East Anglia, UK
Joined Sep 2002
29,687 Posts

Discussion
Understanding prop efficiency.
Been a lot of hooha about what is the best prop to use and at what RPM for best efficiency etc. etc.. So on with the thinking cap, and time for analysis.
First of all, some definitions
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PITCH SPEED.
The speed at which a prop spinning at a given RPM generates no thrust whatsoever, and no drag along the axis of the aircraft. Yes, I know there are other definitions, but this is the one in use here, today, to avoid endless discussions about what pitch speed is.
PITCH
The above speed divided by the RPM. This may or may not bear any resemblance to what is marked on the prop.
PROPELLOR EFFICIENCY
Defined as the power output (airspeed times thrust) divided by the power input (torque times RPM). Note that at zero airspeed, which is where we bench test motors, propellor efficiency is zero, as there is no airspeed. Such figures as grams per watt may provide some indication of the quality of a propellor at generating thrust at zero airspeed, but they have little if anything to do with propellor efficiency at a given airspeed.
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Finally, the analysis is concerned with getting the best out of a prop: cases which are sub optimal will not be considered in great depth. The aim is not to throw vast mathematics about either, but simply to indicate broadly what sort of propellor and RPM you should be aiming for, and why. It also deliberately avoids any issues of motor (or the rest of the power train) efficiency. Today is prop day.
Drag elements
If the propellor is (justifiably) considered as a series of small wing sections, then standard wing aerodynamic theory applies. WE know for example that the best lift to drag will be, for normal sections, at a little over a couple of degrees angle of incidence. .
And any section greater than about 15 degrees angle of attack to the incident airflow will be into stall, and will produce less lift (thrust) and generate higher drag.
So ideally our propellor over its working speed range at a given RPM, should be somewhere between a little below pitch speed (best efficiency: least drag) and wherever the majority of it is into stall.
Static conditions, and hovering
One thing is easy to check. At what pitch to diameter ratio the tips are greater than 15 degrees, so the whole BLADE is stalled at zero airspeed. That's PI x tan (15).
It turns out to be 0.84 give or take, so a 10x9 is fully into the stall regime running static for example. NO part of the blade is less than 15 degrees angle of attack. That doesn't means its producing no thrust at all, just that most is being produced at fairly high cost, in terms of drag, by the tips alone.
If we want a bit better than that, to say have at least the outer half of the prop disc unstalled, (which probably generates at least 75% of the thrust in fact) than we need to halve the pitch. so a 10x4 prop (taking a ten inch prop as something that is familiar) is almost completely unstalled at zero airspeed. And its tips will be at about 7.25 degrees, which although not the best angle of attack, is still fairly good.
Now this leads to the conclusion that the best static thrust is produced by a blade set at exactly the correct angle for the best lift to drag ratio, and being driven no faster than is necessary ( to minimise profile drag and skin friction) to generate lift, and has no helical twist at all. Voila! its a helicopter blade.
This is an important marker. For the best static thrust for a given power, build a helicopter. At last, the definite answer to 'what is the best static thrust I can get out of XYZ?' The answer. gear it like a helicopter, and use a rotor instead chum, and don't ask stupid questions.
It also has another interesting corollary: Helicopters with single rotors as big as or exceeding the maximum dimension of the rest of the aircraft, need extreme measures to counteract torque. This is extremely relevant when the maximum size of prop for a normal aircraft is considered.
Adding in airspeed
The moment you start adding in airspeed, the helicopter blade becomes seriously deficient. Torque considerations aside, its not helically cut, and so the inner sections will stop producing thrust long before the outer ones do. Likewise because of the forward airspeed, no part of the blade is experiencing less than the actual aircraft airspeed, so magnificently long blades with low airspeed, for low drag, simply wont cut the mustard.
If we do the similar 'take it to the limit' exercise for a blade producing thrust at a fixed airspeed, and optimise that for lowest drag per unit lift, or thrust, what we already know is that for best lift to drag, the blade needs to meet the airstream at about 26 degrees angle of attack, and crucially, it shouldn't be running at an actual airspeed any higher than it needs to do, for this to be the case.
Another factor also comes into play at high blade angles of (geometrical) incidence. The lift vector is no longer along the direction of aircraft travel. Its perpendicular to the blade, so for a given incident airspeed, blade area, the thrust decreases as cos(theta) where theta is the geometrical angle of attack. So at, let's say, 60 degrees, the thrust is NOT what we might expect, but half that (cos( 60)=0.5) After that it drops like a stone. so we can probably say that a helically cut prop with the angle at  say mid blade of greater than 60 degrees is probably simply too coarse to be generating any thrust much at all.
This is useful, because it sets an upper limit on the propellor pitch to diameter. It turns out that for the conditions above, its Pi. That is no more than PI times diameter will be a effective as a prop.
so scrap that putative 10x32 prop. It's not going to work.
This is useful. WE have seen that a helicopter prop, all diameter and no pitch, rapidly runs out of effectiveness at forward airspeeds, and that an ultra coarse pitch prop runs out of thrust at too steep a pitch. That means that 'somewhere in the middle' is the best we can hope for.
But look again at the actual blade speed through the air. It is, for an aircraft operating near the pitch speed (where lift to drag is best as the blade meets the airflow at just a couple of degrees or so)), about 1/sin(theta) times forward airspeed. Where theta is the geometrical pitch angle. So for a 45 degree blade part, its seeing about 1.4 times airspeed. Not a huge increase. We know that more than 3 times the sorts of aircraft speed is likely to push the drag up a lot, so for efficiency we don't want to have tip of the blade doing more than 3 times the aircraft airspeed. This turns out to be a pitch to diameter of about 1.11
This is getting interesting, because it says that broadly speaking, for an aircraft operating at a fixed airspeed and operating with the right sized prop, and RPM., we don't want a coarser pitch than 3 times the diameter, because we are losing thrust and we dont want it much less than about 1:1 because we are spinning the prop much faster than it needs to go to generate the lift. So in a fixed speed cruise or a climb, we might use a prop around a 10x10 to a 10x20 say, to get best efficiency.
