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.
http://naca.central.cranfield.ac.uk/reports/1920/naca-tn-4.pdf
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.
Replies
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!
Love this stuff, please keep it up!
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:
http://www.amazon.com/Principles-Helicopter-Aerodynamics-Cambridge-...
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.
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. :)
Very interesting perspective and I'm looking forward to reading more on this..
Cheers..
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