SOLUTION: CONSERVATION OF ENERGY PRE LAB QUESTION Q1. Pressure head will vary along the pipe. Explain why this is the case. The pressure head varies along the pipe because of: (a) The interchange between pressure and kinetic energy

The energy in the flow consists of potential energy, kinetic energy and pressure energy. In this case, the potential energy is constant because the pipe is horizontal. When, the cross-sectional area of the pipe changes, so too does the flow velocity (and therefore the kinetic energy). Therefore, changes in velocity result in changes in pressure energy in order to satisfy conservation of energy. For example, at a contraction, the flow velocity increases, meaning that the pressure head must decrease. (b) Energy losses due to friction Friction from the pipe walls causes a head loss (hL) this loss is seen in the pressure head. It cannot cause a loss in velocity head, as the flow rate through the system is a constant (by conservation of mass). (c) Losses to turbulence at points where the pipe geometry changes At points where the geometry changes, mechanical energy is lost to turbulence. Again, this loss is seen in the pressure head.

(1.5 marks) Pipe Configuration

Table 1. Geometry of Experimental Apparatus

A B C D E

F

G

Example of Experimental Results (NOTE TO MARKERS: Ideally, the students should have a wider range of flow rates than shown here).

Table 2. Example of Experimental Data Run 1 Run 2 Run 3 Run 4

Measuring Tank Volume (L) Distance from Upstream Head to

Piezometer (mm)

50 50 50 50

Discharge Time (s) 16 25 18 21

Upstream Head - Manometer 1 (mm) 0 990 1000 990 995

Piezometer 2 (mm) 340 960 985 965 975 Piezometer 3 (mm) 540 960 985 965 975 Piezometer 4 (mm) 740 960 985 965 975 Piezometer 5 (mm) 940 955 985 960 970 Piezometer 6 (mm) 1140 955 980 960 970 Piezometer 7 (mm) 1340 775 900 815 850 Piezometer 8 (mm) 1540 750 890 795 835 Piezometer 9 (mm) 1740 730 880 775 820 Piezometer 10 (mm) 1940 710 870 760 810 Piezometer 11 (mm) 2140 830 925 860 885 Piezometer 12 (mm) 2340 850 935 875 905 Piezometer 13 (mm) 2540 860 935 880 905 Piezometer 14 (mm) 2600 470 760 560 650 Piezometer 15 (mm) 2800 615 825 680 745 Piezometer 16 (mm) 3000 615 825 680 750 Piezometer 17 (mm) 3200 615 825 680 750 Piezometer 18 (mm) 3400 615 825 680 750

Q = U x A (1) Where, A = pi x D2

4

Cross Section

Description of Cross Section

Distance from Upstream Head

(mm) Inside

Diameter (mm)

Cross-sectional

Area (mm2) Cross-

sectional Area (m2)

A Constant 0 - 1140 74.2 4324.1195 0.0043241 B Contraction 1140 - 1340 - - - C Extended Venturi 1340 - 1940 43.9 1513.6272 0.0015136 D Expansion 1940 - 2140 - - - E Constant 2140 - 2540 74.2 4324.1195 0.0043241 F Orifice Plate 2540 - 2600 44 1520.5308 0.0015205 G Constant 2600 - 3400 74.2 4324.1195 0.0043241

Table 3. Calculation of Velocities

Run 1 Run 2 Run 3 Run 4 Cross-

sectional Area, A

(m2)

Measuring Tank Volume (L) 50 50 50 50 Measuring Tank Volume (m3) 0.05 0.05 0.05 0.05

Discharge Time (s) 16 25 18 21 Flow Rate, Q (m3/s) 0.003125 0.002 0.002778 0.002381

Velocity Section A, UA (m/s) 0.72269 0.462522 0.642392 0.550621 0.004324 Velocity Section C, UC (m/s) 2.064577 1.321329 1.83518 1.573011 0.001514 Velocity Section E, UE (m/s) 0.72269 0.462522 0.642392 0.550621 0.004324 Velocity Section F, UF (m/s) 2.055203 1.31533 1.826847 1.565869 0.001521 Velocity Section G, UG (m/s) 0.72269 0.462522 0.642392 0.550621 0.004324

QUESTIONS Q2. Why is it important to keep the constant head tank full? It is important to keep the constant head tank full so that the water flows through the pipe at a constant flow rate. The force of the water in the constant head tank governs the pressure at the main supply valve that determines the flow rate of the water. Thus, if the level of water in the tank rises or falls, then the flow rate of the water through the pipe will not be constant. Therefore, it is important to keep the constant head tank at an overflowing level for the duration of the experiment in order to ensure that the flow rate of the water through the pipe is constant.

(1 mark) Q3. Plot the hydraulic grade lines for the four discharges. What does this plot tell you about the interchange of different types of energy as the water flows through the different configurations? In Figure 1, hydraulic grade lines are plotted from the data in Table 2.

