Experiment 1 - Friction Losses In Pipes.doc

FACULTY OF CHEMICAL ENGINEERING FACULTY OF CHEMICAL ENGINEERING UNIVERSITI TEKNOLOGI MALAYSIAUNIVERSITI TEKNOLOGI MALAYSIA

FLUID MECHANICS LABORATORYFLUID MECHANICS LABORATORY

TITLE OF EXPERIMENT TITLE OF EXPERIMENT

FRICTION LOSSES IN PIPE (E1)FRICTION LOSSES IN PIPE (E1)

Name Name

Matrix No.Matrix No.

Group / SectionGroup / Section

SupervisorSupervisor

Date of Experiment Date of Experiment

Date of SubmissionDate of Submission

Marks obtained (%)Marks obtained (%)

1

1.0 Objectives

The objectives of this experiment are

i. To measure head loss in pipes for different water flow rates, pipe diameters and pipe roughness.

ii. To estimate the values of loss coefficient for pipes of different flow conditions, diameters and roughness.

iii. To study the effect of the velocity of the fluid, the size (inside diameter) of the pipe, the roughness of the inside of the pipe on the values of loss coefficient.

iv. To study the effect of sudden change in pipe diameter and flow direction on the total energy or head losses in pipes

2.0 Introduction

As an incompressible fluid flows through a pipe, a friction force along the pipe wall is created against the fluid. The frictional resistance generates a continuous loss of energy or total head in the fluid and hence decreases the pressure of the fluid as it moves through the pipe. There are four factors that determine friction losses in pipe

i. The velocity of the fluid. ii. The size (inside diameter) of the pipe

iii. The roughness of the inside of the pipeiv. The length of the pipe

In addition to energy or head loss due to friction, there are always head losses in pipes due to an enlargement or contraction of the flow section, bends, junctions, valves etc., which are commonly known as minor or small losses. When the direction of flow is altered or distorted, energy losses occur which are not recovered are dissipated in eddies and additional turbulence and finally lost in the form of heat. However, this energy must be supplied if the fluid is to be maintained in motion, in the same way, as energy must be provided to overcome friction. In practice, in long pipe lines of several kilometres the

2

effect of minor losses may be negligible. For short pipeline the losses may be greater than those for friction.

3

3.0 Theory

In Bernoulli's equation as shown below, hf represents the head loss due to friction between the fluid and the internal surface of the constant diameter pipe as well as the friction between the adjacent fluid layers

p1/g + V12/2g + Z1 = p2 / g + V2

2/2g + Z2 + hf (1)

This will result in a continuous change of energy from a valuable mechanical form (such as kinetic or potential energies) to a less valuable thermal form that is heat. This change of energy is usually referred to as friction head loss, which represents the amount of energy converted into heat per unit weight of fluid.

The head losses (hf) in pipe due to friction can be determined using Darcy-Weisback equation;

Turbulent flow hf = 4 fLV 2 (2) 2 gD

Laminar flow hf = 32 fLQ 2 (3)2gD5

Where:f = Friction factorL = LengthV = Mean velocity (Q/A)g = GravityD = Constant diameter

The friction head loss for both laminar and turbulent flows can be expressed by similar formulas although the original derivation of each one is different:

(4)

In laminar flow, the friction factor is only a function of Reynolds number while for turbulent flow it is a function of Reynolds (Re) number and the relative roughness of the pipe.

(5)

where : density, V: average velocity, D: pipe inside diameter, : viscosity.

4

Based on the nature of the flow, friction factor (f ) can be estimated using the following correlations

Laminar flow f = 64 (6)

Re

Turbulent Flow f = 0.316 x Re -0.25 (7)

Equation (7) is Blausius Equation and only valid for smooth pipe and 3000 < Re< 105. The value of f for turbulent flow can be obtained experimentally from the Moody Chart.

Moreover, for turbulent flow, the relationship between hf and V takes the form

hf = K. Vn (8)

where K is a loss coefficient and n ranges from 1.7 to 2.0 (depending on the value of Re and ks/D).This equation can be written as

Log hf = Log K + n Log V (9)

in order to find K and n experimentally, using graph

Experimentally, one can obtain the head loss by applying energy equation between any two points along a constant diameter pipe. This is done in Eq. 1 and by noticing that the pipe is horizontal and the diameter is constant. The pressure heads of a fluid between 2 points , h1 and h2, are measured by using Piezometer tubes. The total head loss can be determined experimentally by applying the Bernoulli’s equation as follows:

hf = (P1- P2) /g = h1 - h2 (10)

Energy losses are proportional to the velocity head of the fluid as it flows around an elbow, through an enlargement or contraction of the flow section, or through a valve. Experimental values for energy losses are usually reported in terms of a resistance or loss coefficient K as follows:

(11)

where hL is the minor loss, K is the resistance or loss coefficient, and V is the average velocity of flow in the pipe in the vicinity where the minor loss occurs. The resistance or

5

loss coefficient is dimensionless because it represents a constant of proportionality between the energy loss and the velocity head. The magnitude of the resistance coefficient depends on the geometry of the device that causes the loss and sometimes on the velocity of flow.

