Water Flow in Pipes

The Islamic University of Gaza

Faculty of Engineering

Civil Engineering Department

Hydraulics - ECIV 3322

Chapter 3Chapter 3

2

3.1 Description of A Pipe Flow

• Water pipes in our homes and the distribution system

• Pipes carry hydraulic fluid to various components of vehicles and machines

• Natural systems of “pipes” that carry blood throughout our body and air into and out of our lungs.

3

• Pipe Flow: refers to a full water flow in a closed conduits or circular cross section under a certain pressure gradient.

• The pipe flow at any cross section can be described by:

cross section (A), elevation (h), measured with respect to a horizontal

reference datum. pressure (P), varies from one point to another, for a

given cross section variation is neglected The flow velocity (v), v = Q/A.

4

Difference between open-channel flow and the pipe flow

Pipe flow• The pipe is completely filled with the fluid being transported.

• The main driving force is likely to be a pressure gradient along the pipe.

Open-channel flow• Water flows without completely filling the pipe.

• Gravity alone is the driving force, the water flows down a hill.

5

Types of Flow Steady and Unsteady flow The flow parameters such as velocity (v), pressure (P)

and density () of a fluid flow are independent of time in a steady flow. In unsteady flow they are independent.

0ooo ,z,yxtvFor a steady flow

0ooo ,z,yxtvFor an unsteady flow

If the variations in any fluid’s parameters are small, the average is constant, then the fluid is considered to be steady

6

Uniform and non-uniform flow

A flow is uniform if the flow characteristics at any given

instant remain the same at different points in the

direction of flow, otherwise it is termed as non-uniform

flow.

0ot

svFor a uniform flow

For a non-uniform flow 0ot

sv

7

Examples: The flow through a long uniform pipe diameter at a constant rate is

steady uniform flow.

The flow through a long uniform pipe diameter at a varying rate is

unsteady uniform flow.

The flow through a diverging pipe diameter at a constant rate is a

steady non-uniform flow.

The flow through a diverging pipe diameter at a varying rate is an

unsteady non-uniform flow.

8

Laminar and turbulent flowLaminar flow:

Turbulent flow:

The fluid particles move along smooth well defined path or streamlines that are parallel, thus particles move in laminas or layers, smoothly gliding over each other.

The fluid particles do not move in orderly manner and they occupy different relative positions in successive cross-sections. There is a small fluctuation in magnitude and direction of the velocity of the fluid particles

transitional flowThe flow occurs between laminar and turbulent flow

9

3.2 Reynolds Experiment

Reynolds performed a very carefully prepared pipe flow experiment.

10

Increasing flow

velocity

11

Reynolds Experiment

• Reynold found that transition from laminar to turbulent flow in a pipe depends not only on the velocity, but only on the pipe diameter and the viscosity of the fluid.

• This relationship between these variables is commonly known as Reynolds number (NR)

ForcesViscous

ForcesInertialVDVDNR

It can be shown that the Reynolds number is a measure of the ratio of the inertial forces to the viscous forces in the flow

FI ma AFV

12

Reynolds number

VDVD

NR

whereV: mean velocity in the pipe[L/T]D: pipe diameter [L]: density of flowing fluid [M/L3]: dynamic viscosity [M/LT]: kinematic viscosity [L2/T]

13

14

Flow laminar when NR < Critical NR

Flow turbulent when NR > Critical NR

It has been found by many experiments that for flows in circular pipes, the critical Reynolds number is about 2000

The transition from laminar to turbulent flow does not always

happened at NR = 2000 but varies due to experiments

conditions….….this known as transitional range

15

Laminar flows characterized by:

• low velocities• small length scales

• high kinematic viscosities

• NR < Critical NR

• Viscous forces are

dominant.

Turbulent flows characterized by

• high velocities

• large length scales

• low kinematic viscosities

• NR > Critical NR

• Inertial forces are

dominant

Laminar Vs. Turbulent flows

16

Example 3.1 40 mm diameter circular pipe carries water at 20oC.

Calculate the largest flow rate (Q) which laminar flow can be expected.

mD 04.0

CTat o20101 6

sec/1028.6)04.0(4

05.0. 352 mAVQ

sec/05.02000101

)04.0(2000

6mV

VVDNR

17

3.3 Forces in Pipe Flow

• Cross section and elevation of the pipe are varied along the axial direction of the flow.

