reference, J. Gordon Leishman's Principles of Helicopter Aerodynamics.  He even shows a worked example for your edification:

It is my hope that you're starting to understand the importance of FM, and more generally, how dreadfully awful these model airplane propellers we use really are.  But the fact is, they don't have to be so bad.  More later.

My last message demonstrated the effects of disk loading on hovering efficiency and showed how ideal power is calculated.  I also introduced the concept of the Figure of Merit (FM) as the ratio of Ideal Power to Real Power.

In the given example of the 11" prop quadcopter, the craft total ideal power is 24.2 X 4 or 96.8 watts at the propeller disks in hover.  Don't forget that you're going to need at least a 30% greater capacity than that for control headroom, and the typical BLDC motor is only 75% efficient.  But remember, too, that this is an ideal power - an absolute threshold that a propeller can never exceed.  As a point of fact, the typical model airplane propeller is not very good at converting power to static (not moving forward, or hovering) thrust.  Far from being ideal, they average in the 40 - 60 range, with some as low as 30, depending on the RPM. 

A FM of 30 means for a calculated 100 watt ideal power, the motor(s) will really have to produce 333 watts to hover.  If you add in the 75% efficiency of the motors and the 30% control head room ((333 * 1.3)/0.75), then your real-world battery needs the capability to supply 577 watts, or a non-trivial 41 amps from a 4S lithium polymer (3.5V/cell) pack.  The ideal rotor ship needs only ((100 * 1.3)/0.75) or 12.4 amps from that same pack. 

The difference is ALL in the rotors.  And while disk loading matters, FM matters MORE.

Just as a frame of reference, the average FM for a full size conventional helicopter is around 75.  Let's look at some real data of that 11 X 4.7 APC Slow-Flyer propeller that so many hobby shops sell as the default quadcopter part.  These FMs are calculated from actual windtunnel data gathered from the University of Illinois.  Note that the FM varies according to RPM (angular velocity):

RPM    Ct          Cp          Calculated FM
1666    0.0971    0.0392    54.5875205
2018    0.0989    0.0391    56.2559251
2271    0.1007    0.0396    57.0689143
2556    0.1025    0.0401    57.8751349
2875    0.1043    0.0407    58.5305552
3144    0.1057    0.0411    59.1318205
3423    0.1074    0.0418    59.5498581
3728    0.1097    0.0426    60.3185555
3994    0.111      0.0432    60.54124
4290    0.1133    0.0442    61.0201381
4585    0.1155    0.0452    61.4165097
4853    0.1172    0.046      61.6856544
5175    0.119      0.0468    62.0333421
5450    0.121      0.0477    62.4036895
5710    0.1229    0.0486    62.6963337
6021    0.1239    0.0493    62.562001

Frankly, these FMs aren't bad at all.  Also note that the FM keeps going up with RPM, until past the maximum RPM recommended by APC (5909).  I suspect the fact that FM starts to dip at the final data point may be due to geometric distortion (i.e. don't try this at home).

From basic aerodynamics we know that lift increases with the square (^2) of velocity, but the power required to overcome the resulting drag increases with the cube (^3) of the velocity.  This greater exponential influence of drag must catch up and eventually surpass the gains made in lift - marking a point of diminishing returns where the FM starts to go down with increasing RPM.  Yet we do not see that here at all, even past the maximum RPM rating!  Even so, while this propeller's performance looks good when compared with its model airplane peers, it still can't approach the average for full-size helicopters.  Why is this so?

The answer is Reynolds Number, or Re.  My next message will be about the effects of Re and how they can be addressed.

Love this stuff, please keep it up!

Woow Reynolds number jumped out from deep in my school memories.. yes love it and seriously following !  yes please keep it up  Brad!

of course, viscosity and friends!

Wow!  How serendipitous to see this thread posted on DIYdrones this morning!

I have been recently paper-studying the effects of low-reynolds number flight for my high-altitude glider project and, ironically enough, the problems that I face there and the problem of multicopter lift share some interesting simirlarities.

The Reynolds and Mach numbers for both cases are very similar, where at high altitude, the low air densities and high true airspeeds make for low Reynolds and high Mach numbers simultaneously.  Indeed NASA proposed and began a venture (APEX) to study this flow regime, but it was sadly postponed and then abandoned.

In a multicopter, we again have low chord Reynolds numbers due to the generally narrow chord of the propellors that are available, but the Mach numbers tend to be high, especially at the blade tips, where the chord and thus the Reynolds numbers drop dramatically.

