Moment of Momentum Equation (1)

1
Moment of Momentum Equation (1)

• Moment of a force ? Torque
• Moment of force represents the magnitude of
  force
• applied to a rotational system at a distance
  from the axis
• of rotation.
• The moment arm is key to the operation of the
  lever,
• pulley, gear, and most other simple machines
  capable of
• generating mechanical advantage.
• The SI unit for moment is the Newton meter
  (Nm).
• Moment Magnitude of Force x Perpendicular
  distance
• to the pivot (Fd)
• When torques are important, the
  moment-of-momentum
• equation relates torques and angular momentum

2
Moment of Momentum Equation (2)

• Newtons second law of motion applied to a
  particle of fluid

Taking moment on both sides
r is the position vector from the origin of the
inertial coordinate
Combining all the above
3
Moment of Momentum Equation (3)

• For every particle of the system

where
Since we can change the order of differentiation
and integration without consequence
Combining the above
Time rate of change of the moment -of the
momentum of the system
Sum of external torques acting on the system
4
Moment of Momentum Equation (4)

• For a control volume coincident with the system

For a fixed and non deforming control volume,
RTT leads to
Time rate of change of the moment-of the momentum
of the system
Time rate of change of the moment-of the momentum
of the control volume
net rate of flow of the moment-of the momentum
through the control surface
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Moment of Momentum Equation (5)

• For a system, the rate of change of moment-of
  momentum equals the net torque

Application of moment-of momentum equation
involves machines that rotate or tend to rotate
around a single axis lawn sprinklers,
ceiling fans, lawn mower blades, wind
turbines, gas turbine engines or all
turbomachines.
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Application of Moment of Momentum Equation

• Water enters a rotating lawn sprinkler through
  its base at the steady rate
• of 1000 ml/s. The exit area of each of the two
  nozzles is 30 mm2, and
• the flow leaving each nozzle is in the tangential
  direction. The radius
• from the axis of rotation to the centerline of
  each nozzle is 200 mm.
• (a)Determine the resisting torque required to
  hold the sprinkler head stationary
• (b)Determine the resisting torque associated with
  the sprinkler rotating with a constant speed of
  500 rev/min
• (c) Determine the speed of the sprinkler if no
  resisting torque is applied.

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First Law of Thermodynamics-The Energy Equation
(1)

• Time rate of increase of the total stored
  energy of the
• system net time rate of energy addition by
  heat transfer
• into the system net time rate of energy
  addition by
• work transfer into the system

The total stored energy per unit mass for each
particle in the system, e, is related to the
internal energy u per unit mass, the kinetic
energy per unit mass V2/2, and the potential
energy per unit mass, gz, by the equation,
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First Law of Thermodynamics-The Energy Equation
(2)

• Net rate of heat transfer into the system
• Net rate of work transfer into the system
• Heat and work transfer is ve when going into
• the system, and -ve when coming out
• For control volume coincident with system,

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First Law of Thermodynamics-The Energy Equation
(3)

• 1st Law in terms of RTT, setting b equal to e

Time rate of increase of the total stored energy
of the system
Time rate of increase of the total stored
energy of the control volume
Net rate of flow of the total stored energy out
of the control volume through the control surface
1st Law of thermodynamics for control volume
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First Law of Thermodynamics-The Energy Equation
(3)

• Heat transfer rate, energy exchanged
• between control volume and
• surrounding radiation, conduction
• and convection Zero for adiabatic
• process

Work transfer rate, also called power, positive
when work is done on the control volume by
surroundings. A rising piston, a rotating shaft,
electric wire are all examples of work
Interactions.
Shaft work