Minor losses at sudden enlargement

When a fluid flows from a smaller pipe into a larger pipe through a sudden enlargement, its velocity abruptly decreases, causing turbulence, which generates an energy loss.

where,V1 = velocity at small cross-section (upstream)V2 = velocity at large cross-section (downstream)

The minor loss (hL) due to sudden enlargement of the pipe can be estimated by integrating the momentum, continuity and Bernoulli equations between positions 1 and 2 to give

(12)

Substituting again for the continuity equation to get an expression involving the two areas, (i.e. V2=V1(A1/A2) gives

(13)

Where ,

Minor losses at sudden contraction

When a fluid flows from a larger pipe into a smaller pipe through a sudden contraction, the fluid streamlines will converge just downstream of the smaller pipe, known as vena

6

contraction phenomena, creating a turbulence region from the sharp corner of the smaller pipe and extends past the vena contracta, which subsequently generates an energy loss.

In a sudden contraction, flow contracts from point 1 to point 1', forming a vena contraction. It is possible to assume that energy losses from 1 to 1' are negligible (no separation occurs in contracting flow) but that major losses occur between 1' and 2 as the flow expands again

If the vena contracta area is A1’=Ac, then the minor loss (hL) can be estimated by integrating the momentum , continuity and Bernoulli equations between positions 1 and 2 to give

(14)

7

The above equation is commonly expressed as a function of loss coefficient (K) and the average velocity (V2) in the smaller pipe downstream from the contraction as follows;

(15)

Where

As the difference in pipe diameters gets large (A1/A2 0) then this value of K will tend towards 0.5 which is equal to the value for entry loss from a reservoir into a pipe. The value of K depends upon the ratio of the pipe diameters (D2/D1) as given below;

D2/D1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

K 0.5 0.45 0.412 0.39 0.36 0.33 0.28 0.15 0.15 0.06 0

Minor Losses at elbow or bend pipe

Losses in fittings such as elbow, valves etc have been found to be proportional to the

velocity head of the fluid flowing. The energy loss is expressed in the general form,

(16)

where,

K = loss coefficient (dependent on the ratio of total angle of bending to

radius of bending (R/d) of the curves as the bending occurs)

8

Experimental determination of total head loss

In the experiment the pressure heads before and after a fluid undergoing sudden change in pipe diameter or flow direction, h1 and h2,

are measured by using Piezometer tubes. The total head loss (major and minor losses) can be determined experimentally by applying the Bernoulli’s equation as follows:

P1/g + Vl 2 / 2 g + Z1 = P2/g + V2

2 / 2 g + Z2 + hL (17)

hl + Vl 2 / 2 g + Z1 = h2 + V2

2 / 2 g + Z2 + hL (18)

and since Z1 = Z2 , then (19)

9

4.0 Apparatus

10

Linear Pipe Section Diameter (mm) Length (mm)

1A (rough)

B (smooth)

25.0

23.5

1030

1030

2A (rough)

B (smooth)

14.0

13.3

1030

1030

Note 1: Q (m3/s) = Q (1/min) x 1.667 x 10-5

Note 2 : Reynold Number for linear pipe assumed at room temperature

Pipe 1A: Re = 29.2 x 103 x V

Pipe 1B: Re = 27.5 x 103 x V

Pipe 2A: Re = 16.4 x 103 x V

Pipe 2B: Re = 15.5 x 103 x V

Table of Water Dynamic Viscosity and Density at Different Temperatures

Temperature (oC) (kg/m3) (x 10-3 N.s/m3)

0 999.8 1.781

5 1000.0 1.518

10 999.7 1.307

15 999.1 1.139

20 998.2 1.002

25 997.0 0.890

30 995.7 0.798

40 992.2 0.653

50 988.0 0.547

60 983.2 0.466

70 977.8 0.404

80 971.8 0.354

90 965.3 0.315

100 953.4 0.282

5.0 Experimental Procedure

11

Open all outlet valves of pipes 1, 2 and 4 (valves are in parallel with the pipes). Make certain that the control valve in closed position (turn clockwise). Switch on the pump and slowly open the control valve (turn counter-clockwise) until maximum, and wait for a while in order to remove any air bubble in the flowing pipe.

