# Equilibrium velocity of a propeller-driven aircraft

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1. May 9, 2016

### KarenRei

Hi - I just want to know if I'm approaching this correctly. I'm trying to calculate the equilibrium velocity of a propeller-driven aircraft when supplied a given amount of power. I'm iterating:

min_velocity = 0.01
max_velocity= 200
aircraft_velocity = 5
for i in range(steps):
force = 0.5 * air_density * aircraft_velocity^2 * drag_coefficient
torque = force * propeller_diameter
calculated_efficiency = force * aircraft_velocity / power_in
rotation_rate = power_in / (2 * pi * torque * calculated_efficiency)
advance_ratio = aircraft_velocity / rotation_rate / propeller_diameter
if real_efficiency < calculated_efficiency:
min_velocity = aircraft_velocity
aircraft_velocity = aircraft_velocity * 0.5 + max_velocity * 0.5​
else:
max_velocity = velocity
aircraft_velocity = aircraft_velocity * 0.5 + min_velocity * 0.5​

That is: calculate drag at the current speed, figure out how bad the efficiency would need to be for the given power level to overcome it, calculate the rotation rate, use it to get the advance ratio, lookup the advance ratio in an efficiency table (which I found on some random website), and then either A) increase the speed if the needed efficiency is less than what the propeller should get at that advance ratio, or B) decrease the speed if the needed efficiency is greater than what the propeller should get.

Is this a reasonable way to approach this? Also, I'm calculating a rough wake velocity as:

wake_velocity = 0.5 * (aircraft_velocity + sqrt(4 * force + air_density * square(aircraft_velocity)) / air_density))

.. based on

F = rho * V_wake * (V_wake - V_free)

Does this seem correct? I haven't done much work with airfoils before, so please excuse me if I'm tackling this poorly ;)

2. May 10, 2016

### KarenRei

No airfoil people here? :)

3. May 12, 2016

### KarenRei

Guess not :(

4. May 12, 2016

### Staff: Mentor

I'll move it to engineering, where you may have more luck...

5. May 12, 2016

### SCP

You need to account for the characteristic area here. This area is probably the wing planform area, or it could be the wetted area, depending on how the drag coefficient was determined.
I'm not sure where this comes from. I'm assuming this is the torque of the engine (that's how it appears to be used later). There's no reason to think that the engine torque is the drag acting at an arm of the prop length. The force you're using is the thrust force parallel to the rotation axis of the prop. For torque, you need the force acting on the blade in the plane of rotation. These two forces are related, but they are not the same (under normal cruise conditions for a well designed prop, the in-plane force will be lower than the thrust force). Also, the force is distributed over the blade length, it isn't concentrated at the blade end. In the absence of better aerodynamic analysis, you can probably estimate the average (statically equivalent) force position as half-way along the blade length.

Normally, torque would be calculated from the known power input and the RPM:

torque = (engine_power/rotation_rate)

But if you want to treat rotation rate as an unknown, your method might work if you can get a better estimate of the in-plane force.

If you can improve the drag and torque estimates, the rest of the algorithm looks like it should work. However, it's worth noting that you haven't accounted for induced drag. This may be deliberate if you're assuming flight in the normal cruise range. Otherwise, you should think about how you might account for that.

For the wake, you need to include the area of the wing. Just do a unit check on your equations:

FORCE = DENSITY * VELOCITY^2 --> (kg-m/s^2 == (kg/m^3)*(m/s)^2) = FALSE

or

FORCE = DENSITY * AREA * VELOCITY^2 --> (kg-m/s^2 == (kg/m^3)*(m^2)*(m/s)^2) = TRUE

Cheers,
SCP

6. May 12, 2016

### KarenRei

Oops, that was a transcription error (my code had its variable names in Icelandic, I had to translate :) ); I have the cross section in use in my code already, it just got left out here.

Indeed, rotation rate is an unknown. It's assumed as a given that the motor controller will handle adjusting the rotation rate such that the total motor power output is constant.

You're right that my force above is frontal. Hmm... how would one estimate the in-plane force on the prop halfway down the blade? For some reason my mind is drawing a blank on this one...

What do you mean by induced drag? The drag force (the part where I didn't translate the cross section) is supposed to be the drag for the whole aircraft, prop and all. That's the force that the engine has to exert such that the craft neither accelerates nor decelerates (equilibrium). Do you mean, the drag coefficient of the aircraft changes as the rotation rate of the prop increases, or something of that nature? Note that both top speeds and rotation rates I'm dealing with are low (very large diameter prop).

