deriving prop equations from scratch

[flieslikeabeagle]

Hi folks,
I'm trying to understand the physics of RC planes better, and one of the things I'm trying to understand is the propellor. After an hour of head-scratching and recalling some basic physics formulae, I derived equations for pitch speed (this one is easy) , prop thrust, and motor power. The equations are in SI units, and are in terms of constants like air density, prop diameter, prop pitch, and prop revs/second.

At this point I'm assuming the prop is 100% efficient, once I get everything figured out I can tweak the equations to allow for a less than perfect prop.

Once I know for sure these equations are correct, I can go ahead and uglify them to use Imperial units. But I'd like to check if I got the basics right. My equations predict that:

1) pitch speed is proportional to prop pitch and prop rpm

2) prop thrust is proportional to prop diameter squared times prop pitch squared times prop rpm squared.

3) power needed is proportional to prop diameter squared times prop pitch cubed times rpm cubed

Can anyone verify these dependencies for me?

I'll attach a PDF which shows how I derived my equations.

-Flieslikeabeagle

[vintage1]

Hi folks,
I'm trying to understand the physics of RC planes better, and one of the things I'm trying to understand is the propellor. After an hour of head-scratching and recalling some basic physics formulae, I derived equations for pitch speed (this one is easy) , prop thrust, and motor power. The equations are in SI units, and are in terms of constants like air density, prop diameter, prop pitch, and prop revs/second.

At this point I'm assuming the prop is 100% efficient, once I get everything figured out I can tweak the equations to allow for a less than perfect prop.

Once I know for sure these equations are correct, I can go ahead and uglify them to use Imperial units. But I'd like to check if I got the basics right. My equations predict that:

1) pitch speed is proportional to prop pitch and prop rpm

No brainer on that one. Tick.

2) prop thrust is proportional to prop diameter squared times prop pitch squared times prop rpm squared.

Mmm. Its proportional to volume of air times speed, but volume is proportional to speed times area, so..yes, that sounds right.

3) power needed is proportional to prop diameter squared times prop pitch cubed times rpm cubed

Kinetic energy is volume times speed squared
That is area times speed, times speed squared
Since speed is RPM times pitch, sounds like diameter squared and pitch and RPM both cubed?

Yup. What you said.


Can anyone verify these dependencies for me?

I'll attach a PDF which shows how I derived my equations.

-Flieslikeabeagle

Yes. PDF rings true for 'idealised' propellor.
Now need to fudge in inefficiences due to stalling etc, blade shape and so on..

However I think teh basic dependencies seem aboutr right. Its amazing the difference a little bit of pitch or RPM makes to the power draw.

Next exercise of value, is given lower air density, and need for increased airspeed, what extra pitch and diameter are required at altitude, for the same power? :D

[gkamysz]

2 and 3 don't match my math.

I use W=1.11*D^4*P*kRPM^3 D and P are in feet. 1.11 is a constant to represent different props 1.11 is for APC glow types.

[flieslikeabeagle]

Vintage1, thanks for the sanity checks!

The air density problem is actually trivial: if you look at the PDF I attached, both the thrust and power equations are proportional to the air density. To keep power constant as air density varies, all you have to do is keep density * prop diameter squared * prop pitch cubed * rpm cubed constant. Assuming your prop is fixed pitch and doesn't grow like Pinocchio's nose, that means you have to keep density * rpm cubed constant. A 1% increase in RPM will compensate for a 3% change in air density, more or less (binomial expansion of (1+x)^3 for small x).

I didn't stop to see what thrust change would result, but the thrust equation is also proportional to the air density, so its simple to figure out.

gkamysz, thanks for the feeback. Hmm. Back in college I was taught that one of the first ways to check a classical mechanics equation is to do a dimensional check on it - check that the left and right hand sides of the equation have the same powers of the three basic physical quantities, mass, length, and time. In other words, if an equation that is supposed to predict a length turns out to have dimensions of seconds on the other side of the equal to sign, you know something is wrong! (Please don't bring up relativity and the equivalence of time and space here!).

Power is work done/time ; work is force times distance; and force is mass times acceleration. Put them together, and you find power has the dimensions of mass times (length squared) divided by (time cubed).