18

)(.. '22'11 massfluidfluxmassdVoldVol

Conservation law of mass

Mass enters the control volume

Mass leaves the control volume

QVAVAdt

dSA

dt

dSA

dt

dVol

dt

dVol

.......

..

22112

21

1

'22'11

QVAVA 2211 ..

For Incompressible and Steady flows:

Continuity equation for Incompressible Steady flow

19

Apply Newton’s Second Law:

t

VMVM

dt

VdMaMF

12

xxx WFAPAPF 2211

)(.

)(.

)(.

.

12

12

12

zzz

yyy

xxx

VVQF

VVQF

VVQF

rateflowmassQtMbut

Fx is the axial direction force exerted on the control volume by the wall of the pipe.

)(. 12

VVQF

Conservation of moment equation

20

dA= 40 mm, dB= 20 mm, PA= 500,000 N/m2, Q=0.01m3/sec. Determine the reaction force at the hinge.

Example 3.2

21

3.4 Energy Head in Pipe Flow

Water flow in pipes may contain energy in three basic forms:

1- Kinetic energy,

2- potential energy,

3- pressure energy.

22

Consider the control volume: • In time interval dt:

- Water particles at sec.1-1 move to sec. 1`-1` with velocity V1.

- Water particles at sec.2-2 move to sec. 2`-2` with velocity V2.

• To satisfy continuity equation:

dtVAPdsAP ..... 111111

• The work done by the pressure force

dtVAdtVA .... 2211

dtVAPdsAP ..... 222222

-ve sign because P2 is in the opposite direction to distance traveled ds2

……. on section 1-1

……. on section 2-2

23

• The work done by the gravity force :

).(.. 2111 hhdtVAg

)(...2

1.

2

1.

2

1 21

2211

21

22 VVdtVAVMVM

)(..2

1).(...... 2

12

22121 VVdtQhhdtQgdtQPdtQP

• The kinetic energy:

The total work done by all forces is equal to the change in kinetic energy:

Dividing both sides by gQdt

22

22

11

21

22h

P

g

Vh

P

g

V

Bernoulli Equation

Energy per unit weight of waterOR: Energy Head

24

Energy head and Head loss in pipe flow

25

11

21

1 2h

P

g

VH

22

22

2 2h

P

g

VH

Kinetic head

Elevation head

Pressure head

Energy head

= + +

LhhP

g

Vh

P

g

V 2

22

21

12

1

22

Notice that:• In reality, certain amount of energy loss (hL) occurs when the water mass flow from one section to another.

• The energy relationship between two sections can be written as:

26

Example 3.3 &3.4

ExampleIn the figure shown:Where the discharge through the system is 0.05 m3/s, the total losses through the pipe is 10 v2/2g where v is the velocity of water in 0.15 m diameter pipe, the water in the final outlet exposed to atmosphere.

Calculate the required height (h =?)below the tank

mh

h

hzg

V

g

pz

g

V

g

p

smV

smV

L

AQ

AQ

147.2181.9*2

83.21020

81.9*2

366.60)5(00

22

/366.610.0

05.0

/83.215.0

05.0

22

2

222

1

211

24

24

Without calculation sketch the (E.G.L) and (H.G.L)

30

Basic components of a typical pipe system

31

Calculation of Head (Energy) Losses:

In General:When a fluid is flowing through a pipe, the fluid experiences some

resistance due to which some of energy (head) of fluid is lost.

Energy Losses(Head losses)

Major Losses Minor losses

loss of head due to pipe friction and to viscous dissipation in flowing water

Loss due to the change of the velocity of the flowing fluid in the magnitude or in direction as it moves through fitting like Valves, Tees, Bends and Reducers.

3.5 Losses of Head due to Friction

• Energy loss through friction in the length of pipeline is commonly termed the major loss hf

• This is the loss of head due to pipe friction and to the viscous dissipation in flowing water.

• Several studies have been found the resistance to flow in a pipe is:

- Independent of pressure under which the water flows

- Linearly proportional to the pipe length, L- Inversely proportional to some water power of the

pipe diameter D- Proportional to some power of the mean velocity, V- Related to the roughness of the pipe, if the flow is

turbulent

33

Major losses formulas• Several formulas have been developed in the past.

Some of these formulas have faithfully been used in various hydraulic engineering practices.