Important Note:To identify which inlet flowing pressure (H1) and outlet flowing pressure (H2) during installation of water manometer rubber tube, determine the direction of water inflow and outflow through the pipe.

A) Experiment with Pipe 2A: Rough Surface

1. Connect rubber tube of water manometer at inlet flowing pressure (H1) and outlet flowing pressure (H2) for rough surface of Pipe 2A.

2. Reduce the flow rate (Q) by slowly closing the control valve (turn clockwise) until flow rate of 26 liter/minute is achieved. Then, rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time close the outlet valves of pipes 1 and 4 (turn clockwise) but let only the outlet valve of pipe 2 open . At this moment the flowing system is for pipe 2A of rough surface. During the process, if air bubbles present in the flowing pipe, the air will move through the water manometer rubber tube. Air bubbles will move to the peak of the higher tube. Remove the air bubbles up to the manometer glass tube.

3. Readjust the flow rate to 26 liter/minute, and determine 5 (five) flow rates Q from value of 26 liter/minute to the lowest value 12 liter/min (let the increment as large as possible). Record the values of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

B) Experiment with Pipe 2B: Smooth Surface

1. Move manometer rubber tube from the outlet flowing pressure (H2) of the rough surface of pipe 2A to inlet flowing pressure (H1) of smooth surface of pipe 2B. The system is now flowing through pipe 2B (smooth surface).

2. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches 26 liter/minute. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

12

3. Determine 5 (five) flow rates (Q), similar to Pipe 2A (rough surface). Record the values of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

C) Experiment with Pipe 1A: Rough Surface

1. Move both manometer rubber tubes of inlet (H1) and outlet (H2) flowing pressures of pipe 2B (smooth surface) to the section of pipe 1A (rough surface).

2. Open the outlet valve of pipe 1 (turn counter-clockwise), and close the outlet valve of pipe 2 (turn clockwise). The system is now flowing through pipe 1A (rough surface).

3. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches maximum value 42 liter/min. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

4. Readjust the flow rate to appropriate maximum value 42 liter/min, and determine 5 (five) different flow rates Q from manimum value to the lowest value 12 liter/min (let the increment as large as possible). Record the values of H1 and H2 in (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

D) Experiment with Pipe 1B: Smooth Surface

1. Move the manometer rubber tubes from outlet (H2) flowing pressures of pipe 1A (rough surface) to inlet (H1) flowing pressure of pipe 1B (smooth surface). The system is now flowing through pipe 1B (smooth surface).

2. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches maximum value 42 litter/min. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

3. Determine 5 (five) different flow rates Q similar to pipe 1A (rough surface). Record the values of H1 and H2 in (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

13

E) Experiment with Pipe 4: Sudden Enlargement

1. Move both manometer rubber tubes of inlet (H1) and outlet (H2) flowing pressures of pipe 1B (smooth surface) to the section of pipe 4 (Sudden Enlargement).

2. Open the outlet valve of pipe 4 (turn counter-clockwise), and close the outlet valve of pipe 1 (turn clockwise). The system is now flowing through pipe 4 (Sudden Enlargement).

3. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches 30 liter/minute value. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

4. Readjust the flow rate to 30 liter/min, and determine 5 (five) flow rates Q from value of 30 liter/minute to the lowest value 12 liter/min (let the increment as large as possible). Record the values of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

F) Experiment with Pipe 4: Sudden Contraction

1. Move the manometer rubber tubes from the inlet flowing pressure (H1) of pipe 4 (sudden enlargement) to the outlet flowing pressure (H2) of pipe 4 (sudden contraction). The system is now flowing through pipe 4 (sudden contraction).

2. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open control valve (turn counter-clockwise) until flow rate Q reaches value 30 liter/minute. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

3. Readjust the flow rate to appropriate value 30 liter/minute, and determine 5 (five) different flow rates Q from value 30 liter/minute to the lowest value 12 liter/min. (let the increment as large as possible). Record the values of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressures as Q is changed.

14

G) Experiment with Pipe 4: 90o Bend

1. Move the manometer rubber tube from the inlet flowing pressure (H1) of pipe 4 (sudden contraction) to the outlet flowing pressure (H2) of pipe 4 (90o bend). The system is now flowing through pipe 4 (90o bend).

2. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches value 30 liter/minute. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

3. Readjust the flow rate to appropriate maximum value 30 liter/min, and determine 5 (five) different flow rates Q from value 30 liter/min to the lowest value 12 liter/min. (let the increment as large as possible). Record the value of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressure as Q is changed.

