Abstract

The main aim of this experiment was to show how the friction factor varies with Reynolds number by manipulating the flow rate of the fluid in a pipe. The main principle used in the experiment was the Bernoulli’s equation, taking major head losses into account. These major head losses were normally due to wall friction in the pipe and viscous forces between layers in a fluid.

Generally the results obtain from the experiment do agree with the theoretical prediction and the discrepancy of the points as shown in the moody chart was relatively small. In the laminar flow and turbulent flow region where the frictional factor values are similar to the values in the moody chart.

The relative roughness value obtained for the pipe was 0.0008, based on the points plotted on the moody chart. There were a lot of sources of errors in this experiment such as systematic errors, human errors, equipment limitations and some other factors that were not taken into consideration during analysis of the data.

The main errors involved human and equipment limitations, which caused the data points obtained to vary from the expected theoretical values. The factors and assumptions made were that the minor head losses were negligible, which might not have been the case. The density and viscosity values obtained from the data sheet might have also introduced error into the results.

Introduction

Theory and Principles

In this experiment, the theory and principles in used is the loss of energy and total head of fluid due to frictional resistance of real or viscous fluid. In fully developed straight pipe flow, energy loss or head losses occurs due to wall friction. These losses are usually known as the major head losses (hLmajor).

Other than major head losses, minor head losses (hLminor) too occur due to losses due to bends, contractions, valves, and others. To calculate major head loss, the formula in used is

Where is known as the friction factor

Is the length of the pipe

To determine the type of flow whether in laminar or turbulent region, Reynolds number is then used. For a laminar flow, the Reynolds number could be up to 2000 while for a complete turbulent flow, the Reynolds number could be up to 4000. Reynolds number can be determined using the formula, Re =

Where ρ= density of the fluid

V=average fluid velocity flowing through pipe

D= diameter of pipe

µ= dynamic viscosity of the fluid

Next, to determine the friction factor for laminar flow, it is a function of Reynolds’s number and the formula used is f= For turbulent region, f is a function of Reynolds’s number and relative roughness (/D or k/D) and these values can be determined from the Moody Chart and Table 8.1 (Munson et. al., 2006) History

The Darcy-Weisbach equation was founded and named after two great hydraulic engineers of the middle 19th century.

One of them is Julies Weisbach, a German engineer who proposed in 1845 the equation in the form that we are using today where. However, he did not provide adequate data for the variation in f with velocity. Thus, his equation performed poorly compared to the empirical Prony equation, which was widely used at that time, where a and b are empirical friction factors for the velocity and velocity squared.

Before Weisbach, in about 1770, Antoine Chézy, published an equation for flow in open channels but Chézy’s work was lost till 1800 when his former student, Prony published an account to describe it. Darcy, who is Prony’s student, then published new relations for the Prony coefficients. The equation is

Where c, d and e are empirical coefficients for a given type of pipe. Darcy thus introduced the concept of the pipe roughness scaled by the diameter; which is known as the relative roughness when applying the Moody Chart today. Ways of Reducing Pipe Losses

Using the equation of major head losses,, Pipe losses can be reduced by various ways such as Increasing the size of pipe or using a pipe with larger diameter. A larger diameter will cause the hL to be smaller while other parameters remain constant. Small size of pipe will often contribute to higher pipe loss as the pipe size is reduced, the pressure at the pump inlet increases.

Reducing the flow rate of the fluid. By reducing flow rate, the parameter V in the equation will reduced and this will result in a smaller hL.

Using a smoother pipe which is also an alternative to reduce head loss as smoother pipe means smaller f. With a smaller f the fluid will experience less friction when flowing in the pipe and thus head loss is reduced. Other than that, the surface of the pipe can also be waxed to reduce the friction between the fluid and the pipe.

Experimental Procedures

1) First of all, the water is fed through a straight tube.

2) The flow rate of water is then controlled using a flow control valve facing the volumetric tank and measured by collecting the water in the tank and then divide the amount of water collected over time taken.

3) The inlet pipe is then connected directly to bench supply for higher flow rates and a Hoffman clamp is clamped to each of the water manometer

connection tubes.

