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**

- a bucket with thin (approximately) vertical sides
- a yard stick
- a stopwatch
- 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**

- 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.
- 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.
- Place the yard stick vertically in the bucket and measure the initial depth.
- 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.
- Stop recording when the depth of water over the orifice is less than 2 orifice diameters.
- 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 *h *(see diagram).

Denote the orifice area as *A*_{o} and the bucket cross sectional area as *A*_{b} (*>>A*_{o}) 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+Q*_{out}-Q_{in}=0.

The volume of water in the bucket is *V=A*_{b}h and there is no inflow so this can be simplified to

*dh/dt=-Q*_{2}/A_{b}=-(A_{o}/A_{b})u_{2}

Write Bernoulli’s equation

*p*_{1}/γ +u_{1}^{2}/2g+z_{1}= p_{2}/γ +u_{2}^{2}/2g+z_{2.}

Assuming the free surface and free jet pressures are zero and that *u*_{1}≈0 the equation simplifies to

*h=u*_{2}^{2}/2g or *u*_{2}=(2gh)^{1/2}

Substituting into conservation of volume leads to

*dh/dt=-(A*_{o}/A_{b})( 2gh)^{1/2 }(1)

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

*h*^{1/2}– H^{1/2}=-(A_{o}(2g)^{1/2}/2A_{b})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 (*C*_{d}<1). Therefore (2) can be written as

*h*^{1/2}=H^{1/2}– (C_{d} A_{o}(2g)^{1/2}/2A_{b})t. (3)

**Data analysis**

The goal of the data analysis is to establish the value of *C*_{d} 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 *h*^{1/2} is a linear function of time. Therefore, if you plot *h*^{1/2} 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=- C*_{d} (A_{o}(2g)^{1/2}/2A_{b}) or C_{d}=-2mA_{b}/( A_{o}(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.