# Videos of “Radial flow, Bernoulli, and levitating an index card”

Here are the videos of the “Radial flow, Bernoulli, and levitating an index card” demonstration. The higher resolution videos are linked from the titles.

Foam plate version

Index card version

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.

# Radial flow, Bernoulli, and levitating an index card

This experiment was suggested by Dr. John Foss from Michigan State University. He presents a more detailed write up, along with a discussion of Reynolds number effects in Experiments in Fluid Mechanics, R.A. Granger, Ed. Holt, Rienhart and Winston, (1988). The basic idea is that a radial outflow between parallel plates has a velocity that decreases with increasing radial distance. Therefore, if a high Reynolds number situation exists such that the shearing effects are small and a nominal balance exists between the pressure gradient and the acceleration, the pressure will increase with radial distance. If the outflow boundary condition is atmospheric pressure then the pressure between the parallel plates must be a vacuum pressure. This is surprisingly easy to do with very simple equipment.

Equipment:

There are a couple of methods for doing this. Two are described below. The index card version is the easiest to do in large numbers.

• Foam Plate Version
• 1 Foam plate
• 1 Straight pin
• A can of compressed air
• 1 spool of thread
• Tape
• Index Card Version
• 1 straw
• 1 3”x5” index card
• 1 straight pin
• 1 spool of thread
• Tape
• Adhesive putty (optional, may help hold the straw in the spool)

Procedure:

• Foam Plate Version
1. Place the straight pin through approximately the center of the foam plate. Tape the pin to the bottom of plate to stabilize it and to keep it perpendicular to the plate.
2. Have someone hold the plate so that the pin is horizontal
3. Hold the spool of thread to where it covers most of the pin but is not touching the plate
4. Take the can of compressed air and aim it through the spool of thread. The nozzle does not need to be placed directly in the spool. Just aim it so that the air will flow through the spool.
5. When the air is released from the can the plate should move towards the spool of thread and should stay there without support as long as the compressed air can is blowing

*CAUTION:  Do not aim the compressed air can downward. The air will become very cold and could possibly burn someone

• Index Card Version
1. Place the straight pin through approximately the center of the index card. Tape the pin to the bottom of the card to stabilize it and keep it perpendicular to the card
2. Place the index card on the table with the pin pointing straight up
3. Hold the spool of thread directly above the pin but not touching the index card
4. Insert the straw into the spool. The adhesive putty can be used to attach the straw to the spool so that you only need one hand.
5. Blow through the straw and, if the spool is close enough to the card, you should be able to lift the card off the table. The card should stay as long as there is a steady flow of air

** Trial and error may need to be used to in both experiments to determine the gap distance needed between the spool and card/plate in order to pick the object up.

Foam Plate Version                             Index Card Version

Analysis

Consider an incompressible fluid flowing horizontally and radially out from a point source between two parallel plates separated by a distance T. At any arbitrary radial distance r from the source the area of the flow is

A= 2π r T

(see diagram below). For a constant volume flow rate Q the velocity is given by

U(r)=Q/2 πr T

Writing Bernoulli’s equation from r to the outlet at a radial distance R and taking the outlet pressure to be atmospheric leads to

Pr = (Q/2 π T)2 (R-2-r-2)

Therefore, given that r<R, Pr<0 and the plate/card will be pushed toward the spool. However, for this to work the gap width needs to be small. If T is too large the pressure vacuum pressure over the card will no be enough to overcome the weight of the card.

In reality life is a little more complex and a more detailed analysis of this problem is given by Prof. Foss in Experiments in Fluid Mechanics, R.A. Granger, Ed. Holt, Rienhart and Winston, (1988).

Thanks to John Foss for suggesting the demonstration and helping with the write up. Thanks to Meredith and Alex for testing the procedure, putting together pictures, and making the videos. Videos to follow soon.

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.

# Videos of “Discharge coefficient calculation and data presentation” demonstration

Here are the videos from the “Discharge coefficient calculation and data presentation” demonstration. The titles link to the full videos. If you really want to watch the whole thing (12 minutes) it is here.

Initial release

Depth measurement

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.

# Discharge coefficient calculation and data presentation

This is another simple activity that allows one to demonstrate how to calculate the discharge coefficient for a simple orifice. This activity is used in our introductory fluid mechanics lab and is very cheap and easy to setup. The activity also demonstrates some basic Bernoulli equation modeling, data analysis, and data presentation.

Equipment

1. a bucket with thin (approximately) vertical sides
2. a yard stick
3. a stopwatch
4. small rubber plug

Drill a small hole in the side of the bucket and plug it with the rubber plug. you can also drill the hole in the base of the bucket. This avoids the problem of having a head that varies across the orifice (see here) but means that you need to mount the bucket so that water can drain out the bottom (that is, you can’t just do this on a table). Also, provided that you do not collect data when the water level is close to the orifice it makes virtually no difference if the orifice is in the side or the base of the bucket.