Hydraulic Grade Line

0

200

400

600

800

1000

1200

0 500 1000 1500 2000 2500 3000

Distance Along Pipe Length (mm)

Hyd

rau

lic H

ead

(m

m)

Run 1 Run 4 Run 3 Run 2

Figure 1. Hydraulic Grade Line for Each Run (1.5 marks)

As the water flows into contractions (e.g. 6-7 or 13-14), the hydraulic head drops dramatically due to the increase in velocity head (and therefore the drop in pressure head). Conversely, at expansions (e.g. 10-11), the hydraulic head increases dramatically.

(1.5 marks) Q4. Plot the total energy lines for the four discharges. Compare the slopes of the total energy line in the larger and smaller pipes. Why are they different? The total energy lines are presented in Figure 2 (data in Table 4).

Table 4. Total Head Calculations

Piezometer Number

Hydraulic Head (mm) Velocities (m/s) Run 1 Run 2 Run 3 Run 4 Run 1 Run 2 Run 3 Run 4

1 990 1000 990 995 0.72269 0.462522 0.642392 0.550621 2 960 985 965 975 0.72269 0.462522 0.642392 0.550621 3 960 985 965 975 0.72269 0.462522 0.642392 0.550621 4 960 985 965 975 0.72269 0.462522 0.642392 0.550621 5 955 985 960 970 0.72269 0.462522 0.642392 0.550621 6 955 980 960 970 0.72269 0.462522 0.642392 0.550621 7 775 900 815 850 2.064577 1.321329 1.83518 1.573011 8 750 890 795 835 2.064577 1.321329 1.83518 1.573011 9 730 880 775 820 2.064577 1.321329 1.83518 1.573011 10 710 870 760 810 2.064577 1.321329 1.83518 1.573011 11 830 925 860 885 0.72269 0.462522 0.642392 0.550621 12 850 935 875 905 0.72269 0.462522 0.642392 0.550621 13 860 935 880 905 0.72269 0.462522 0.642392 0.550621 14 470 760 560 650 2.055203 1.31533 1.826847 1.565869 15 615 825 680 745 0.72269 0.462522 0.642392 0.550621 16 615 825 680 750 0.72269 0.462522 0.642392 0.550621 17 615 825 680 750 0.72269 0.462522 0.642392 0.550621 18 615 825 680 750 0.72269 0.462522 0.642392 0.550621

Piezometer Number

Velocity Head (mm) Total Head (mm) Run 1 Run 2 Run 3 Run 4 Run 1 Run 2 Run 3 Run 4

0 26.61985 10.90349 21.03297 15.45279 1016.62 1010.903 1011.033 1010.453 340 26.61985 10.90349 21.03297 15.45279 986.6199 995.9035 986.033 990.4528 540 26.61985 10.90349 21.03297 15.45279 986.6199 995.9035 986.033 990.4528 740 26.61985 10.90349 21.03297 15.45279 986.6199 995.9035 986.033 990.4528 940 26.61985 10.90349 21.03297 15.45279 981.6199 995.9035 981.033 985.4528 1140 26.61985 10.90349 21.03297 15.45279 981.6199 990.9035 981.033 985.4528 1340 217.2517 88.9863 171.6557 126.1144 992.2517 988.9863 986.6557 976.1144 1540 217.2517 88.9863 171.6557 126.1144 967.2517 978.9863 966.6557 961.1144 1740 217.2517 88.9863 171.6557 126.1144 947.2517 968.9863 946.6557 946.1144 1940 217.2517 88.9863 171.6557 126.1144 927.2517 958.9863 931.6557 936.1144 2140 26.61985 10.90349 21.03297 15.45279 856.6199 935.9035 881.033 900.4528 2340 26.61985 10.90349 21.03297 15.45279 876.6199 945.9035 896.033 920.4528 2540 26.61985 10.90349 21.03297 15.45279 886.6199 945.9035 901.033 920.4528 2600 215.2834 88.18009 170.1005 124.9718 685.2834 848.1801 730.1005 774.9718 2800 26.61985 10.90349 21.03297 15.45279 641.6199 835.9035 701.033 760.4528 3000 26.61985 10.90349 21.03297 15.45279 641.6199 835.9035 701.033 765.4528 3200 26.61985 10.90349 21.03297 15.45279 641.6199 835.9035 701.033 765.4528 3400 26.61985 10.90349 21.03297 15.45279 641.6199 835.9035 701.033 765.4528

Total Energy Lines

0

200

400

600

800

1000

1200

0 500 1000 1500 2000 2500 3000

Distance Along Pipe Length (mm)

To

tal

Hea

d (

mm

)

Run 1 Run 2 Run 3 Run 4

Figure 2. Total Energy Line for Each Run (1.5 marks)

The slope of the total energy line is greater in the smaller pipe than in the larger pipe. From the Darcy-Weisbach equation, it can be seen that the slope of the total energy line is given by

gU

Df

Lh

dxdh L

2

2

=

= .

In the smaller pipe, (i) D is lower and (ii) the velocity head is greater. These both contribute to a greater slope in the smaller pipe (ignoring small differences in friction factor).

