Euler's pump and turbine equation


The Euler’s pump and turbine equations are most fundamental equations in the field of turbo-machinery. These equations govern the power, efficiencies and other factors that contribute in the design of Turbo-machines thus making them very important. 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.

Another consequence of Newton’s second law of mechanics is the conservation of the angular momentum (or the “moment of momentum”) which is of fundamental significance 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 are acting on an impeller or a diffuser.

Since no pressure forces are created on cylindrical surfaces in the circumferential direction, the conservation of momentum equation can be written as shown here.

Related formulas


Mexternal torque (N*m)
Mrfriction moments due to shear stresses (N*m)
ρdensity (kg/m3)
Qvolumetric flow rate (m3/s)
c2flow velocity circumferential component at the outlet (m/s)
r2outer radius of the impeler (m)
c1flow velocity circumferential component at the inlet (m/s)
r1inner radius of the impeler (m)