# Matching Motor and Propeller to Airplane

There are some generalizations, rules of thumb, that help estimate prop and motor sizes for free flight rubber powered model airplanes.
There is a relationship between the swept disk (circular) area of the prop and the wing area. The wing generates lift per unit area and the prop generates thrust per unit area. The two are related. McCombs recommends the the prop diameter should be from 1.0 to 1.5 times the square root of the area of the wing. Pitch is best determined by test. Or you can use John Barker’s Prop Picker. Go here, scroll down about 2/3 and click on ‘Prop Picker’.
In horizontal flight, Lift equals Weight.
Trim attack angle establishes a trim Lift to Drag ratio L/D. That and Lift establish Drag. We have some reasonable ballpark guesses for that L/D, usually based on glide tests. For typical rubber models, L/D is probably in the range of 4 to 6. (Trim L/D is not necessarily maximum L/D, usually isn’t, shouldn’t be.)
In horizontal flight, Thrust equals Drag. I know of no way to measure either in flight. L/D under power is not the same as L/D in glide for several reasons; different trim attack angles, prop slipstream. But if that is all you have, try it. It won’t be too far off.
For typical rubber model propellers, there is a range of typical Thrust to Torque ratios. It is difficult to establish values because Thrust is unknown. At the end, we will estimate a value based on typical values for the other numbers.
It is possible to measure level flight torque Q; wind the motor until the plane makes a level circle from launch to catch. Measure the torque at begin and end. The numbers will be close and the average is a good estimate for level flight torque. For a fast, steep climb, torque must be much greater. You can weigh the plane, with motor, and calculate a Torque to Weight ratio Q/W.
Torque relates to motor cross section. Q = Kq x S^2/3 where Q is torque, Kq is a torque coefficient specific to a batch of rubber and S is the cross sectional area of the motor. S^2/3 is the two thirds power of cross sectional area. Use the exponentiation function on your calculator. This allows you to calculate how many strands of what width you will need when you know the required torque. I am measuring torque in gram centimeters and cross section in square inches. Level flight Kq ranges from about 21,000 to 24,000. Adjust as necessary for your units.
This chain of relationships implies a ratio of level flight torque to weight Q/W, which can be measured, as above. For my planes, mostly simple stick models, I find a level flight torque to weight ratio ranges from 0.767 for the Dandiflyer (stick ROG, cambered single tissue wing surface, moderately high aspect ratio) to 1.24 for the AMA Cub and 1.38 for the Squirrel (stick models with flat plate wings of low to moderately low aspect ratio). The Sig Tiger was 1.63 (box fuselage, two surface airfoil wing of moderately high aspect ratio). The Big Pussycat got 0.952 in one test and 1.106 in another (also box fuselage, single tissue surface cambered wing of moderate aspect ratio). I am measuring torque in gram centimeters and weight in grams, so torque to weight ratios are in centimeters. Convert as necessary.
Put this together and we get
Q = (Q/T) (D/L) W
Rearranging gives Q/W = (Q/T) (D/L) and T/Q = (D/L) (W/Q)
This would suggest for a typical value of D/L of 1/5 and W/Q of 1/cm a T/Q around (1/5) 1/cm = 0.2/cm.
For a start, select a cross section that gives a level flight torque in gmcm about equal to the weight of the plane, including weight of motor, in grams, with Kq of around 21,000 to 24,000. This torque coefficient falls in the mid range of the torque curve. The spike will typically be about 5 times higher. The cross section is calculated from rearranging Q = Kq x S^3/2  to S = (Q/Kq)^2/3. Remember that S = n t w where n = number of strands, t = strip thickness and w = strip width. So the number of strands would be n = S/tw. Typical t is 0.042″.
If you want a faster climb, use a greater cross section.