530.327 - Introduction to Fluid Mechanics - Su. Second experiment: Flow in pipes
. The (individual) reports for this lab will be due Tuesday, November 17.
530.327 - Introduction to Fluid Mechanics - Su Second experiment: Flow in pipes The (individual) reports for this lab will be due Tuesday, November 17. The reports are to conform to the guidelines already distributed for Lab 1.
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Background and objectives.
We have been discussing in class how viscous fluid flow in pipes is subject to energy losses through frictional effects, and also because of area changes or bends in the piping, or the presence of fittings such as valves, etc. In this lab we will make explicit measurements of these energy loss effects in the flow of water through a pipe flow network. The energy loss between two points 1 and 2 in a pipe flow is described by the following equation: 2 2 V1 V2 p2 p1 + α1 + gz1 − + α2 + gz2 = hl (1) ρ 2 ρ 2 where V 1 and V 2 are averaged velocities, α1 and α2 are correction factors that allow the flow kinetic energy to be written in terms of the averaged velocities, and hl is the total energy loss per unit mass, also called the head loss. The head loss over a given length of straight pipe can be predicted given knowledge of the pipe diameter and wall roughness, the fluid properties, and the flow rate. For a pipe section with inner diameter D and length L, we have 2
LV hl = f D 2
(2)
where f is the friction factor, which depends on the Reynolds number of the flow, and, if the flow is turbulent, also on the roughness of the pipe walls. Appropriate values of f for different parameters are given by the Moody diagram (on the inside back cover of the text). In the laminar flow regime (low Reynolds number), f is known analytically; in turbulent flow (high Reynolds number), f must be found from experiments. The head loss through bends, valves etc. is often tabulated in terms of equivalent lengths of straight pipe. For example, from Table 8.4 in the text, an ideal, fully open ball valve imposes a pressure loss equivalent to a section of pipe with length equal to three times its diameter. In this experiment, you will measure the energy loss, in terms of the pressure drop, across different lengths of different pipes, and across different bends and valves. A major objective is to confirm that you observe the appropriate qualitative effects, for example that the pressure drop is proportional to pipe length. You will also use your results in more quantitative analyses (discussed in §4). 2
Experimental apparatus.
The pipe network apparatus is located in Krieger B-4 (down the hall from the wind tunnel lab). The room is rather claustrophobic, so it would be an excellent idea to minimize your time in there by familiarizing yourself with the lab procedure in advance. A photograph of the apparatus is given in Fig. 1, while a schematic of the flow network is given in Fig. 2. The pump on the left side of the facility drives the water flow through the network. The system of butterfly valves (shown as rectangles in the schematic) allows the flow to be directed through each of the horizontal pipes separately. The schematic indicates the sizes (inner diameters) of the different lengths of pipe.
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supply tank
manometer
flowmeter flow control valve pump
Figure 1: Photograph of the pipe flow network.
These are nominal sizes only, that correspond to standards that have developed who-knows-how.1 For accuracy we will measure these pipes explicitly. The fifth row of pipe consists of nominal 1”, schedule 40 cast iron piping and is equipped with a butterfly valve, a gate valve, a globe valve, two bends and a strainer, to allow us to assess the effects of these fittings. The water returns to the supply reservoir through the bottom pipe. A valve just upstream of the outlet to the reservoir allows control of the total flow rate. (Depictions of the different valve types are given in Fig. 3). A variable area flowmeter (‘rotameter’) in the bottom pipe measures the volumetric flow rate of the water. The rotameter (shown in Fig. 4a) consists of a weight (the ‘float’), which sits in a vertical tube that has a larger cross-sectional area at the top than the bottom. The flow enters the bottom of the rotameter and exits the top. The drag on the float, which acts vertically upward (the positive z-direction), is dependent on the fluid velocity in the rotameter at the position of the float. You will observe that when then pump is off and the fluid isn’t flowing, the float sits at the bottom of the rotameter (which is the situation depicted in Fig. 4a). When you turn the pump on, the upward drag on the float initially overcomes the weight of the float, and it rises. Since the tube area increases in the upward direction, but the flow rate has to be constant, the flow velocity, and thus the drag, decrease as the float rises. At some height, the drag balances the weight of the float, and by noting this position, we can determine the volume flow rate of the fluid. Pressure taps are indicated by ‘X’s on the schematic. To measure the pressure difference between any two pressure taps, flexible tubes can be used to connect the taps to opposite sides of the mercury manometer. Figure 4b shows a close-up view of the manometer. One end of each flexible tube is more-or-less permanently connected to one side of the manometer, while there is a quick-connect fitting on the free end, so the tube can simply be plugged into a given pressure tap. With the tubing in place, a screw-driven valve at each pressure tap can be opened so that the static pressure of the water in the pipe can be read. Figure 4c shows a detailed view of the valves and quick-connect fittings. Notes on the pressure measurements. If you’re not careful with your procedures in this lab, there’s a good chance that you’ll get pretty wet, and a non-zero chance that you’ll spill mercury 1
Generally, in the plumbing trade, ‘pipe’ refers only to cast iron and steel. Our cast iron pipe is 3/4”, schedule 40 (referring to the wall thickness). The term ‘tube’ usually refers to copper and brass. Our copper tubes are 3/4”, 1/2”and 1/4”, type L ( again referring to the wall thickness). In this write-up we’ll just call all of these ‘pipes’ for simplicity.
