Thanks Chris. Hmmm ... my aim is just to increase the efficiency of the wheel, not to replace it with a turbine.
Thanks for explaining the pipes. I see the elegance of the solution, but I also see its inflexibility. It's one of the things that put me off a turbine. If one or both pipes need to be constantly full to maintain its operation, you're going to be in a situation a lot of the time where you're only using the smallest nozzle pipe because there's not enough to fill the second. Hence all that potential power is wasted, and although the efficiency within the system itself is high and remains constant, the efficiency of the overall set-up - ie. power inherent in the potentially usable water flow / actual power generation - is low. Efficiency all depends on where you draw the line around your 'system' ...
OK. I gave some of the practical reasons for my choice of the wheel, but some are a matter of foundational principle.
There is this general incongruence between human engineering and nature. The natural world comprises interdependent open systems which are in a continual state of dynamic change, and we use them by constraining them to work in closed systems within narrow parameters in an inflexible way. This is fine where there's overabundance of primary system input, eg. where there's gallons of water, billions of barrels of oil, thousands of square miles of fertile forest soils to clear to grow food in, and billions of fish in the sea, but we're now at the point where that overabundance no longer exists. We've used it all up by exploiting it in incredibly wasteful ways (overall efficiency <1%).
We need to adapt. We need to adapt both our thinking and our engineering to work with nature, not against it. This means studying how natural systems work and imitating them, because natural systems are self-maintaining and have a robust flexibility that can deal with an enormous amount of perturbation without compromising overall system integrity or efficiency. Our systems are the opposite: they're fragile, inflexible, and enormously vulnerable - remove that word 'unlimited' from the primary energy input, and the whole house of cards collapses. So to me, because our water supply is marginal, this means engineering a generating system that's open and can accommodate continual variability, so what we lose in the specific efficiency of a closed system, we gain overall.
When I was thinking about using a turbine here, these considerations were continually nagging away at me. I knew there had to be a more open, flexible, simple, elegant and efficient way to do it. So when an engineer friend and neighbour started talking to me about his experience with turbines and the thought process that went into the wheel-based system he was designing, I knew he was on the same track. When he completed it and I saw it in operation, it was one of those Aha!! moments. His wheel works beautifully.
Because it's the same size as mine with similar gearing, I can use his rpm and power output to gauge what I can reasonably expect from mine. This is why I've been playing around with various equations to come up with something that accurately reflects the behaviour and output of these wheels, rather than use the conventional head x flow / efficiency, because that doesn't work with the engineer's wheel. His power output is greater than that calculation predicts. I'm presuming this is because the weight of water in the buckets has something to do with it.
So to me, because what we have is essentially an open system - although the chute only has 3m of head, the water entering it has much more even allowing for substantial friction losses, and there's no break in flow - the calculations for wind turbines, which are acknowledged as open systems in the maths, also have relevance. Taking the actual speed of the bulk of water hitting the wheel means that head, flow and friction have already been accounted for since they're implicit in the speed of the water, but I think I need to more accurately model the volume rather than taking the cube of the linear velocity.