H) Experiment with Pipe 4: Elbow

1. Move the manometer rubber tube from the inlet flowing pressure (H1) of pipe 4 (90o bend) to the outlet flowing pressure (H2) of pipe 4 (elbow). The system is now flowing through pipe 4 (elbow).

2. Rise both water manometer rubber tubes at inlet flowing pressure (H1) and outlet flowing pressure (H2) while at the same time slowly open the control valve (turn counter-clockwise) until flow rate Q reaches value 30 liter/minute. During the process, if air bubbles present in the flowing pipe, the air will move through the higher end of water manometer rubber tube. Remove the air bubbles up to the manometer glass tube.

3. Readjust the flow rate to appropriate maximum value 30 liter/minute, and determine 5 (five) different flow rates Q from value 30 liter/minute to the lowest value (let the increment as large as possible). Record the value of H1 and H2 in millimeter (mm) of the inlet and the outlet of water manometer flowing pressure as Q is changed.

15

6.0 Experimental data and analysis

PipeQ

(1/min)Q x 10-4

(m3/s)h1

(mm)h2

(mm)A

(m2x10-4)V

(m/s)Re

(x 103)

ftheo

(Eq 6 or Eq. 7 or Moody

diagram)

hf.theo

(Eq. 4)

hf.exp

h=h1-h2)

(m)

fexp

(Eq. 10)

2A

26 4.33 920 35 4.91 0.882 14.46 0.162 0.473 0.885 0.303

22 3.67 770 155 4.91 0.747 12.25 0.168 0.352 0.615 0.294

18 3.00 660 240 4.91 0.611 10.02 0.177 0.248 0.420 0.300

14 2.33 565 320 4.91 0.475 7.790 0.189 0.160 0.245 0.290

12 2.00 530 350 4.91 0.407 6.670 0.196 0.122 0.180 0.289

2B

26 4.33 915 185 4.34 0.998 15.47 0.159 0.625 0.730 0.186

22 3.67 780 245 4.34 0.846 13.11 0.166 0.469 0.535 0.189

18 3.00 660 300 4.34 0.691 10.71 0.174 0.328 0.360 0.191

14 2.33 570 350 4.34 0.537 8.320 0.185 0.211 2.220 0.193

12 2.00 535 370 4.34 0.461 7.150 0.193 0.162 0.165 0.197

16

PipeQ

(1/min)Q x 10-4

(m3/s)h1

(mm)h2

(mm)A

(m2x10-4)V

(m/s)Re

(x103)

ftheo

(Eq 6 or Eq. 7 or Moody

diagram)

hf.theo

(Eq. 4)

hf.exp

h=h1-h2)

(m)

fexp

(Eq. 10)

1A 28 4.67 390.2 300 4.91 0.951 28.33 0.137 0.260 0.09 0.047

24 4.00 373 315 4.91 0.815 24.29 0.142 0.198 0.06 0.043

20 3.33 360 330 4.91 0.678 20.20 0.149 0.144 0.03 0.031

16 2.67 353 340 4.91 0.544 16.21 0.157 0.098 0.013 0.021

12 2.00 349 345 4.91 0.407 12.13 0.169 0.059 0.004 0.011

1B 28 4.67 384 305 4.34 1.076 29.59 0.135 0.349 0.079 0.031

24 4.00 370 315 4.34 1.168 32.12 0.132 0.402 0.055 0.018

20 3.33 365 317 4.34 1.402 38.56 0.126 0.553 0.048 0.011

16 2.67 363 320 4.34 1.749 48.10 0.120 0.820 0.043 0.006

12 2.00 365 325 4.34 2.335 64.21 0.111 1.352 0.04 0.003

Table for Data of Sudden Enlargement Pipe

17

Q(1/min)

Qx10-4

(m3/s)h1

(mm)h2

(mm)h(m)

A1

(m2x10-4)A2

(m2x10-4)V1

(m/s)V2

(m/s)

hL,theo

(m)Eq. 2

hL,exp

(m)Eq. 9

K

hL,exp /(V12/2g)

28 4.67 5356 525 0.01 1.39 4.26 3.360 1.096 0.261 0.524 0.911

24 4.00 490 485 0.005 1.39 4.26 2.878 0.939 0.192 0.382 0.905

20 3.33 445 440 0.005 1.39 4.26 2.396 0.728 0.133 0.266 0.909

16 2.67 405 400 0.005 1.39 4.26 1.921 0.627 0.085 0.173 0.920

12 2.00 365 363 0.002 1.39 4.26 1.439 0.469 0.048 0.096 0.910

Table for Data of Sudden Contraction Pipe

Q(1/min)

