The classic equation used in hydro power applications is

$latex p=qhg$

where $latex p$ is the power output, $latex q$ is the flow rate, $latex h$ is the available hydraulic head across the machine, and $latex g$ is acceleration due to gravity. This equation assumes the fluid is water. Specific gravity $latex \lambda=1.0$ is therefore omitted for clarity. [1] Note that this equation is applicable to pump or turbine configurations as they are the same machine. In crude oil service where the density is not equal to that of water, $latex \rho$ is included:

$latex p=qhg\rho$

where $latex \rho$ is the fluid density. A common adaptation of these equations is

$latex p=\frac{qh\gamma}{3960}$

where $latex p$ is the power output in horsepower (convenient for pump prime mover sizing), $latex q$ is flow rate in gallons per minute and $latex \gamma$ is the specific gravity. Expressing $latex p$ in kW (convenient for turbine electrical power output calculations) yields

$latex p=\frac{qh\gamma}{5310}$

These equations do not account for inefficiencies of the energy conversion process. The power available to recover from a hydraulic power recovery turbine in crude oil service accounting for fluid density, turbine and generator inefficiencies is

$latex p=\frac{qh\gamma\eta_m\eta_e}{5310}$

where $latex p$ is the power output in kW, $latex q$ is flow rate in gallons per minute, $latex h$ is the differential hydraulic head in feet, $latex \eta_m$ is turbine efficiency, $latex \eta_e$ is the generator efficiency and $latex \gamma$ is the specific gravity of the fluid.

## References

[1] H. Addison, *A Treatise on Applied Hydraulics*, Fifth. London: Chapman and Hall Limited, 1964.

Note: This is a revised, WordPress friendly adaptation of content from Brendon Bruns’ Petroleum Engineering Masters project: https://scholarworks.alaska.edu/handle/11122/10947