4) The test rig flow control valve is then closed and zero flow reading is taken from mercury manometer. The flow control valve is then opened fully and head loss shown by mercury manometer is then measured.

5) The flow rate and the temperature of water are then measured.

6) For lower flow rates, the inlet pipe is connected to the outlet at the base of the constant head tank while the inlet to the tank is connected to the beach supply. The pressure drop is then measured using water manometer instead of mercury manometer.

7) The experiment is then repeated six times to get six different flow rates for each high and low flow rates with lowest value of approximately 30mm height difference in the manometer reading.

Pictures of Experimental Rig

Inlet Pipe

Water Manometer (For small

Pressure drop, low flow rates)

Mercury Manometer (For larger

Pressure differences, high flow

Rates)

Flow control valve

Measuring Cylinder

Thermometer

Stopwatch

Results

Step 1: The following formula have been used to calculate Re and friction factor for every observation

/ kg/m3 value obtain from tables of water properties at t/ oC value obtain from tables of water properties at t/oC

Diameter of the pipe, D/m (0.003m)

Velocity of flow in the pipe, V/ ms-1

Where velocity of the flow, V/ ms-1= Rate of flow / Area of pipe

=

Calculation of head loss, hL on every observation

As and (due to arrangement of the manometer which cancel out the height difference)

Pressure change for water monometer

Pressure change for mercury manometer

; where L is the length of pipes in meters, m (0.5m)

Tables of result

For low flow rate

Volume of water, V

Time, t/s

Temperature, T/oC

V1/ml

V2/ml

average V/m3

t1

t2

tave

T1

T2

Tave

water

148.00

149.00

1.49×10-4

30.13

30.15

30.14

27.00

27.00

27.00

134.00

134.00

1.34×10-4

30.15

30.25

30.20

27.00

28.00

27.50

126.00

124.00

1.25×10-4

30.20

30.10

30.15

28.00

27.00

27.50

112.00

113.00

1.13×10-4

30.10

30.30

30.20

28.00

27.00

27.50

96.00

96.00

9.60×10-5

30.20

30.30

30.25

28.00

27.00

27.50

66.00

65.00

6.55×10-5

30.25

30.10

30.18

28.00

27.00

27.50

hi/mm

hf/mm

change in height, ∆h/m

h1

h2

average hi

h1

h2

average hf

210.00

210.00

210.00

89.00

89.00

89.00

0.12

200.00

200.00

200.00

93.00

93.00

93.00

0.11

195.00

195.00

195.00

96.00

96.00

96.00

0.10

188.00

188.00

188.00

103.00

103.00

103.00

0.09

180.00

180.00

180.00

109.00

109.00

109.00

0.07

173.00

173.00

173.00

125.00

125.00

125.00

0.05

density of water,ρwater

viscosity of water,µwater

density of air, ρair

996.59

8.520E-04

1.23

996.45

8.610E-04

1.23

996.45

8.610E-04

1.23

996.45

8.610E-04

1.23

996.45

8.610E-04

1.23

996.45

8.610E-04

1.23

Diameter of pipe, d = 0.003 m;Length of pipe, L = 0.50 m Volumetric flow rate, m3s-1

Velocity, V/ms-1

Re

Pressure change, ∆P/Pa

Head loss, HL/m

frictional factor, f

4.93E-06

0.70

2445.96

1181.50

0.12

0.0293

4.44E-06

0.63

2179.41

1044.65

0.11

0.0319

4.15E-06

0.59

2036.40

966.55

0.10

0.0338

3.73E-06

0.53

1829.73

829.86

0.08

0.0360

3.17E-06

0.45

1558.79

693.18

0.07

0.0414

2.17E-06

0.31

1066.19

468.63

0.05

0.0598

Gravitational acceleration, g = 9.81 m/s2;Area of pipe, A = πd2/4 = 7.07×10-6 m2 For high flow rate

