hes2340 fluid mechanics 1, semester 1, 2012, assignment 2

SWINBURNE UNIVERSITY OF TECHNOLOGY (SARAWAK CAMPUS)

FACULTY OF ENGINEERING AND INDUSTRIAL SCIENCE

HES2340 Fluid Mechanics 1 Semester 1, 2012

ASSIGNMENT 2

By

Stephen, P. Y. Bong (4209168)

Lecturer: Professor Alexander Gorin

Due Date: Week 12

ASSIGNMENT 2 Stephen, P. Y. Bong (4209168)

HES2340 Fluid Mechanics 1, Semester 1, 2012 of 13

ASSIGNMENT PROBLEM

Two big reservoirs A and C having a difference level of 15.5 m are connected by a pipeline ABC

and the elevation of point B being 4.0 m below the level of water in reservoir A. The length AB of

the pipelines is 250 m, the pipe being made of mild steel having a friction coefficient of f1, while the

length of BC is 450 m, the pipe is made of cast iron having a friction coefficient of f2. The pressure

head at B is 0.5 m. Both the lengths AB and BC have a diameter of 200 mm. A partially closed

valve is located in the length BC at a distance of 150 m from reservoir. Taking into account head

loss at entrance and exit points of the pipeline:

(a) Find the friction coefficients f1 and f2 if the discharge through the pipeline is 3m3/min, and the

head loss at the valve is 5.0 m.

(b) Calculate the value of K for the valve if the discharge from task (a) above is reduced by one

half.

Draw the hydraulic and energy grade lines of the pipeline and indicate on the diagram head loss

values at significant points for both tasks, (a) and (b).

All your assumptions and information from external texts, books, handbook and etc. should be

listed and commented in the report with relevant references. Your report with all schematics and

diagrams must be prepared by your own using a computer. For part (b), you can use whether the

Moody Chart or any other relevant experimental correlation (clear explain and show in your report

what and how you use them). If you use the Moody Chart, it is allowed the chart to be copied from

texts.



SOLUTION

Problem Statement: The properties (diameters & materials) of pipes AB and BC, the elevations of

water level in reservoir A and C, and the discharge are given, the friction coefficient f1, f2, and the

loss coefficient, K, of the partially closed valve are to be determined.

Schematic Diagram:

Assumptions:

1. The flow is steady and the fluid is incompressible and Newtonian with constant properties.

2. The entrance effects are negligible, and thus the flow is fully developed and turbulent (to be

verified) with kinetic energy correction factors of α1 = α2 = 1.

3. The flow in each sections of the pipeline is adiabatic and thus there is no heat transfer.

4. Turbine and pump are not involved in piping system.

5. Water levels at the reservoirs and the discharge remain constant.

6. Both the reservoirs are open to atmosphere.

7. The pipe inlet at point A is a sharp-edged entrance with loss coefficient of 0.50 (Cengel, Y. A.

& Cimbala, J. M., 2006, pp. 350).

8. The pipe exit at point C is a sharp-edged exit with loss coefficient of α (Cengel, Y. A. &

Cimbala, J. M., 2006, pp. 350).

9. The horizontal sections of the pipeline (AB and the section from the partially closed valve to

point C) are connected with an oblique pipeline by two 45° threaded elbow with loss

coefficient of 0.4 (Cengel, Y. A. & Cimbala, J. M., 2006, pp. 351).



Physical Laws: The First Law of Thermodynamics (Conservation of Energy Principle):

Ein – Eout = ∆E

Properties

Properties of Water at 20 °C:

According to TABLE A-3: Properties of Saturated Water (Cengel, Y. A. & Cimbala, J. M., 2006,

pp. 888), the properties of water at temperature of 20 °C are:

Density: ρ = 998.0 kg/m3

Dynamic Viscosity: µ = 1.002 × 10-3

kg/m·s

Properties of Piping System:

Pipe Length

Length of Pipe AB LAB = 250 m

Length of Pipe BC LBC = 450 m + 150 m = 600 m

Loss Coefficients of Various Pipe Components

A sharp-edged entrance at point A KL = 0.50

A sharp-edged exit at point C KL = α = 1

Two 45° threaded elbow KL = 0.40

According to TABLE 8.1: Typical Values of Absolute Roughness for New, Clean Pipe (Rennels, D.

