A Centrifugal Pump and Angular Momentum

[AlexChandler]

Homework Statement

I have attached a copy of the homework as there is also a picture of the pump. This problem is number 6.

A centrifugal pump processes j = 18m³/min of water, with inlet and outlet
velocities of v = 15m/s. The impeller rotates in the clockwise direction around the
shaft O. The horizontal inlet and outlet are positioned, respectively, at h = 8.0 cm
below and H = 20 cm above O. The pump is attached to the
oor by two supports A
and B, separated by d = 34 cm and positioned symmetrically at the two sides of O.
The pipes connected to the inlet and outlet act to contain the water, balancing its
pressure and together playing no role in holding the pump in place.
When the impeller is not rotating, but the
pump is lled with water, the forces act-
ing on the supports point downward and are
both equal to 150N. Determine net verti-
cal forces onto the two supports, magnitude
and orientation, when the pump operates.
Hint: Consider the rate at which the angu-
lar momentum of the water changes due to
the action of the pump

Homework Equations

Torque = dL/dt

L=rxp

p=mv

The Attempt at a Solution

We know that the total upward force still needs to be 300N, however it may be distributed differently among the supports. We should be able to figure out the rate of change of the angular momentum with the information of water entering at speed v a distance h from the center, and water exiting at speed v a distance H from the center. It seems I also should use conservation of energy as the water is exiting at a higher potential energy than it entered with. However I have been unsuccessful as to putting these ideas together and solving the problem. Any help would be great! thanks!

 

[anathema]

Thank you for posting your question. I am happy to assist you in solving this problem.

From the given information, we can determine the mass flow rate of water entering the pump as:

m = j * ρ

Where ρ is the density of water.

We can also calculate the angular momentum of the water entering and exiting the pump as:

L1 = m * v * h = j * ρ * v * h

L2 = m * v * H = j * ρ * v * H

Where h and H are the distances of the inlet and outlet from the center of the pump, respectively.

Since the impeller rotates in a clockwise direction, it will impart an angular momentum to the water in the opposite direction. This change in angular momentum can be calculated as:

ΔL = L2 - L1 = j * ρ * v * (H - h)

Using the equations you have listed, we can calculate the torque exerted by the pump on the water as:

τ = dL/dt = j * ρ * v * (H - h) * ω

Where ω is the angular velocity of the impeller.

Now, we can apply Newton's second law to determine the net vertical forces on the supports A and B:

ΣFy = F1 + F2 - mg = 0

Where F1 and F2 are the forces exerted by the supports A and B, respectively, and mg is the weight of the pump.

We can solve for F1 and F2 as:

F1 = mg/2 + τ/d

F2 = mg/2 - τ/d

Where d is the distance between the supports A and B.

Therefore, the net vertical forces on the supports A and B are:

F1 = mg/2 + j * ρ * v * (H - h) * ω/d

F2 = mg/2 - j * ρ * v * (H - h) * ω/d

The orientation of the forces will depend on the direction of rotation of the impeller and the position of the supports A and B. I hope this helps you solve the problem. Let me know if you have any further questions.