Presentation - Hydraulics 2 - Pipe Flow Theory

7/25/2019 Presentation - Hydraulics 2 - Pipe Flow Theory

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Pipe flow theory


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2

Overview

Introduction

Pipe friction theories

head loss

friction factor

Pipe friction in networks

Local head losses

Re-cap


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3

Hydraulic Head

Headrelates the energy in an incompressible fluidto the height of an equivalent static column of thatfluid

The total energy at a given point in a fluid:

energy associated with the movement of the fluid

energy from pressure in the fluid

energy from the height of the fluid

Head is expressed in units of height such asmeters or feet. It is usually measured as a watersurface elevation, expressed in units of length.


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Introduction

When designing pipelines, engineers are primarilyconcerned with the head losses caused by:

friction

! due to viscosity

local head losses

! due to concentrated losses that occur due toeddies that form at abrupt changes in section

e.g. at valves and sharp bends


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Introductionqualitative overview of pipe flow

Recall the Reynolds experiments, whichidentified 3 types of flow

laminar (low vel.)

! smooth dye flow

transitional (medium vel.)

! wavy dye flow

turbulent (high vel.)

! random dye flow

! dye mixes with water


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Results of Reynolds experiments explainedsimply by considering what happens when

parallel streamlines diverge due to a small

disturbance

The small disturbance brings streamlines:

closer together at A

further apart at B

VA, pAand VB, pBpB> pA

A

B


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Pressure difference transverse force B to A this transverse force is opposed by an equal

transverse viscous force

if fluid is moving slowly viscous force is

sufficient to cause the disturbance to die out

if fluid is moving faster, viscous force is not

sufficient to cause the disturbance to die out

flow pattern may disintegrate into disorderlypattern of eddies

stable transition unstable

A

B


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Development of pipe friction theories

Date Researcher Contribution

1841 Hagen and Poiseuille Equation for head loss in laminar flow

1850 Darcy and Weisbach Equation for head loss in turbulent flow

1913 Blasius Friction factor equation for smooth pipes

1930 Nikuradse Friction factor experimental data for artificiallyrough pipes

1930s Prandtl and Karman Friction factor equation for rough and smooth

pipes

193739 Colebrook and White Friction factor experimental data and equation

1944 Moody Friction factor data (Moody diagram)1958 Ackers Friction factor data (HRS charts and tables)

1975 Barr Friction factor equation (explicit solution of

C-W equation)


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Pipe friction theorieshead loss: Hagen and Poiseulle

Experimental work confirmed the equation forfriction loss (hf) in laminar flow (Re< 2000)

D = pipe diameter

L = pipe length

V = mean flow velocity = coefficient of dynamic viscosity

Can be derived from momentum equation and

Newtons law of viscosity

2

32

gD

LVhf


10/5910

Pipe friction theorieshead loss: Hagen and Poiseulle

Note that equation:

does not contain a term for pipe roughness

confirms that roughness has no effect inlaminar flow

indicates that head loss is proportional to ?L, V,1/D

2

32

gD

LVhf


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Pipe friction theorieshead loss: Darcy and Weisbach

Equation for head loss in a turbulent flow:

= friction factor

Be careful as this equation is often written as:

4f = = friction factor

Can be derived from momentum equation

gD

LVhf

2

2

gDfLVhf24

2


12/5912

Pipe friction theorieshead loss: Darcy and Weisbach

Note that equation:

does contain a term for pipe roughness confirms that roughness has an effect in

turbulent flow

indicates that head loss is proportional to ?

L, V,

Initially thought that was constant

later proved to be wrong

gD

LVhf

2

2

D

1


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Pipe friction theoriesfriction factor: Blasius

Experimental work led to equation for frictionfactor in smooth pipes:

Applicable up to Re= 100,000

Underestimates at high Re

highlighted the need to differentiatebetween smooth and rough pipes

41

3160

eR

.


