Sizing a propeller to match an electric motor

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Ok, I think I did the math properly. I finally tracked down a paper (Thank you Heimfried) by Newton and Rader who describe a series of propellers. I started with the 3 blade, 48% blade area ratio, 2.08 pitch to diameter ratio. I determined that my 5kw motor would put out about 20 newton meters of torque at the wheel after losses. I had initially planned on a propeller of .24 meter diameter and a maximum RPS of 41.67. However, at peak efficiency at a J of 1.8, this would require nearly 30 kW of power input. Obviously this is unrealistic. I see four options:

1. Keep the propeller same as above, and just operate at very low slips. (I think this was my subconscious plan before doing the math.) This would result is pathetic efficiencies -- a J of 2.25, the highest with data -- would require a torque input of 31.7 Nm for a mere 107.5 N of thrust. This wouldn't work unless vessel drag was unrealistically low.
2. Reduce the propeller diameter and increase the RPS, but maintain the pitch ratio. This looks like it would work quite well -- if my lower unit just doesn't blow up, which I have not much faith in!
3. Maintain the same RPS, reduce the diameter, and increase the pitch ratio: This seems like it would be the best option, but I can't find any data on P/D ratios higher than 2 -- despite the fact that the highest ratio almost universally has the highest efficiency!
4. Reduce the pitch: Dropping down to 1.68 P/D actually doesn't really help me much. The speed loss is just too great compared to the other options.

So, I guess I need to get a better idea of what my vessel drag curve will look like, and then I can tweak the propeller dimensions to intersect it at a reasonable efficiency while maximizing speed.

Some questions I still have:

• Does Kt account for friction and other losses at the wheel? (I assume yes)
• How do I estimate my t and w? I have found some information in the literature about w, but it is all centered around displacement vessels. I have not yet turned up anything empirical about outboard legs or hydrofoils or flat bottomed planing vessels, and I am hesitant to use a purely theoretical method.

 

I decided to go with a compromise, 16x33.3 cm turning at a 50% overspeed. Here are the plots for torque and thrust. The J values 1.5 and 2.05 correspond to the upper and lower 70% efficiency, while 1.8 is peak 78% efficiency.

However, I ran into more issues when it came time to model the propeller. I transcribed the offsets given in the paper, but when I plotted them there was a strange result. The authors had modified the original sections to reduce cavitation by removing part of the leading edge and pressure face, but my plots make it look like they had actually moved parts of the section, which I'm pretty sure is incorrect. All I did was scale the table of offsets by desired fraction of the nominal size. I'm not sure if I did something wrong here, or if this issue is somehow innate to the chord line and thickness presentation of the sections.





Also, I don't understand if these sections are supposed to be on flat planes or cylindrical. The wrong assumption could result in too much or too little blade area, especially in the tips. The paper doesn't mention, would they have expected readers to know based on the methods in use at the time?

Thank you in advance to @jehardiman or anyone else who may be able to point me in the right direction.

 

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3rd Edit: Ignore this cut section

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Edit to add: Check the paper again, as I am not sure that delta, Yt and Yc non-dimensionalized...i.e. the table values are for a real 5" radius propeller.

2nd Edit: I'm plotting this out...hold on a few.

 

Ok, I plotted it out, assuming the data given is inches for a 5"R wheel. Yes, there is a reverse, but only on the 0.3R section, and it is not as bad as you show. What I think they did is something I also have done at the root where there is a large difference between angle of attack and geometric pitch angle. They twisted the LE into the flow to gain pressure and prevent immediate separation.

 

Not saying I don't believe you, but I don't understand. Wouldn't this cause seperation at practically any positive aoa?

 

No, there are different effects playing off against each other here. First is the fact that we are dealing with the root. See the figure below.

The geometric pitch is fixed. However the actual pitch varies with omega*r and Va. Because r is so small near the root, to achieve large Va either the Geometric pitch, and therefore the hub, must become large or alpha goes negative. When alpha goes negative...

...pressure is built up on the suction face LE and the pressure face goes into negative pressure, losing all thrust development as effectively as if you backed the wheel. To prevent this you twist just the LE into the relative velocity, Vr, always maintaining a pressure on the pressure face.
OK you say, then why don't you just increase omega?...Because then the tip, which at large r, would be moving very fast. This has bigger issues because general cavitation is a function of Vr^2. Considering that Vr at the tip is 3.33 times Vr at the root, and that thrust per unit area is also proportional to Vr^2, it is better to make the root a slightly less efficient lifting surface than cavitating the tip. This is why most blades are widest at ~0.7r, because you also have to play with the tip to prevent the AoA from becoming too large.

Really, there are whole courses on propeller design about these topics.

 

I noticed I didn't answer this part of the question. Think about the sections being laid out flat on a strip of paper 2*pi*r wide and geometric pitch high. The line from a lower corner to the opposite upper corner is the cord line for development (which corner depends on whether it is RH or LH wheel). From this and geometry you can lay out the 3D shape. Note that Pitch does not have to be, and usually isn't, constant across the radius; usually it is reduced at the root (due to the wake fraction) and tip (due to large omega r) for reasons mentioned in my previous posts.