Volume of water, V

Time, t/s

Temperature, T/oC

V1/ml

V2/ml

average V/m3

t1

t2

Tave

T1

T2

Tave

mercury

156.00

156.00

1.56E-04

5.19

5.20

5.20

28.00

28.00

28.00

174.00

174.00

1.74E-04

6.20

6.20

6.20

28.00

28.00

28.00

186.00

187.00

1.87E-04

7.15

7.20

7.18

28.10

28.10

28.10

157.00

156.00

1.57E-04

7.20

7.20

7.20

28.00

28.00

28.00

143.00

141.00

1.42E-04

8.20

8.06

8.13

29.00

29.00

29.00

137.00

136.00

1.37E-04

15.30

15.20

15.25

29.10

29.10

29.10

hi/mm

hf/mm

change in height, ∆h/m

h1

h2

average hi

h1

h2

average hf

362.00

362.00

362.00

21.00

21.00

21.00

0.34

345.00

345.00

345.00

36.00

36.00

36.00

0.31

323.00

323.00

323.00

57.00

57.00

57.00

0.27

287.00

287.00

287.00

92.00

92.00

92.00

0.20

256.00

256.00

256.00

122.00

122.00

122.00

0.13

209.00

209.00

209.00

166.00

166.00

166.00

0.04

density of water,ρwater

viscosity of water,µwater

density of air, ρair

density of mercury, ρmercury

996.31

8.330E-04

1.23

13559.53

996.31

8.330E-04

1.23

13559.53

996.28

8.483E-04

1.23

13559.29

996.31

8.483E-04

1.23

13559.53

996.02

8.100E-04

1.23

13557.10

995.99

8.133E-04

1.23

13556.86

Diameter of pipe, d = 0.003 m;Length of pipe, L = 0.50 m Volumetric flow rate, m3s-1

Velocity, V/ms-1

Re.

Pressure change, ∆P/Pa

Head loss, HL/m

frictional factor, f

3.00E-05

4.25

15243.24

42026.62

4.30

0.0280

2.81E-05

3.97

14246.10

38082.77

3.90

0.0291

2.60E-05

3.68

12956.20

32782.67

3.35

0.0292

2.17E-05

3.08

10834.67

24032.82

2.46

0.0306

1.75E-05

2.47

9115.27

16512.04

1.69

0.0326

8.95E-06

1.27

4652.17

5298.55

0.54

0.0398

Gravitational acceleration, g = 9.81 m/s2;Area of pipe, A = πd2/4 = 7.07×10-6 m2

Step 2: Plotting of the f over Re graph (Refer to moody graph) Does low Re value matches with the theoretical line? ()

Yes, generally the laminar and turbulent flow matches with the theoretical line in the moody graph. Does the high Re value case matches with a particular relative roughness, line? Yes, the high Re case matches with a particular relative roughness line in the moody graph. What do you think the relative roughness, value is for the pipe in your experiment? The relative

roughness value for the pipe in this experiment is 0.0008.

The friction factor in the turbulent regime estimated)

From table 8.1-Introduction Fluid Mechanics MYO, the value if relative roughness, ε = 0.0015mm

The diameter, D of the pipe= 3mm

Therefore the relative roughness value ε/D= (0.0015)/(3)

= 0.0005

From the moody chart, the friction factor value relative to the Reynolds’s no. in the turbulent regime is as follow: Reynolds’s No.

Friction factor form theoritical data

Friction Factor of experimental data

Percentage error, %

15243.24

0.029

0.0280

3.45%

14246.10

0.030

0.0291

3.00%

12956.20

0.031

0.0292

5.81%

10834.67

0.032

0.0306

4.38%

9115.27

0.035

0.0326

6.86%

4652.17

0.038

0.0398

-4.74%

The percentage error is calculated by applying the following formula : (ftheory-fexperimental)/ftheory x 100%

Step 4

Assuming that the value of viscosity, . The value of viscosity, can be determined base on for the cases where there is a low Re value. a) Use data taken where there is a low Re value. ()

Velocity, V/ms-1

Re

Pressure change, ∆P/Pa

0.63

2179.41

1044.65

0.59

2036.40

966.55

0.53

1829.73

829.86

0.45

1558.79

693.18

0.31

1066.19

468.63

b) ; as

c) Plot the graph of pressure change, (P1- P2) versus velocity, V

d) Find the gradient of the graph, the viscosity value, can be determine based on the gradient of the graph The gradient of the graph is 1634 Pa/ms-1

Where gradient of the line =

as length of pipe, L/m and diameter of pipe, D/m are pipe geometries

Therefore the value of determined from the graph is :