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pressure taps
butterfly valve
3/4" cast iron pipe
3/4" copper pipe
1/2" copper pipe
1/4" copper pipe gate valve
bends
globe valve
strainer
to supply tank flow control valve flow direction flowmeter
from supply tank pump
Figure 2: Schematic of the pipe flow network. The rectangles shown represent butterfly valves (which act as on-off valves), and the ‘X’s are pressure tap locations.
somewhere it doesn’t belong. It is important to understand how the pressure measurement works. Figure 5 illustrates the connection between the flow in the pipes and the manometer. When a given flexible tube is connected to a pressure tap, and the valve on the taps is open, the water inside the flexible tube right at the tap essentially feels the higher-than-atmospheric pressure inside the pipe. Using the static pressure relation (since the water in the flexible tube isn’t moving), we can then determine the pressure that pertains inside the manometer. With a little thought, you can see how it is important that water completely fills the tubes right up to the mercury in the manometer. If there’s an air bubble in the tube, the relation of pressure to change in height is thrown off. In case air bubbles of noticeable size (say, an inch long or so) appear in one of your flexible tubes, we have a way of getting rid of them. Figure 4c is a close-up view of the connections at the top of the manometer. The two relief valves allow you to vent the flexible tubes to atmospheric
butterfly valve
gate valve
globe valve
Figure 3: Schematics of the butterfly, gate and globe valves.
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(a)
(b)
relief valves
to pressure tap to pressure tap
manometer
(c) quick-connect fitting
screw-driven valve
flexible tubing
Figure 4: Close-up views of: (a) the rotameter; (b) the mercury manometer; (c) the valves on the pressure taps, and the quick-connect fittings that connect the valves to the flexible tubing.
pressure. When you do this, the bubbles (and a small amount of water) will slowly make their way out of the lines. If you need to do this venting, always open both valves in tandem, carefully – otherwise, you can generate pressure differences that make it possible for mercury to work into the flexible tubes or, worse yet, out through the vent lines onto the floor. Since the water in the tubes is at elevated pressure during the measurements, we need to be careful that it doesn’t spew out when you disconnect the flexible tubes. To disconnect the either one of the tubes, follow this procedure: • Close the valves on both pressure taps to which the tubes are connected. • At this point the water in the lines is still at elevated pressure. Relieve the pressure by opening and closing both valves above the manometer. Again, always relieve the pressure in both lines simultaneously to avoid pressure differences that can cause mercury to get out of the manometer. • Now you can disconnect either of the flexible tubes, and only minimal water should leak out.
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flow direction
pressure taps
flexible tubing
manometer
mercury
Figure 5: The connection, by means of the flexible tubing, between the flow in the pipes and the mercury manometer.
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Experimental procedure.
