Hello! Not sure if I can post a link here, but will try. I was just reading about typical configurations here:
https://www.alternative-energy-tutorials.com/hydro-energy/waterwheel-design.html
Check out the breastshot water wheel design. You can see that the wheel is nearly twice the height of the head. You can also see that the vanes are buckets to hold the water as the wheel rotates, not just paddles that the water hits and splashes away. There is a combination of factors that you can use to get more power: increase the wheel diameter, convert to buckets (or troughs, if you will), extend the width of the wheel so that the entire flow of water falls within the buckets instead of splashing away, direct the flow of water right into those buckets and ensure that you are getting as much of the head height as possible (don't let the water fall off the edge and strike the bucket/paddle farther down as you effectively lose advantage of your limited head height here, and finally, tighten up the bottom channel to minimize or eliminate any water bypassing the bottom of the wheel by flowing around it - forcing the pitch-back water at the bottom through the wheel will drive up the efficiency. The comment about using the outer edge of the wheel vs. the gearing is not correct. Your wheel generates so much torque at a certain RPM. The torque is greatest at the center. When you hear up to increase RPM, you trade power for speed (and lose some energy in the conversion). You will want to minimize the change here to match whatever you are driving so that you don't suffer any more conversion loss then necessary.