µ value form data sheet = 8.610 x 10-4 kg/m.s

µ value form gradient of the plotted graph = 9.19125 x 10-4 kg/m.s

The percentage of difference between the viscosity, µ value obtains from the data sheet and the value obtains from the plotted graph is:

Discussion

The sources of errors were mainly due to human error, and equipment limitations. Human error could have occurred due to shaking of our hands when holding the pipe. This would have affected the fluid flow which might cause the reading on the manometer to fluctuate about the actual value. Besides that, the time measured using the stopwatch might be wrong due to reaction of human when using the stop watch. A time lag or excess time could have easily been introduced into the time recorded.

The volume of water collected might be more due to human reaction time in removing the measuring cylinder away from the pipe. Other than that parallax error could have occurred when taking reading of the manometer and the volume of water in the measuring cylinder. Equipment limitations cannot be neglected because it would have caused significant errors to the results obtained. One of the possible errors was that minor losses in the pipe due to pipe fittings were assumed to be negligible.

However, in reality, these minor losses will be significant. As a result, errors were introduced into our calculations which would affect the results of the experiment. Other than that, air bubbles were seen to develop inside the pipe. This could have resulted in fluctuations of the liquid level in the manometer. The value taken might have varied from the actual value. Furthermore, the temperature measured might also be inaccurate, and could have given rise to wrong estimations for the values of density and viscosity of water. In conducting the experiment, the pipe was not held straight and at a constant level.

This means that the assumption made in using the Bernoulli’s equation that both ends of the pipe were at the same level was inaccurate. It was also assumed that the fluid was incompressible for the Bernoulli’s equation to be used. The equation, , used in the calculations was a rough estimate; and this could have introduced some error into our calculations.

The data recorded for the laminar flow was shown to be quite accurate as all the points only deviated slightly from the laminar flow line on the moody chart. However, some of the points recorded a Reynolds number in the transition range. With that said, the data recorded for the transition stage of the flow did not agree with what was expected.

The friction factor for the data points in the trasition stage was expected to increase slightly, before a further rise when the flow eventually becomes turbulent. This result could have been caused by the error in the viscosity at certain temperatures, leading to an error in the Reynolds number calculated. The data recorded for the turbulent flow showed precision. Most of the points, except for one, were lying on the same line which correspond to a relative roughness of 0.0008.

As we do not know the actual relative roughness of the pipe used in the experiment, we were unable to make comparisons between this theoretically calculated value and the actual value. The viscosity calculated is 9.19×10-4 and the value from data sheet is 8.610 x 10-4. The percentage error calculated was about 6.75%.

This discrepancy could be due to the slight error in temperature recorded where the experiment was conducted. Other than that, the rounding up of values throughout the calculations might have intorduced a certain percentage error to the final results. The error between the estimated friction factor in the turbulent regime and the data from the experiment might be cause by the above error when conducting the experiments besides that in the moody chart the values plotted taken into account the viscosity effect which causes the discrepancies of 3%-6% between the 2 datas.

Conclusion

This experiment was conducted to study the resistance to flow in a pipe and to establish a critical Re value for the flow in the pipe. Through the plotting of the first set of data on the moody charts, it was shown that the flow was laminar up to a Reynolds number of about 2446. This was close to the expected critical Reynolds number of 2300. From the second set of data, the relative roughness of the pipe was estimated to be 0.0008.

Using , the viscosity of water calculated was 9.19×10-4 kg/ms. This value has an error of 6.75% from the value given in the data sheet. Slight discrepancies in the values were expected as there were a few errors associated with the carrying out of the experiment. Some of the errors involved in the experiment include human errors in taking measurements, equipment limitations and precision and the accuracy of the assumptions and basis on which the equations and analysis were based.

References

B. R. Munson, D. F. Young, T. H. Okiishi, Fundamentals of Fluid Mechanics, 5th Ed., 2006, John Wiley and Sons Inc.

Darcy-Weisbach equation 2008. Retrieved October 3, 2008, from http://en.wikipedia.org/wiki/Darcy-Weisbach_equation

Glenn Brown (2000), The History of the Darcy-Weisbach Equation. Retrieved October 4, 2008, from http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm