Jet propulsion, - for the mathematically (and musically) minded.

When the turbine was off its plinth last summer for its annual overhaul, I took the opportunity to photograph one of the jets in operation, - something which wouldn't be possible in most Powerspout installations and something you don't normally get to see.  Here's what it looked like:

The distance the jet travelled was 15 metres.

Quite early after emerging from the orifice the jet can be seen to begin breaking up, probably due to a tiny irregularity in the plastic of the nozzle at the orifice.  Better transfer of power to the pelton runner would be achieved if the jet remained more compact. Tidy orifices are important.

The net head recorded on the test gauge is 53.3 metres, and with the orifice diameter being 7.46 mm, the calculated flow should be 1.28 litres per second. 

The formula for calculating this is: Q (m³/s) = CD x Anoz(m²) x √ (2g x Hn),  where CD is 0.91

Also by calculation, the velocity of the water at the orifice is 31.6 metres per second, which is 70.6 miles per hour !  The formula is vjet = Cv x √(2g x Hn), where Cv is 0.98

To get optimum energy transfer to the pelton wheel, the cups of the wheel should be moving away from the jet at a velocity just a little bit less than half (0.46 to be exact) of the velocity of the jet.  This wheel velocity needs to be measured at the circumference of an imaginary circle defined by where the jets strike the runner, - a circle which has a diameter, in the case of a Powerspout pelton, of 220 mm, - what is often referred to as the pcd or "pitch circle diameter".  Putting all this together, it can be calculated that the optimum speed at which my runner should turn is 1260 rpm, or 21 revs per second.

This speed of revolution is relevant because it defines the frequency of the sound emanating from a turbine, at least from the hydraulic side but not the SmartDrive side, which tends to have a higher pitched sound. 

The frequency of the sound can be calculated knowing there are 20 cups on a Powerspout runner: each jet will therefore 'see' 20 x 21 cups passing it each second. The resulting sound should have a frequency of 20 x 21 ie 420 Hz, - not far off Standard Concert pitch A, 440 Hz, the note used for tuning musical instruments to in an orchestra.  See what you think from the video clip below; I think it's turning more slowly than 1260 rpm; you can 'tune in' your ear here.

I have found it really interesting getting to grips with the maths which underpins the operation of a micro hydro.  Jeremy Thake's book on the subject is easy to read. I recommend it.