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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 Ve relative to the nozzle.
A control volume with its surface around the turbine is also shown in the fig below.
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering
Fig 12.1    A Sprinkler like Turbine

Application of Moment of Momentum Theorem (Eq. 10.20b) gives
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering    (12.1)

When Mzc  is the moment applied to the control volume. The mass flow rate of water through the turbine is given by
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering

The velocity  Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering  must be referenced to an inertial frame so that
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering   (12.2)

The moment Mz acting on the turbine can be written as
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering  (12.3)

The power produced by the turbine is given by
Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering   (12.4

 

The document Application of Moment of Momentum Theorem | Fluid Mechanics for Mechanical Engineering is a part of the Mechanical Engineering Course Fluid Mechanics for Mechanical Engineering.
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FAQs on Application of Moment of Momentum Theorem - Fluid Mechanics for Mechanical Engineering

1. What is the moment of momentum theorem?
Ans. The moment of momentum theorem, also known as the angular momentum theorem, states that the rate of change of angular momentum of a system is equal to the net torque acting on the system.
2. How is the moment of momentum theorem applied in real-life situations?
Ans. The moment of momentum theorem is applied in various real-life situations, such as understanding the motion of rotating objects like wheels, gyroscopes, or tops. It is also used in engineering applications involving rotating machinery, such as turbines, motors, and flywheels.
3. Can you explain the mathematical equation for the moment of momentum theorem?
Ans. The mathematical equation for the moment of momentum theorem is L = Iω, where L represents the angular momentum, I is the moment of inertia, and ω is the angular velocity. This equation shows the direct proportionality between angular momentum and angular velocity for a rotating object.
4. What are some examples of moments of momentum in everyday life?
Ans. Moments of momentum can be observed in everyday life in several instances. Examples include the spinning motion of a figure skater during a spin, the rotation of a bicycle wheel, or the movement of a spinning top.
5. How does the moment of momentum theorem relate to Newton's laws of motion?
Ans. The moment of momentum theorem is an extension of Newton's laws of motion to rotational motion. It provides a way to analyze the rotational dynamics of objects by considering the torque acting on the system and its effect on the angular momentum.
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