Activity / Demonstration

1. Measure the cross sectional area of the bucket, the diameter of the hole (orifice) and the height of the center of the orifice above the base of the bucket.
2. Fill the bucket so that the water level is close to the top but not so deep that it overflows when you put the yard stick in.
3. Place the yard stick vertically in the bucket and measure the initial depth.
4. Remove the plug and measure the depth every 15 seconds as the water drains out the orifice. you may need to record data more frequently if the bucket is small or the orifice is large. The goal is to get 10-20 depth measurements for the data analysis.
5. Stop recording when the depth of water over the orifice is less than 2 orifice diameters.
6. plot depth of water above the center of the orifice versus time

Theoretical Analysis

Draw a diagram of the setup and draw a control volume around the water in the bucket. Denote the water at the free surface as point (1) and the water leaving the orifice as point (2). Define the origin (z=0) to be at the center of the orifice and the depth of water above that point to be (see diagram).

Denote the orifice area as Ao and the bucket cross sectional area as Ab (>>Ao) which is independent of height (given the vertical sides of the bucket). As the orifice area is relatively small we assume that the velocity in the bucket far from the orifice is negligibly small. We can now write conservation of volume

dV/dt+Qout-Qin=0.

The volume of water in the bucket is V=Abh and there is no inflow so this can be simplified to

dh/dt=-Q2/Ab=-(Ao/Ab)u2

Write Bernoulli’s equation

p1/γ +u12/2g+z1= p2/γ +u22/2g+z2.

Assuming the free surface and free jet pressures are zero and that u1≈0 the equation simplifies to

h=u22/2g          or         u2=(2gh)1/2

Substituting into conservation of volume leads to

dh/dt=-(Ao/Ab)( 2gh)1/2                                                                                 (1)

This is a separable first order ODE which can be solved to give

h1/2– H1/2=-(Ao(2g)1/2/2Ab)t                                                        (2)

where H is the initial water depth and t is the time elapsed since the plug was pulled.

In reality, the pressure in the free jet is not zero but is only zero some distance later after the flow has contracted somewhat. This, combined with minor energy losses means that equation (1) needs to be modified by multiplying the right hand side by a discharge coefficient (Cd<1). Therefore (2) can be written as

h1/2=H1/2– (Cd Ao(2g)1/2/2Ab)t.                                                   (3)

Data analysis

The goal of the data analysis is to establish the value of Cd for the orifice tested. The experiment produces a set of depth (h) and time (t) measurements (see plot of raw data above). Examining equation (3) indicates that h1/2 is a linear function of time. Therefore, if you plot h1/2 vs t then you should get a straight line (see figure below). A least squares fit through that data will give you the slope of the line (m). This can be used to calculate the discharge coefficient as

m=- Cd (Ao(2g)1/2/2Ab)              or         Cd=-2mAb/( Ao(2g)1/2)

where everything on the right hand side is known.

I like this lab because it illustrates a slightly different method of looking at the data. There is all too often an urge in undergraduate labs to plot the raw data and then run through the basic curve fitting tools to see what gives a good fit and report that. This activity shows that you can get a really great fit if you pay a little attention to the underlying physics. It also allows you to calculate the discharge coefficient from a really simple experiment and a little analysis.

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.

# Take Home Lab – Measuring the mass flow rate from a compressed air can

The mass flow rate of a fluid appears many times in an introductory fluids class (pump power, conservation of mass, momentum, etc.). This take home lab requires students to use at least two different approaches to measure the mass flow rate out of a compressed air can. My experience with this is that some students struggle with finding a second way to do the measurement. This has led to some rather creative, if not physically appropriate, measurement methods.

It is a nice experiment because there are substantial difficulties in taking accurate measurements. For example, it is hard to accurately measure the diameter of the straw connected to the can as it is so small. It is not unusual to have a measurement uncertainty of 50-100% in the diameter leading to an error/uncertainty of up to 400% for the straw area. These errors  can make a substantial difference to the resulting calculated mass flux. There can, therefore, be substantial differences between the two sets of measurements that are still within the bounds of uncertainty. There is also a repeatability problem with this lab that is easily observed and explained if one is paying attention to the data.

As with other take home labs that I use later in the semester, the students are required to do some basic error analysis to explain the differences between their two measurements. This is a very useful complement to their fluids lab class that runs in parallel with the main lecture class. The students need to estimate the uncertainty in each measurement they take and then use that data to estimate the uncertainty in their measured mass flow rate.

As with all the take home labs I will not publish details on specific methods for conducting the tests as I still use them in class and want my students to figure it out on their own.

Instructions to students

Introduction

In this class we have looked at a range of different flow analysis techniques. In this lab you need to use 2 different approaches to estimate the mass flow rate coming out of a compressed air can such as are used for cleaning computer keyboards.

1. Run a series of experiments to establish the mass flow rate out of a compressed air can. You can borrow an air can from me when you are ready to do your testing. You may use an electronic scale and a stopwatch but otherwise only non-lab equipment is to be used without permission. If you would like to use something else you need to check with me.
2. Write a brief report that
1. Is a maximum of 3 pages including photos of you running your experiments.
2. Describes the experiment(s) you used to establish your result including
1. How the test was run
2. What data you collected including estimates of your measurement uncertainty
3. How you performed your calculations including diagrams (with control volumes), equations, and relevant theory
4. A quantitative discussion of the uncertainties in your measurements and calculations including an analysis of the differences between your two sets of measurements.

Due in 2 weeks

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.