# Video of “Hydrostatics under solid body acceleration: dropping leaky bottles”

Here is a link to a  slow motion video (gif below)of the “Hydrostatics under solid body acceleration: dropping leaky bottles” demonstration. The video shows the dyed water flowing out of the base of the bottle. It stops flowing the instant I let go of the bottle.

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.

# Hydrostatics under solid body acceleration: dropping leaky bottles

I saw this on a late night chat show in Australia while on sabbatical. It was done by Brian Greene from Columbia University though in his version the demonstration was about relativity. It is incredibly simple, potentially dramatic, and also gives you the opportunity to get soaked in front of your students which is often memorable and fun (for them anyway).

Equipment

1. One 2 liter soda bottle with cap (remove the label)
2. Water
3. Food coloring for visualization
4. A knife or scissors for cutting a small hole in the base of the bottle
5. Tape to temporarily cover the hole
6. A stairwell or step ladder for additional height to drop from

Demonstration

1. Cut a small hole in the side of the bottle near the base. The hole needs to be small enough that the bottle does not empty too rapidly but large enough that so that the water can be seen as it shoots out.
2. Tape over the hole from the outside to seal it while setting up the demonstration
3. Fill the bottle with water and food coloring and put the cap back on loosely (if it is tight then air cannot get in and the water will pulse out the base of the bottle rather than flow smoothly
4. Hold the water bottle at a great height (stairwell, stepladder, etc.) Make sure there is no way for anyone to get underneath the bottle when you drop it.
5. Remove the tape so that water is smoothly pouring out the base of the bottle
6. Drop the bottle so that it falls freely. The water should stop flowing out as soon as you release it.

Analysis

The water flows out of the base of the bottle because there is a pressure difference between the water inside the bottle (due to hydrostatic pressure) and atmosphere outside. Under solid body acceleration the hydrostatic pressure changes. Below is a very basic derivation of the vertical hydrostatic pressure equation under solid body acceleration.

Start by drawing a small element of fluid with vertical height dz and horizontal area dA (see figure 2a). As the fluid is accelerating as a solid body there are no shear stresses. Therefore, the forces acting on the element are its weight (mg=ρdzdA g) the pressure force up (PdA) and the pressure force down ((P+dP)dA) where we allow the pressure to vary vertically and take vertically upward as the positive z direction. The complementary kinetic diagram (figure 2b) simply has the mass times vertical acceleration (ma= ρdzdA)

(a)                                                     (b)

Figure 2. (a) Force diagram showing the elements weight and pressure forces in the vertical direction. (b) Kinetic diagram showing the mass times acceleration. Up is positive.

Writing the equation of motion for the element in the z direction we get

ΣF=PdA – ρdzdA g – (P+dP)dA = ma = ρdzdA a

this simplifies to

dP/dz = – ρ (a + g)

again where a is positive upwards. For a static fluid (a=0) the standard hydrostatic pressure equation is recovered (dP/dz=-ρg). This hydrostatic pressure gradient produces the pressure difference between the water in the bottle and the atmosphere outside when you are holding the bottle.

When the bottle is released it initially accelerates at a rate a=-g (there is zero velocity and, therefore, no vertical drag force). In this case a+g=0 and the pressure in the water instantaneously drops to zero (atmospheric) everywhere. This means there is no longer a pressure difference across the hole in the base of the bottle and the water immediately stops flowing.

Discussion

I did this in class down a 4 story stairwell. The students all stood around the stairwell when I dropped it. However, I did not use food coloring as I did not want to get students wet and stained. Also, when I did it in class the hole was too small so it was not easy to see on the video a student took. I repeated it later in the lab with a larger hole and food coloring. See photos below. The demonstration is essentially fool proof. As long as the cap is loose and the hole is big enough the only thing that can go wrong is that you get wet. One cautionary note., if you are dropping the bottle down a stairwell you need to have something for it to land in or it may disintegrate and cause a big mess. I dropped it into a large tub filled with sand.

figure 3. Images of the leaky bottle just before (left) and just after (right) dropping.

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.

# Desktop pipe flow and pipe network demonstrations

In class demonstrations on pipe flow head loss and pipe networks have been a challenge but I think I have found a way using kid’s straw construction kits (e.g. 1, 2, & 3). These kits can be used to build any number of different pipe networks. The T connectors can also be used to add in piezometer tubes to measure the local static pressure in the straw. This, in turn, can be used for measuring head loss along the system.

Equipment needed

2. As many straw construction kits as you desire.
3. Tape measure and calipers to measure the straw dimensions.
4. Measuring cylinder and stopwatch to measure flow rate
5. Imagination

Photograph of the equipment needed including the desktop constant head tank system, straw ‘pipe fittings’, calipers, tape measure, stop watch, and measuring cylinder. In the experiment the water was collected in a plastic cup and then transferred to the cylinder for measuring.

Example demonstration: Head loss along a pipe and local losses in bends

An easy use of these straws is to measure the head loss in a pipe and around bends. You will need all the equipment listed above.

