Euler's pump and turbine equation


The Euler pump and turbine equations are the most fundamental equations in the field of turbomachinery. These equations govern the power, efficiencies and other factors that contribute to the design of turbomachines. With the help of these equations the head developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by Leonhard Euler in the eighteenth century. These equations can be derived from the moment of momentum equation when applied for a pump or a turbine.

Conservation of angular momentum

A consequence of Newton's second law of mechanics is the conservation of the angular momentum which is fundamental to all turbomachines. Accordingly, the change of the angular momentum is equal to the sum of the external moments. Angular momentums ρ×Q×r×cu at inlet and outlet, an external torque M and friction moments due to shear stresses Mτ act on an impeller or a diffuser.
Since no pressure forces are created on cylindrical surfaces in the circumferential direction, it is possible to write:
ρ Q = M + Mτ

Velocity triangles

The color triangles formed by velocity vectors u,c and w are called velocity triangles and are helpful in explaining how pumps work.
'a' and 'b' show impellers with backward and forward-curved vanes respectively.

Euler's pump equation

Based on Eq., Euler developed the equation for the pressure head created by an impeller:
Yth : theoretical specific supply ; Ht : theoretical head pressure ; g : gravitational acceleration
For the case of a pelton turbine the static component of the head is
zero, hence the equation reduces to:

Usage

Euler’s pump and turbine equations can be used to predict the effect that
changing the impeller geometry has on the head. Qualitative estimations can
be made from the impeller geometry about the performance of the
turbine/pump.