(1.5 marks) BONUS MARK: Does your total energy line increase anywhere? Why is/isn't this possible? Suggest a reason why your measurements indicate a rise in total energy along a certain portion of the pipe. The total energy line should not increase in the absence of pumps. It should only decrease, due to the effects of pipe friction and losses at the Venturi & orifice. However, a slight increase in the total head is observed between 11-13 (after the Venturi expansion). This is due to flow separation; the flow exits the contraction as a high-velocity jet, without encompassing the entire cross-sectional area of the pipe. That is, the effective area of the pipe there is less than the area of the larger pipe (A). So, by estimating the velocity from Q/A, we underestimate the velocity head at 11 at 12. This leads to an apparent increase from 11-13.

(1 mark)

Q5. Using the Darcy-Weisbach equation, determine the friction factor f of the smaller pipe (stations 7-10) for each run. Also, calculate the Reynolds number in the smaller pipe for each run. Plot your values on a Moody diagram and use them to obtain an

estimate for the roughness () of the smaller pipe. The friction factor is found using the Darcy Weisbach equation, using the hydraulic head loss (which equals the total head loss) between piezometers 7 and 10. The calculation and results for each run are shown in Table 5.

Table 5. Calculation of Friction Factor Using Head Loss

Run 1 Run 2 Run 3 Run 4 Hydraulic Head at Piezometer 7

(mm) 775 900 815 850 Hydraulic Head at Piezometer 10

(mm) 710 870 760 810 Change in Hydraulic Head between

7 and 10 (mm) 65 30 55 40 Head Loss between 7 and 10, hL (m) 0.065 0.03 0.055 0.04

Diameter, D (m) 0.0439 0.0439 0.0439 0.0439 Length, L (m) 0.6 0.6 0.6 0.6

Velocity, U (m/s) 2.064577 1.321329 1.83518 1.573011 Friction Factor, f 0.021891 0.024667 0.023443 0.023206

Table 6. Calculation of Reynolds Numbers

Run 1 Run 2 Run 3 Run 4 Diameter, D (m) 0.0439 0.0439 0.0439 0.0439 Velocity, U (m/s) 2.064577 1.321329 1.83518 1.573011

Reynolds Number, Re 90634.93 58006.36 80564.39 69055.19

(2 marks) If the values of f and Re are plotted on the Moody diagram, they fall (roughly) on the line for

which /D = 0.001. That is 0.001 x 44 mm = 0.04 mm (very roughly). [NOTE TO MARKERS: You will get a wide range of results here]

(1 mark) Q6. Determine the discharge coefficient for the Venturi meter (CV) using the pressure

drop between stations 6 and 7 for PHV. Is CV a true constant, or does it vary with Reynolds number?

Table 9. Calculation of Coefficients of Discharge of Venturi Meter

Run 1 Run 2 Run 3 Run 4 Flow Rate, Q (m3/s) 0.003125 0.002 0.002778 0.002381

Throat Diameter, DT (m) 0.0439 0.0439 0.0439 0.0439 Throat Cross-sectional Area, AT (m2) 0.001514 0.001514 0.001514 0.001514

Original Diameter of Pipe, DP (m) 0.0742 0.0742 0.0742 0.0742 Piezometer 6 (mm) 955 980 960 970 Piezometer 7 (mm) 775 900 815 850

Head Loss between 6 and 7, hV (m) 0.18 0.08 0.145 0.12 Venturi Discharge Coefficient, CV 1.029109 0.987945 1.019204 0.960302

(1 mark)

CV is not a true constant. It increases with decreasing Reynolds number (this is more apparent when the students have a wide range of Reynolds numbers)

(0.5 marks) Q6. Determine the head loss coefficients KV and KO using the total head loss between stations 6 and 13 for the Venturi, and stations 13 and 14 for the orifice plate. Which is greater and why?

Table 11. Calculation of Loss Coefficients KV

Run 1 Run 2 Run 3 Run 4 Velocity Section C, uT (m/s) 2.064577 1.321329 1.83518 1.573011

Piezometer 6 (mm) 955 980 960 970 Piezometer 13 (mm) 860 935 880 905

Head Loss between 6 and 13, HV (m) 0.095 0.045 0.08 0.065 Venturi Loss Coefficient, KV 0.437281 0.505696 0.466049 0.515405

Table 12. Calculation of Loss Coefficients KO

Run 1 Run 2 Run 3 Run 4 Velocity Section F, uO (m/s) 2.055203 1.31533 1.826847 1.565869

Piezometer 13 (mm) 860 935 880 905 Piezometer 14 (mm) 470 760 560 650

Head Loss between 13 and 14, HO (m) 0.39 0.175 0.32 0.255 Orifice Loss Coefficient, KO 1.811565 1.984575 1.881241 2.040461

(1 mark)

The loss coefficient is much greater for the Orifice than for the Venturi. This is because the change in geometry at the Orifice is much more abrupt, meaning that more energy will be lost to turbulence there.

(1 mark)