1. Darcy-Weisbach formula

2. The Hazen -Williams Formula

3. The Manning Formula

4. The Chezy Formula

5. The Strickler Formula

34

The resistance to flow in a pipe is a function of:

• The pipe length, L

• The pipe diameter, D

• The mean velocity, V

• The properties of the fluid ()

• The roughness of the pipe, (the flow is turbulent).

Darcy-Weisbach Equation

25

22

8

2

Dg

QLf

g

V

D

LfhL

Where: f is the friction factorL is pipe lengthD is pipe diameterQ is the flow ratehL is the loss due to friction

It is conveniently expressed in terms of velocity (kinetic) head in the pipe

The friction factor is function of different terms:

D

eVDF

D

eVDF

D

eNFf R ,,,

Renold number Relative roughness

36

Friction Factor: (f)

• For Laminar flow: (NR < 2000) [depends only on Reynolds’ number and not on the surface roughness]

RN

64f

• For turbulent flow in smooth pipes (e/D = 0) with 4000 < NR < 105 is

4/1

316.0

RNf

37

2.51log2

1 fN

fR

510e

7.3log21

RNfor

D

f• Colebrook-White Equation for f

fND

e

f R

51.2

7.3ln86.0

1

For turbulent flow ( NR > 4000 ) with e/D > 0.0, the friction factor can be founded from:• Th.von Karman formulas:

There is some difficulty in solving this equationSo, Miller improve an initial value for f , (fo)

2

9.0

74.5

7.3log25.0

R

oND

ef

The value of ffoo can be use directly as ff if: 26

83

101101

101104-

R

D e

N

Friction Factor f

e7.1'

ee 7.108.0 '

RNf

64

pipe wall

e

51.2log2

110

fN

fR

e

pipe wall

transitionallyrough

e

pipe wall

rough

f independent of relative roughness e/D

f independent of NR

f varies with NR and e/D

turbulent flow

NR > 4000

laminar flow

NR < 2000

e08.0'

e

D

f7.3log2

110

fN

De

f R

51.2

7.3log2

110

Colebrook formula

The thickness of the laminar sublayer decrease with an increase in NR

Smooth

Moody diagram

• A convenient chart was prepared by Lewis F. Moody A convenient chart was prepared by Lewis F. Moody and commonly called the Moody diagram of friction and commonly called the Moody diagram of friction factors for pipe flow,factors for pipe flow, There are 4 zones of pipe flow in the chart:

• A laminar flow zone where f is simple linear function of NR

• A critical zone (shaded) where values are uncertain because the flow might be neither laminar nor truly turbulent

• A transition zone where f is a function of both NR and relative roughness

• A zone of fully developed turbulence where the value of f depends solely on the relative roughness and independent of the Reynolds Number

40

Laminar

Marks Reynolds Number independence

42

Typical values of the absolute roughness (e) are given in table 3.1

43

Notes:

• Colebrook formula

is valid for the entire nonlaminar range (4000 < Re < 108) of the Moody chart

12

3 7

2 51

f

e D

f

log

/

.

.

Re

In fact , the Moody chart is a graphical representation of this equation

Problems (head loss)

Three types of problems for uniform flow in a single pipe:

Type 1:Given the kind and size of pipe and the flow rate head loss ?

Type 2:Given the kind and size of pipe and the head loss flow rate ?

Type 3:Given the kind of pipe, the head loss and flow rate size of pipe ?

45

The water flow in Asphalted cast Iron pipe (e = 0.12mm) has a diameter 20cm at 20oC. Is 0.05 m3/s. determine the losses due to friction per 1 km

1.59m/sm0.2π/4

/s0.05mV

22

3

56

26

1015.33148521001.1

2.059.1

0006.0200

12.0

12.0

/sm101.01υ20

VD

N

mm

mm

D

e

mme

CT

R

o

f = 0.018 Moody

m

m/s.

.

m.

m,.

g

V

D

Lfh f

55.11

8192

591

200

00010180

2 2

22

Example 1

Type 1:Given the kind and size of pipe and the flow rate head loss ?