Demonstration

1. Measure the internal diameter (D) of the straws (they should all be the same in a given set).
2. Build a horizontal pipe with piezometers at the start and immediately before and after each bend (see figure below)
3. Connect the start of the pipe to the upper constant head tank and have the end drain into the overflow tank (see here for details of the constant head tank).
4. Fill the constant head tank and turn on the pump so that water flows along the pipe and also recirculates within the constant head tank system.
5. Have students measure the height of water in each of the piezometer tubes and the length of each pipe section.
6. Have students use the measuring cup to capture a known volume of water over a measured time and calculate the volume flow rate
7. Calculate the average velocity in the pipe (U=Q/A) and Reynolds number (Re=UD/ν).
8. Calculate the head loss hlp along each section of pipe based on the change in piezometer height measurements.
9. Calculate the head loss around the bend (hlB) based on the difference in piezometer heights.
10. Calculate the friction factor for the pipe (f=hlpD2g/U2L) (based on the Darcy-Weisbach equation)
11. Calculate the loss coefficients for the various bends (Kl=hlB2g/U)
12. Compare the pipe friction factor (f) and local loss coefficient (Kl) to standard values.

figure 2. (Left) Photograph of the pipe flow setup. The head loss was measured from the piezometer just downstream from the inlet to the tube just upstream of the bend. The outlet is pointed upward to reduce the total head difference along the pipe and to ensure that the piezometer tubes filled up to a height above the red solid T fittings.  (Right) photograph of a T fitting used to insert a piezometer tube into the pipe system.

Discussion

When I did this test (see photograph above), I got a flow rate of 1.75 ml/s with a straw diameter of 4.4 mm. This led to a mean velocity of 1.15 cm/s and a Reynolds number of 506 (laminar). I measured the head loss over 660 mm length of pipe to be 11 mm leading to a calculated friction factor of f=0.109. This is quite close to the theoretical value of f=64/Re=0.126. The head loss around the 180o bend was 0.6 mm which led to a calculated local loss coefficient of 0.89 which is within the range of values quoted for 180o bends. It is fiddly trying to get the system level with the piezometer tubes vertical. I would suggest using a more stable platform than piles of books. That said, the results were encouraging.

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 “A \$19 desktop constant head tank”

Here is a link to a video from the “A \$19 desktop constant head tank” post. The video shows the water being pumped from the lower tank up into the constant head tank, overflowing into the funnel and draining back into the lower tank. The outflow is bent upward to prevent water flowing out, but can be connected to tubing to provide constant flow rate over prolonged periods of time.

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.

# A \$19 desktop constant head tank

I have been planning on doing some pipe flow demonstrations in class using straws and rubber connectors. However, to do that I need a constant head tank that will drive the flow. Ideally the entire system would sit on a desktop so that no special mounting would need to be brought into the classroom. I am still working on the pipe flow demonstration but I thought that the budget desktop constant head tank design may be helpful to share.

Equipment

You will need:

1. A small aquarium pump \$8
2. A funnel \$1
3. 2 plastic tubs with straight sides (I used plastic shoe boxes) \$4
4. A small tube of silicone sealant \$3
5. A binder clip \$1
6. A short length of tubing to connect to the pump \$1
7. A hard plastic straw \$1

Figure 1: Materials needed for construction (binder clip missing).

Design

The basic idea is that there is a lower reservoir tank that feeds the upper tank via a pump. The upper tank contains a constant height weir overflow, with return to the reservoir tank, and an outlet below the overflow that will have a constant head. Provided the weir length on the overflow is large and the flow rate through the pump is substantially larger than the flow rate out of the constant head outlet then there will always be water flowing over the weir and the head over the weir will be relatively constant. In this budget design the reservoir and upper tank are plastic shoe boxes and the weir overflow is a funnel. Water is pumped using an aquarium pump from the lower to the upper tank and returns through the funnel to the lower tank. The upper tank rests on the lower so that the entire system can sit on a desk.

Construction

1. Drill a hole in the center of the base of one of the shoe boxes with a diameter equal to that of the middle of the funnel neck.
2. Drill a hole in the side of the same shoe box with a diameter a fraction smaller than the straw.
3. Push the straw through the side hole (it should be a tight fit) and then seal around the hole with the silicone sealant on both sides
4. Place the funnel inside the same shoe box with the neck protruding through the hole and seal around the funnel neck on both sides of the hole. The top of the funnel should be below the rim of the box so that water will flow into the funnel before it overflows out of the box.
5. Attach the tubing to the aquarium pump and place it in the second shoe box.
6. Attach the binder clip to the box with the funnel and use it as a mount for the tubing such that the tubing is pointed into the box but not into the funnel
7. Place the box containing the funnel on top of the box containing the pump with the funnel outlet draining into the lower box.

Figure 2: (a) fully assembled constant head tank system. The yellow straw is the constant head outlet. (b) close up of the upper tank showing the inflow tube mounted (from the pump) through the binder clip and the funnel overflow back into the lower reservoir tank. (c) alternate view of the entire system.

Operation

1. Block the end of the outlet straw or connect it to the test rig to be used.
2. Fill the lower box until it is almost overflowing and the upper box until it is about to overflow into the funnel.. This is most easily done by pouring water into the upper tank and allowing it to overflow through the funnel into the lower tank.
3. Turn on the pump. The water will be pumped into the upper box and drain through the funnel back into the lower box. The head in the upper box will remain essentially constant provided there is water overflowing into the funnel.