Fully Developed Pipe Flow And Friction Loss Engineering Essay

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Abstract

The aims of these experiments were to measure the fully developed velocity profiles for both laminar and turbulent flow regimes inside a smooth pipe and to study the frication losses at different Reynolds number. The measured velocities profiles were compared with the theoretical profiles. Using the measured velocity profile, the volume flow rate were also calculated and compared with the measured flow rate. The other part of the experiment measured the friction coefficient for both laminar and turbulent flows through pipes and these values were compared to the theoretical coefficients profiles.

The test rig used for this study consisting primarily of a perfectly smooth pipe in a closed circuit of oil flowing from the oil tank using a gear pump through the pipe and back again to the tank. A Pitot tube installed near the end of the pipe was used to measure the velocity profile through the pipe cross section. Pressure tapping are distributed along the pipe to help read the pressures at certain locations. The flow rate was adjusted using a bypass valve to make it laminar or turbulent. Flow rate was measured using rotameter. Pressure readings helped in calculating the velocity profile and friction coefficients.

The results showed that the measured velocity profiles were close to the theoretical profiles yet there were some differences especially near the pipe walls. The volume flow rate calculated for laminar and turbulent was close to the measured one. The friction coefficients calculated had same behavior like the theoretical one but with some differences.

Introduction and Background

Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers.[1] At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents perpendicular to the direction of flow, nor eddies or swirls of fluids.[2] In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls.[3] In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion and low momentum convection. [2]

In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Feynman described turbulence as "the most important unsolved problem of classical physics."[1] Flow in which the kinetic energy dies out due to the action of fluid molecular viscosity is called laminar flow. While there is no theorem relating Reynolds number to turbulence, flows with high Reynolds numbers usually become turbulent, while those with low Reynolds numbers usually remain laminar. For pipe flow, a Reynolds number above about 4000 will most likely correspond to turbulent flow, while a Reynolds number below 2100 indicates laminar flow. The region in between (2100 < Re < 4000) is called the transition region. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.

Mathematical Model

When a uniform flow enters a straight pipe and due to viscosity, the fluid velocity at the pipe wall must be zero. The fluid near the wall thus gradually slows down the fluid adjacent to it, which, in turn, gradually slows down the faster moving fluid layer next to it, and so on. Finally, at some distance downstream of the pipe inlet, the effect of viscosity is felt across the entire pipe, and the flow is termed fully developed viscous flow.

Figure Velocity profiles for laminar and turbulent flow

The actual velocity distribution in the pipe depends on the type of flow, i.e., whether it is laminar or turbulent. Laminar flow is characterized by smooth and steadiness. However, when the flow becomes fluctuating and agitated it is called turbulent flow. Theoretical analysis shows that the velocity profile of a fully developed laminar pipe flow is parabolic and is expressed in the form:

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Where u is the velocity at any radius r, and umax is the velocity at pipe centerline, and D is the inner diameter of the pipe. In turbulent flow, extensive mixing of the fluid particles tends to even out the velocity profile, and accordingly the velocity distribution becomes more uniform, see Figure 1. The shape of the velocity profile is dependent on the turbulent intensity. Some empirical relations were derived from experimental results for the velocity distribution on fully developed turbulent flow. For flow through a smooth pipe at high Reynolds number, the velocity profile may be represented by the following empirical equation:

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The actual velocity profile inside the pipe can be measured by traversing a total pressure tube (Pitot - static tube) across the diameter of the pipe. The static pressure is constant over the cross-section of the pipe and can be measured using a pressure tap. By measuring the total pressure po and the static pressure ps, the velocity at any point can be calculated from the following equation:

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Where ρ is the fluid density, the above equation can be rearranged to give the velocity at a point as follows:

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Using the measured velocity profile, the volume flow rate is calculated by integrating the velocity over the cross sectional area of the pipe:

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When u is plotted against r2 the area under the curve represents. Numerically, the above equation is expressed as follows:

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Where is the average velocity between two consecutive values of r whose difference is Δr.