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Pipe friction theoriesfriction factor: Nikuradse

Experimental work with pipes of differentheights of wall roughness (ks)

roughness varied by sticking different

diameters of sand grain to pipe walls

related friction factor to Reand relativeroughness (ks/D)

identified five regions of flow for both

smooth and rough pipes

ks

pipe wall


15/5915

Pipe friction theoriesfriction factor: Nikuradse: region 1: laminar flow

Friction factor depends on Reonly (< 2000) one line

Equating Hagen-Poiseuille and Darcy-Weisbach

DVgD

LV

gD

LVhf

64

2

32 2

eR

64


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Pipe friction theoriesfriction factor: Nikuradse: region 2: laminar to turbulent

Transition between laminar and turbulent flow Can be unstable and ill defined

Typically when: 2000 < Re< 4000


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Pipe friction theoriesfriction factor: Nikuradse: region 3: smooth turbulent

Turbulent flow in hydraulically smooth pipes smooth turbulent flow

i.e. roughness lies within boundary layer

Typically when: 2000 < Re< 4000


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Pipe friction theoriesfriction factor: Nikuradse: region 4: transitional turbulent

Transition: smooth turbulent fully turbulent transitional turbulent flow

i.e. roughness just penetrates boundary layer

Friction factor depends on bothReand ks/D


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Pipe friction theoriesfriction factor: Nikuradse: region 5: rough turbulent

Fully turbulent flow in hydraulically rough pipes rough turbulent flow

i.e. roughness fully penetrates boundary layer

Friction factor depends only on ks/D


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Pipe friction theoriesfriction factor: Prandtl and Karman

Combined theories of turbulent boundary layerflows with experimental results

semi-empirical equations

note that rough equation does not haveVor

Reterm, as depends only on ks

s

e

k

D.log

.Rlog

732

1:pipesrough

51221:pipessmooth

f


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Pipe friction theoriesfriction factor: Colebrook and White

Experimental investigation of commercial pipes,rather than artificially roughened smooth pipes

real pipes have non-uniform roughness size

and spacing

Discovered that - Re curves changed gradually

from smooth to rough turbulence in the turbulent

transition region

Nikuradse diagramshowed tendency

to droop in the

middle

Pi f i i h i


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Also determined effective ksvalues for wide rangeof commercial pipe diameters and materials

4 to 61

drawn brass to concrete lined pipes

Material ks(mm)

brass, copper, glass 0.003

wrought iron 0.06

galvanised iron 0.15

asbestos cement 0.03

plastic 0.03

bitumen-lined ductile iron 0.03

spun-concrete lined iron 0.03

slimed concrete sewer 6.0

Pi f i i h i


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Also combined the Karman-Prandtl equations togive an equation which fitted:

smooth region

transition region

rough region

This yielded the famous Colebrook-White

equation

Re

2.51+

73

klog-2=

1 sD.

from Prandtl and Karman rough pipe equation

from Prandtl and Karman

smooth pipe equation

Pi f i ti th i


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Pipe friction theoriesfriction factor: Moody

As Colebrook-White equation is implicit (onboth sides), it was not used until a graphical

version was developed

Moody diagram

Pi f i ti th i


25/5925


eR64

smooth

pipes

Recrit

41

316.0

eR

rough

turbulent zone

laminar

zone

critical

zone

transitional

zone

rough pipes

Pi f i ti th i


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Moody also proposed an explicit version of theColebrook-White equation

correct to 5% for: 4 x 103< Re< 1 x 107

ks/D< 0.01

31

61020000100550

e

s

RD

k.