The power equation I derived has the same dimensions, i.e, M L^2 T^(-3).
(I didn't type the full equation into the text above - if you look at the PDF I attached, there is a factor of rho - the air density, which has dimensions of mass divided by length cubed - in front of it. With that factor of density, my power equation has the right dimensions.)

This alone doesn't guarantee the equation is entirely right, but does it suggest I got the important parts right.

The power equation you quoted, W = 1.11 D^4 P rpm^3 does not at first sight seem to have the right dimensions. The trouble is, I don't know what dimensions might have been buried inside the magic number of 1.11! Even so, I suspect that this equation you are using is wrong. Here's why: if you double the pitch, you move the air twice as fast. This means it has four times the kinetic energy, since energy is proportional to velocity squared. This alone would mean that doubling the pitch would require FOUR times more power from the motor - but the equation you're using doesn't predict that, saying you need only a doubling of motor power.

There's more: doubling the pitch not only doubles the speed of the air sucked through it, but also means that twice as much air will go through it each second! So now you have twice the mass of air, accelerated to twice the former speed. Put both factors together, and it will take EIGHT times as much power to turn a prop with twice the pitch at the same rpm.

Can you look up the original source of your equation, to see if there's a typo or something? It would be nice to get this sorted out.

-Flieslikeabeagle

[gkamysz]

The power equation I've posted is one that has been used by modelers for many years. I've used it countless times and it works. Motorcalc and Ecalc use it in their calculations as do many of the online calculators.

I used your formula to compare to the ones I have been using and the results weren't a few percent off they were 500% off.

Put both factors together, and it will take EIGHT times as much power to turn a prop with twice the pitch at the same rpm.

Um, no. I know that if I want to double flow velocity of any given prop by doubling RPM this is true. But turning a prop with double the pitch at the same RPM requires only twice the power.

Your equation is true of the power required to move the amount of air as described by the diameter and pitch. Everything that I've read about props seems to indicate the velocity at the prop is only 1/2 the final velocity of the air acted upon.

My experince with this is practical, not theoretical. I do have books that explain it all but there is way to much typing involved.

I just remembered a good link for props.

http://www.mh-aerotools.de/airfoils/propuls4.htm

Greg

[vintage1]

well..this promises to be interesting.

I THINK that beagles math is is correct IF the prop accelerates the air from standstill to effectively pitch speed....over the diameter of the prop.

Is that a valid assumption?

[hul]

the first way to check a classical mechanics equation is to do a dimensional check on it

step 2 would be to check a known case. Doing this for my E3D gives a thrust of about 80 Newtons, which is way too high (it's about 20N).

I think the problem is your assumption that air gets accelerated to vp from zero speed.

Found a very similar formula on Martin Hepperle's site, here:
http://www.mh-aerotools.de/airfoils/propuls4.htm

Hans

[vintage1]

too late for me to plough through that one, but it looks as tho a more complex analaysis is taking place.

All yours beagle!
I'm for bed...

[gkamysz]

My fan calculations do say that if I double velocity for a fan power required will be eight times. BUT, my fan calculations also say that pressure also increases by 4 times. If I set pressure to a fixed value then I get double the power.

If you would like to go through the formulae from the basics that's your call. It's much safer and easier for me to look up the information. Propeller theory isn't exactly a new science. It's been studied for over a hunderd years now and the basics are well known.

http://www.grc.nasa.gov/WWW/K-12/airplane/propth.html
http://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html

Greg

[flieslikeabeagle]

Greg, just to make it clear, I'm not trying to prove you wrong - I'm trying to learn what is right. By doing it from scratch, I expect to make mistakes, but like most people, I learn the most from my mistakes. (If I already knew enough to get it right the first time, I wouldn't really learn anything new, would I?). The intent is not to develop something new to the world, but for me to really understand what is going on.

How are you doing the fan calculations? The equation you posted for prop power is linear in pitch, so if you're seeing eight times the power for a doubling of velocity, then something more than that equation is going on. Does the software(?) you use allow you to force the pressure behind the prop to any arbitrary value? Because if so, the missing factor of (pitch squared) may be hidden inside that pressure variable.