The water flow in commercial steel pipe (e = 0.045mm) has a diameter 0.5m at 20oC. Q=0.4 m3/s. determine the losses due to friction per 1 km

sm

A

QV / 037.2

45.0

4.02

013.0

109105.0

045.0

10012.110006.1

037.25.0

10006.15.4220

10497

5.42

10497

53

66

65.1

6

5.1

6

f

D

e

N

T

Moody

R

kmmh f / 5.581.92

037.2

5.0

1000013.0

2

Example 2

Type 1:Given the kind and size of pipe and the flow rate head loss ?

fRD

k

f e

s 51.2

7.3ln86.0

1

Use other methods to solve f

01334.0

10012.1

74.5

7.3

109log25.0

74.5

7.3log25.0

2

9.06

52

9.0

e

so

R

Dkf

678.866.8

01334.0

51.2

7.3

109ln86.0

01334.0

1 5

eR

1- Cole brook

kmmh f / 5.581.92

037.2

5.0

100001334.0

2

Cast iron pipe (e = 0.26), length = 2 km, diameter = 0.3m. Determine the max. flow rate Q , If the allowable maximum head loss = 4.6m. T=10oC

10135.0

81.923.0

20006.4

2

2

2

2

fV

Vf

g

V

D

LfhF

00009.01067.8103.0

26.0

210296.21031.1

3.0

1031.15.4210

10497

5.42

10497

53

66

65.1

6

5.1

6

D

e

VV

N

T

R

Example 3

Type 2:Given the kind and size of pipe and the head loss flow rate ?

02.0

1067.8

10668.2

m/s 16.101.0

4

52

1

f

D

e

N

Vf

Moody

Req

eq

021.0

1067.8

10886.1

m/s 82.002.0

4

52

1

f

D

e

N

Vf

Moody

Req

eq

10135.02 f

V

210296.2 6 VNR

Trial 1

Trial 2

V= 0.82 m/s , Q = V*A = 0.058 m3/s

Example 3.5Compute the discharge capacity of a 3-m diameter, wood stave

pipe in its best condition carrying water at 10oC. It is allowed to have a head loss of 2m/km of pipe length.

hf fL

D

V 2

2g

V 2ghf

L

1/ 2D

f

1/ 2

fV

Vf

12.0

)81.9(23

10002 2

2

Table 3.1 : wood stave pipe: e = 0.18 – 0.9 mm, take e = 0.3 mm

Solution 1:

0001.03

3.0

D

e

At T= 10oC, = 1.31x10-6 m2/sec VVVD

NR .1029.21031.1

3 6

6

Type 2:Given the kind and size of pipe and the head loss flow rate ?

• Solve by trial and error:

• Iteration 1:• Assume f = 0.02 sec/45.2

02.0

12.02 mVV

66 106.545.2.1029.2 RN

From moody Diagram: 0122.0f

Iteration 2:update f = 0.0122 sec/14.3

0122.0

12.02 mVV

66 102.714.3.1029.2 RN

From moody Diagram: 0122.00121.0 f

0 0.02 2.45 5.6106

1 0.0122 3.14 7.2106

2 0.0121

Iteration f V NR

Convergence

Solution:

/sm 2.2724

3.15.3

m/s 15.3

3

2

2

VAQ

V

flow rate ?

Determines relative roughness e/D

2/12/3 2

L

ghDfN f

R

Type 2. Given the kind and size of pipe and the head loss

Given and e/D we can determine f (Moody diagram)fNR

Use Darcy-Weisbach to determine velocity and flow rate

Alternative Method for solution of Type 2 problems

Because V is unknown we cannot calculate the Reynolds number

However, if we know the friction loss hf, we can use the Darcy-Weisbach equation to write:

hf fL

D

V 2

2g

V 2ghf

L

1/ 2D

f

1/ 2

We also know that:

Re VD

Re 1

f 1/ 2

D 3 / 2

2ghf

L

1/ 2

unknowns

2/12/32/1 2

L

ghDfN f

R

Can be calculated based on available data

Quantity plotted along the top of the Moody diagram

Moody Diagram

Lam

inar flo

w

Smooth pipes

Transitionally rough pipes

Fully rough pipes

Res

ista

nce

Co

effi

cien

t f

Reynolds number

Rel

ativ

e ro

ug

hn

ess

e/D

2/12/32/1 2

L

ghDfN f

R

Example 3.5Compute the discharge capacity of a 3-m diameter, wood stave pipe in its best

condition carrying water at 10oC. It is allowed to have a head loss of 2m/km of pipe length.

5

6

232/12/3

1062.91000

)3)(81.9(2

1031.1

)3(2

L

ghDfN f

R

Table 3.1 : wood pipe: e = 0.18 – 0.9 mm, take e = 0.3 mm

Solution 2:

0001.03

3.0

D

e

Type 2: Given the kind and size of pipe and the head loss flow rate ?

