# 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
• Tape
• Index Card Version
• 1 straw
• 1 3”x5” index card
• 1 straight pin
• 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 experiment – flow rate from a hose

I typically use this lab as the 3rd or 4th lab of the semester. The lab is simple enough, they have to use two different methods to measure the flow rate out of a hose. They can use kinematics, conservation of volume, or even momentum (though this is a little more tricky). This is one of the labs where I ask my students to use their estimates of measurement uncertainty, through some basic linear error analysis, to estimate their calculated flow rate uncertainty. If their 2 measurements are not the same (within the bounds of uncertainty they calculated) they have to discuss why not. It is a great experiment for discussing errors because, even though the measurements are simple to make, they often have significant percentage errors that propagate into very large percentage error in their calculated flow rates. For example, measuring the diameter of the hose outlet can be tricky and a 10% error in the measurement becomes a 20% error in the hose area. There are also challenges with repeatability of the experiments as it is hard to get the hose to have the same flow rate each time you turn it on. I do not explicitly ask them to discuss repeatability but rather I discuss it when I return the graded reports and ask them to think about repeatability as part of their next take home lab.

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

Introduction

In this class we have looked at a range of different flow analysis techniques (conservation of mass, kinematics, Bernoulli, momentum, etc.). In this 3rd lab you need to use 2 different approaches to calculate the flow rate from a garden hose.

1. Run a series of experiments to establish the flow rate our of the flow from a regular garden hose. There is a hose in the fluids lab that you could use. You can use buckets, measuring tapes, and stopwatches. If you wish to use anything other than that you will need to check with me first. You are not to use laboratory flow rate measurement devices such as the venturi meter.
2. Write a brief report (3 page max) that:
1. Includes photos of you running your 2 experiments.
2. Describes how the test was run
3. Includes diagrams showing what you measured
4. Presents the theory and equations you used in your calculations
5. Lists what data was collected and estimates of your measurement error (in a table)
6. Error analysis (see class notes) to estimate your uncertainty in your calculation of flow rate.
7. A table listing the two calculated flow rates and your uncertainty estimation.
8. If the two measurements to not agree (within the error range you calculated) then discuss why not.

Due date 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.