
Feb 12, 2013, 12:40 PM  
Joined Jun 2012
811 Posts

Question
Why does prop thrust increase with fourth (not third) power of diameter?
Quick theory question,
I'm pretty sure I understand how all different effects increase the size of the indices in the equation, but I can't think of what the last effect is. As diameter increases, To start off, prop size increases linearly with D, which linearly more prop doing the same thing, i.e. linearly more thrust therefore T∝ D Because the blade isn't moving straight through the air, but rather is on a pivot, that added prop is going linearly faster  Aerodynamic forces scale with the square of speed (e.g. drag equation, lift equation), so that gives us two more powers: T∝ D*(D*D) = D^3 But all the equations say that thrust increases by the fourth power of diameter, which means either I am missing one modifier, or I am misunderstanding part of the last two. I can't think of what it is? Can anyone let me know? The only thing I can think of right now is that the equation is assuming the chord of the prop is increasing with its diameter, and that linearly increases thrust and gives us D^4. Someone halp me learn D: 
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Feb 12, 2013, 02:11 PM  

For geometricallysimilar props:

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Feb 12, 2013, 03:32 PM  
Massachussets
Joined Nov 2005
233 Posts

Quote:


Feb 12, 2013, 11:37 PM  
Joined Jun 2012
811 Posts

So in conclusion, some of you have mentioned it is due to pitch increasing for P/D to stay the same, and some have mentioned it is due to chord increasing as the whole prop scales up. Either way it is because the formula assumes more than just D changes. Unfortunately, both of the points are valid but there is only room for one. So I did some thinking and I figured out  actually there is room for both due to another effect:
Just as the average speed of the prop decreases as D increases because the new section of prop moves at linearly higher speed  the average AOA of the prop is actually decreasing as the new section of prop has linearly decreasing AOA (due to twist). This is important  another application being that the whole reason aero forces increase by speed squared is that objects moving through air at higher speed move both more air, and move it faster. We are no longer moving it faster if we are reducing the AOA as we increase the speed. So, the effects of changing prop geometry and speed are: 1) Thrust from increased RPM, changes ∝ n^2 1) Thrust from Increased average speed due to diameter, changes ∝ D^2 2) Thrust from Increased length due to diameter, changes ∝ D 3) Thrust from Increased Chord, (not sure the symbol but calling it C) changes ∝ C. We assume this is proportional to diameter, therefore changes ∝ D. 4) Thrust from decreased average AOA, changes ∝ D^1 5) Thrust from increased pitch, changes ∝ P. So in total we get this: Thrust ∝ N^2*D^2*D*D*D^1*P = N^2*D^3*P The only way we get the conventionally used N^2*D^4 relationship is if we, as Martin Mckinney pointed out, assume that P/D ratio is constant and thus P∝D, so we get: Thrust ∝ N^2*D^3*D = N^2*D^4 Does this sound reasonable, everyone? I hope so because it makes me feel much better as this has been nagging at me for weeks if not months. @Jrb, I read everyone's posts in bed this morning on my mobile because I couldn't wait to read them, and thought about this response in my head at the time. At the time, with my stillasleep brain, I couldn't really comprehend your post, so I haven't had a chance to read, digest, and respond to you yet. It looks interesting though. Just give me some time :P 
Latest blog entry: Some mathematical relationships for...


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