At T= 10oC, = 1.31x10-6 m2/sec

From moody Diagram: 0121.0f

sec/15.32

2

2/12/12

mf

D

L

ghV

g

V

D

Lfh f

f

/sm 2.2724

3.15.3,

3

2

VAQ

f = 0.0121

Example (type 2)

H

L

H = 4 m, L = 200 m, and D = 0.05 m

What is the discharge through the galvanized iron pipe?

Table : Galvanized iron pipe: e = 0.15 mm e/D = 0.00015/0.05 = 0.003

= 10-6 m2/s

We can write the energy equation between the water surface in the reservoir and the free jet at the end of the pipe:

Lhg

Vh

p

g

Vh

p

22

2

22

2

2

11

1

g

V

D

Lf

g

V

2200040

22

2

fDL

f

gV

40001

5.78

1

422

1

2

Example (continued)Assume Initial value for f : fo = 0.026

Initial estimate for V: m/sec 865.0026.040001

5.78

V

Calculate the Reynolds number 44 103.4105 VDV

NR

Updated the value of f from the Moody diagram f1 = 0.029

m/sec 819.0029.040001

5.78

V

442 101.4105 VDV

NR

0 0.026 0.865 4.3104

1 0.029 0.819 4.1104

2 0.0294 0.814 4.07104

3 0.0294

Iteration f V NR

Convergence

Solution:

V2 0.814 m/s

Q VA 0.814 0.052

41.60 10 3 m3 /s

e/D = 0.003

Initial estimate for fA good initial estimate is to pick the f value that is valid for a fully rough pipe with the specified relative roughness

fo = 0.026

Solution of Type 3 problems-uniform flow in a single pipe

Given the kind of pipe, the head loss and flow rate size of pipe ?

Determines equivalent roughness e

Problem?Without D we cannot calculate the relative roughness e/D, NR, or fNR

Solution procedure: Iterate on f and D

1. Use the Darcy Weisbach equation and guess an initial value for f2. Solve for D3. Calculate e/D4. Calculate NR

5. Update f6. Solve for D7. If new D different from old D go to step 3, otherwise done

Example (Type 3)A pipeline is designed to carry crude oil (S = 0.93, = 10-5 m2/s) with a discharge of 0.10 m3/s and a head loss per kilometer of 50 m. What diameter of steel pipe is needed? Available pipe diameters are 20, 22, and 24 cm.

From Table 3.1 : Steel pipe: e = 0.045 mm

Darcy-Weisbach:

g

V

D

Lfh f 2

2

2

2

542

22

2

2

1614

22 g

fLQ

DDg

Q

D

Lf

gAQ

D

Lfh f

5/1

2

2

2

16

fhg

fLQD

5/15/1

5/1

2

2

440.05081.92

10.0100016ffD

Make an initial guess for f : fo = 0.015

D 0.440 0.0151/ 5 0.190 m

Now we can calculate the relative roughness and the Reynolds number:

33

2108.66

1107.12

144

DD

QD

D

QD

A

QVDNR

00024.010045.0 3

DD

e

update ff = 0.021

e/D = 0.00024

Updated estimate for f

f1 = 0.021

0 0.015 0.190 66.8103 0.00024

Iteration f D NR e/D

Example Cont’d5/1440.0 fD

update f

DNR

1107.12 3

From moody diagram, updated estimated for f :

f1 = 0.021 D = 0.203 m3105.62 RN

00023.0D

e

1 0.021 0.203 62.5103 0.00023

2 0.021 Convergence

Solution:

D = 0.203

Use next larger commercial size:

D = 22 cm

Example 3.6Estimate the size of a uniform, horizontal welded-steel pipe installed to carry 14

ft3/sec of water of 70oF (20oC). The allowable pressure loss is 17 ft/mi of pipe length.

From Table : Steel pipe: ks = 0.046 mm

Darcy-Weisbach:

hL fLD

V 2

2g

Q VA

hL fLD

QA

2

2gf

LD

Q 2

2g42

2D 4 1

D 5

16fLQ 2

2g 2

5/1

2

28

Lhg

fLQD

afff

D 5/15/1

5/1

2

2

33.41781.9

1452808

Let D = 2.5 ft, then V = Q/A = 2.85 ft/sec

Now by knowing the relative roughness and the Reynolds number:

55

10*6.610*08.1

5.2*85.2

VDNR

0012.05.2

003.0

D

e

We get f =0.021

Solution 2:

A better estimate of D can be obtained by substituting the latter values into equation a, which gives

ftfD 0.2021.0*33.433.4 5/15/1

A new iteration provide V = 4.46 ft/secNR = 8.3 x 105 e/D = 0.0015f = 0.022, andD = 2.0 ft.More iterations will produce the same results.