C. & Hudson, H. M., 2012, pp. 83), and TABLE 8-2: Equivalent Roughness Value for New

Commercial Pipes (Cengel, Y. A. & Cimbala, J. M., 2006, pp. 341), the roughness height for mild

steel (or carbon steel) and cast iron are:

Roughness Height for Mild Steel (Carbon Steel) ε = 0.045 mm = 0.045 × 10-3

m

Roughness Height for Cast Iron ε = 0.26 mm = 0.26 × 10-3

m

Analysis:

(a) The discharge through the pipeline is given by:

sm 0.053=⋅=

s 60

min 1

min

m3

3

V&

Since the diameters of pipe AB and pipe BC are the same, DAB = DBC = D = 200 mm, therefore,

they have the same cross sectional area. Hence, the average velocity can be determined as

follows:

sm1.5915======⇔= sm5

m) 2.0(

)sm050(44

4

2

3

22 ππππ

.

D

V

D

V

A

VVVAV

C

C

&&&&



(a) (Continued)

Based on the average velocity computed above, the Reynolds number is given by:

3317029.341=⋅×

==− smkg 101.002

m) 2.0)(sm59151)(mkg 0.998(Re

3

3 .DV

µ

ρ

Since Re > 4000, therefore, the flow is fully turbulent. The friction factors f1 and f2 for pipe AB

and BC respectively can be determined by using an approximate explicit relation for f which

was given by S. E. Haaland in 1983 (Cengel, Y. A. & Cimbala, J. M., 2006, pp. 341):

+

−≅

Re

9.6

7.3log8.1

111.1

D

f

ε

Making f the subject yields:

211.1

Re

9.6

7.3log8.1

1

+

−

≅

D

f

ε

The friction factor, f1 for pipe AB can be calculated by substituting εAB = 0.045 × 10-3

m, DAB =

0.2 m into the explicit form of Colebrook equation above:

0.01616≈

+

×−

≅−

211.1

3

1

3413.317029

9.6

7.3

m 2.0m 10045.0log8.1

1f

Likewise, for pipe BC, εBC = 0.26 × 10-3

m, DBC = 0.2 m, thus the friction factor f2 is:

0.02167≈

+

×−

≅−

211.1

3

2

3413.317029

9.6

7.3

m 2.0m 1026.0log8.1

1f



(b) If the discharge in part (a) is reduced by one half, then the new volumetric flow rate becomes:

sm 24

1 3=⋅==−=s 60

min 1

min

m 5.2minm 5.2minm 5.0minm 3

3333

V&

Likewise, the average velocity for the fluid flow through the pipeline is determined as follows:

sm 1.3263==

==== sm 6

25

)m 2.0(

sm 24

14

4

4

2

3

22 ππππ D

V

D

V

A

VV

C

&&&

Based on the average velocity obtained above, the Reynolds number is:

8264201.077=⋅×

==− smkg 101.002

m) 2.0)(sm 3263.1)(mkg 0.998(Re

3

3

µ

ρ DV

Since Re > 4000, thus, the flow is turbulent. The friction factor f1 and f2 for pipe AB and BC

are determined by the explicit relation for f used in part (a):

0.01648≈

+

×−

≅−

211.1

3

1

0778.264201

9.6

7.3

m 2.0m 10045.0log8.1

1f

Likewise, for pipe BC, εBC = 0.26 × 10-3

m, DBC = 0.2 m, thus the friction factor f2 is:

0.02179≈

+

×−

≅−

211.1

3

2

0778.264201

9.6

7.3

m 2.0m 1026.0log8.1

1f

From the First Law of Thermodynamics, the general form of energy equation is given by:

eLu hhzg

V

g

phz

g

V

g

p,TurbineTotal ,2

2

2

2

2

,Pump1

2

1

1

1

22++++=+++ α

ρα

ρ

Term-by-term analysis:

• 021 == pp because the pressure at top of reservoirs A and C is patm = 0 gage.

