Power Calculations :
In SI units, (metric)
The energy available in falling water is expressed as, E = mgh where...
e = energy = ( m g h ) Joules (a Joule is one watt for one second)
m = mass, the product of density (p) x volume (V)
g = gravitational constant of acceleration, 9.8 meters per second per second
h = head in meters
Note: time is not a factor in this relationship for energy.
Power is defined as energy released over a specific time, so substituting in the values above,
Power = ( p V g h ) Joules / second (same as watts)
The density of water is 1000 kg/m^3, (or 1 kg/liter), and the gravitational constant of acceleration on earth is 9.8 m/s^2, (the force that gives 'mass' its weight). In hydro applications, the letter 'Q' is used to represent the volume of water combined with its density.
Substituting in gives gross power equal to,
Power = ( 1000 x Q x 9.8 x h ) watts, or in kW Power = (1000 x Q x 9.8 x h ) / 1000 kilowatts, and futher reduced to...
Power = (Q x 9.8 x h ) kilowatts, ( with Q in cubic meters per second )
Note: as mentioned above, see how the "V" for volume has been replaced by "Q" for quantity representing 1 cubic meter of water.
Note: See how this is similar to the energy equation above, energy = m(Q) x g(9.8) X h
Overall system efficiency is often around 55 - 65 % in systems under a kW, so you can round off the above to the following for rough working.
Power = ( Q x h x 10 x 0.6 ) kW ... for small systems where Q is volume in cubic meters per second, and h is head in meters.
For plants in the multi kW range, efficiencies reach 70 percent or higher. Actual efficiency depends on the water flow characteristics, nozzle design, turbine design, coupling, and generator. Power line losses, battery charge / discharge losses and inverter effiency must also be considered. Example:
You have a stream with a flow of 3 liters per second and a drop of 50 meters available.
What is the gross power potential in watts ?
3 liters per second is 0.003 cubic meters per second)
Power = ( 1000 x Q x h x 9.8 x efficiency) watts
Power = ( 1000 x 0.003 x 50 x 10 ) watts
Now multiply this result by the expected overall efficiency. (say 60%)
Power = ( 3 x 50 x 10 ) x 0.6 = 900 watts (rounded off)
In imperial units:
Total energy available is expressed as,
energy = Q x 62.4 x h
Q = cubic feet of water per minute
62.4 = density of water, @ 62.4 lbs. per cubic foot
h = net head in feet
This is really foot pounds of potential energy per minute, (so many pounds at so many feet head.)
One horsepower is equivalent to 33,000 foot pounds of work done in 1 minute, so to figure in HP,
Q X 62.4 X h / 33,000 or Q x h / 529 Since 1 Hp is equivalent to 745 electrical watts, convert to kW by Power in kW =(Q x 62.4 x h / 33,000 ) X 0.745
or,
Power in kW = (Q x H / 709) where Q is in cubic feet per minute.
In all cases it is important to multiply the gross power result by the system efficiency, again use 55- 65 % for small systems, and perhaps 70 % for larger system. On larger sites, it is often easier to think of cubic feet in one second, so convert the above to...
Power in kW = ( Q x h / 11.8 ) where Q is now in cubic feet per second.
For general work on small systems, the following formula is easy to remember and works well. It assumes an overall efficiency of 55 %, so the result is close to the power you will get.
Power in watts = (net head in feet x flow in USGM) / 10 (again, at an overall efficiency of 55 %)
Volumes and conversions. 1.0 litre (1 kg)
1.0 cubic ft (62.4 pounds)
1.0 us gallon (8.3 pounds)
1.0 imp gallon (10 pounds)
1.0 cubic foot (62.4 pounds)
1.0 cubic meter (1000 kg)
| = US gallons * 0.26442
= litres * 28.313
= litres * 3.785
= litres * 4.5459
= US gallons * 7.49
= 1000 litres |