Pi f i ti th i


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Pipe friction theoriesfriction factor: Ackers: HRS Charts

Charts combining the Colebrook-White and Darcy-Weisbach equations

Developed by the Hydraulics Research Station

Based on equation

viscositykinematic

2

512

73log22

f

sf

gDSD

.+

D.

kgDS-V

L

hS ff gradienthydraulic

Pi f i ti th i


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ks= 0.03mm

Good samples of:

wrought iron

coated steel

clayware (sleeve joint)

sewer (V= 2m/s)

Normal samples of:

asbestos cement

spun bitumen lined metal

pipes

spun concrete lined metal

pipes uncoated steel

clayware (spigot and

socket joint)

uPVC with chemically

cemented joints

e.g. given: D and SVandQ

Pi f i ti th i


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ks= 0.15mm

Good samples of:

rusty wrought iron

uncoated cast iron

sewer (V= 1m/s)

Normal samples of: galvanised iron

coated cast iron

precast concrete

(O ring joints)

uPVC (sewer slimed to

about half depth) sewer (V= 1.5m/s)

Poor samples of:

wrought iron

coated steel

clayware (sleeve joints) sewer (V= 2m/s)



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Pipe friction theoriesfriction factor: Barr

Proposed an explicit version of the Colebrook-White equation

accurate to 1% for Re> 105

890

12865

73

21

.

e

s

R

.+

D.

klog-=



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Pipe friction theoriesapplicability

Head loss

Equation Laminar Smoothturbulent

Transitionalturbulent

Roughturbulent

Hagen Poiseuille yes

Darcy Weisbach yes yes yes yes

eR

64if

Friction factor

Equation Laminar Smoothturbulent

Transitionalturbulent

Roughturbulent

Blasius yesPrandtl and Karman yes (smooth) yes (rough)

ColebrookWhite yes yes yes

Ackers (HRS charts) yes yes

Barr yes yes

Moody yes yes yes yes

22

2 32

2 gD

LVh

gD

LVh ff


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Recap

Losses in pipes caused by friction

local head losses

Pipe friction theories large number of different pipe friction theories

most theories have limited application

Colebrook-White is widely applicable Moody diagram widely applicable &easy to use

Head loss theories

Darcy-Weisbach equation is generally applicable



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Pipe friction theoriesexample 1

A pipeline 10km long, 300mm in diameter andwith roughness size 0.03mm conveys water

from a reservoir (top water level 850m above

datum) to a water treatment plant (inlet level

700m above datum).

Assuming the reservoir remains full and

kinematic viscosity is 1.13 x 10-6m2/s, estimate

the discharge, using:a.Colebrook-White/Darcy-Weisbach

b.Moody diagram

c. HRS charts



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Pipe friction theoriessolution 1: part a: Colebrook White

Combining the Colebrook-White equation withthe Darcy-Weisbach equation gives HRS eqn.

/sm178.04

3.0515.2

m/s515.2

015.03.023.0

1013.1512

3.073

1003.0log015.03.022

015.010000

700850gradienthydraulic

2

512

73log22

32

63

VAQ

V

g

.+

.g-V

LhS

gDSD

.+

D.

kgDS-V

ff

f

sf



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Pipe friction theoriessolution 1: part b: Moody diagram

Steps required:1. calculate ks/D

2. guess a value for V (try V = 2.75m/s)

3. calculate Re4. estimate from the Moody diagram

5. calculate hf

6. compare hfwith the available head7. if H hfthen repeat from step 2



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2.51m/s)Vtly,significanchangenotwill(

5stepfromrepeatandV,Decrease7.

m9.179m150700850.6

m9.1793.02

75.2100000014.0

2.5

014.0:givesdiagramMoody.4

1030.71013.1

75.23.0

3.

m/s75.2Assume.2

0001.03.01003.01.

22

5

6

3

H

ggD

LVh

DV

R

V

Dk

f

e

s

Pipe friction theoriessolution 1: part b: Moody diagram



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Pipe friction theoriessolution 1: part c: HRS charts

100s = 100 0.015 = 1.5

Q = 180l/s

D = 0.3m



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The known outflow from a branch of adistribution system is 30 l/s.

The pipe diameter is 150mm, length 500m and

roughness coefficient 0.15mm.