To validate the calculated value of the flow rate,, a rotameter is used to separately measure the flow rate. The percentage error in the calculated value compared with the measured one is given as follows:

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When a fluid flows through a pipe of constant cross-section, a loss of pressure head due to friction is observed. Energy degradation due to friction is known as major loss. It is found that the pressure drop Δp depends on the average velocity of the flow , the fluid density ρ, the pipe diameter D and the pipe length L, and the friction factor ƒ. The relation between pressure drop Δp and other flow parameters is given by:

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In terms of Δp, the friction factor ƒ is expressed as follows:

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For laminar flow, theoretical analysis showed that the friction factor depends only on Reynolds number, Re and is given by:

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For turbulent flow, the friction factor depends on Reynolds number and the relative roughness ε/D of the pipe. The variation of ƒ with Re at different values of ε was obtained experimentally and represented in a chart called Moody Chart, see Figure 2. For turbulent flow (3000 ≤ Re ≤ 100,000) in a smooth pipe, ƒT is given by the following empirical formula obtained by Blasius:[1]

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Figure Moody Chart

Experimental Setup and Procedure

The experimental apparatus used in this experiment is shown in Figure 3. The test setup consists of a closed circuit through which the fluid is circulated continuously by means of a gear pump. The selected fluid is oil in order to give Reynolds number clearly in the laminar region.

Figure Experimental setup

Oil is drawn from the reservoir and delivered through a Rotameter to the Perspex settling chamber. The oil passes from this chamber through a parabolic bell mouth into the upper horizontal pipe in which the observations are taken. This pipe is made of aluminum with a 19 mm bore and an overall length of 4.4 m it's adjusted on its supports to be accurately straight and horizontal. The arrangement of smoothing screens in the settling chamber ensures that the oil enters the pipe in an adjustable flow disturber downstream of the bell mouth may be used to induce turbulence.

Eighteen pressure tapping are spaced along the pipe in order to determine the pressure gradient. The velocity profile towards the downstream end of the pipe is observed by a Pitot tube of diameter 1.2mm traversed by a micrometer. Leaving the pipe, the oil is collected in a Perspex deflector, permitting observation of a laminar transition or turbulent jet. The oil returns to the reservoir through the lower pipe that forms the spine of the apparatus. The flow quantity is varied by means of an adjustable bypass valve. The maximum pipe delivery is 5000 L/h at a pressure of about 2 bars, corresponding to a velocity in the test pipe of 5 m/s. Pressure are observed by means of a transducer, with a range from 0 to 3 bars, in conjunction with a 20 way selector switch for scanning the various pressure tapping.

Experimental Procedure:

It is desirable to run the apparatus for a few minutes to warm up the motor and ensure steady conditions. Care must be taken to bleed air from the settling chamber and from all the tubes to avoid errors in pressure measurements. The difference between laminar and turbulent flow is clearly indicated by the nature of the discharge. If the flow rate is increased or, alternatively, the flow disturber is pushed in progressively, periodical 'slugs' of turbulence will appear; eventually, at higher rates of flow with sufficient disturbance the flow will be continuously turbulent. The oil flow rate is measured using the Rotameter.

The Pitot tube is attached to a sliding carrier which is spring loaded to bear against the micrometer, and stops are provided so that the Pitot tube cannot be moved into contact with the wall of the pipe. In all cases, the temperature of the oil in the system should be recorded, since this has a great effect on viscosity.

Velocity Profile Measurements Procedure:

Start the pump and bleed air from the settling chamber and the tubes that are connected to the pressure taps and the Pitot tube section (static and total pressure taps).

In case of laminar flow, make sure that the flow disturber is pulled out. In case of turbulent flow, make sure that the flow disturber is pushed in.

Measure the oil temperature and record it. Use this temperature to determine the properties of the oil from the Appendix (Shell Tellus Oil R5).

Adjust flow rate and watch the flow to get the required flow type (laminar or turbulent flow).

Read the flow rate on the Rotameter and record it.

Record the static pressure reading at the test section of the Pitot tube.

