Conservation of Momentum using Control Volumes

Conservation of Linear Momentum

Recall the conservation of linear momentum law for a system:

In order to convert this for use in a control volume, use RTT with B = mV, beta = V

we get:

NOTE: Recall that at any instant of time t, the system & CV occupy the SAME physical space.

So, the forces of the system are the same at the forces of the control volume at a given instant.

Example Problems

Given: A water jet of velocity Vj and thickness Dj impinges on a turning block which is held in place by a force Fx, as shown in the sketch.

As the water leaves the block, the round jet flattens out and slows down due to friction along the wall. The water turns a full 180 degrees and flattens into a rectangle shape of thickness Dj/6 and width 10Dj in cross section. The flow is steady.

a) Find: Ve, the exit velocity of the jet.


b) Find: Fx, the force required to hold block in place.


Momentum Flux Correction Factor

Most Useful Form of the Momentum Equation

Example Problem

Given: Consider incompressible flow in the entrance of a circular tube.

The inlet flow is uniform u1 = U0. The flow at section 2 is developed laminar pipe flow.U0, p1, p2, R, and , are also known. At section 2, .

Find: Total friction force on the fluid from 1 to 2.


More Example Problems

Problem # 3.58 in the text:

Given: Cart with water jet, deflector, as shown

Known in this problem are the jet area A, the average velocity Vav, the jet deflection angle, , and the momentum. flux correction factor of the jet, . Also, frictionless wheels are assumed.

Find: Tension in cable at time t=0.


Example A water-mounted fire pump (an old exam problem.)

Given: A pump is anchored to the ground as shown, with Vj=35.0 m/s and dj=3.00 cm. Assume the jet has a fully developed turbulent pipe flow profile at its exit.
picture wat_pump.gif

Find: Horizontal force required to hold platform in place.


Problem # 3.51 in the text:

Given: A turbine wheel, powered by a water jet, as shown in the sketch (at time t = 0)

The turbine is spinning at a constant rotational speed.


(a) Find: The force of the turning bucket on the turbine wheel at this instant of time.


(b) Find: The power, P, delivered to the wheel at this instant of time (t = 0).


(c) Find: The angular velocity which provides the maximum power to the wheel.