Understanding Fluid Behavior in Turbines

EULER TURBINE EQUATIONS

• The fluid velocity at the turbine entry and exit

• the fluid velocity can have three components in the tangential, axial and radial directions of the rotor

• the fluid momentum can have three components at the entry and exit

• the tangential force only can cause the rotation of the runner and produce work.

• The axial component produces a thrust in the axial direction  taken by suitable thrust bearings.

• The radial component produces a bending of the shaft  taken by the journal bearings

• The work done or power produced by the tangential force equals the product of the mass flow, tangential force and the tangential velocity

• As the tangential velocity varies with the radius, the work had done also will be vary with the radius

• moment of momentum theorem is used: the torque on the rotor

equals the rate of change of moment of momentum of the fluid as it

passes through the runner

• V

u1

= the tangential component of the absolute velocity at inlet

• V

_u2

=the tangential component of the absolute velocity at exit.

• r

1

and r

2

be the radii at inlet and exit.

• The tangential momentum of the fluid at inlet = m V

_u1

• The tangential momentum of the fluid at exit = mV

u2

• The moment of momentum at inlet = mV

_u1

r

₁

• The moment of momentum at exit = mV

u2

r

2

• Torque

• the Euler Turbine equation:

Components of Power

• V₁, V₂ = Absolute velocities, V_r1, V_r2 = relative velocities , u₁, u₂ = tangential velocities, Vu1, Vu2 = tangential component of absolute velocities at inlet and outlet

• From outlet velocity triangle, (Vu2 = V2 cos a2))

• Substituting in Euler equation,

• Power per unit flow rate (here the V_u2 is in the opposite to V_u1

• The first term only will be present in Pelton or impulse turbine of tangential flow type.

• In pure reaction turbines, the last two terms only will be present.

• In impulse reaction turbines of radial flow type, all the terms will be present. (A Francis turbine is of

• In impulse reaction turbines, the degree of reaction is defined by the ratio of energy converted in the rotor and total energy converted this type).

• The degree of reaction is considered in detail in the case of steam turbines where speed reduction is necessary. Hydraulic turbines are generally

operated of lower speeds and hence degree of reaction is not generally considered in the discussion of hydraulic turbines

FREE VORTEX THEORY

• The origins of free vortex law from the law of conservation of angular momentum. The primary conditions like irrotational flow and

constant axial velocity need to be satisfied for this law.

• The final form of the free vortex law:

• The runner design is developed for a gross head of 1.75 m and a flow of 75 l/s.

• A moderate working hydraulic efficiency of 75% is assumed

• the Euler shaft head of 1.3 m is obtained,