Carefully use the total pressure tube to traverse the pipe cross-section and record the reading corresponding to each radius.

Push in the flow disturber and increase the flow until the flow becomes continuously turbulent (observe it).

Repeat steps 5 to 7 for turbulent flow velocity profile.

Friction Coefficient Measurements Procedure:

laminar flow

Record oil temperature.

Adjust the flow rate to obtain the maximum flow rate to give laminar flow.

Determine the flow rate.

Record the pressure readings at taps 12 and 18.

Repeat steps 3 and 4 for different flow rates in the laminar range. Make sure that the flow in the pipe is laminar by observing the discharging jet from the pipe.

Record oil temperature at the end of the laminar flow test.

Turbulent Flow

Insert the flow disturber all the way into the pipe.

Reduce the flow rate using the bypass valve to obtain the minimum flow rate in the turbulent region. Make sure that the flow in the pipe is turbulent by observing the discharging jet from the pipe.

Determine the flow rate.

Record the pressure readings at taps 12 and 18.

Repeat steps 3 and 4 for different flow rates in the turbulent range.

Record oil temperature at the end of the test.

Collected Data Tables

Table Data for laminar flow velocity profile

r (mm)

P0 (bar)

-8.1

0.04

-6.5

0.04

-5.0

0.05

-3.5

0.05

-1.5

0.06

0.0

0.06

1.5

0.06

3.5

0.06

5.0

0.05

6.5

0.04

8.1

0.04

Table Data for turbulent flow velocity profile

r (mm)

P0 (bar)

-8.1

0.09

-6.5

0.14

-5.0

0.19

-3.5

0.23

-1.5

0.28

0.0

0.3

1.5

0.3

3.5

0.26

5.0

0.22

6.5

0.16

8.1

0.1

Table Data for laminar flow friction coefficient

V

P12

P18

18

0.07

0.03

19

0.07

0.03

20

0.08

0.03

21

0.08

0.03

22

0.08

0.03

23

0.08

0.03

24

0.08

0.03

25

0.085

0.03

Table Data for turbulent flow friction coefficient

V

P12

P18

50

0.15

0.04

52

0.15

0.04

54

0.16

0.04

56

0.16

0.05

58

0.17

0.05

60

0.175

0.05

62

0.18

0.05

70

0.20

0.05

72

0.25

0.05

Sample Calculation

Calculations of velocity profile for laminar flow, from Table 1:

For r = 6.5 mm

For r = 5 mm

Calculation of the friction coefficient, for laminar flow from Table 3

For run no. 7:

A similar procedure can be used to calculate the friction coefficient for turbulent flow from Table 5, the theoretical friction coefficient for run no. 8

Uncertainty Analysis

Uncertainty in friction coefficient due to the uncertainties in both pressure and flow measurements can be evaluated as follow, for run no 7, from Table 4:

For run no. 8 (turbulent):

With confidence 95%

With confidence 95%

Assume the bias error of all devices is 1%

The total uncertainty in the friction factor due to the precision and bias errors in pressure and volume flow measurements can be evaluated as

Tables of Results

Table Results for laminar velocity profile

T

25

oC

D

0.019

mm

Re

3350.6

 



870

kg/m3

A

2.84E-4

m2

 



8.0E-6

m2/s

U

1.411

m/s

Umax

2.63

m/s

V

24

L/min

P18

0.03

bar

 

 

 

r (mm)

P0 (bar)

u (m/s)

uth (m/s)

Å«

(m/s)

r2

(m2)

r2

(m2)

ū r2 (m3/s)

 

-8.1

0.04

1.52

0.72

1.52

6.561E-05

2.336E-05

3.542E-05

 

-6.5

0.04

1.52

1.40

1.83

4.225E-05

1.725E-05

3.157E-05

 

-5.0

0.05

2.14

1.90

2.14

2.500E-05

1.275E-05

2.734E-05

 

-3.5

0.05

2.14

2.27

2.39

1.225E-05

1.000E-05

2.385E-05

 

