Gary Webster (EAA 837 Vice President) and I were having coffee and the conversation eventually came around to the difficulty of deciding on a the proper prop for a LSA or Experimental aircraft. I mentioned that making a propeller could not be all that hard because the FAA allows an A&P to make or repair a wood propeller for certified aircraft (or at least use to allow it 15 years ago).

So, right after I said it was not hard, Gary then said, "... he sure wished someone would at least show the math involved...". This was at least the second time we had talked about propellers in the last few weeks and he mentioned a desire to "know the math". This article will address that in as simple terms as I know how.

45" pitch | |

100% efficiency | 85% efficiency |
---|---|

1000rpm =43 mph 1500rpm =64 mph 2000rpm =85 mph 2500rpm =107 mph 3000rpm =128 mph 3500rpm =149 mph 4000rpm =170 mph 4500rpm =192 mph |
1000rpm =36 mph 1500rpm =54 mph 2000rpm =72 mph 2500rpm =91 mph 3000rpm =109 mph 3500rpm =127 mph 4000rpm =145 mph 4500rpm =163 mph |

Propellers were originally called "air screws" because a laymen immediately understood that the same principle that drives a wood screw into a piece of lumber caused a propeller to move through the air. The two factors acting on a propeller that we concerned with are diameter and pitch.

In choosing or designing a propeller we want the propeller to extract the maximum amount of energy at the rpm limit provided by the manufacture. We are left with adjusting the propeller diameter and pitch. Because the rpm is "fixed", our discussion is about pitch and diameter.

**Diameter**

The longer and narrow a propeller, the more efficient it is at extracting
energy with a minimum amount of drag. The diameter of a propeller for
any aircraft is usually limited by two factors; gear (ground clearance)
and weight. For aircraft with a maximum rated horsepower around 2700rpm the
industry standard is usually a diameter of about 60 to 65" for 50-100hp,
and 75-80" for 150-200hp.

**Pitch**

Given the longest possible propeller diameter the aircraft and engine
combination can use, we must adjust the pitch to extract the rated
horse power at the proper rpm.

There are two things we should note about our picture of the aircraft pitch. The first thing is that the physical (or theoretical) pitch of the is greater than the actual pitch due to "slip". In other words, as the prop screws through the air it does not perfectly cut the air, rather it slips about 10-20%. So a propeller with a 70 inch pitch does not actually move the airplane forward 70 inches with each turn, instead it will only move forward about 60 inches. This "slip" is also called propeller efficiency.

As a rule of thumb, wooden propellers usually have 80-85% efficiency (15-20% slip) due to thicker less efficient airfoils. Metal propellers typically have 85-90% efficiency (10-15% slip). The very best constant speed metal props rarely achieve better than 92% efficiency.

The second thing we need to note is the pitch of a propeller is not measured at the prop tip. Industry standard has found from testing that the most accurate measurement is at the 75% position of the radius. So a prop with a 70 inch diameter would have a 35 inch radius, and the pitch would be measured 75% from the center of the hub towards the tip, or about 26 inches (35in x .75) from the center of the hub.

Here are a few on line calculators to help with propeller design.

http://personal.osi.hu/fuzesisz/strc_eng/

http://www.csgnetwork.com/directhpthrustcalc.html

Professional design software is also available such as the JC Propeller
Designer for $200

http://user.tninet.se/~trz012v/JCPropellerDesign/index_gb.html

Suppose we have a 65hp engine and we know that we need a 60" diameter prop with a 45" pitch. What will the angle of the prop be at the 75% radius position? For the answer we need a simple trigonometry formula. But first lets illustrate this by turning it into a 2 dimensional problem.

1) To solve this problem we need to use the 75% dia radius mark for our pitch
and diameter.

r = 75% of actual radius

= dia / 2 * .75

= 60 inch / 2 * .75

= 22.5 inches from hub

c = circumference (distance of one rotation at the 75% mark

= pi * r * 2

= 3.14 * 22.5 * 2

= 141 inches

Now we need to use a trig formula (inverse sine) to solve for angle(A).
The answer will be in radians and we can convert that to degrees.

Given:

P(pitch) = 45"

C(circumference) = 141"

Solution:

A = sine^{-1} (P/C)

= sine^{-1} (45/141)

= sine^{-1} (.319)

= 0.325(radians)

To convert radians to degrees, multiply radians by 57.29

A(deg) = A(rad) * 57.29

= 0.325 * 57.29

= 18.6 deg

So that's our answer!

The prop should have a 18.6° deg angle at the 75% radius position.

So what about the angles for the rest of the prop? That is where the talent and art of prop making happens. The ideal prop has a constant load all the way from the hub to the tip. This is not practical to achieve because the first 25-30% of the prop is not spinning fast enough to ever produce usable thrust.

I have not found any information about calculating intermediate angles.
I have heard (but not confirmed) that a rule thumb may be applied.
Use twice the angle at 30% and 1/2 the angle at the prop tip. For example
above, we have about...

47 deg at 30%

23 deg at 60%

12 deg at 100%

Another possibility would be to calculate the circumference for different distances from the hub - and we could do this. This is not as straightforward a solution as we might think because the airfoil changes and so the load on the prop will not necessarily be proportional to the angle at the 75% reference. Propeller airfoils are usually thinner and more efficient at the tip, and they become very thick and rounded the closer we get to the hub. Estimating the intermediate airfoils is considered more of an art than a science and likely is chosen more from previous experience and testing than from math calculations.

Perhaps the best way to come up with intermediate pitch angles from our 75% reference would be to take a prop from a manufacture that did lots of testing and see how they did it. I would suggest looking at a metal prop from a Cessna or Piper. These folks would have spent plenty of time matching a propeller design to a large production run.

How can we compare the performance of two different props. Is one prop more efficient than another?

Here is a formula for comparing propellers. It is commonly used by racers.

Given:

S1 & RPM1 are with a new prop

S2 & RPM2 are with a old prop

S1 = new speed

S2 = old speed

RPM1 = new rpm

RPM2 = old rpm

efficiency = 100 * ((S1/S2)^3 / (RPM1/RPM2)-1)

Example: Here is an example reported by Bob Bryson after a prop modification to his biplane.

Given:

S1 = 211 (new speed)

S2 = 201 (old speed)

RPM1 = 3250 (new rpm)

RPM2 = 3100 (old rpm)

e = efficiency

e = 100 * ((211/201)^3 / (3250/3100)-1)

e = 10.3%

For this airplane, it would have required 15.7% more horsepower to increase to the new speed, 4.8% hp came from the engine's increased rpm, so the remaining 10.3% hp came from the propeller's increased efficiency.