Hi
Well you have the motor completely stripped down so starting from scratch isn't a problem! You have room for a lot more magnets than that, and you certainly don't want to use 22 Gauge wire.
When comparing relatively thin magnets, the performance does increase in proportion to the thickness of the magnets (ie. twice the magnets can double the power). Thicker magnets than those ones may give you more current, but the return isn't proportional any more. Some zones of the stator may be close to saturated with magnetic flux. Thicker magnets just sends them further. A 50% thicker magnet won't get you 50% more current. You definitely DO need more of those magnets, though, to "fill in" the empty spaces on each pole.
You seem to have decided on a 12V system, so I won't question that, except by saying that my experience is that higher voltage makes a big difference.
Roughly starting with the size of the motor it is comparable to other 3HP examples I have, so maybe my experience will be helpful.
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Take a 4-pole rotor as an example. With 36 slots, it's a very common layout so I can say that you will have 7 teeth in each winding. With 42 turns of thin wire in that pole you can expect a 12V cut-in around 60 to 80 RPM. Way too low, considering the destination windmill will have a 10-12 foot diameter prop. (The numbers are pulled out of some bench tests I've been doing on a similar motor - but your case may be close).
Since the windmill wants a cut-in speed of (say) 180 RPM, you can do with 1/3 fewer turns of wire. So use a heavier wire gauge or same gauge and wind "3-in-hand" - 3 strands in parallel. This gets you a wire gauge closer to about 17 Gauge. The resistance per phase of your result is still very high. The output will be better than your first, but not better than an axial machine like Dan's.
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Changing the number of poles will not affect this result (much). Think about what frequency of AC you want to receive - if it doesn't matter, then go for whatever you think it easiest.
Since I was making an estimate, I think you won't know what you have until you have tested it yourself. You're probably in a position to assemble the rotor, run a single turn of wire through all poles and phases, connect them up, and calculate the cut-in speed as a ratio of the number of turns. Measure RPM accurately.
That's what Zubbly did on a number of occasions, and you can see more detailed explanation of that if you look through his old postings. I think Peter Dingemans did that too, in his most recent re-wind.
Good luck! Please keep us in the loop (no pun intended) as you make progress.