Oh - if you mean in relation to the L/D ratio of the craft, I should probably mention, these calculations are for a simple airship - there's no aerodynamic lift.

Yep, clearly missed that ;) Hmm... I know my prop length but I don't know its area. I can't just make up an area for it because I'm sure that the table I found listing the correspondence between advance ratio and efficiency involves a specific-shaped prop for each condition. Let's see... hmm, the table has three entries: 15°/angle=0.75R; 30°/angle=0.75R; and 45°/angle=0.75R. Hmm... I suppose I should probably try to figure out what that means...

7. May 12, 2016

### SCP

Ok, so no induced drag. That makes more sense.
Ok. I though you were trying to find the wake of the aircraft. I was actually thinking in terms of downwash, but that's not applicable if there's no lift being produced. For the wake of the prop, use the area of the prop disk (so your blade length would be the radius).
For this, I got: $$V_{wake}=\frac{1}{2}\left [ V_{aircraft}+\sqrt{V_{aircraft}^2+4\frac{F}{\rho A}} \right ]$$

8. May 12, 2016

### KarenRei

Sounds like I just made an error in solving for the wake velocity then :) So the only thing missing, then, is:

Thanks a bunch!

9. May 16, 2016

### KarenRei

Sorry to bring this back up again, but, any thoughts on how to estimate the in-plane force on the prop halfway down the blade?

10. May 16, 2016

### rcgldr

You may want to consider power as thrust x speed, rather than torque x angular velocity. When dealing with propeller efficiency versus power = thrust x speed, the speed term is probably aircraft speed, as opposed to the induced flow speed through the propeller, which would be a higher speed than the aircraft speed.

11. May 16, 2016

### SteamKing

Staff Emeritus
This is what a propeller is designed to do: convert the torque of the engine into a thrust force which can overcome the drag of the rest of the airplane (or ship).

The efficiency of a propeller is based on how well this conversion is accomplished:

$$η = \frac{propulsive\, power\, out}{shaft\, power\, in}=\frac{thrust ⋅ axial\, speed}{torque ⋅ shaft\, rotational\, speed}$$

https://en.wikipedia.org/wiki/Propeller_(aeronautics)

12. May 17, 2016

### KarenRei

Torque is require to calculate angular velocity, and angular velocity is required to calculate the advance ratio, and the advance ratio is used to look up the efficiency. That is to say, a fast-turning high-diameter prop facing a slow airspeed has a high advance ratio, while a slow-turning low-diameter prop facing a fast airspeed has a low advance ratio. According to my tables here, a 15° 0.75R prop peaks in efficiency at an advance ratio around 0.65; a 30° 0.75R prop has its efficiency peak at around 1.25; and a 45° 0.75R prop at around 1.9. The latter two are more efficient than the former.

Still leaves open the question, of course, how to roughly estimate what the torque would be - I'd mistakenly been applying the frontal force the craft has to overcome at the end of the prop. SCP recommended the in-plane force on the prop applied at half the prop length, which makes better sense. But for some reason I'm drawing a blank on how to figure out the in-plane force.

Last edited: May 17, 2016
13. May 17, 2016

### SteamKing

Staff Emeritus
I'm not sure what SCP is, but the torque a propeller absorbs and converts into thrust comes from the output of the motor driving the shaft, after it passes through any gearboxes on its way to the propeller after leaving the motor. That's the way it's done with marine propellers, and I'm sure that that procedure is analogous to what occurs with aero propellers.

A motor spins a shaft while operating, making a certain amount of power at a certain RPM, from which the torque acting on the propeller shaft can be calculated. If there are any reduction gearboxes between the motor and the propeller, the propeller RPM is reduced by the gearbox ratio and the torque is multiplied by that same ratio, so that power into the gearbox is the same as the power coming out, minus a few percent for losses in the gearbox itself.

The angular velocity of the propeller is derived from the RPM at which the prop is being turned by the engine.

This is a discussion of basic screw propeller theory for aircraft from MIT:

http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html

Aircraft propellers are designed so that the tip speeds remain subsonic. Marine propellers don't usually encounter this restriction, since the speed of sound in water is greater than the speed of sound in air, and marine propellers don't need to turn as fast to generate useful amounts of thrust to propel the vessel, again because of the density of the working fluid (water).

Now, if you are designing an aircraft from scratch, you have two choices:
1.) pick a powerplant and then select a propeller to drive the plane, or
2.) select a propeller, and then try to design a powerplant to furnish the required torque.
I think the vast majority of propeller-driven aircraft are designed using 1.), since selecting or designing a propeller is cheaper and quicker than trying to design a powerplant.

