440 report_1 copy

Title

Author

Group No.

Date

ii

Table of Contents

Contents Table of Contents ................................................................................................................................................................. ii

List of Figures ....................................................................................................................................................................... iv

List of Tables .......................................................................................................................................................................... v

1. Executive Summary .................................................................................................................................................... 1

2. Introduction ................................................................................................................................................................... 2

3. Methodology .................................................................................................................................................................. 3

3.1 Governing equations .......................................................................................................................................... 3

3.1.1 Continuity Equation: ............................................................................................................................. 3

3.1.2 Energy equation: .................................................................................................................................... 3

3.2 Underlying assumptions ................................................................................................................................... 4

4. Experimental Procedure ........................................................................................................................................... 5

4.1 Material: ................................................................................................................................................................... 5

4.2 Procedure ................................................................................................................................................................ 5

5. Data Collected ................................................................................................................................................................ 7

5.1 Straight Pipe trials ............................................................................................................................................... 7

5.2 Contraction ............................................................................................................................................................. 7

5.3 Bend ........................................................................................................................................................................... 8

6. Calculation and Results ............................................................................................................................................. 9

6.1 Friction Factor results: ...................................................................................................................................... 9

6.2 Kc results: ................................................................................................................................................................ 9

6.3 Kb Results: ............................................................................................................................................................ 10

7. Discussion and Analysis .......................................................................................................................................... 11

7.1 General Errors ..................................................................................................................................................... 11

7.2 Friction Factor .................................................................................................................................................... 11

7.3 Contraction loss coefficient ........................................................................................................................... 11

7.4 Bend Loss Coefficient ....................................................................................................................................... 12

8. Conclusions .................................................................................................................................................................. 12

Acknowledgments ............................................................................................................................................................. 13

iii

References ............................................................................................................................................................................ 14

Appendix ............................................................................................................................................................................... 15

iv

List of Figures

Figure 1 Measuring time needed to fill 10 L of water in the tank .................................................................. 5

Figure 2 Placing the sensors on the end points of the segment under study ............................................ 6

Figure 3 Reading pressure head values from the software .............................................................................. 6

Figure 4 Technician fixing pressure censors ......................................................................................................... 12

v

List of Tables

Table 1 Data collected from straight pipe trials ..................................................................................................... 7

Table 2 Data collected from contraction trials ....................................................................................................... 7

Table 3 Data collected from 90 degree bend trials ............................................................................................... 8

Table 4 Calculations of Reynold’s number, friction factors and percentage error ................................. 9

Table 5 Calculation of Kc and the percentage error ............................................................................................. 9

Table 6 Calculations of Kb and percentage error ................................................................................................ 10

Table 7 Kc values for contraction of different angles ........................................................................................ 15

Table 8 K values for threaded pipe fittings ............................................................................................................ 15

Table 9 Ks values for pipes made of different materials .................................................................................. 15

Table 10 Moody Diagram ............................................................................................................................................... 16

1

1. Executive Summary

The purpose of this report is to show the results of calculating experimental values of friction factors and loss coefficients of a contraction and a 90 degree threaded bend. Our experiment involved 3 phases and 3 trials per phase:

1) Friction factor of a straight pipe 2) Kc value of the contraction 3) Kb value of a 90 degree threaded bend

The experiment resulted in large errors in determining the different factors relative to the theoretical ones. The friction factor found had an average error of 42.45% relative to the theoretical friction factor. Also, the Kc value, having the largest error in calculation, had an average error of 672% relative to the theoretical Kc. Finally, the Kb value had an average error of 35 % from the theoretical Kb. The errors calculated were a result of several factors including:

1) Malfunction and inaccuracy of the pressure head sensors 2) Assuming of Ks instead of determining the actual value 3) Not measuring again the flow after the transition from one segment to another

2

2. Introduction

In hydraulic engineering practice, it is necessary to estimate the head loss incurred by

a fluid as it flows along a pipeline. For example, it may be desired to predict the rate of

flow along a proposed pipe connecting two reservoirs at different levels. Or it may be

necessary to calculate what additional head would be required to double the rate of flow

along an existing pipeline. Thus it’s necessary to calculate the head losses for the above

calculations to be accurate.

