Euler's pump and turbine equation

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.

${\displaystyle c_1}$ and ${\displaystyle c_2}$ are the absolute velocities of the fluid at the inlet and outlet respectively.

${\displaystyle w_1}$ and ${\displaystyle w_2}$ are the relative velocities of the fluid with respect to the blade at the inlet and outlet respectively.

${\displaystyle u_1}$ and ${\displaystyle u_2}$ are the velocities of the blade at the inlet and outlet respectively.

${\displaystyle \omega }$ 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:

${\displaystyle Y_{th}=H_t\cdot g=c_{2u}{u}_2-c_{1u}{u}_1}$ (1)

${\displaystyle Y_{th}=1/2(u_2^2-u_1^2+w_1^2-w_2^2+c_2^2-c_1^2)}$ (2)

Y_th : theoretical specific supply; H_t : 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:

${\displaystyle H=\frac{1}{2g}(V_1^2-V_2^2).}$

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:

${\displaystyle I=h_0-u{c}_u}$

where ${\displaystyle I}$ is constant across the rotor blade.