-1.5

0.06

2.63

2.56

2.63

2.250E-06

2.250E-06

5.909E-06

 

0.0

0.06

2.63

2.63

2.63

0.000E+00

0.000E+00

0.000E+00

 

1.5

0.06

2.63

2.56

 

 

 

 

 

3.5

0.06

2.63

2.27

 

Vc

3.898E-04

m3/s

5.0

0.05

2.14

1.90

 

Vm

4.000E-04

m3/s

6.5

0.04

1.52

1.40

 

e

2.541%

 

8.1

0.04

1.52

0.72

 

 

Table Results for laminar friction coefficient

V

Å« (m/s)

P12

(bar)

P18

(bar)

p

(bar)

f

Re

fL

18

1.058

0.07

0.03

0.040

7.208E-02

2512.97

0.02547

19

1.117

0.07

0.03

0.040

6.469E-02

2652.58

0.02413

20

1.176

0.08

0.03

0.050

7.298E-02

2792.19

0.02292

21

1.234

0.08

0.03

0.050

6.620E-02

2931.80

0.02183

22

1.293

0.08

0.03

0.050

6.032E-02

3071.41

0.02084

23

1.352

0.08

0.03

0.050

5.518E-02

3211.02

0.01993

24

1.411

0.08

0.03

0.050

5.068E-02

3350.63

0.01910

25

1.470

0.085

0.03

0.055

5.138E-02

3490.24

0.01834

Table Results for turbulent velocity profile

T

25

oC

D

0.019

mm

Re

9772.7

 



870

kg/m3

A

2.84E-4

m2

 



8.0E-6

m2/s

U

4.115

m/s

Umax

7.58

m/s

V

70

L/min

P18

0.05

bar

 

 

 

r

(mm)

P0 (bar)

u (m/s)

uth (m/s)

Å«

(m/s)

r2

(m2)

r2

(m2)

ū r2 (m3/s)

 

-8.1

0.09

3.03

6.26

3.79

6.561E-05

2.336E-05

8.855E-05

 

-6.5

0.14

4.55

6.75

5.11

4.225E-05

1.725E-05

8.816E-05

 

-5.0

0.19

5.67

7.03

6.05

2.500E-05

1.275E-05

7.717E-05

 

-3.5

0.23

6.43

7.24

6.85

1.225E-05

1.000E-05

6.852E-05

 

-1.5

0.28

7.27

7.45

7.43

2.250E-06

2.250E-06

1.671E-05

 

0.0

0.3

7.58

7.58

7.58

0.000E+00

0.000E+00

0.000E+00

 

1.5

0.3

7.58

7.45

 

 

 

 

 

3.5

0.26

6.95

7.24

 

Vc

1.065E-03

m3/s

5.0

0.22

6.25

7.03

 

Vm

1.167E-03

m3/s

6.5

0.16

5.03

6.75

 

e

8.684%

 

8.1

0.1

3.39

6.26

 

 

Table Results for turbulent friction coefficient

V

Å« (m/s)

P12 (bar)

P18

(bar)

p

(bar)

f

Re

fT

50

2.939

0.15

0.04

0.11

2.569E-02

6980.48

0.00917

52

3.057

0.15

0.04

0.11

2.375E-02

7259.70

0.00882

54

3.174

0.16

0.04

0.12

2.403E-02

7538.92

0.00849

56

3.292

0.16

0.05

0.11

2.048E-02

7818.14

0.00819

58

3.409

0.17

0.05

0.12

2.083E-02

8097.36

0.00790

60

3.527

0.175

0.05

0.13

2.027E-02

8376.58

0.00764

62

3.645

0.18

0.05

0.13

1.975E-02

8655.80

0.00739

70

4.115

0.2

0.05

0.15

1.787E-02

9772.67

0.00655

72

4.232

0.25

0.05

0.20

2.253E-02

10051.89

0.00637

Analysis and Discussion

The velocity profiles of the laminar and turbulent flow are shown in Figures 4 and 5 were for both actual and theoretical values are presented. For the laminar flow, the measured and theoretical velocity profiles were matched over the entire radius of the pipe. Maximum measured flow velocity occurred at the center line as the theoretical one. For turbulent flow, the measured velocity profile differs from the theoretical one especially near the pipe wall, where it was very difficult to measurement exactly at the wall due to the thickness of the Pitot tube used. Another difficulty faced us during the experiment which is the pressure indicator shows only two digits of reading which increased the error associated with such measurements.

