The Euler
pump
A pump is a device that moves fluids (liquids or gases), or sometimes Slurry, slurries, by mechanical action, typically converted from electrical energy into hydraulic or pneumatic energy.
Mechanical pumps serve in a wide range of application ...
and
turbine
A turbine ( or ) (from the Greek , ''tyrbē'', or Latin ''turbo'', meaning vortex) is a rotary mechanical device that extracts energy from a fluid flow and converts it into useful work. The work produced can be used for generating electrical ...
equations are the most fundamental equations in the field of
turbomachinery
Turbomachinery, in mechanical engineering, describes machines that transfer energy between a Rotor (electric), rotor and a fluid, including both turbines and gas compressor, compressors. While a turbine transfers energy from a fluid to a rotor, ...
. These equations govern the power, efficiencies and other factors that contribute to the design of turbomachines. With the help of these equations the
head
A head is the part of an organism which usually includes the ears, brain, forehead, cheeks, chin, eyes, nose, and mouth, each of which aid in various sensory functions such as sight, hearing, smell, and taste. Some very simple ani ...
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
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
(or the “moment of momentum”) which is fundamental to all turbomachines. Accordingly, the change of the angular momentum is equal to the sum of the external moments. The variation of angular momentum
at inlet and outlet, an external torque
and friction moments due to
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
es
act on an
impeller
An impeller, or impellor, is a driven rotor used to increase the pressure and flow of a fluid. It is the opposite of a turbine, which extracts energy from, and reduces the pressure of, a flowing fluid.
Strictly speaking, propellers are a sub-clas ...
or a diffuser.
Since no pressure forces are created on cylindrical surfaces in the circumferential direction, it is possible to write:
:
(1.13)
:
:
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.
:
and
are the absolute velocities of the fluid at the inlet and outlet respectively.
:
and
are the relative velocities of the fluid with respect to the blade at the inlet and outlet respectively.
:
and
are the velocities of the blade at the inlet and outlet respectively.
:
is angular velocity.
Figures 'a' and 'b' show impellers with backward and forward-curved vanes respectively.
Euler's pump equation
Based on Eq.(1.13), Euler developed the equation for the pressure head created by an impeller:
:::
(1)
:::
(2)
Y
th : theoretical specific supply; H
t : theoretical head pressure; g: gravitational acceleration
For the case of a
Pelton turbine
The Pelton wheel or Pelton Turbine is an impulse-type water turbine invented by American inventor Lester Allan Pelton in the 1870s. The Pelton wheel extracts energy from the impulse of moving water, as opposed to water's dead weight like the t ...
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.
This equation can be written as
rothalpy invariance:
where
is constant across the rotor blade.
See also
*
Euler equations (fluid dynamics)
In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity ...
*
List of topics named after Leonhard Euler
*
Rothalpy
References
{{DEFAULTSORT:Euler's pump and turbine equation
Turbines
Pumps
Gas compressors
Ventilation fans
Fluid dynamics
Leonhard Euler