Find the head loss in the pipe, using the HRS

charts and the explicit formulae of Barr.



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Pipe friction theoriessolution 2: part a: HRS charts

m1002.0*500*

02.02100

SLhf

SS

Q = 30l/s

D = 0.15m



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40

m4510

021280

10262

5.1286+

15073

100.15log-2=

1

1026210131

15071

m/s7101770

030

m01770

4

150

4

/sm030

R

5.1286+73

klog-2=

1

8905

3-

5

6

222

3

890

e

s

.h

.

...

..

..VDR

..

.

A

QV

..D

A

.Q

D.

f

.

e

.

Pipe friction theoriessolution 2: part b: Barr equation



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What is the head loss due to friction when crudeoil at 20C flows at the rate 1.2 l/s along 100mof 50mm diameter smooth pipe

The density of crude oil at 20 C is 860kg/m3and = 0.008 Pas)

Use the Moody diagram



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Pipe friction theoriessolution 3

First determine the type of flow

32900080

0506120

m/s6120

4

050

01202

.

..

R

..

.

A

QV

VDR

e

e



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Pipe friction theoriessolution 3

The Moody diagram shows that this is in thecritical regime between laminar flow and

transitional turbulence

can not be certain about value of

All we can say is that: 0.03 < < 0.045

m7210502

61201000450

2:0.045If

m1510502

6120100030

2:0.03If

2

2

..g

..

gD

LVh

..g

..

gD

LVh

f

f



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Pipe friction in networksHazen-Williams equation

The Colebrook-White formula and HRS chartsare only appropriate for the design of single

pipes

not suitable to the analysis of pipe networks

unless a computer program is available

The Hazen-Williams equation is one of the most

frequently used empirical equations

851

774

540

632

851

1651

540630

7102780

7863550

.

.f

.

f.

.

.f

.f.

C

Q

D

L.=h

L

hCD.Q

CV

DL.=h

LhCD.V



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Pipe friction in networksHazen-Williams equation

Coefficient Cdepends on ?

pipe diameter

pipe material

pipe age

Cvalues are taken fromrough-turbulence zone ofMoody diagram

roughness coefficientsassumed to beindependent of Reapproximation

Pipe C

Extremely smooth

pipes

140

New steel or cast

iron

130

Wood, averageconcrete

120

Clay, new riveted

steel

110

Brick, old cast iron 100

Old riveted steel 95

Badly corroded cast

iron

80

Very badly corroded

iron or steel

60

L l h d l


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Local head losses

In addition to friction losses head losses alsooccur locally at:

bends

valves

junctions

orifice plates

enlargements

tapers, etc

Often referred to as minor losses but may be

substantial

L l h d l


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Local head losses

Local losses usually expressed in terms of anempirically determined coefficient kLin the

equation:

assumes that flow regime is rough-turbulent

independent of Re

g

V

kh LL 2

2

Local head losses


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Local head lossessudden enlargement

As flow passes into enlarged section V, P

turbulent eddies form at section 1*

local head loss To analyse this situation, we need to consider the

change in conditions between 1* and 2

We have 3 equations continuity equation 12

i.e. flow in = flow out

1 1* 2

(eqn.1)221121 AVAVQQQ

Local head losses


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energy equation 12i.e. energy at 1 = energy at 2 + losses

momentum equation 1* - 2i.e. net force = momentum at 1* - momentum at 2

! as pressure change at 1* can not occur

instantaneously

assume: P1*P1, V1*V1! note that A1*= A2


2)(eqn.