14. May 17, 2016

### KarenRei

SCP is a user on this forum who replied above.

It seems you're working the problem in reverse. See the comments above. The motor supplies a fixed power output, not fixed torque, nor a fixed angular velocity from which the torque could be calculated given the power.

The in-plane force is related to aerodynamics, as by changing the prop shape you could get the same torque from a high speed / low drag prop or a low speed / high drag prop. The parameters that control this are encapsulated in the details of the prop in my efficiency table, which covers props from 15-45° / 0,75R. But I don't know how to use those numbers to calculate the in-plane force. Or, if you'd rather, the ratio of torque to angular velocity. Or really, just anything that will give angular velocity, because that's the figure that's needed to give the advance ratio. But it's going to depend on the prop shape.

If given the in-plane force acting on the propeller. The missing piece above that I'm seeking. It cannot be simply calculated from the angular velocity because the angular velocity itself is not fixed. To reiterate: neither torque nor angular velocity are fixed variables.

A tip speed just below the speed of sound is not a given in this scenario, it can be well less than that. Angular velocity is not a fixed variable. The fixed variables are power, prop diameter, drag area, atmospheric parameters (such as density), and a table of efficiencies for different propellers at different advance angles (although I'm open to using a different approach than the latter to get prop efficiency if there's a better way).

Here's where things currently stand, after SCP's feedback above:

min_velocity = 0.01
max_velocity= 200
aircraft_velocity = 5
for i in range(steps):
force = 0.5 * air_density * aircraft_velocity^2 * drag_coefficient * cross_section
calculated_efficiency = force * aircraft_velocity / power_in
in_plane_force_on_propeller = ???
torque = in_plane_force_on_propeller * (0.5 * propeller_diameter)
angular_velocity = power_in / (2 * pi * torque * calculated_efficiency)
advance_ratio = aircraft_velocity / angular_velocity / propeller_diameter
if real_efficiency > calculated_efficiency:
min_velocity = aircraft_velocity
aircraft_velocity = aircraft_velocity * 0.5 + max_velocity * 0.5​
else:
max_velocity = velocity
aircraft_velocity = aircraft_velocity * 0.5 + min_velocity * 0.5​

wake_velocity = 0.5 * (aircraft_velocity + sqrt(aircraft_velocity^2 + 4 * force / density / prop_cross_section))

In plain English: Assume a starting speed and iteratively do the following.

1) Calculate the force that the engine needs to provide to the aircraft by calculating the drag force of the aircraft.
2) Calculate what sort of efficiency the prop would need to have in order for the fixed power output from the engine to overcome this drag.

Once we have that, we need to figure out the actual efficiency we'd get when trying to fly at this speed. So....

3) Calculate the in-plane force on the spinning prop by some unknown means.
4) Estimate the torque that the engine needs to provide to equal this force.
5) Calculate the angular velocity from the power, torque, and calculated efficiency.
6) Calculate the advance ratio from all this.
7) Look up the real efficiency for the given advance ratio.
8) If the prop would be more efficient at this advance ratio then our calculated efficiency, then we can go faster - increase the aircraft velocity.
9) If not, then we can't actually go this fast - decrease the aircraft velocity.

When we have the aircraft velocity, use it to determine the approximate wake velocity.

When the function that does this is complete, it will be used in an optimization loop for a weighted combination of maximizing efficiency, maximizing aircraft velocity, minimizing prop mass, minimizing engine mass and power consumption (related), minimizing wake velocity, and maximizing the mass flow rate in the wake, with various constraints on the parameters, and under arbitrary air conditions that are not guaranteed to be optimal.

Last edited: May 17, 2016
15. May 17, 2016

### SteamKing

Staff Emeritus
You apparently are not aware that power, shaft rotational speed, and torque are intimately related.

https://en.wikipedia.org/wiki/Torque

If you know two out of the three quantities listed above (power, torque, and RPM), you can calculate the third.
Once the plane has taken off and reaches a steady speed and altitude, the torque and propeller speed will be constant. If the plane is accelerating or maneuvering is some fashion, then there will be some variability. However, planes, like most vehicles, are designed to operate at some set condition or speed. The transient conditions are generally left alone.

If your powerplant is putting out a fixed amount of power, it won't be speeding up and slowing down at random, unless the throttle or other controlling mechanism is broken. Powerplants tend to be most efficient when operated at constant RPM, which suggests the angular velocity of the propeller is also going to be constant.

