The Euler’s pump and turbine equations are most fundamental equations in the field of turbomachinery. These equations govern the power, efficiencies and other factors that contribute in the design of Turbomachines 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.
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Conservation of angular momentum
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, it is possible to write Eq. (1.10) as:


 M + Mr = \rho.Q (c_2u.r_2  c_1u.r_1) (1.13)^{[1]}

 c_2u=c_2cos\alpha_2\,
 c_1u=c_1cos\alpha_2.\,
Triangle velocity
The color triangle formes
by velocity vector u,c,w call 'triangle velocity" this is an important role in old academic, this rule was helpful to detail Eq.(1) become Eq.(2) and wide explained how the pump works.
 c_1\, and c_2\, are the absolute velocities at the inlet and outlet respectively.
 w_1\, and w_2\, are the relative velocities at the inlet and outlet respectively.
 u_1\, and u_2\, are the velocities of the blade at the inlet and outlet respectively.
 \omega is angular velocity.
Fig 1 shows triangle velocity of backward curved vanes impeller ; it's illustrate rather clearly energy add to the flow (shown in vector c) inversely change upon flow rate Q (shown in vector c_{m}).
Euler's pump equation
Based on Eq.(1.13), Euler developed the equation for the pressure head created by the impeller (see Fig.1).


 Yth.g=H_t= c_2u.u_2  c_1u.u_1 (1)



 Yth=1/2(u_2^2u_1^2+w_1^2w_2^2+c_2^2c_1^2) (2)

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 (See Fig.2).
 H = {1 \over {2g}} ( V_1^2  V_2^2 ). \,
Usage
Euler’s pump and turbine equations can be used to predict the impact of changing the impeller geometry on the head. Qualitative estimations can be made from the impeller geometry about the performance of the turbine/pump. For the design of an aeroengines and the designing of power plants, the equations assume prime significance. Thus for the design aspect of turbines and pumps, the Euler equations are extremely useful.