Usually, head loss is incurred by fluid mixing which occurs at fittings such as bends or

valves, and by frictional resistance at the pipe wall. Major losses are due to bends,

contraction-‐expansion and valves. Friction losses are between fluid and contact surface.

Where there are numerous fittings and the pipe is short, the major part of the head loss

will be due to the local mixing near the fittings. For a long pipeline, on the other hand,

skin friction at the pipe wall near will predominate. In the experiment described below,

we investigate the losses due to frictional resistance in a straight pipe with rough walls,

fluid mixing in a bend and a contraction.

3

3. Methodology

3.1 Governing equations

In this experiment two main equations were used: Continuity and energy equations.

3.1.1 Continuity Equation:

The continuity equation states that the net rate of mass inflow to a control volume is

equal to the rate of increase in mass within fixed boundaries.

Therefore, 𝜌1 * A1*V1=𝜌2 * A2*V2 Since the fluid is incompressible then 𝜌1 =𝜌2 therefore

A1*V1=A2*V2 Then Q1=Q2

Where:

𝜌 = density (Kg/m3)

V: velocity (m/s)

A= area in m2

Q= flow rate (m3/s)

3.1.2 Energy equation:

!!! + 𝑧! + 𝛼1

!!!

!!+ ℎ𝑚 = !!

!+ 𝑧! + 𝛼2

!!!

!!+ ℎ!

Where:

§ !! : Pressure Head (m)

§ 𝑧! : Elevation Head (m)

§ !!

!! : Velocity head (m)

§ HM: Head of the machine (m). There is an absence of pumps or turbinesà HM =0

§ ℎ! Summation of all major and minor losses

Assuming turbulent flow (as will be determined from Re in the experiment) 𝛼1=𝛼2=1

4

Let’s define the hydraulic head as 𝐻 = !! + 𝑧 + !

!

!!

The equation becomes: H! = H! + h!

Head Losses: ℎ! = ∑𝑓 !!× !!

!!+ 𝑘 !

!

!!

Where:

§ 𝑓: Friction loss coefficient

§ 𝐿: Length of the pipe (m)

§ 𝐷: Diameter of the pipe (m)

§ !!

!!: velocity head (in m)

§ 𝑘: minor loss coefficient for each fitting

Friction Coefficient 𝑓 = !.!"

!"# !"!.!!!

!.!"!"!.!

!

Reynolds Number: 𝑅𝑒 = !"!

Where:

§ 𝜈: Kinematic Viscosity =10-‐6 m2/s at 20°C

§ Ks= relative roughness (in m).

§ D= diameter of pipe (in m).

3.2 Underlying assumptions

There are 3 main assumptions in this experiment:

1) For the bend and contraction, the head loss 𝑓 !!× !!

!! is neglected because L is

small

2) 𝛼 = 1

3) Assume Ks= 0.3mm

5

4. Experimental Procedure

4.1 Material:

§ Hydraulic Bench with adjustable flow

§ Pipes of different sizes and fittings equipped with valves including:

o 1 m straight pipe

o A contraction with an angle of 60o

o A threaded 90o bend

§ PC interface connected to two sensors

§ Stop watch

§ Measuring tape

4.2 Procedure

a) Use a measuring tape to measure the straight pipe length

b) Record the necessary dimensions of the segments under study.

c) Open the valve for the rough pipe

d) Turn on the pump and adjust initial flow

e) Using a stopwatch, record the time needed to fill 10 Liters in the tank as done in

figure 1.

Figure 1 Measuring time needed to fill 10 L of water in the tank

6

f) Place the sensors at the start and end points of the segment under study as done

in figure 2.

