Most manufacturers design propellers in the same way and they read the old books and reports and the prop gets no more than 80-85% efficiency at best. It is written in books that propeller efficiency will be about that at best and it is left often open how low it can be at worst.
Here is an interesting article about a guy that made a prop that was 90 percent efficient by not abiding the "old truths" but thinking out of the box:
http://www.eaa.org/experimenter/articles/2020-02_elippse.asp
Having a strong taper certainly makes sense since the propeller tip travels very much faster than the root through the air. Also the old saying that single blade prop is most efficient does not make sense if you think it in detail: the air that enters in the next blade is not the same air that went through the previous blade because of the forward movement of the aircraft. This could be extrapolated in a such way, that the faster the aircraft travels, the more blades the propeller can have without sacrificing the propeller efficiency. This should not actually require very high mathematics, but I am quite sure that it could be estimated with simple calculations where the downwash of the previous blade goes in relation to the next blade on the speed range intended for the aircraft being designed.
High altitude propeller will require some additional thinking for the tip chord because the Reynolds number will become low if the chord is this short. The TAS is much higher at high altitude, therefore the air travels faster through the prop, that would mean that the prop could have more blades. The high altitude propeller does not require full efficiency at low altitude because to be able to operate at high altitude, there needs to be a lots of excess thrust available regardless.
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