The speed of advance of the propeller is going to change with the speed of the plane, so given a set of performance curves for a propeller along with the power output of the powerplant, one can usually determine the operating point of the propeller, so that the speed of advance gives enough thrust to exactly balance the drag of the plane at that speed. It usually requires an iterative process to arrive at a steady state solution which satisfies all or the requirements.
What you have outlined above is a good start, but I would compare the thrust of the propeller to the drag of the aircraft at the given cruising speed. If you have a set of propeller performance curves, this should be easy to do, without worrying about mysterious 'in-plane' forces, which no one seems to understand.

1. If the propeller is too lightly loaded, i.e., it can produce more thrust at a given RPM than is needed to overcome drag, then the propeller RPM will tend to increase, unless the powerplant is throttled back to reduce RPM and bring thrust into balance with drag.

2. If the propeller is overloaded, i.e., it cannot produce enough thrust to counteract drag, and assuming you don't want to increase powerplant RPM, then the plane must slow down until thrust and drag are in balance.

16. May 17, 2016

### KarenRei

I'm not understanding how you've been missing this: we don't know two of the three. I've mentioned this many times in this thread, including in the post you're directly replying to, where I mentioned it repeatedly.

Of those three, we only know power. The other two are not supplied.

As has been mentioned many times, and is in the title of this thread, the goal is to solve for the speed. It is not a supplied variable.

Not only is that optimal angular velocity not known for a given set of aircraft parameters, but it will vary depending on the flight conditions (such as air density).

I'll reiterate: angular velocity is not a supplied variable.

I know this problem would have been much easier if you could fix it at some given value. It cannot. I wouldn't be asking it if it was that easy ;)

Except that the advance ratio changes with both the speed of the aircraft and the (unknown) angular velocity.

I'll reiterate what I wrote before: by changing the prop shape you could get the same torque from a high angular velocity / low resistance prop or a low speed / high resistance prop. The angular velocity - and thus the advance ratio - depend on the details of the shape of the prop. Which involves the prop angle, but I don't know how to go from a prop angle to anything that can give me the advance ratio.

I have a set of curves of the efficiency at different advance ratios, for different prop angles.

SCP clearly understood it, as they recommended that approach. I understood what SCP meant. I think you're the only one here who doesn't understand the concept. In-plane: the prop is rotating in a plane. Force: exactly what it says on the tin. The force resisting the rotation of the prop in its plane of rotation. Which is what torque is needed to overcome.

The loading on the prop is the aforementioned in-plane force resisting it.

17. May 17, 2016

### KarenRei

To make what I need very simple:

"You have a set of arbitrary aircraft parts sitting in front of you. You know all possible useful parameters of these individual parts individually - name it and you know it. You know nothing of how they'll perform once put together (except the CdA). They're going to be put together into an aircraft and flown at under atmospheric conditions X, which you have no control over, and is in no way guaranteed to be "optimal". Calculate the speed at which the aircraft will fly."

That is the problem I need to solve. I know you want to solve a different, simpler problem. Sorry, but that isn't an option. :) Because the parts are constrained by all sorts of other things and I'm dealing with a wide range of atmospheric conditions. I need to be able to get datapoints in any of them, for any possible combination of parts.

You have a prop, with a prop angle, a diameter, and whatever else you want to know about it.
You have a motor with a given maximum power output
You have a motor controller which will adjust the RPM to ensure that it stays at its maximum power output
You have an airframe to attach everything to which will result in a given CdA.
You're being told, "we're going to fly it under X conditions".

Solve for flight speed.

Last edited: May 17, 2016
18. May 17, 2016

### rcgldr

Why is advanced ratio (based on propeller diameter) used instead of slip ratio (based on propeller pitch)?

19. May 17, 2016

### KarenRei

Because the primer on propeller-powered flight that I read only went into the advance ratio and only gave a table of efficiencies for advance ratios? ;) hehe

I take it I'd have better luck if I dug into slip ratios instead?

20. May 17, 2016

### rcgldr

Slip ratio might make more sense if trying to derive the data as opposed to using a table. The tables I've seen for efficiency include advance ratio and a series of curves based on the blade angle at .75 radius, so both propeller diameter and pitch are being taken into account. Take a look at figure 11.28 in section 11.7.4.5 of this article:

http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html

A bigger version of what appears to be the same image:

The general trend is that propellers are about 80% to 85% efficient when operating close to an ideal speed range for a particular combination of diameter and pitch. This is offset by the fact some of the prop wash is diverted by the fuselage.