Preliminaries. (Target time: 5 minutes.) 1. The first thing to do is to measure the inner diameters of the various pipes. Short samples of the different pipe sizes are hanging by the manometer. Measure the inner diameters using the calipers provided. We don’t have a sample of the 3/4” cast iron pipe - for that one, use a value for the inner diameter of 0.824”. 2. Next, measure the distances between the pressure taps along the top four pipes using the yardstick provided. 3. Before turning anything on, you should also satisfy yourself that you understand how to control the flow in the apparatus using the array of butterfly valves, and also that you know in which direction the flow is moving at all locations. NOTE: In order to avoid damaging the pump, make sure that there is an open flow passage at all times. For example, to investigate a different flow path, open that path first, then close the old one. 4. Now open the appropriate valves to allow flow through the top tube. Turn on the pump. Try varying the flow rate by adjusting the valve just upstream of the outlet in the bottom pipe, and noting how the rotameter float comes to rest at different positions. To read the rotameter, match the position of the raised flange around the float with the scales printed on the outside of the tube. Measurements. (Target time: 45 minutes.) 1. Set the butterfly valves to allow flow through the top pipe only. 2. Set the flow rate at 15 gallons per minute. 3. Measure the pressure difference between (A) the first two taps, (B) the second and third taps, and (C) the first and third taps. 4. Repeat step 3, for a flow rate of 10 gpm. 5. Set the valves to allow flow through the second pipe from the top only. Repeat step 3 for flow rates of 15 gpm and 10 gpm. 6. Set the valves to allow flow through the third pipe only. Repeat step 3 for flow rates of 12 gpm and 6 gpm.
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7. (Do this step only if time permits.) Set the valves to allow flow through the fourth pipe only. Repeat step 3 for flow rates 4 gpm and 2 gpm. 8. Now make sure the different valves on the fifth pipe are all in the fully open position. Set the butterfly valves to allow flow through this row of valves. Set the flow rate to 15 gpm. 9. Measure the pressure difference across each of the following fittings on the fifth pipe: the butterfly valve, the gate valve, and the globe valve. 10. Now consider the 45◦ bends on the fifth pipe. Measure the pressure difference between two taps that span both bends. Also measure the total length of ‘clean’ pipe (i.e. no fittings) between the two pressure taps you’re using, and finally measure the height difference between the taps. 11. When finished, shut down the pump.
Report.
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Guidelines for the formatting and style requirements of the formal report will again follow the handout distributed for Lab 1. The report should include the following analyses. i) Do your data verify the proportionality between pressure drop in a pipe, and the pipe length? Think of the best way to display the data. ii) The theoretical pressure drop on an uninterrupted length of pipe is given by 1 2L f, ∆p = ρV 2 D
(3)
where V is the average flow velocity, L and D are the pipe length and diameter, and f is the friction factor as tabulated in the Moody diagram. The proper friction factor will depend on the Reynolds number, and the roughness of the pipe. You can find V , and thus the Reynolds number, from the flow rates measured by the rotameter. Determine f for each of your ∆p measurements along the uninterrupted pipes, and obtain an averaged f value at each Reynolds number for each pipe. iii) For the averaged f data from ii), use the Moody diagram (not the so-called Colebrook relation in the text) to estimate the relative roughness, e/D, for each measurement. Can you determine the appropriate e/D for each pipe? Do the results agree with Table 8.1 in the text2 ? It may happen that you’ll find that you can’t use the Moody diagram to get a plausible value of e/D. (This could be true even if you perform the experiment exactly correctly.) If this happens to you, explain what might cause this to happen, and whether you think your data are reasonable. Note that you can comment on the comparative roughnesses between different pipe materials even if the e/D values don’t match the tabulated ones. iv) Now look at the pressure drops along the valves and bends (also called ‘elbows’). The relevant equation is ∆p =
1 2 Le ρV f, 2 D
(4)
where the ratio Le /D, the normalized equivalent length, is tabulated for valves and elbows in Table 8.4 in the text3 . To get the appropriate value for f , the friction factor, use the Moody 2 3
The ‘drawn tubing’ mentioned in the table refers to copper or brass pipes. For butterfly valves, which aren’t covered in the table, use Le /D = 45.
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diagram, with the roughness figure for cast iron from Table 8.1. Determine Le /D for each of your measured pressure drops. For the two 45◦ bends, also take into account the effect of the length of clean pipe, and of the elevation change between the taps, when determining Le /D. (Explain how you account for these effects.) Do these values agree with Table 8.4? In the event your values don’t agree with the tabulated values, explain why this might be the case, and whether you think your data are reasonable.
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