Everything that I've read about props seems to indicate the velocity at the prop is only 1/2 the final velocity of the air acted upon.

Could you clarify that for me? I don't understand the sentence. Are you saying the air leaves the prop at half the pitch speed?

Hul, when you tried my formula out, did you use the full formula from the PDF I attached, including the air density in kg/m^2, the factor of pi, and the constant 4? Also, could you tell me the actual numbers for your E3D? (rpm, prop pitch and dia, thrust, and anything else you know?) You're right, my calculations assumed the propeller was static (since most modellers measure their prop thrust, rpm, and motor watts on a stationary test stand). But the full-blown equation should reduce to mine with v=0, and I don't think it does, meaning I still may have a mistake in my formula.

Thanks for the links - I'll scratch my head over those and see if I can figure out what I'm missing.

-Flieslikeabeagle

[flieslikeabeagle]

Ah-ha! I may have realized the mistake I made. I was treating the air as stationary in front of the prop, and moving at pitch speed behind it - that seemed like a reasonable assumption since the air sufficiently far ahead of the prop is in fact stationary. But evidently that analysis obscured something else - the factor of one half that Greg mentioned. Okay, back to the equations for a closer look...

-Flieslikeabeagle

[gkamysz]

My fan calculations come from a book I found to design fans. The formulae are based simply on mass flow and pressure rise. Pressure rise is important in fans, but I'm not sure how it applies to props. That said, I don't use the prop formula for fan calculations. Nor do I design props.

Greg

[vintage1]

A hint from one of those sites posted up..

In order for pressure to remain the same throughout the slipstream, the air has to change volume. In order for air to move into the prop at all its already being accelerated. Haven't puzzled it hrugh yet.

[Bill Mixon]

Good thread topic
I have just recently started trying to understand all this myself. Looks like this is just the info I've been looking for.

[flieslikeabeagle]

I've been looking through the formulae and derivations on those two websites, and there are two things that don't match my formula:

1) I did not allow for the "contraction of the stream tube passing through the propeller disk". I totally overlooked this issue. The effect of it will be to reduce the area of cross section of the propellor disc from pi D^2/4 to something smaller, which in turn means my equations (2) and (3) will aquire a numerical constant in front of them, with value less than 1.

2) The thrust equation on http://www.mh-aerotools.de/airfoils/propuls4.htm , i.e equation (1) on that page, is identical with my thrust equation if you set v=0 and delta v = vp, which are the assumptions I made. Unfortunately, neither website gives a formula to connect the change in velocity through the prop (delta v) to the pitch and rpm of the prop itself!

On the minus side, I missed two important factors the first time round: one, the contraction of the stream tube, and two, the entering air velocity on the front of the prop. On the plus side, both of these merely add a numerical multiplicative constant to my equations...they don't change the major dependencies on prop constants. So all I really need to do is get good measured data for a couple of high-efficiency props, and use those to figure out the numerical "fudge factor".

I have two books on model aircraft aerodynamics in the mail on the way to me, maybe I will get this figured out once I get them into my hands.

-Flieslikeabeagle

[Tom Harper]

The equation for thrust given by Chevalier in "Model Airplane Propellers..." is:

Thrust= (K) x (Cf) x (RPM^2) x ( D^4)

Cf is coefficient of thrust for this propeller at a known J (advance ratio)

Ct~.1 to .15 are reasonable typicals for static thrust (advance ratio=0)

Thrust is in pounds
K=3.1876 x 10^-11 (normalizes everything to feet, pounds and inches)
Assumes sea level air density
D is diameter in inches

J=(V/(Pi x RPM x D))
V is velocity in MPH

Chevalier provides a disk with short programs that provide performance charts based on your input parameters. They are very handy.

[vintage1]

mmm. so diamter to the fourth power eh?

More and more intruiging!

[gkamysz]

The calc programs use a very similar formula for thrust. I use the same one they do. I didn't volunteer it here as I know it has issues, such as not accounting for stalling at all.

Tha above formula is great assuming you have perfomance data for the prop you want to model. No such thing exists.

