# Power

Power is the rate at which work is done. The SI units are as follows:

Power is measured in Watts (W), which is one Joule (J) per second; distance or displacement is measured in metres, and time is measured in seconds.

Power (W) = Work (J) / Time (s)

Because Work (J) = Force (N) x Displacement (m), this can be rewritten as:

Power (W) =  Force (N) x Displacement (m) / Time (s)

And because displacement travelled over time is called speed, or velocity, v (m/s), we can simplify this further as follows:

Power (W) = Force (N) x velocity (m/s)

The force exerted on the turbine is as follows:

Force (N) = Net Head, Hnet (m) x density of water, 𝛾 (N/m³) x flow area, A (m²)

so Power (W) = Hnet (m) x 𝛾 (N/m³) x A (m²) x velocity (m/s)

But A (m²) x velocity (m/s) = flow rate, Q (m³/s), so this can be written:

Power (W) = Hnet (m) x 𝛾 (N/m³) x Q (m³/s)

This is the power supplied to a turbine. If the turbine were 100% efficient, it would also be the power delivered by the turbine to the turbine shaft. No mechanical device is 100 % efficient, but water turbines have high efficiencies.

If the turbine efficiency is η, the turbine output power is

Pt = 𝛾 H Q η

Watts are a rather small measure for hydropower, so this value is divided by 1e3, 1e6 or 1e9 to get kilowatts, megawatts or gigawatts.

(1e3 denotes 10³, for example)