Application of Moment of Momentum Theorem

Let us take an example of a sprinkler like turbine as shown in Fig. 12.2. The turbine rotates in a horizontal plane with angular velocity ω. The radius of the turbine is r. Water enters the turbine from a vertical pipe that is coaxial with the axis of rotation and exits through the nozzles of cross sectional area ’a’ with a velocity V_e relative to the nozzle.

A control volume with its surface around the turbine is also shown in the fig below.

Fig 12.1    A Sprinkler like Turbine

Application of Moment of Momentum Theorem (Eq. 10.20b) gives

(12.1)

When M_zc  is the moment applied to the control volume. The mass flow rate of water through the turbine is given by

The velocity must be referenced to an inertial frame so that

	(12.2)

The moment M_z acting on the turbine can be written as

(12.3)

The power produced by the turbine is given by

(12.4)