Calculating Water Flow Duration in Cylinder Reservoir

[Pzi]

Hello.

Homework Statement

There is a vertical cylinder-shaped reservoir full of water:
Height h = 18 meters
Radius R = 2 meters
If suddenly a hole appeared on the bottom (radius r = 0.25 meters) how long would it take to empty the reservoir?

Homework Equations

Probably related to Bernoulli's principle somehow someway.

The Attempt at a Solution

To be honest with you I just reposted this problem from elsewhere. Some girl tried to solve it and since I am a pure mathematician I pretty much do not have a clue about those things. Tried to google it, but without proper knowledge did not succeed.
Some kind of powerful formula and basic steps would be appreciated.

Thanks.

 

[gneill]

Bernoulli is the key. His principle gives you the speed of the water exiting the orifice, hence the flow rate and rate of change of velocity in the tank. [EDIT: I meant change of VOLUME, not change of velocity!].

Essentially, for a depth of water h above the opening, the velocity of the water will be given by $\sqrt{2 g h}$ .

With the flow rate and opening size you can construct the differential equation for the volume of water in the tank.

 

[Pzi]

Bernoulli is the key. His principle gives you the speed of the water exiting the orifice, hence the flow rate and rate of change of velocity in the tank.

Essentially, for a depth of water h above the opening, the velocity of the water will be given by $\sqrt{2 g h}$ .

With the flow rate and opening size you can construct the differential equation for the volume of water in the tank.

So it seems that I am supposed to solve this
[PLAIN]http://img708.imageshack.us/img708/6712/eqn9284.png

Can you confirm?

 

[gneill]

Yes, that is one differential equation that fits the bill!

 

[asteriatic]

Dear student,

Thank you for your question. I would approach this problem by first understanding the physical principles involved. In this case, we are dealing with the principles of fluid dynamics, specifically the Bernoulli's principle which states that in a flowing fluid, the sum of its kinetic energy, potential energy, and pressure energy remains constant.

To solve this problem, we need to consider the conservation of mass and energy. The rate of change of water level in the cylinder reservoir can be calculated using the continuity equation, which states that the volume of water flowing out of the hole in a given time is equal to the volume of water in the cylinder at that time. We can also use the Bernoulli's equation to calculate the velocity of the water at the hole, which is related to the rate of flow.

Once we have the velocity, we can use the equation of motion to calculate the time it takes for the water level to reach the hole. This involves calculating the acceleration of the water as it flows out of the hole, which can be determined using the equation for the conservation of energy.

In summary, to calculate the water flow duration in the cylinder reservoir, we need to consider the principles of fluid dynamics, specifically the continuity equation, Bernoulli's equation, and the equation of motion. By applying these equations and solving for the unknown variables, we can determine the time it takes for the reservoir to empty through the hole.

I hope this explanation helps. If you have any further questions, please do not hesitate to ask. Good luck with your homework!

Sincerely,