Updated: Nov 21, 2020
This blog used to start with the calculations of hull drag, but as I was asked about this topic a couple of times, it became apparent that the two issues were best separated. Hence Hull Drag is now in a separate post.
To summarize for those who just need the answer, the force to drive a narrowboat hull through the water rises by the square of the speed, and is made up of various components:
The question we address in this post is this. If I want to generate a force of, say, 1kN to travel at about 6kts, how much power do I need to put into the propeller?
The difficulty is that a propeller has many parameters which are all interdependent. There are diameter and pitch (the distance the blades will travel through the water if the propeller turns once, like a corkscrew). Then the area of each blade and the number of blades. The speed of rotation and the torque. When the boat is moving, hull drag causes the water entering the propeller to be slower than the actual boat speed, but because the propeller is sucking water in, there is a corresponding increase in drag.
Let's get back to first principles. The whole point of Perseverance is to cruise quietly. Smoothly. Without fuss. Therefore I opted for a 4-bladed propeller as more blades give a smoother ride. I also found that standard propellers have a limited range of area ratios, so that reduces the number of unknowns further.
Knowing the thrust needed to push the hull through the water at different speeds it is possible to use charts to find out the propeller performance. When I say “charts” there are some fiendish ones out there. The common ones look like this:
or like this:
or even this:
But the one I found I could understand was this sort. For a 4-bladed propeller with 70% area ratio:
Also it's very colourful !
From the drag equations, using Hans Otto Kristensen’s formula for wake fraction and Harvald’s formula for drag inflow, it is possible to compute the value of the a parameter shown as the blue numbers. They come out to have values between 1 and 2.5. The thick blue line is the maximum efficiency line for that range of conditions, and from the red solid lines we can find out the optimum pitch:diameter ratio which is about 0.8. We can also read off the advance coefficient, J and torque coefficient, Kq, from the axes. Throw in the odd density of water and we get:
This tells us the fact well known to all boaters, that it takes a lot more power to go a little bit faster.
Electric motors give practically the same maximum torque for any speed (unlike diesel engines where the torque varies significantly with engine speed). The motor I selected for Perseverance gives a maximum of 70Nm torque, and the chart above goes beyond the available torque from the motor. That is, my motor won't drive Perseverance at 7.5kts.
It is possible to compute the maximum speed for the available torque as a function of propeller diameter thus:
So you can have propellers that are too small or too big, given a maximum available torque.
Now, as well as the speed that can be achieved, there is a handling aspect to propellers. If you want to stop quickly, or punch the boat out of a lock, you do just need to move lots of water. Also, I am sure that the wake caused by a large propeller moving more water more slowly will be less turbulent, and hence quieter, than a small propeller moving a little water more quickly. For this reason, I opted for an 18in propeller. The maximum efficiency line from the multi-colour chart above was 0.8, corresponding to a 14in pitch.
Am I Being Daft?
As a sanity check, I asked lots of people for advice, used on-line and downloadable propeller sizing tools, and plotted all the information I could gather on a chart of diameter vs pitch. Fortunately my 18x14 choice was in the middle of the range which provides reassurance.
The actual propeller and sterngear can be seen in Sterngear Vlog, and Why Waste Money addresses the different types of power plant which could be used.
Just one last thing. The very good chart I used was from an article in the Journal of Marine Science and Engineering, entitled "Surprising Behaviour of the Wageningen B-Screw
Series Polynomials" by Stephan Helma. Who said engineering was dull?