EnergySmart School Inventors
Miniature Hydroelectric Power Plant
Advanced Peltonology (Metric Units)
By MHEPP staff
In the ideal Pelton wheel, a water jet impinges on a bucket which is curved to reverse the direction of the water's flow without reducing the water's speed. In practice, a bucket can't precisely reverse the jet, else the water leaving would crash into the water entering or other buckets, but the reversal is nevertheless adequate. In order to get the maximum power from a jet of water, a motor should be able to reduce the water's speed to zero. If the bucket is moving in the same direction as the water, but half as fast, then when the water is reversed by a bucket, the water's speed will be zero relative to the frame of reference of the laboratory.
Assume the nozzle is 0.0032 m in diameter for a first guess. What is the pressure drop through the nozzle if the flow rate is 0.126 l/sec (1.26E-4 stere per second), and the nozzle is 100% efficient?
v = 1.26E-4/((pi/4)*(0.0032)^2) = 15.7 m/sec
velocity pressure = rho*v^2/2 = ((1000 kg/stere)*(15.7 m/sec)^2)/2 = 1.23E5 N/m^2 (pascals)
A good nozzle will have almost 100% efficiency and so the actual pressure required to deliver 0.126 l/sec through our 0.0032 m nozzle won't be more than a very few percent higher than this 1.23E5 Pa.
As shown above, jet speed is 15.7 m/sec. Wheel pitch speed will be half that or 7.83 m/sec. We have two wheels; one has a pitch diameter of 0.102 m, the other 0.0508 m. So one has a circumference of 0.102*pi = 0.32 m, and the other half that. Shaft speed will then be 7.83/0.32 = 25 rps = 1500 rpm. The smaller wheel will have twice that shaft speed.
If our wheel speeds are 1500 and 3000 rpm, and we want, say, 10 V out of our generator, then we need servo motors with 150 and 300 rpm per volt, or 6 and 3 volts per 1000 rpm.
Now for wheel or shaft torque. Force is mass flow rate times change in velocity. (This can be derived from the impulse- momentum equation which can be derived from F = m*a.) Change in velocity also includes the sign (direction) of the initial and final velocities.
F = m-dot*dv
m-dot = q*rho = 0.126 l/sec * 1 kg/l = 0.126 kg/sec
dv is the aforementioned 15.7 m/sec in either lab frame or bucket frame.
So F = m-dot*dv = 0.126 kg/sec * 15.7 m/sec = 1.98 N
If the pitch diameter of the wheel is 0.102 m, the torque will be 0.100 N-m, and if it's -.0508 m, 0.050 N-m.
Power = 2*pi*T*n = 2*pi*(0.100N-m)*25 rps = 16 watts
Hydraulic power = q delta P = 1.26E-4 m^3/sec * 1.23E5 N/m^2 = 16 W
So we have calculated the power coming out of the wheel to be 16 watts by two independent methods.
What can we say about the general characteristics of a Pelton wheel? If the wheel is not turning, the water will have its direction reversed, ending up traveling at full velocity in the opposite direction. This doubles the dv in the m-dot*dv equation. So if the wheel is not turning, torque is doubled as compared to the torque at optimum wheel speed. If there is no load on the wheel, the wheel will try to go as fast as the water jet, with the result that there will be no change in the water's speed and hence no torque. This top speed is double the optimum speed. We can see that the Pelton wheel is a very forgiving machine, tolerating load abuse without causing disasters such as blowing up or breaking its shafting.
The Pelton wheel is historically important because it is low-tech, and easily made, while having high efficiency. Higher efficiency means that it delivers more power from a given water supply than another low-tech machine.