The fundamental lessons I have learned from my reading is that aerodynamics at low Reynolds numbers is inherently draggy and not very lifty.  On top of that, compressibility factors of high Mach numbers seem to be always deleterious.  Actually, it makes me wonder how insects can fly!

There is a very interesting PhD dissertation by a Mr, presumably now Dr Peter Kunz where he looks specifically at ultra-low Reynolds number flight and winds up in the end building three quadcopters of varying scales, from 15g flying mass to 150g.  Ï'm guessing he may be famous in aerodynamics circles these days and maybe he even lurks here, but I don't know for sure.

My current views of multicopter propellors then is to reduce the speed and diameter (lower Mach numbers) and increase the chord substantially (higher Reynolds numbers) to avoid the worst of the draggy effects the small scale.  A quad-copter made from toilet ventilation fans may work surprisingly well, at least from an aerodynamic perspective...

Awesome thread!  Being an engineer working on UAV propulsion systems, I have been really wanting to dive into the multirotor propeller issue more thoroughly. I don't get to often, but I will definitely be following this thread.  keep up the good work!

Hi Brad - loving this - keep it up. On the ducted fan issue I can't speak for others but my interest in ducted fans for large scale multi-rotors is primarilly safety. I want to build a "moller skycar." :) Obviously it's not working for him though...

@Everyone: Thank you for your kind words of support.

The Reynolds number is an expression of the behavior of a fluid.  Air is classified as a fluid for the purposes of dynamic analysis, but as a mixture of gases, it doesn't seem much like one.  Gases literally do change their behavior depending on the scale of observation, starting out as a viscous fluid and, after a while, morphing into a collection of molecules with inertial tendencies.  The Re number is a "dimensionless" expression for the ratio of viscosity to Newtonality (yes, I invented that word, but is seems to fit) in a given scenario.

If you want a more complete explanation, here's the Wiki link:

WikiPedia on the Reynolds Number

For aircraft, the Re varies according to the scale; the larger the scale, the larger the number.  For regular air, the numbers are all rather large.  The wing on a 747 will have a Re in the 10 million range, whereas the wing on a small balsa glider might only be around 10,000.  This variation in ratio is important because it significantly changes the nature of drag forces.  A viscous goop of cohesive molecules opposes anything that attempts to move through it, and the thicker the object, the more the viscosity opposes the motion.  A Newtonian-dominated kinetic collection of particles merely wants to preserve inertia, and if the moving object is shaped properly, it can minimize the deflective impedance.  It therefore follows that once the air is separated into the lower and upper flows over a wing surface, the longer you keep it flowing over the wing without creating turbulence, the more it starts to act kinetically, raising the Re.  This is why the chord (or flow distance) is a factor in calculating the Re.

It follows, then, that really thin wings (and therefore, propeller blades) are best for low Re applications.  Unfortunately, very thin propeller blades have other obvious issues.  In a scale model hobbyist market, ideally-shaped low Re propellers would not look like their full size counterparts.  Because they're thin, they would be more fragile, and they would be much better at slicing through almost anything with high viscosity, like your finger.  :-)  That's why high-performance, low Re propeller blades are very difficult to find (impossible).

Dr. Paul Pounds, now at Yale university, decided to make his own low Re quadcopter blades.  Calculating that his X-4 Flyer would have a 75% blade radius (most stats on a propeller blade are given at the 3/4s of the length point) Re of about 70K, he quickly realized that there were no commercial options that would give him the efficiency required.  He had his own carbon-fiber layup molds machined, and based his design on a low Re airfoil with a nearly-optimal taper and twist.  The result of his effort yielded an astounding FM of 77! 

However, there are issues with using thin airfoils, as I alluded to above.  But there is big one unrelated to aesthetics or product liability litigation; the angle of attack range can be very limited.  This means that there's a very narrow range of speeds (advance ratios) in which such blades can operate.  To explain that, however, I need to introduce the whole concept of an "airfoil polar", which I shall do in my next message.  Oh, and if you still believe that old dogma about lift being produced by the Bernoulli Principle sucking wings upward, prepare to get your proverbial paradigm shifted.

Here's a link to Dr. Pounds' original X-4 paper, including a picture of his custom blade:

Paul Pounds X-4 Flyer Paper

A very interesting thread indeed, hopefully would translate to a new platform ;)

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