Empirical Formulas 1• Hazen-Williams

UnitsSISRCV hHW54.063.085.0

tCoefficien iamsHazen Will

44

P wetted

A wetted Radius hydraulic

2

CL

hS

D

D

D

R

HW

f

h

UnitsSI

0.71 852.1

87.4852.1 QDC

Lh

HW

f

Sim

pli

fied

UnitsBritishSRCV

mVcmD

hHW54.063.0318.1

sec/0.35

tCoefficien iamsHazen WillHWC

68

tCoefficien iamsHazen WillHWC

69

081.0

sec/0.3

V

VCC

mVWhen

oHoH

Where:CH = corrected valueCHo = value from tableVo = velocity at CHo

V = actual velocity

Empirical Formulas 2

70

Manning Formula

• This formula has extensively been used for open channel designs

• It is also quite commonly used for pipe flows

71

• Manning

tCoefficien M

4 P wetted

A wetted Radius hydraulic

anningnL

hS

D

R

f

h

Sim

pli

fied

UnitsSI

0.3133.5

2

D

nQLh f

2/13/21SR

nV h

72

• n = Manning coefficient of roughness (See Table)• Rh and S are as defined for Hazen-William

formula.

Vn

R Sh1 2 3 1 2/ /

3/16

223.10

D

QLnh f

2233.1

35.6 VnD

Lh f

73

74

The Chezy Formula

V C R Sh 1 2 1 2/ /

2

4

C

V

D

Lh f

where C = Chezy coefficient

75

• It can be shown that this formula, for circular pipes, is equivalent to Darcy’s formula with the value for

[f is Darcy Weisbeich coefficient]

• The following formula has been proposed for the value of C:

[n is the Manning coefficient]

Cg

f

8

C S n

S

n

Rh

230 00155 1

1 230 00155

.

(.

)

76

The Strickler Formula:

V k R Sstr h 2 3 1 2/ /

2

33.135.6

strf k

V

D

Lh

where kstr is known as the Strickler coefficient.

Comparing Manning formula and Strickler formula, we can see that

1

nkstr

77

Relations between the coefficients in Chezy, Manning , Darcy , and Strickler formulas.

nkstr

1

6/1hstr RkC

g

Rfn h

8

3/1

Example

New Cast Iron (CHW = 130, n = 0.011) has length = 6 km and diameter = 30cm.

Q= 0.32 m3/s, T=30o. Calculate the head loss due to friction using:

a) Hazen-William Method

b) Manning Method

33332030130

6000710

710

85218748521

8521

8748521

m . .

. h

Q DC

L.h

...f

.

..HW

f

m

.

.. .h

D

nQ L h

.f

.f

47030

32001106000310

3.10

335

2

335

2

79

Minor losses

It is due to the change of the velocity of the

flowing fluid in the magnitude or in

direction [turbulence within bulk flow as it moves through and

fitting] Flow pattern through a valve

80

• The minor losses occurs du to :

• Valves • Tees• Bends• Reducers• Valves• And other appurtenances

• It has the common form

2

22

22 gA

Qk

g

Vkh LLm

can be the dominant cause of head loss in shorter pipelines

“minor” compared to friction losses in long pipelines but,

Losses due to contractionA sudden contractionA sudden contraction in a pipe usually causes a marked drop in pressure in the pipe due to both the increase in velocity and the loss of energy to turbulence.

g

Vkh cc 2

22Along centerline

Along wall

Value of the coefficient Kc for sudden contraction

VV22

83

Head Loss Due to a Sudden Contraction

h KV

gL L 22

2

g

VhL 2

5.02

2

Head losses due to pipe contraction may be greatly reduced by introducing a gradual pipe transition gradual pipe transition known as a confusor confusor

g

V'k'h cc 2

22

'kc

85

Head Loss Due to Gradual Contraction (reducer or nozzle)

g

VVKh LL 2

21

22

100 200 300 400

KL 0.2 0.28 0.32 0.35

A different set of data is :