Figure Actual and theoretical velocity profile for laminar flow

Figure Actual and theoretical velocity profile for turbulent flow

The measured velocity profile was used to calculate the volume flow rate through the pipe which represented by the area under the curve in u-r2 plot, Figure 6. In Figure 6, the shaded area under the curve represents the volume flow rate. The volume flow rate is represented by the area summation of both curves. However, as the calculation of the area under the curve is not easy to be done, the measured velocity at each radius are used in equation 6, where the integration is transformed into summation. A spread sheet was used to perform this summation procedure as shown in the results, Tables 5 and 7. For laminar flow, the calculated volume flow rate differs by only 2.54% from the measured one, while for the turbulent this difference was 8.68%.

Figure Local velocity versus radius for laminar and turbulent flow

The variation of the friction coefficient with Reynolds number is presented in Figure 7. Experimental and theoretical frication coefficients for lamina and turbulent flow are shown in the graph. For the laminar flow, and as expected the frication coefficient decreased with the increase of Reynolds number for both experimental and theoretical results. Increasing Reynolds number tends to reduce the viscous effect in the boundary layer around the pipe surface which in turn reduces the pressure drop and hence the friction coefficient. These results match with those represented in Moody chart, Figure 2. Same behavior for the friction coefficient was found for the turbulent flow region.

Figure Experimental and theoretical friction coefficient versus Reynolds number for laminar and turbulent flow regions

The uncertainty of the friction coefficient was calculated and found to be; which means that the friction coefficient for run no 7 in Table 4 lie in the range . Comparing this uncertainty with the actual friction coefficient (0.01787) a percentage of uncertainty was found to be 17%.

Conclusions

The velocity profile for laminar and turbulent flow in a smooth pipe was measured using a Pitot tube and compared to the theoretical profiles and the matched together with small error especially for turbulent flow

The measured velocity profile was used to calculate the volume flow rate with maximum error less than 9% for the turbulent flow case.

Experimental and theoretical friction coefficient for laminar and turbulent flow was evaluated and the trend was identical, where the friction factor decreases as the Reynolds number increases.

The uncertainty in evaluating the friction factor due to the uncertainties in pressure and volume flow rate was evaluated and found to be

Nomenclature

A

Cross-sectional area

Bias error in f

Bias error in ∆p

Bias error in Q

D

Diameter of the pipe

e

Percentage error

f

Friction coefficient

fT

Turbulent friction coefficient

fL

Laminar friction coefficient

g

Gravitational acceleration

L

Distance between taps 12 and 18

Po

Total pressure

P

Static pressure

∆p

Pressure drop

Pf

Precision error in f

Precision error in ∆p

Precision error in V

V

Volume flow rate

Re

Reynolds number

r

Radius of pipe

S

Standard deviation of N repeated measurements

T

Temperature of the oil

Ti

Initial temperature of the oil

Tf

Final temperature of the oil

u

Local velocity of flow

umax

Maximum velocity of flow

Average velocity

U

Total uncertainty

Measured flow rate

Calculated flow rate

Kinematic viscosity

References

Potter, M. C., & Wiggert, D. C. (2001). Mechanics of Fluids. 3rd e. United States of America: Bill Stenquist. 298-305.

Incropera, F. P., Dexitt D. P., Bergman T. L., & Lavine A.S. (2007). Fundamenetals of Heat and Mass Transfer. 6th ed. United States of America: Daniel Sayre. 488-491.

Elsayed M. M, and Chakroun W. (1997), An Experimental Coursed in Thermal Engineering. Kuwait: The Authorship. 27-50, and 119-136.

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