Qx10-4

(m3/s)h1

(mm)h2

(mm)h(m)

A2

(m2x10-4)A2

(m2x10-4)V1

(m/s)

V2

(m/s)

hL,theo

(m)Eq. 5

hL,exp

(m)Eq. 9

K

hL,exp /(V22/2g)

28 4.67 845 280 0.565 4.26 1.39 1.096 3.360 0.288 0.051 0.089

24 4.00 765 300 0.465 4.26 1.39 0.939 2.878 0.211 0.088 0.208

20 3.33 635 335 0.300 4.26 1.39 0.782 2.396 0.146 0.039 0.133

16 2.67 560 340 0.220 4.26 1.39 0.627 1.921 0.094 0.052 0.276

12 2.00 485 360 0.125 4.26 1.39 0.469 1.439 0.053 0.031 0.294

Table for Data of 90o Bend Pipe

Q Qx10-4 h1 h2 h A V hL,theo hL,exp K

18

(1/min) (m3/s) (mm) (mm) (m) (m2x10-4) (m/s)(m)

Eq. 6(m)

Eq. 9 hL,exp /(V2/2g)

28 4.67 825 2201 0.605 1.27 3.677 0.262 1.294 1.878

24 4.00 735 250 0.485 1.27 3.150 0.192 0.991 1.960

20 3.33 640 295 0.345 1.27 2.622 0.133 0.695 1.983

16 2.67 550 330 0.220 1.27 2.102 0.086 0.445 1.976

12 2.00 475 355 0.120 1.27 1.575 0.048 0.246 1.946

Table for Data of Elbow Pipe

Q(1/min)

Qx10-4

(m3/s)h1

(mm)h2

(mm)h(m)

A(m2x10-4)

V(m/s)

hL,theo

(m)Eq. 6

hL,exp

(m)Eq. 9

K

hL,exp /(V2/2g)

28 4.67 860 80 0.780 1.27 3.677 0.262 1.469 2.132

24 4.00 755 155 0.600 1.27 3.150 0.192 1.106 2.187

20 3.33 640 230 0.410 1.27 2.622 0.133 0.760 2.169

16 2.67 545 295 0.250 1.27 2.102 0.086 0.475 2.109

12 2.00 475 340 0.135 1.27 1.575 0.048 0.261 2.064

19

20

7.0 Laboratory report

1. See handout (Laboratory Report Format)2. Additional report requirement

i. Plot a graph of ftheo and fexp versus Re on the same graph and comment on the results.

ii. Plot a graph between experimental hf and V on a log-log paper to obtain the values of K and n in eq. (9) for turbulent flow in a pipe. Use log-log paper and remember that n, the slope of the straight line, is given as n = (log hf 1 - log hf 2 ) / (log V1 - log V 2 ). The y-intercept gives the value of log K.

a. Calculate the value of n. Theoretically, the head loss due to friction is proportional to the velocity of the flow (i.e. hf = kV2/2). Comment on the obtained value of n.

b. Discuss the effect of fluid velocity, pipe roughness and pipe diameter on the value of loss coefficient (K) and hence friction loss in pipe.

iii. Plot (hL)th and (hL )exp versus Q on the same graph. Compare the difference between the experimental and theoretical results and discuss on the effect of fluid flowrate on energy loss.

iv. Plot on the same graph paper graphs of (hL )exp versus V21/2g (sudden

enlargement), V22/2g (sudden contraction), V2/2g (90o bend) and

V2/2g (90o elbow)

a. Estimate the value of loss coefficient (K) , i.e. slope of the graph, for each flow condition

b. Compare the K values and briefly discuss the effect of pipe geometry on the value of loss coefficient and hence energy loss in pipe.

c. Compare the average value of K with the theoretical value for the experiments involving sudden pipe enlargement and 90o bend.

v. Briefly discuss factors contributing to errors or inaccuracy in experimental data and propose recommendation to improve the results

21

GRAPH RESULT:

22

23

24

25

26

1.value of n:

n = (log hf 1 - log hf 2 ) / (log V1 - log V 2 )

pipe 2A: (log 0.84 – log 0.32)/ (log 0.84 – log 0.46)

= 1.4688

pipe 2B: (log 0.54 – log 0.16)/ (log 0.82 – log 0.44)

= 1.0525

pipe 1A: (log 0.075 – log 0.01)/ (log 0.9 – log 0.5)

= 0.1612

pipe 1B: (log 0.05 – log 0.04)/ (log 01.35 – log 2.1)

= 0.0234

27

28

29

30

31

32

Result and discussion:

1.

33