Losses due to Enlargement

g

VVhE 2

)( 221

A sudden EnlargementA sudden Enlargement in a pipe

Note that the drop in the energy line is much larger than in the case of a contraction

abrupt expansion

gradual expansion

smaller head loss than in the case of an abrupt expansion

88

Head Loss Due to a Sudden Enlargement

h KV

gL L 12

2

KA

AL

1 1

2

2

h

V V

gL 1 2

2

2

or :

Head losses due to pipe enlargement may be greatly reduced by introducing a gradual pipe gradual pipe transition transition known as a diffusor diffusor

g

VV'k'h EE 2

22

21

90

Head Loss Due to Gradual Enlargement (conical diffuser)

g

VVKh LL 2

22

21

100 200 300 400

KL 0.39 0.80 1.00 1.06

91

Gibson tests

92

Loss due to pipe entranceGeneral formula for head loss at the entrance of a pipe is also expressed in term of velocity head of the pipe

g

VKh entent 2

2

Different pipe inlets

increasing loss coefficient

94

Head Loss at the Entrance of a Pipe (flow leaving a tank)

Reentrant(embeded)KL = 0.8

Sharpedge

KL = 0.5

Wellrounded

KL = 0.04

SlightlyroundedKL = 0.2

h KV

gL L2

2

95

Another Typical values for various amount of rounding of the lip

96

Head Loss at the Exit of a Pipe (flow entering a tank)

hV

gL 2

2

the entire kinetic energy of the exiting fluid (velocity V1) is dissipated through viscous effects as the stream of fluid mixes with the fluid in the tank and eventually comes to rest (V2 = 0).

KL = 1.0 KL = 1.0

KL = 1.0 KL = 1.0

97

Head Loss Due to Bends in Pipes

R/D 1 2 4 6 10 16 20

Kb 0.35 0.19 0.17 0.22 0.32 0.38 0.42

g

Vkh bb 2

2

98

Miter bends

For situations in which space is limited,

99

Head Loss Due to Pipe Fittings (valves, elbows, bends, and tees)

h KV

gv v2

2

100

101

The loss coefficient for elbows, bends, and tees

Loss coefficients for pipe components (Table)

Minor loss coefficients (Table)

Minor loss calculation using equivalent pipe length

f

DkL l

e

Energy and hydraulic grade lines

Unless local effects are of particular interests the changes in the EGL and HGL are often shown as abrupt changes (even though the loss occurs over some distance)

Example In the figure shown two new cast iron pipes in series, D1 =0.6m , D2 =0.4m length of the two pipes is 300m, level at A =80m , Q = 0.5m3/s (T=10oC).there are a sudden contraction between Pipe 1 and 2, and Sharp entrance at pipe 1.Fine the water level at B

e = 0.26mmv = 1.31×10-6Q = 0.5 m3/s

exitcentffL

fBA

hhhhhh

hZZ

21

g

Vk

g

Vk

g

Vk

g

V

D

Lf

g

V

D

Lfh exitcentL 22222

22

22

21

22

2

22

21

1

11

01800170

000650000430600

26.0

102211018

sec98340

4

50sec771

604

50

21

11

6222

5111

222

211

.f .f

,.D

, .D

,.υ

DV R , .

υ

DVR

, m/..

π.

A

Q, V m/.

.π

.

A

QV

moodymoody

ee

1 ,27.0 ,5.0 exitcent hhh

Solution

m.g

.

g

..

g

..

g

. .

. .

g

. .

. .h f

36132

983

2

983270

2

77150

2

983

40

3000180

2

771

60

3000170

222

22

ZB = 80 – 13.36 = 66.64 m

g

Vk

g

Vk

g

Vk

g

V

D

Lf

g

V

D

Lfh exitcentL 22222

22

22

21

22

2

22

21

1

11

Example

A pipe enlarge suddenly from D1=240mm to D2=480mm. the H.G.L rises by 10 cm calculate the flow in the pipe

smAVQsmV

g

V

g

VV

g

V

g

V

VV

VV

AVAV

g

VV

g

V

g

V

zg

pz

g

ph

g

V

g

V

hzg

V

g

pz

g

V

g

p

L

L

/103.048.057.0/57.0

1.02

6

1.02

4

22

16

4

48.024.0

1.0222

22

22

324222

22

2

222

22

2

21

242

241

2211

2

212

22

1

11

22

22

21

2

222

1

211

Solution

111

• Note that the above values are average typical values, actual values will depend on the make (manufacturer) of the components.

• See:– Catalogs – Hydraulic handbooks !!