Figure 2 Placing the sensors on the end points of the segment under

study

g) Reset the computer program and record the values for the pressure heads as

done in figure 3.

Figure 3 Reading pressure head values from the software

h) Open the valve of the contraction then close that of the rough pipe to avoid

damaging the equipment.

i) Repeat steps e to g after changing the flow

7

j) Open the valve of the pipe line including the bend then close that of the

contraction

k) Repeat the steps e to g after changing the flow

l) Open the valve of the straight pipe and close that of the bend.

m) Repeat the procedure for a total of 3 different flow rates.

5. Data Collected

5.1 Straight Pipe trials

Trials Flow Calculated

(m3/s)

hu (cm) hd (cm)

1 4.33E-‐04 106 70

2 6.60E-‐04 217 151

3 7.71E-‐04 271 190

Length (cm) Diameter (mm)

100 17

Table 1 Data collected from straight pipe trials

5.2 Contraction


(m3/s)

hu (cm) hd (cm)

1 4.33E-‐04 65 59

2 6.60E-‐04 128 116

3 7.71E-‐04 155 141

Diameter 1 (mm) Diameter 2(mm)

40 25

Table 2 Data collected from contraction trials

8

5.3 Bend


(m3/s)

hu (cm) hd (cm)

1 4.33E-‐04 78 70

2 6.60E-‐04 158 143

3 7.71E-‐04 193 175

Diameter 1 (mm) 𝚫𝒛(cm)

17 7

Table 3 Data collected from 90 degree bend trials

9

6. Calculation and Results

6.1 Friction Factor results:

Trial V(m/s) Re fexperimental ftheoretical Error (%)

1 1.9056 32395 0.03306 0.04817 31.37

2 2.9080 49437 0.02603 0.04761 45.32

3 3.3968 57746 0.02341 0.04746 50.67

Table 4 Calculations of Reynold’s number, friction factors and percentage error

Sample Calculation based on trial 1:

1) Velocity : 𝑉 = !!= !.!!∗!"!!

!!∗ !.!"

! = 1.9056!!

2) Experimental friction factor: 𝑓 = !!" !!!!! ∗!"!!

!!∗!= !∗!.!"#∗!.!"∗ !"#!!" ∗!"!!

!.!"#$!∗!= 0.03306

3) Reynold’s number: 𝑅𝑒 = !∗!!= !.!"#$∗!.!"#

!"!!= 32395

4) Theoretical friction factor: 𝑓 = 0.25

log 𝐾𝑠3.7𝐷+

5.74𝑅𝑒0.9

2 =0.25

log 0.33.7∗17+

5.7426678.40.9

2

5) 𝐸𝑟𝑟𝑜𝑟 = !experimental!!theoretical!theoretical

∗ 100 = !.!"#$%!!.!""!#!.!"#$%

∗ 100 = 31.37%

6.2 Kc results:

Trial Vu(m/s) Vd(m/s) Kcexperimental Kctheoretical Error (%)

1 0.3442 0.8811 0.6688 0.06 1014.7

2 0.5253 1.3448 0.4547 0.06 657.8

3 0.6136 1.5707 0.2660 0.06 343.3

Table 5 Calculation of Kc and the percentage error

10


1) Velocities:

a. V1 =𝑄𝐴 =

!!!!!

= !∗!.!!∗!"!!

!(!"∗!"!!)!= 0.3442

m

s

b. V1 =𝑄𝐴 =

!!!!!

= !∗!.!!∗!"!!

!(!"∗!"!!)!= 0.8811

m

s

2) !!! + !!

!

!!= !!

!+ !!

!

