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, it is possible to write Eq. (1.10) as:
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.
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 cm).
Based on Eq.(1.13), Euler developed the equation for the pressure head created by the impeller (see Fig.1).
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