Flow out an opening

The velocity of water out an opening under h feet of water is given by

where h is in feet.

The rate of flow of water out the opening is

where A is in square feet.

The same is true for the rate of flow of water out a pipe in a dam, under h feet of water. This is general, it predicts the flow of ice tea out an ice tea jug or water out a discharge pipe in a dam.

The following calculator is helpful in predicting the discharge rate out a circular opening of known diameter and under a known depth of water.

Discharge Rate Calculator

First, select the units you want to use from the pull down menus. Then enter the values on the left and the calculator will give the value on the right

Experiment - Flow from holes in bottle

This experiment allows you to see the variations in flow out of three holes placed in the side of a plastic bottle, such as a 2 liter pop bottle. The holes are drilled, cut or poked in the plastic and cleaned up so that they do not have a lot of debris around them. A strip of electrical tape is then placed up and down the side of the bottle to cover all three holes. The tape is folded over at the top so that we can get hold of it to pull it off. The bottle is filled with water and placed in a sink so that the water will not make a mess when it comes out.

Pull off the tape and see what happens.

Why you saw what you did: The pressure in the water is proportional to the depth of the water from the surface. For the three holes the lower hole has the greatest pressure in the water just inside the bottle. The lower hole has a greater pressure pushing water out the hole. One might expect it to go it faster than out the top hole.

The velocity that water leaves a hole is proportional to the square root of the depth of the water. The velocity the water leaves the bottom hole is going to be greater than the top hole. The stream having the highest velocity will travel the farthest in the same time. The bottom stream goes farthest. The difference in velocity would be more obvious if the bottle was placed on a table edge and the water flowed out and dropped 3 feet onto the floor. Even so, the bottle sitting in the sink will also show the difference in the velocity of the streams leaving the bottle. The size of the arcs of the streams is an indication of their velocity in leaving.

The rate of flow out each hole is the cross sectional area of the hole times the velocity. Again, the bottom stream has the highest rate of flow out of the bottle. In a given time, more water flows out of it than out of either of the two holes.

The following calculator gives the time required for water to drain from an open top container with an opening on the side, at the bottom, of known size and depth from the top of the stored water. To run it, you input the volume of stored water, the diameter of the opening, and the depth of the water (distance between the top of the water and the opening). The calculator gives the time for the water to drain out the opening.

Experiment - Draining a Tank

Question: How long does it take a tank of water to drain out an opening near the bottom?

Finding: The water depth drops much more quickly when the opening is first opened as compared to when the water is nearly gone. The total time was measured.

Reason: The rate the water leaves the opening depends on the height of water in the tank above the opening and the cross-sectional area of the opening. The rate is given by

The rate varies since the water height continues to go down. The time to drain is related to the volume of water, the initial height of the water and the cross-sectional area by

where t_d is the time to drain, A_o is the cross-sectional area of the opening and h_o is the initial height of the water above the opening. This also shows that a good guess for an average flow rate would be 1/2 of the discharge rate at the start. This is the average rate from the mathematics. The average rate is used to find the drain time by

Drain Tank Time Experiment

Assume you have an open topped tank which holds 100 gallons of water. You can change the shape of the tank by changing the initial height of the water in the tank. You can also change the diameter of the drain opening near the bottom of the tank. This experiment allows you to see the effects of changing the dimensions of the tank and opening on the time required for all the water to drain out of the tank. You can click on the pointer to change the initial height of the water in the tank over the range 2 to 10 feet. You can also use another pointer to change the diameter of the drain opening from 1 to 4 inches. As you change these things the time to drain is computed, displayed, and entered into a table.

Please use this experiment to see if you can determine how the drain time is related to the initial water height in the tank and also how the drain time is related to the drain opening diameter. You can do this by holding one of the constant say the height at 10 feet and use different opening diameters to see how the drain time changes. Now do the same by holding the drain diameter constant and vary the initial water height to see the effects on the drain time.

Findings: From looking at the values of the initial water height, the drain diameter and the drain time you should come up with the idea that the drain time varies with 1 over the square root of the initial drain height. The greatest initial drain height has the smallest drain time for the same diameter drain opening.

You also should have seen that the drain time varied with 1 over the square of the drain diameter. The largest diameter hole, for a constant initial water height, will have the shortest drain time.