Calibration of a Flowmeter

The overall purpose of this experiment is to calibrate and use a flowmeter, which is a bulk flow measuring device. The results of this experiment, as well as the analysis that follows, are meant to give greater insight into how flow rate relates to both manometer deflection and voltage output. They are also meant to help give greater insight into the discharge coefficient and its relationships, which altogether allows us to assess the reliability of the paddlewheel flowmeter used in the lab.

While there are not really any "supplies" used for this experiment, there is a one apparatus that you will need to be familiar with to successfully complete a lab session. Though it may seem complicated at first, it is actually very straightforward! The setup includes two different flowmeters, one being a paddlewheel and the other a hydraulic flowmeter. Both flowmeters are in a pipe that is connected to the ceiling of the laboratory. The setup also includes a weighted tank in the laboratory basement.

Both of the flowmeters are used to help track the flowrate of water, Q, throughout the experiment. However, both of them have to be read differently. The hydraulic flowmeter consists of multiple different parts, with the readable part being the mercury manometer. Though I expect you to know this, you take pressure readings from the manometer by taking the difference between the mercury's height in each of the tubes. The heights of liquid should be allowed to stabilize before readings are taken. If a significant amount of time has passed and the liquid doesn't stabilize, an average should be taken. The paddlewheel flowmeter is much easier to read, as it has a digital display that provides pressure readings directly. For the purpose of this experiment, you don't need to know exactly why or how these work.

Although this also isn't a "supply", LabVIEW software is also used in this experiment to obtain and organize results. Many of the calculations needed for results and analysis post-lab can be calculated easily with this software, rather than being done by hand.

Before starting anything else, make sure that the main discharge valve is closed. The valve is a black wheel at about chest height, located just northeast of the manometer setup. Make sure you turn it until you can no longer hear water flowing. At this point, you should check the manometer to ensure that the mercury levels in both tubes are equal. If they aren't, you should open and close the manometer drain valves until the mercury levels settle. This allows for excess air in the pipes to be released, providing a more accurate measurement.

Start by zeroing the transducer output on the interface box right next to the computer. Open the manometer bleed valve labeled "CAL VALVE", which will reduce pressure in one of the two manometer pipes. While doing this, you should use the LabVIEW software to take readings of transducer output (V) and manometer levels (cm) and to record the obtained results. Obtain at least five separate data points, with the two extreme points being where the bleed valve is fully opened and fully closed. This ensures that a wide range of pressure differentials are included in the calibration. If the maximum voltage exceeds 10V (which doesn't allow it to be correctly read and recorded), come speak to me directly about the matter. My new office is in room 110.

Make sure the Gain Adjust control for the paddlewheel flowmeter is set to 3 turns for P3, and to 6.25 turns for P1 and P4. You should then use the Zero Adjust control to set the paddlewheel flowmeter input to 0. Then, begin opening the discharge valve very slowly. Stop when the valve is completely open or when the maximum allowable manometer deflection is reached. During this process, record both the paddlewheel and differential pressure voltage readings at the moment when the paddlewheel voltage becomes a meaningful nonzero number.

Once the maximum flow rate is reached, record the readings from the paddlewheel flowmeter as well as the manometer. Pay attention to the maximum manometer deflection you can observe. Take a weight-time measurement and record the time-averaged voltages. Repeat these steps for progressively lower flow rates. The rates should be determined based on the manometer deflection, delta h; that is, the flow will be gradually reduced to match certain delta h values. These values are h_max * (0.9)^2, h_max * (0.8)^2, and so on, with the final value being h_max * (0.1)^2. Always wait until the mercury in each tube has settled before proceeding with a trial.

Following the conclusion of the experiment, you should compile and organize your data in a few specific ways in order to allow for easier analysis of experimental results. I've included results from one of my previous labs to give you an example of high-quality data, and to give some insight as to what you should be looking for.

Before doing anything else, you should create two graphs that plot flow rate (Q) as a function of the manometer deflection (delta_h). One of these graphs should use a linear scale, while the other should use a logarithmic scale. Examples of both are shown in the images. Both trendlines (curves) can serve as calibration curves for the flowmeter. As you can see, the linear scale demonstrates an exponential relationship between the two variables while the logarithmic scale makes the data more linear. In fact, the trendline is essentially a straight line. This indicates that a power-law relation of the type Q = K(delta_h)^m could certainly apply. The equation on the top left of the logarithmic scale graph demonstrates this, reading y = 0.0012x^0.7204. In this scenario, Q = y, K = 0.0012, x = delta_h, and m = 0.7204. You will need to determine whether a power-law relation exists for your own data as well.

The next step of analysis is to plot the discharge coefficient Cd vs Reynolds number Re. This should once again be done on a linear and logarithmic scale. It is important to note that Re = v_i*D/v, where v_i is the velocity of fluid in the pipe, D is the pipe diameter, and v is viscosity. For this experiment, viscosity can be obtained by plugging the temperature of the water into LabVIEW. An example of the discharge coefficient vs Reynolds number plot, both in a linear and logarithmic scale, are shown in the pictures. As you can see, the linear scale plot has a linear relationship while the logarithmic scale plot has an exponential relationship. Values can vary of course, but this is roughly how your plots should look.

The next step of proper analysis is to plot a voltage output vs discharge rate graph. When it comes to these types of graphs, rising and falling cutoff flow rates can be a useful tool for analysis. In a system with steadily increasing flow, the rising cutoff flow rate is the flowrate at which the first nonzero quantity of voltage is produced. The falling cutoff flow rate is the exact opposite; in a system with steadily decreasing flow, the falling cutoff flow rate is the flowrate at which the first quantity of 0 voltage is produced. For this particular experiment, there will only be a rising cutoff flow rate since the rate never decreases again to a point where voltage = 0. Since the graph's y-intercept is = 0.0022, the rising cutoff flow rate must be 0.0022m^3/s. Using the formula v = Q/A, where A = (pi/4)*(0.1016m)^2= 0.0081m^2, we can find the rising cutoff and maximum fluid velocities. Plugging in the known Q values, the rising cutoff velocity is 0.27m/s and the maximum velocity is 2.62m/s.

The discharge coefficient varies significantly over the range of Reynolds numbers tested, with Cd values spread well between 0.45 and 0.95. The ideal value of unity is = 1, and the obtained Cd values are generally not close to this value. The average of all of the data points comes between 0.70 and 0.75, which is significantly lower than 1. There are two data points with Cd values over 0.9 (which are considered close to the ideal value of unity, but those points alone are not enough to say that the values as a whole were close to 1. To get better Cd values in the future, accounting for energy lost through head loss could make a significant impact. Another possible solution could be to consider both shear stress and viscosity, which can have a significant impact on non-ideal fluids (real-world fluids).

Based on the data, the paddlewheel flowmeter can be considered only somewhat reliable for this experiment. While the voltage vs discharge rate graph (shown in step 6) has a relatively linear relationship, there are a few data points that are non-negligible outliers. In scenarios where a greater degree of precision is necessary, there are likely many better options than the paddlewheel flowmeter. Interestingly, the more extreme data points seemed to cluster closer to the trendline while the two most significant outliers were more towards the middle of the data. However, higher flowrates in general gave more accurate voltage readings. This is due to the fact that the experimental setup is not an ideal system. Friction is not negligible, and the water must be above a certain flowrate to start spinning the paddlewheel. A flowrate that may record a voltage on an already spinning wheel may not be able to get the wheel to move if it starts out stationary. This issue simply doesn't come up at higher flowrates, and this discrepancy gives high flowrates the edge in terms of accuracy.