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) • Newton’s 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

5. 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.

6. 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.

7. 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,

8. 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,

9. 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

10. 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