!!+ 𝑘𝑐à

𝑉𝑑2

2𝑔𝑘𝑐 =

2𝑔𝑉22

ℎ𝑢 +𝑉𝑢2

2𝑔 − ℎd−𝑉𝑑2

2𝑔 = 2 ∗9.81

0.8811265 ∗ 10−2 +

0.34422

2 ∗ 9.81 − 59 ∗ 10−2 +

0.88112

2 ∗ 9.81

= 0.6688

3) 𝐸𝑟𝑟𝑜𝑟 = !"experimental!!"theoretical!"theoretical

∗ 100 = 1014.7%

6.3 Kb Results:

Trial V(m/s) Kbexperimental Kbtheoretical Error(%)

1 1.906 0.8101 0.9 9.999

2 2.908 0.5104 0.9 43.29

3 3.397 0.4251 0.9 52.77

Table 6 Calculations of Kb and percentage error


1) Kbexperimental: !!! + 𝑧1 = !!!+ 𝑧2 + 𝑘𝑏

!!!

2𝑔 à

Kb = !!(!!!!!! ∆!)!!

=!∗!.!" !"!!"!! ∗!"!!

!.!"#!= 0.8101

2) 𝐸𝑟𝑟𝑜𝑟 = !"experimental!!"theoretical!"theoretical

∗ 100 = !.!"#"!!.!!.!

∗ 100 = 9.999 %

11

7. Discussion and Analysis

The flow, in this lab, could be measured in two different ways. The first one is obtained

from the software on the computer while the second one can be measured manually.

The ones obtained by the software were disregarded.

It’s notable that there were some large errors in the determination of f, Kc and Kb.

7.1 General Errors

General errors include:

• Inaccurate measurement of the flow rate manually ( pressing stop watch +-‐ 1-‐2 s

from the actual time needed to fill 10 L)

• Precision of measuring the pipe length is to the nearest mm.

7.2 Friction Factor

An average error of 42.45% is a rather noticeable one (The average error was calculated

by referring to the table comparing the experimental and textbook f). This error might

be due to the fact that we assumed a roughness of 0.3 mm whereas it could have been a

different value.

7.3 Contraction loss coefficient

An average error of 672% is alarming and implies the occurrence of major errors. These

errors could be due to several factors including:

• Neglecting the frictional loss f along the contraction pipe

• Malfunctioning of the sensors in calculating the pressure heads as seen in figure

4 where the technician fixes them after they start dripping water.

• The valve of the pipe including the contraction might be causing increased flow

of fluid than the valve of the straight pipe. Thus, the actual value of Q is greater

than the one calculated initially for the first pipe. Doing the calculation for 1

percent increase in Q leads to an error of 964.9% for first run. Therefore, the

difference in error is 49.8 %.

12

Figure 4 Technician fixing pressure censors

7.4 Bend Loss Coefficient

An average error of 35.35% is noticeable but it’s more acceptable than the errors faced

in contraction and friction loss trials. This error could be due to the following factors:

• Neglecting the frictional losses along the length of the bend.

• Also the flow might be varied in the transition to the new pipe containing the

bend.

8. Conclusions

The calculated friction and loss coefficients for the pipe, bend and contraction had large

errors relative to the theoretical ones exceeding 10-‐14 percent. Therefore, there is a

need to question the quality of the sensors being used to measure the pressure heads

especially when it comes to the contraction section.

The errors calculated were partially a result of several factors: malfunction and

inaccuracy of the pressure head sensors, assuming of Ks instead of determining the

actual value, and not measuring again the flow after the transition from one segment to

another.

13

Acknowledgments

Special thanks for Mr.Rabih our lab assistant for explaining thoroughly the procedure of

the experiment.

Special thanks for Dr. Basha for explaining the continuity and energy equations in class,

and explaining their usage in this experiment.

14

References

Basha, H. (2013). Hydraulics & Laboratory. Beirut, Lebanon.

15

Appendix

Table 7 Kc values for contraction of different angles

Table 8 K values for threaded pipe fittings

Table 9 Ks values for pipes made of different materials

16

Table 10 Moody Diagram

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