# Video of “Bernoulli and the discharge velocity from a water pistol”

Here are some videos of the “Bernoulli and the discharge velocity from a water pistol” demonstration. The full videos are linked from the GIF titles.

Vertical injection

Injection at 45 degrees

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Bernoulli and the discharge velocity from a water pistol

The goal of the demonstration is to illustrate the use of Bernoulli’s equation to compute the discharge velocity of a water pistol.

Equipment

1. A water pistol or a syringe
2. A tape measure
3. Level
4. OHP with a sheet with parallel lines drawn on it (helpful but not essential)
5. Protractor or Set Square (optional)
6. A few student volunteers

Leveling the vertical scale created by the OHP slide

Measuring the vertical scale

Demonstration

The demonstration is fairly simple in concept but it can be quite hard to maintain both a smooth flow rate and steady angle. Simply project the water vertically upward and measure the height to which it rises. This can be a little challenging as the height can be quite large depending on how hard you press the plunger. An alternative is to project it at some angle say 20o-30o from the horizontal. You will need the Protractor or Set Square for this. Firing at an angle will make the maximum height lower but possibly more challenging to measure. It might help to draw a series of horizontal lines on a board, or project a set of horizontal lines using an overhead projector. You can get the lines horizontal using the level. After you have completed the demonstration you will have the change in height from the discharge to the maximum height (H2-H1) and the angle of discharge (θ) measured from the horizontal.

Analysis

The analysis for the vertical case is the simplest. Start by writing Bernoulli’s equation from the release point (1) to the rise height (2)

P1/γ+H1+(u12)/2g=P2/γ+H2+(u22)/2g

The pressure everywhere is atmospheric and u2=0 at the point of maximum vertical rise. Therefore, the discharge velocity is given by

u1=(2g(H2-H1))1/2

The angled release is a little more complex. Combining Pythagoras theorem and Bernoulli leads to

[P/γ+H+(ux2+uz2)/2g]1=[P/γ+H+(ux2+uz2)/2g]2

Where  x and  z denote the horizontal and vertical directions. Assuming there is negligible drag or energy loss then then the horizontal velocity  will remain a constant. Therefore Bernoulli’s equation can be simplified to

uz1 = u1 sinθ = (2g(H2-H1))1/2

as the vertical velocity is zero at the maximum rise height. Therefore, the discharge velocity is given by

u1 = (2g(H2-H1))1/2/ sinθ

Note that this equation can be applied to the vertical release case as sin90=1.

Discussion

There is a lot of uncertainty in the measurements so, as with so many other demonstrations, it is a good opportunity to talk about experimental errors and repeatability. It would be fairly straight forward to run this test for a few different release angles to test the relationship in the final equation. However, the main repeatability problem is likely to be getting the jet release velocity constant between tests.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Video of “Pump performance curves” demonstration

Here are some videos of the “Pump performance curves” demonstration. The full videos are linked from the GIF descriptions.

measuring the flow rate

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Pump performance curves

I sometimes use this as an in class demonstration and sometimes as a lab test toward the end of the semester. It is good for illustrating experimental method and measurement uncertainty while demonstrating pump performance characteristics. I first saw a version of this demonstration at an APS DFD meeting about 8 years ago though I am having trouble recalling who presented it.

Equipment

1. Bucket
2. Stopwatch
3. Tape measure
4. Measuring cup
5. Aquarium pump
6. Extension cord
7. Tube connected to the outlet of the aquarium pump

Demonstration

1. Fill the bucket with enough water to fully cover the pump, attach the tube to the pump outlet and place the pump in the bucket.
2. Hold the tube vertically above the pump and turn it on (typically there is no on – off switch so plugging it in turns it on). The water should rise up the tube and then stop. Measure the head difference between the top of the water in the tube and the top of the water in the bucket. This is the shut off head.
3. Lower the tube outlet until water starts to flow. Holding the outlet steady measure the distance from the bucket water surface to the outlet and the time taken to fill the measuring cup. Calculate the flow rate.
4. Repeat step 3 until you have 6 to 8 different head – flow rate data pairs ranging from the shut off head to negligible elevation difference.
5. Write up the head flow rate pairs on the board and plot the data by hand with elevation on the vertical axis and flow rate on the horizontal axis.

Analysis The data plotted should show an increase in flow rate with decreasing elevation like a typical pump performance curve. However, the data needs to be corrected to account for head loss in the tubing.

Draw a sketch of the pump – bucket – tube system with a control volume around the whole system.

Write down the work energy equation from the water surface and the tube outlet.

Z1 + u12/2g + p1/g + EP= Z2 + u22/2g + p2/ g +  hl

All the terms on the left hand side are zero except the pump head EP while the pressure at the outlet is p2=0. Using the Darcy–Weisbach equation for the head loss then Eis given by

EP= h+ (u22/2g)(1+ fL/D)

where f is the friction factor, L is the tube length, and D is the tube diameter. The exit velocity can be calculated based on the flow rate u2=Q/A=4Q/πD2. The main problem here is that the friction factor f varies with the Reynolds number and, therefore, the flow rate. Therefore, you need to calculate the friction factor and exit velocity for each data point. This can be given to the class as an in class exercise. Once the actual EP values have been calculated they can be plotted on the same graph. Some aquariums pumps actually come with a pump performance curve that can be compared to the measured data. In that case you can print the manufacturer curve on an overhead transparency. My experience with this is that cheap pumps rarely behave exactly as given in the manufacturer curve.

The demonstration presents a great opportunity to discuss experimental error. The first major source of error is the head measurement because it is hard to hold the tube steady, the water level in the bucket drops while you are filling the measuring cup, and you need to keep the tape measure vertical (though small angles away from the vertical will not make much difference). The second major error is in the measurement of the time taken to fill the measuring cup. Even if you use a 4 cup measure, it still fills in a few seconds for the larger flow rates. Therefore, a small error in timing of say half a second can lead to 10-20% error in the flow rate calculation. Both of these errors can be reduced by making multiple measurements at each height. It is also possible to estimate the individual errors and use them to place error bars on the pump performance curve data.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.