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Why does a propeller with a low pitch have better acceleration/take-off and climb characteristics vs. a high pitch propeller?

And conversely, why is a propeller with a high pitch not as good during a climb, but is good for a cruise phase?

I am looking for a „physics and maths” explanation, I like the bicycle and car gears analogy and they are really intuitive, but they do not explain why does it works like that.

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    $\begingroup$ One way to look at it if you ride a bicycle: how hard is it to get started in high gear, or to climb a hill in high gear? How is cruising on a flat stretch in a low gear vs. a high gear? It is very similar for propeller pitch in airplanes. A single speed bike is much like an airplane with a fixed-pitch propeller -- a compromise. $\endgroup$
    – Fred Larson
    Commented Jun 30, 2021 at 16:39
  • $\begingroup$ So, in reality, you want the prop to take the same "bite" of air (at its optimal AOA), based on relative wind created by forward airspeed (higher in cruise) and rpm (lower in cruise). $\endgroup$
    – Robert DiGiovanni
    Commented Jul 2, 2021 at 10:30

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Low pitch advantage

Power

A lower pitch allows the blade to spin faster ($\omega$) for the same torque ($T$). The relationship between power ($P$) and speed and torque is $P = T \omega$, so this means that for the same $T$ we get more power out of the prop when it spins at a faster $\omega$. More power means a higher climb rate.

Stall

At low airspeed, significant sections of a high-pitch propeller will be stalled. This translates into lost thrust, and explains why a plane with a climb prop has a shorter ground roll than the equivalent with a cruise prop.

High pitch advantage

Lower tip speed

Efficiency goes down at high tip speeds, and basically falls off a cliff when the tip speeds reach the speed of sound. So a higher pitch allows for creating the same thrust at a lower tip speed, and thus gives higher efficiency.

Engine RPM

A secondary effect is that a prop which is spinning more slowly typically leads to a more efficient engine, and reduces wear. So while this isn't directly related to your question about the prop being more efficient, it is a very important effect in the overall system efficiency.

Additional reading

Do propeller coefficients go to zero at the same value of advance ratio? gives a good explanation of propeller behavior for different pitch.

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  • $\begingroup$ And by a torque you mean engine output, right? Because on some propeller forces schemes there is a force, perpendicular to axis of rotation, also called torque, what is weird to me, because I always thought that torque = force x arm $\endgroup$
    – Konrad
    Commented Jul 1, 2021 at 8:32
  • $\begingroup$ That's correct, I mean engine torque. This second torque you're referring to could be gyroscopic precession or p-factor, both of which have important impacts on the plane's flight dynamics. $\endgroup$
    – Kenn Sebesta
    Commented Jul 1, 2021 at 10:19
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    $\begingroup$ Engine torque is constant-ish. wildnordics.com/wp-content/uploads/2020/06/… gives torque for a Rotax. If you look at the y axis, you can see that it varies +-10% across the 3000-6000RPM range. At any throttle setting, the prop will speed up until the load torque equals the output torque. A higher pitch prop has a higher load torque for the same RPM, which conversely means that for the same engine torque it won't spin as quickly. $\endgroup$
    – Kenn Sebesta
    Commented Jul 1, 2021 at 17:12
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    $\begingroup$ A load is a general term for the thing which is being driven. So in your house, the refrigerator and lightbulbs are "loads" on the electric grid. In a bus, the "load" is the wheels. In a plane, the "load" is the propeller. The word "torque" serves to differentiate the load from other load types, for instance a voltage load or a pressure load. $\endgroup$
    – Kenn Sebesta
    Commented Jul 2, 2021 at 2:42
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    $\begingroup$ @niels nielsen gives a lay description in his answer, which might be what you're after. If you want a little more formal of an answer, a higher pitch has a higher angle of attack. A higher AoA produces more lift. More lift produces more induced drag, and more induced drag makes it require more torque for the same RPM. $\endgroup$
    – Kenn Sebesta
    Commented Jul 2, 2021 at 17:26
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A low-pitch or "climb" propeller takes smaller "bites" of the air and allows the engine to run all the way up to its maximum speed, and therefore produce its maximum rated power setting (don't make the pitch too fine, or the engine will overspeed!). The fuel burn rate will then be at a maximum and you'll get maximum acceleration and climb rate.

A high-pitch or "cruise" propeller takes bigger bites and therefore loads the engine down to the point where it is turning slower than its redline limit and developing less power at wide-open throttle. This lower power setting lets you obtain the most economical cruise in level flight by reducing the fuel burn rate- but it prevents the engine from running at its maximum power setting during a climb. Therefore a cruise prop gives you a lower acceleration rate, and lower rate of climb.

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