A propeller's job is to move a large mass of air by a small increment and turn that into thrust: twice the air density times disk area times induced velocity times the sum of flight speed and induced velocity — momentum theory. How well it does that job is propeller efficiency, which splits into three multiplied factors — induced efficiency (the Froude efficiency, flight speed over flight speed plus induced velocity), profile efficiency (what the blades lose to airfoil drag), and swirl efficiency (what's lost spinning the slipstream instead of pushing it straight back). At a standstill thrust is huge but induced efficiency is exactly zero — the propeller's oldest paradox. A bigger disk lowers disk loading and raises efficiency, but tip Mach — set by diameter and RPM together — limits how large and how fast the disk can spin. Set the diameter, RPM, flight speed, shaft power, and the blade's profile and swirl efficiency, then watch it all play out live. It all runs on your device.
You are in the Physics lab.
Shaft power in, induced, profile and swirl losses out — and how much of it survives as thrust.
What the propeller schematic, the loss-split bar and the two plots are telling you — then what the model says about diameter.
The schematic shows the rotating disk accelerating a streamtube of air, which narrows downstream as it speeds up into the slipstream. The gauge beside it tracks overall efficiency — how much of the shaft power actually survives as thrust power once induced, profile and swirl losses are subtracted (the same split the loss bar shows).
The loss-split bar breaks shaft power into three losses that never show up as a single number — induced loss, profile loss, and swirl loss — plus what's left as delivered thrust power. Widen any slice and overall efficiency falls, since it's the product of induced, profile, and swirl efficiency together.
The first plot ramps flight speed from a standing start up to cruise speed, tracking thrust falling and induced efficiency rising together — the static-thrust paradox made visible. The second plot draws overall efficiency and tip Mach against propeller diameter and marks where the current design sits, showing why a bigger disk trades efficiency against the tip-Mach limit.
Three efficiency factors multiplied together, three losses that eat into them, and a diameter dial that runs into a speed-of-sound limit.
Momentum theory gives thrust as twice the air density times disk area times the induced velocity times the sum of flight speed and induced velocity. A propeller's whole job is accelerating a large mass of air through the disk by a small velocity increment — that's why a wide, slow-turning disk delivers the same thrust more efficiently than a narrow, fast one.
Induced efficiency is flight speed divided by the sum of flight speed and induced velocity — it is zero at a standstill and climbs as induced velocity shrinks relative to flight speed. Lowering disk loading (thrust per unit of disk area) by using a bigger diameter lowers the induced velocity needed for the same thrust, which is why larger propellers are inherently more efficient at a given thrust.
Profile efficiency and swirl efficiency multiply onto induced efficiency to give overall efficiency, capturing blade airfoil drag and energy left spinning in the wake. But tip speed is pi times rotational speed times diameter, so a bigger disk spun at the same RPM raises tip Mach; once the tips approach the speed of sound, compressibility drag and noise climb sharply, forcing designers to cut RPM as diameter grows.
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