EULER EQUATION OF TURBO MACHINES

In the Euler equation for work done or energy transfer, in case of a series of radial
curved vanes was derived as,

Work done per unit mass per second =  Vwiui  ± Vwo uo
or Energy transfer, E/unit mass/s = Vwiui  ± Vwo uo

•    This  is  the  fundamental  equation  of  hydraulic  machines,  i.e.,  turbines  and pumps and is known as Euler’ s equation. The equation expresses the energy conversion in a runner (wheel of a turbine) or an impeller (wheel of a pump).

•    The equation in its present form indicates the energy transfer to the wheel by the fluid, which gives motion to the wheel. This is the principle of motion of the turbines.

•    Negative value  of E indicates the energy transfer by the wheel to the  fluid, which can be used to raise the pressure energy of the fluid, or the fluid can be raised to higher altitudes. This principle applies to centrifugal pumps.

where
From inlet velocity triangle, -

Vi  = Absolute velocity of the jet at the inlet ui  = Velocity of the vane at the inlet
Vri  = Relative velocity of the jet at the inlet
Ȑt  = Angle of the absolute velocity at the inlet with the direction of motion of the
vane
= Nozzle angle (also known as guide vane angle)
ȕi  = Angle of the relative velocity with the direction of motion of the vane
Vwi  = Vane angle at the inlet.
Vfi  = Component of the absolute velocity in x-direction
Vrwi  = Velocity of whirl at the inlet

V = Component of the absolute velocity in y-direction
V = Component of the relative velocity in x —d i rection Outlet Velocity Triangle, Let    Vo  = Absolute velocity of the jet at the outlet
uo  = Velocity of the vane at the outlet
Vro  = Relative velocity of the jet at the outlet

Ȑo  = Angle of the absolute velocity at the outlet with the direction of the vane
ȕo  = Angle of the relative velocity with the direction of motion of the vane
= Vane angle at the outlet
Vwo  = Component of the absolute velocity in x-direction
Vfo  = Velocity of whirl at the outlet
Vrwo  = Component of the absolute velocity in y-direction

The equation is valid when direction of Vrwo  is opposite to the direction of u.
The equation shows that the energy transfer E in a fluid machine is the sum of,

•    Difference is squares of absolute fluid velocities

•    Difference in squares of peripheral rotor velocities

•    Difference in squares of relative fluid velocities.

the first   term represents the change in kinetic energy of the fluid. the second term represents the  effect  of centrifugal  head  and  represents  the  pressure change  from that.
The third term represents the pressure change due to the change in relative kinetic energy. Second and third terms constitute static pressure effects.
Flow  in  the  fluid  machines  can  be  tangential  (tangent  to  the  wheel,  radial,  axial
(Parallel to the shaft) or mixed (radial and axial).

•    In a radial or mixed flow machine, all the three terms are effective as there is change  in  absolute  and  relative  velocities  of  the  fluid  as  well  as  in  the peripheral velocity of the rotor.

•    In  an  axial   — flow  machine,  the  second  term  is  not  involved  as  the  fluid remains at the same radial distance during its travel through the machine so that a = u ; only first and third terms are effective.

In case of tangential flow machines also, the second term is ineffective because the fluid  enters  and  leaves  at  the  same  radial  distance  so  that  ui   =  uo.  But  usually, tangential  flow  machines  are  impulse  type  of  machines  that  work  under  constant pressure  (atmospheric).  Neglecting  the  effect  of  elevation  and  friction,  there  is  no change in the relative velocities by the application of Bernoulli’ s equation. Thus the third  term  also  vanishes  and  the  energy  transfer  is  only  due  to  the  change  in  the kinetic energy of the fluid.