Greg

[flieslikeabeagle]

The plot thickens! At the suggestion of another RC Groups member, I tried this: let's treat the prop blade as an aerofoil, and assume it has the same lift equation as any other wing, i.e.,

Lift force = 1/2 rho C_L S V^2

where S is the area of the wing, V is its velocity, C_L the coeff of lift, and rho the air density.

Applying this to a propellor, the lift force is, of course, the prop thrust. To go from V to prop rpm, I substituted V = r * omega where r is the radius and omega is the rotation rate in radians/sec. Leaving out numerical constants I end up with something of this form:

Prop thrust = constant x D^3 x rpm^2 x (width of blade) (eqn 4)

The width of the prop blade (mean aerodynamic width, actually) comes in because of the area of the wing in the lift formula - area of the prop blades is width x D, more or less.

Now, since prop diameter and blade width are related for any given planform of propellor blade, it makes sense to say that

width = Diameter x some numerical constant.

Put this back in my equation, and sure enough, you get

Prop thrust = const x D^4 x rpm^2

Now the formula agrees with Chevaliers :D ,and of course his "J" term is my "C_L", which varies with advance ratio. So there is actually a prop pitch term in that constant, i.e, prop thrust is proportional to pitch * diameter^4 * rpm^2.

My tentative conclusion: my initial attempt to derive the prop equations based on momentum and energy conservation were inaccurate because I missed a few important things: one, the Bernoulli effect, which lowers the pressure, and therefore the density, and therefore the mass, in the fast-flowing parts of the slipstream; and two, the acceleration of the air ahead of the prop, as well as ongoing acceleration behind it. But I continue to be baffled by the fact that the Chevalier equation has a length^5 dependence (D^4 , times the hidden pitch in the "J" advance ratio), which means the Chevalier equation does not work out to the correct dimensions for force (which has dimensions of mass x distance^2 / time^2).

I'd say that the best thing to be done now is to use the Chevalier equations, and use any available experimental data to figure out the numerical constant for any given prop model (eg. GWS Slowflyer props).

Incidentally, the approach I used of treating the prop blade as a wing should work for drag force just as well as lift force...just replace C_L with C_d...which means we should be able to derive a formula for the the torque required to turn a prop, since torque = r x force.

I ran through this, and came up with

Torque ~ rho * C_d * rpm^2 * D^5
(using the same assumption that prop blade width is proportional to D).

Since power is proportional to torque x rpm,

Power ~ rho C_d rpm^3 * D^5

As before, there's a "pitch" or "J" term in C_d.

Geez, D^5, can that be true??

-Flieslikeabeagle

[JMP_blackfoot]

Power ~ rho C_d rpm^3 * D^5

As before, there's a "pitch" or "J" term in C_d.

Geez, D^5, can that be true??

-Flieslikeabeagle

Yes, that is correct. I have designed and carved many props for our 328 km/h (French) record straight line IC speed model. Some of them were single blade. Taking a successful two-bladed prop as a basis, I made the single-bladed props the same pitch, but of larger diameter by a factor of 1.15, which I derived as the 1/5th power of 2 (2 being the ratio of the number of blades 2^.2 = 1.1487) on the assumption that the same power at the same rpm was required for both props. It worked out quite well, the speeds being indeed very similar.

[flieslikeabeagle]

JMP, 328 km/hr, wow! 205 mph from a propellor-driven model!

It's great to have you experimental results confirm the D^5 dependence.

-Flieslikeabeagl

[JMP_blackfoot]

JMP, 328 km/hr, wow! 205 mph from a propellor-driven model!

It's great to have you experimental results confirm the D^5 dependence.

