Challenge accepted! A website member has come up with a fascinating solution to my challenge from two weeks ago.
Where There’s a Will
I recently challenged website members to help me come up with a simplified method for matching the right motor and propeller to a model airplane. Long time website member, Bill “aeromodel03″ Garner, accepted the challenge.
Now, Bill is not exactly new at doing calculations like these. In fact, he has been working for the last few years on breaking a long-standing model airplane flying record. We have exchanged several emails on that subject, and believe me, he knows what he is doing. Rest assured, when the time is right, there will be full write-up on his project on this website.
The method described below is sound. You can always argue the details and the various assumptions made, but that is the point of me posting it here.
I did not come up with this method, so I will leave it up to Bill to answer your questions. I plan to put the method through its paces soon. Please give it a try and report back on your results.
Everything else that follows in this article was written by him.
Proposed Method (by Bill)
Finding a combination of electric motor and propeller that matches a particular airplane and its desired performance is complex if done in an exact manner. To do so usually requires the use of a computer program that searches through many combinations of motors and propellers before providing an answer. The purpose of this document is to describe a simplified method suitable for implementation in a single spreadsheet without the need for complex computations. Empirical data is used as the basis for the method, supplemented by some measured propeller data. The results do require interpretation and some iteration by the user. The method takes advantage of several empirical observations.
Watts Per Pound
One observation is that performance is proportional to watts per pound. There are several lists of W/lb ranges for different classes of aircraft and desired performance in existence. The minimum motor power required is obtained by multiplying the weight of the plane by the selected W/lb. Here is such a list:
- 50 – 80 W/lb: powered gliders, basic park fliers and trainers
- 80 – 120 W/lb: general sport flying & basic/intermediate aerobatics
- 120 – 180 W/lb: more serious aerobatics, pattern flying, 3D and scale EDF jets
- 180+ W/lb: All-out performance
Pmotor = Weight * W/lb
Propeller Power Unloading
Another observation is that maximum power loading occurs under static conditions, slowly decreasing as air speed increases. Hence it can be assumed that power and thrust calculations done at zero airspeed will satisfy the motor not-to-exceed conditions for any airspeed. Wind tunnel measured thrust and power coefficients, Cto and Cpo, as a function of pitch/diameter ratio for APC Thin Electric propellers can be used to calculate prop input power and thrust for candidate combinations of prop diameter and pitch.
Pprop = Cpo * rho * n^3 * D^5 * 746/550 watts
Tprop = Cto *rho * n^2 * D^4 Lbf
Cpo = -0.0116+.0957*p/D
Cto = -.2179*(p/D)^2+.359*p/D-.0356
rho = .002378 air density
There are three variables in these equations:
n is revolutions per second
D is diameter, in feet
p is pitch, in feet
Another observation is that the thrust goes to zero as a function of p/D ratio and D at specific advance ratio (J) values. The resulting revolution rate, nm, can be approximated by the following function:
nm = Vmax/(0.2*D+0.74*p)
Vmax is the maximum desired air speed in ft/sec
Propeller RPM Unloading
The next observation is that the loaded RPM at zero airspeed is about 90% of that at maximum airspeed, Vmax. Therefore the revolution rate at zero airspeed is approximately 0.90*nm. Using this value of n and calculating Po and T for a range of diameters and pitches will result in a listing such that the calculated prop power and the available motor power can be compared to find a match.
Matching Motor Kv and Battery Voltage
The other issue is insuring that the motor Kv rating and battery voltage are sufficient to support the selected revolution rates. Empirical evidence indicates that the maximum supportable rps when the thrust goes to zero (maximum air speed) can be approximated by the following formula:
Max rps motor = Kv * Vbattery/1.1 rev/sec
Therefore the selected motor should have a Kv rating based on the required rev rate for the particular prop.
Kv = nm*60*1.1/Vbattery
In summary, the method consists of generating a table with combinations of diameter and pitch in order, calculating the p/D, nm, Cpo, Po, Cto , thrust, Kv and Pmotor for each combination. Then sort the table in ascending order of Pmotor. Select a motor that meets the motor power requirement and also meets the calculated Kv requirement. Note that the thrust values generated provide a check on the desired performance capability relative to the aircraft weight.
Attached is an example Excel worksheet implementing the method. Except for the input cells the sheet is protected without a password. The wanted Pm is 300 watts. The following combinations of diameter and pitch approximately meet this requirement:
- 10X5 290 W 1112 Kv 3.52 lbf
- 11×7 293 W 893 Kv 3.6 lbf
- 10×4 324 W 1277 Kv 3.8 lbf
A table line entry result is for a particular propeller and battery voltage combination that matches the desired maximum air speed. It does not include any specific motor parameters other than assuming that the motor efficiency is 75% under static full load. Nor does it take into account any airplane properties directly. Hence the results are applicable to any airplane whose weight is known and the Watts/lb value has been selected. For instance, assume that the plane weighs 2 pounds and 50 W/lb is desired. Then the required motor power is 50 * 2 = 100 watts. The table shows that this requirement could be approximately met by any of these combinations:
|Prop||Kv||Thrust – lbf|
Changing the maximum air speed requirement or the battery voltage will change the results and therefore the prop and motor selection needed to match the desired motor power.
The method uses wind tunnel measured results for APC Thin Electric propellers in sizes ranging from 7 to 14 inches in diameter. Other model propeller lines have different prop coefficients as a function of advance ratio, J, and therefore might not give similar results. It is not recommended to use this specific model for those other propeller types until there is evidence that the results are comparable.