Propeller Head

The other day on the highway, we passed a trailer carrying a motorboat with two outboard motors on it: one with a very small propeller, and the other with a larger one.  It made me wonder how the two would behave differently – a larger propeller clearly would give more thrust, but would also require more power.  Do the two cancel each other out?

We can find the forward force on the boat by relating it to the change in momentum of the water:

where m is the mass of water accelerated, Δv is its change in velocity, and Δt is the time it takes to accomplish the acceleration.  Normally, this should be a differential, but we're only interested in the average force, so we can treat it as a difference quotient.  Let's define some parameters for each propeller:

The blue circle represents the rotation of the propeller at an angular velocity ω, or with period p.  The red circle represents the angle the blades are twisted to, θ.  The length of the blades is r, and their apparent width, as viewed from the side, is w.  Using all this, we can rewrite the force as

where n is the number of blades, and ρ is the density of water.

Next, we want to figure out the power required to turn the blades.  We'll use a similar technique to the above, but this time using torque and angular momentum.  The water will be accelerated to a velocity given by

That means at a radius a, the water will have an angular momentum of

or resolving the cross product,

To get the total angular momentum, we integrate over all radii:

As with the force above, we can express the torque as a change in angular momentum:

And from this, we can find the power output of the motor:

Interestingly, this means that both the power required and the force obtained are proportional to r/p.  The implication then is that the size of the propeller doesn't matter — a smaller one needs to move faster, but it's easier.  I'm not sure why one size might be preferable to another, but it may have to do with the amount of noise or wake created.