-Flieslikeabeagl

Our model was repeatedly timed at over 360 km/h during training, but for the record day, it was very important to follow the rules as outlined in §7.5 of the FAI Sporting Code, General Rules for CIAM activities, competitions and records section, hence the lower official speed
ftp://www.fai.org/sporting_code/sc4/sc4-abr.04.zip

The N^3*D^5 formula can be used to extrapolate a successful gear ratio/prop to another gear ratio.
As an example :
Given a successful combination of 3:1 gearing and 10 x 6 prop, what size prop should I use with the same motor, battery and model if I change to a 6:1 gearing ?
Initial combination : No, Do, Po (rpm, diameter, pitch)
New combination : N1, D1, P1
Pitch : rpm is halved so the pitch will have to be doubled. P1 = 2*Po
Diameter : (Do^5*No^3) = (D1^5*No^3)/2^3
or D1^5 = Do^5*2^3 or D1 = Do*2^3/5 = Do*2^.6 = Do*1.516

So here we are :
3:1 gearing -> 10x6 prop
6:1 gearing -> 15x12 prop

Checking with Motocalc, I find that they recommend a somewhat smaller prop :

AP-29, 6 cells, 8 m/s :

10x6 prop, 3:1 gear : 15.9A, 5952 prop rpm, 17857 motor rpm, 343g thrust, 44.3% prop efficiency
14x11 prop, 6:1 gear : 15.9A, 2981 rpm, 17884 motor rpm, 431g thrust, 57.9% prop efficiency.

According to Motocalc, the 15x12 prop would have pushed the current to 17.5A at the same speed, or the speed to 12 m/s at the same current and rpm.

Do you see the better thrust and prop efficiency with a higher gear ratio ?

As you may guess, props are one of my pet subjects :)

[flieslikeabeagle]

JMP, got it. Very cool. I'd say that props are a good choice for a "pet subject", after all, they are what make most of our models fly! (with apologies to the sailplane and model jet folks).

I assume the improved efficiency comes from operating the propellor blades, essentially, at a better L/D (lift/drag) ratio? Since the motor is operating at the same rpm and current draw, there is no improvement in efficiency to be had from the motor itself.

Is there a way to tell for a specified pitch speed and thrust what prop will be operating at optimum L/D, i.e, optimum propellor efficiency?

Part of the reason I am trying to understand propellors is because I have been interested in the "high voltage, low current" approach used by Vintage1, DeanInMilwaukee, Gene Bond, and others on this forum to increase the efficiency and output power available from brushed motors. If I can also understand what goes into improving prop efficiency, that will allow yet a little more tweaking.

-Flieslikeabeagle

[vintage1]

Well if you gear and go for a big prop you can improve thrust at the expense of pitch speed for the same horsepower...that much is sure.

But thats easily understood from comparsin of momentum and Kinteic energy eqations.

i.e. static thrust is mv^2, but power is 1/2mv^3. Power requirement goes up faster than thrust. as the size of the acclerated air mass decreases.

As far as prop efficiency goes, it has to be a function of lift to drag on a per blade basis.

Which is why you try and opearate the prop at something fairly close to pitch speed, to get AoA down and drag down with it.

[kallend]

All this stuff was done in the 1930s by people like Locke and Glauert.

The problem with a simple equation is that it doesn't account for inflow, both axial and rotational.

I once wrote a program to do the blade element theory calculations with inflow (pre-Windows days); to do that you have to have a good idea of the model's drag characteristics because the motor/prop do not exist in isolation from the rest of the model. The program gave very reasonable predictions, in line with experience, and was incorporated in later versions of the the old Dave Brown flight simulator.

I thought I could curve fit some simpler equations to the output, but nothing fit very well.

So on the whole I don't susbscribe to the dia^n * pitch^m * rpm^j type approach.

[flieslikeabeagle]

I think for most modellers without accurate telemetry equipment, about the only drag characteristic estimate we can come up with is a rough eye-balled idea of the glide ratio - you can't tell if its 15:1 or 16:1 that way, but you can definitely tell if its closer to 5:1 or 10:1. I'm using a guesstimate for glide ratio to figure out the motor power needed to sustain level flight.

Kallend, how far off were the simple dia/pitch/rpm equations from your measurements? Are we talking 10%, or 100%?

I ask because I think there is great value in an estimate, even if its 10% or 20% off. I'm hoping to develop a method to narrow down the hundreds of possible combinations of motor, gearbox, prop, and battery to a few. Those last few can always be tried out and chosen based on real-world performance.

And yeah, I know I'm not inventing anything new, but trying to figure this stuff out is helping me to learn more about the physics behind flight.

-Flieslikeabeagle