Todorov - Elementary Laboratory Procedures Partial Report

Figure 1: A Bourdon gauge and a differential manometer, two devices used to measure pressure

Greetings. The following are instructions on how to conduct elementary lab procedures when it comes to fluid mechanics. Pay close attention.

Figure 2: The setup on which you will test the skills described in this instructable

The supplies needed to design the setup shown above for these measurements are as follows:

1. Water supply
2. Piping
3. Balance beam with a scale and weight
4. Tank with drain system and valve
5. 2 Bourdon Gauges
6. Differential manometer
7. Ruler
8. Timer
9. Data recording device

Figure 3: How to read a differential manometer

Figure 4: A Bourdon gauge

Figure 5: Diagram of a balance beam

There are several things that you will need to know how to measure when it comes to fluids. A few examples are mass flow rate, pressure, and the change in pressure throughout the system. Many physical constants are required for meaningful measurements, such as fluid density (ρ=1000kg/m^3), and gravitational acceleration (g=9.8m/s^2). We'll start with mass flow rate.

This measurement will require the balance beam and weight, tank, and timer. Open the drain valve and water source all the way until it reaches a steady flow. This is 100% flow rate. Mass flow rate can be defined as the change in mass in the tank over the change in time. Overbalance the scale by a bit, sliding the cursor until the scale arm hits the bottom stop. With a timer ready, close the drain and start the timer when the beam rises above the halfway point between the stops. To measure 100 pounds of water, select a 1 pound weight and place it on the balance pan. Wait until the balance arm passes the balance point again, at which point stop the timer. The time it displays is how long the system took to flow 100 pounds of water. Because we are an engineering firm, we use the metric system so convert 100lbs to kg, by dividing by 2.2. Technically, we first have to convert 100 pounds of force to mass, but since we're on Earth it is just 100 pounds mass. The change in mass over change in time is 45.35kg divided by the seconds recorded from the timer. This is how to find mass flow rate in kg/s, which is the only sensible system of units.

Secondly, you will need to measure pressure and determine the change in pressure throughout a system. This particular facility is equipped with two Bourdon gauges which each read a pressure at their location, which allows us to make one comparison throughout the system. The differential manometer is connected to two different points in the system and requires more math. Let's start with reading a Bourdon gauge.

The gauge reads pressure at the center of the gauge, which is some height above the pipe it is connected to. This creates a difference in pressure, since there is more weight and therefore more force on the water in the pipe. This can be defined as hydrostatic pressure, which is calculated by multiplying the water density by gravitational acceleration by the height of the gauge above the pipe. In the diagram, the pressure at A (pA), is the reading given at p1 plus ρgh where h is the difference in height, a1-aA (.810m-.613m). This can be repeated with p2 and b2-bB (2.376m-2.183m), to find pB.

For a differential manometer, the device has a U-tube, usually with mercury and ruled measurements to give pressure in mmHg. The manometer connects to two points, in this case A and B, where you will try to determine the pressure. The right and left points of interface between mercury and water in the manometer have heights R and L, respectively. The pressure is equal on either side of the manometer at a given height, but you will use height R for simplicity. This pressure, pR, is equal to the pressure at A, minus the hydrostatic pressure of the water column of height R-aA. A simplification of hydrostatic pressure is γ=ρg. The ratio of γ(mercury) to γ(water) is 13.55:1. Similarly, pR is equal to pB plus the hydrostatic pressure of mercury of height L-R plus the hydrostatic pressure of water of height bB-L. Doing some math and cancelling the pR term, the difference in pressure from A to B can be calculated by:

pA-pB = γ(water)*(bB-aA) + (γ(mercury)-γ(water))*(L-R)

Ultimately this means it is a function of L and R, since all other variables are constant in the system. As such, once you have done the math you can determine the pressure difference from the height difference on either side of the manometer. This pressure has a quadratic relationship with flow rate, so an 80% flow rate would have 64% of the height difference in mercury.

Collect five samples of data first using the described method, aiming for 20% intervals of flow rate. Remember that this can be determined by squaring the % of the maximum flow rate to find the expected percent of the maximum height difference, so adjust the flow rate until the height stabilizes at 64%, 36%, 16%, 4%. Repeat the mass flow measurement once the system stabilizes each time too, and record all the data. To verify your results, the mass flow rates should linearly decrease by 20% of the maximum each time, and plotting pressure against flow rate should yield the right side of a quadratic curve. Sources of error could come from human delay when timing during the mass flow rate portion, so multiple recorders is ideal to obtain a consistent reading. Rounding, or incorrect physical constants like density could also cause errors in the math, so double check your work. At the end, you will evaluate the precision of your measurements and aim to fit them within the 10% precision dictated by the company.

This figure plots the difference in pressure at points B and A, given at 5 different flow rates. The x-axis is the measurements made using the differential manometer method, while the y-axis plots the data from the Bourdon gauges. The line across the entire figure with orange points is a line of slope y=x, to show how close the measurements from each method are to each other. The closer to y=x the slope of our data is, the better we can say that we can provide comparable data from either method.

This figure establishes a trend between the change in pressure across the setup and volumetric flow rate. The different points are taken at 5 even increments from the maximum volumetric flow rate. The relationship is not linear, and seems accurately quadratic with the difference in pressure increasing proportional to the square of flow rate. The R-squared value does not identify a method that was significantly more reliable for pressure measurement, as the difference is less than a percent. However, it seems that at the higher values of the graph the trendlines are closer to the data points, which would make sense since the margin of error is much less significant with larger values. Although there isn't a significant difference between each method, the needle on a Bourdon gauge oscillates whereas the fluid in a manometer is quite stable, which can make it difficult to obtain an accurate reading. Try to see the maximum swing in either direction when reading a gauge to estimate the average pressure it is reading.

To determine the precision of your findings, use the equation e shown in Figure 8. Calculate Qs and Ql using the volumetric flow rate from your shortest and longest time measurements to. As you can see, the precision of this particular trial is 10.5%, but when we take it to 2 significant figures it reaches 11%. This is not within the typical acceptable range, and as such you must aim to do better. These skills take time to learn, so practice and do not be afraid to revisit this instructable when questions come up. Best of luck!