• V1 = V2 ≈ 0 because the level of the reservoirs is constant or changing very slowly.

• 0== tp hh because there are no pumps or turbines in the system.

• m 5.151 =z and 02 =z

Through term-by-term analysis, the energy equation above becomes:

Total ,1 Lhz =



Part (b) (Continued)

The term hL, Total in the energy equation above is the total head loss due to friction and pipe

fittings and it can be computed as follows:

∑∑ +=+=j

j

jL

i

i

i

i

iLLLg

VK

g

V

D

Lfhhh

22

2

,

2

Minor ,Major ,Total ,

Since the average velocity of the fluid flow throughout the entire pipeline are constant due to

constant diameters of pipes AB and BC, therefore, the head loss equation above can be

expressed as:

+= ∑∑

j

jL

i i

i

iL KD

Lf

g

Vh ,

2

Total ,2

Since the piping system consists of a sharp-edged entrance and sharp-edged exit at the inlet and

outlet of pipe AB and BC respectively, and the oblique section is connected to the horizontal

sections by two 45° elbow, and there is a partially closed valve located in the length BC at a

distance of 150 m from pipe outlet, therefore, the equation for head loss above can be written

as:

++++

+= ° Valve ,Elbow@45 ,Exit ,Entrance ,

BC

BC

2

AB

AB1

2

Total , 22

LLLLL KKKKD

Lf

D

Lf

g

Vh

Substituting the values of the loss coefficients of various pipe components listed in the

properties above yields:

( )Valve ,

Valve ,2

2

Total ,

88.27m 08966.0

)4.0(2150.0m 0.2

m 60002179.0

m 0.2

m 25001648.0

)sm 81.9(2

)sm 3263.1(

L

LL

K

Kh

+=

++++

+

=

Substituting ( )Valve ,Total , 88.27m 08966.0 LL Kh += into the simplified energy equation above

gives:

( )Valve ,1 88.27m 08966.0 LKz +=

Since z1 = 15.5 m, thus, the value of K of the valve is

79.0287=−= 27.88m 08966.0

m 51Valve ,LK



The Hydraulic and Energy Grade Lines of the piping system are plotted and as shown in Figure

below:

Figure: HGL and EGL of the piping system



APPENDIX

Table 8-4: Loss Coefficients for Various Pipe Components (Cengel, Y. A. & Cimbala, J. M., 2006,

pp. 350)



Table 8-4 (CONCLUDED) (Cengel, Y. A. & Cimbala, J. M., 2006, pp. 351)



Table A-3: Properties of Saturated Water (Cengel, Y. A. & Cimbala, J. M., 2006, pp. 888)



TABLE 8-2: Equivalent roughness value for new commercial pipes* (Cengel, Y. A. & Cimbala, J.

M., 2006, pp. 341)

TABLE 8.1: Typical Values of Absolute Roughness for New, Clean Pipe (Rennels, D. C. &

Hudson, H. M., 2012, pp. 83)



REFERENCES

Cengel, Y. A. & Cimbala, J. M. (2006), Fluid Mechanics – Fundamentals and Applications,

McGraw-Hill Companies, Inc., Singapore.

Cengel, Y. A. & Boles, M. A. (2007), THERMODYNAMICS – An Engineering Approach (SI

Units), 6th

Edition, McGraw-Hill Companies, Inc., Singapore.

Crow, C. T., et. al., (2010), Engineering Fluid Mechanics, 9th

Edition, John Wiley & Sons (Asia)

Pte Ltd, Asia.

Munson, B. R., et. al., (2008), FUNDAMENTALS OF FLUID MECHANICS, 6th

Edition, John

Wiley & Sons, Inc., United States of America.

Rennels, D. C. & Hudson, H. M. (2012), PIPE FLOW – A Practical and Comprehensive Guide, pp.

83, John Wiley & Sons. Inc., Canada.

hes2340 fluid mechanics 1, semester 1, 2012, assignment 2

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