Speed range
If you remmeber earlier, I estimated that a prop with a pitch to diameter of 1:2  say a 10x5 or so, was a prop that by and large wasn't stalled at zero speed . So thrust was generated effectively at zero airspeed, and in fact thrust will peak at zero airspeed or thereabouts, and fall away fairly sharply to the pitch speed, where it is by definition, zero.
The 1:1 or coarser prop may well be (considerably) more efficient at airspeeds around the pitch speed, but its a bit of a dog at lower airspeeds. In particular its pretty much fully stalled for a 1:2.5 to something like 70% of pitch speed. Such a prop will be slow to get airborne, and will 'come in' as the model gets up to speed, loading up and 'biting' as the aircraft gets close to pitch speed. In fact the coarser the pitch, the narrower the range over which its likely to deliver its best, But the better its best is, up to around 2.5:1 pitch to diameter, after which its downhill all the way. Whereas a 10x4 or 10x5 will be capable of efficient hovering, and rapid takeoffs, but wont be efficient flying on the wing at all. It will rapidly unload and run out of thrust. This may be in fact what you want for a 3D plane, where ability to hover, and punch out effectively is what is required, and top speed or efficient cruising is not the issue.
There is one more issue to consider, that is relevant to a speed model. If the blade tips get near mach 1, drag goes up completely out of proportion to the normal equations.
If you want a 250mph model, a 1:1 pitch to diameter is going to take you WELL into supersonic blade tips. You have no choice BUT go coarser. The limiting speed on the very coarsest pitch we might consider, about 2.5:1 pitch to diameter, should be enough for 400mph before the tips approach Mach 1.
What RPM should I use?
Ok, lets say we settle on a 1:1 pitch to diameter, and want a 50mph pitch speed. Should we run a 10x10 at 5000 RPM or a 5x5 at 10k RPM? Or a 20x20 at 2500 RPM? Which is more efficient, which has more thrust? All have the blade tips running at similar airspeed..
It is pretty obvious though, that the bigger the prop the more blade AREA it has running at that tip speed, so the more thrust its going to generate. And likewise, because its operating at similar thrust to drag (a function of the geometry, really), the more power its going to take to generate that thrust. So comparing props based on pitch speed alone is a bit meaningless. .
What we want to know is what, for a given power, is the most efficient generator of thrust..large, slow coarse, or medium, or fine less diameter and faster. and what RPM to operate at.?
The reality is for a given power, we would more likely be looking at a choice between say a 10x5 at 10K RPM and a 15 x 15 at 3.3k RPM.
Here I defer to the Beagles amazing webocalc, which tells me that I can get about 41 oz of static thrust out of a 10x5 at that RPM from 222 watts into the prop.
A 14x10 at 5000 RPM takes the same POWER but static thrust is actually HIGHER. So if Beagle has got his sums right, there is absolutely NOTHING better about a 10K driven 10x5 than a 5k driven 14x10. We would expect that, because the blade tip speed on a 10x5 is unnecessarily high.
Beagle doesn't have a 15x15 in his database..but I patched one in..result! About the same as a 12x8 or 14x10 power and static thrust wise..but lower in RPM of course.
Now webocalc doesn't tell the whole story, but what it does say is that a slow revving 15x15 pitch to diameter isn't going to sacrifice much, if any, static thrust compared with a higher revving 14x10 or 12x8. All of which outperform a high revving 10x5. My guess is, that because actual blade airspeed is slower, it will be less draggy per unit thrust as well, up at pitch speed.
The actual target model in this case was a sort of middling 50" sporty trainer. 4lb weight, 250+ watts but delivered from between a 10" and 15" prop. between 4000 and about 1000 RPM.
Final rules of thumb
A prop much less than 0.5 pitch to diameter is a waste of space and power, unless you want a helicopter. Or simply cannot get your RPM down because you picked the wrong motor! Or are constrained by diameter as in an EDF. It may rev a little faster though, so its good for fast response. You are looking at 10K plus RPM for modest pitch speeds.
Over the range 0.7:1 to 1:1 props are broadly similar at low end thrust. You aren't sacrificing much due to blade stall and you are getting much better efficiency at cruise as the pitch coarsens and RPM comes down.. RPM comes down for similar powers into the 35006000 sort of range typically. Sadly outrunners prefer the upper edge of this band, so its a balancing act without a gearbox.
Over 1:1 you are beginning to sacrifice low end thrust for a more efficient cruise, or a higher top speed. RPM may well come down a LOT for low powered models. I think a rubber model typically flies with oversquare props and sub 1000 RPM.
1:2.5 represents the extreme limit of what you probably want to consider, for extremely high top speed, or on a one speed model where RPM is severely limited. Probably the sort of prop you would use to try and reach 300mph, or conversely, for absolute best performance on an indoor rubber model at 5mph! In short it is the ultimate sort of efficiency, but at the cost of hugely unsatisfactory off pitch speed performance.
Some real world full size examples
Sopwith Camel. about 1:1 pitch to diameter, running at about 1200 RPM. Pitch speed about 130mph
Hawker Hurricane Watts fixed, about 11x20 foot running at 1500 RPM. Pitch speed about 360mph
Gypsy Major (Dragonfly) 6'4"x5'3" Probably cruise around 2300 RPM, pitch speed 145 mph.
The only finer pitch prop I found was a Lycoming, roughly 6x4 feet, operating at 2700 RPM. pitch speed around 130 mph.