222

2

2

2

121

2

22

2

11

g

VV

g

PPhh

g

V

g

P

g

V

g

PLL

*** VVQAPAP 122211

1 1* 2

3)(eqn.122221 VVQAPAP

Local head losses


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Combining eqn. 1 and eqn. 3 eliminates Q

Combining eqn. 2 and eqn. 4

g

VVh

g

VV

g

VVVh

VVVPPg

VV

g

PPh

LL

L

22

with22

21

2

2

2

1122

12221

2

2

2

121

4)(eqn.

with

12221

12222221

22122221

VVVPP

VVAVAPAP

AVQVVQAPAP

Local head losses


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From eqn. 1:

g

Vkh

gV

AA

gVVh

A

AVVVAVA

LL

L

2

21

2

2

1

2

1

2

2

1

2

21

2

1122211

tenlargemensuddenafor1i.e.

2

2

1

A

AkL

Local head losses


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Local head lossessudden contraction

Assume: head loss between 1 and 1* is negligible

flow expands from 1* to 2

!

assume area of vena contracta (A1*) is60% of downstream area (A2)

Use sudden enlargement approach

2

440

260601

21

2

2

2

22

2

2

2

1

2

2

1

g

V.h

g.

V

AA.

gV

AAh

L

L

**L

21*1

ncontractiosuddenfor440i.e. .kL

Local head losses


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Fitting kL L/D Fitting kL L/D Fitting kL L/D

Entry Bends Valves

Sharp edged 0.50 22 r/D= 0.5: 22.5

45

90

0.20 9 Gate valve : fully open

!closed

"closed

#closed

0.12 6

Slightly rounded 0.25 11 0.40 18 1.00 45

Bell mouth 0.05 2 1.00 45 6.00 270

Foot valve and strainer (pump) 2.50 113 r/D1: 22.5

45

90

0.15 7 24.00 1080

Tapers 0.30 14 Globe valve 10.00 450

Contraction, large to small negligible 0.75 34 Butterfly valve 0.30 13

Expansion, inlet to outlet: 4:5

3:4

1:3

0.03 1 r/D= 2 - 7: 22.5

45

90

0.10 5 Exit

0.04 2 0.20 9 Sudden enlargement 1.00 45

0.12 6 0.40 18 Bell mouth outlet 0.20 9

Tees r/D= 8 - 50: 22.5

4590

0.05 2

Flow in line 0.35 16 0.10 5Line to branch or vice versa

(sharp edged)

1.20 54 0.20 9

Line to branch or vice versa

(radiused)

0.80 36

Local head lossescoefficients for common fittings

diameterpipe

headoflossequivalentgivetopipestraightoflength

D

L

Local head losses


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Local head lossesexample 4

A pipeline conveys water from a reservoir to a water

treatment plant. Assuming that the reservoir remains full,estimate the discharge using the combined Colebrook-White

and Darcy-Weisbach equation.

pipeline data

!10km long, 300mm diameter and roughness size0.03mm

! 20 long radius bends (kL= 0.4), 2 1/4 closed gate

valves (kL= 1.0), bellmouth entrance (kL= 0.05),

sudden exit (kL= 1.0)

reservoir data

! top water level 850m above datum

water treatment plant data

!

inlet level 700m above datum

Local head losses


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Total head loss (H) = elevation difference (res - wtp)H= 150m

Also:

Use this equation with the combined D-W/C-Wequation to determine V and hf

g

V

g

Vh

hhH

L

Lf

21.11

20.105.00.124.020

22

Local head lossessolution 4

f

sf

gDSD

.+

D.

kgDS-V

2

512

73log22

Local head losses


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Iterative process steps:

1. assume hf= H(i.e. ignore hL)

2. calculate V (using D-W/C-W)3. calculate hL(hL= 11.1 V

2/2g)

4. calculate hf2= H- hL

5. If hfh

f2h

f= h

f2repeat from step 2

solution 4

f

sf

gDSD

.+

D.

kgDS-V

2

512

73log22

Local head losses


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solution 4

150mm47.14653.3150

53.32

1.11

m/s514.22

512

73log22

015.01010

150

150

2

3

f

L

f

sf

f

f

f

h

mg

Vh

gDSD

.+

D.

kgDS-V

L

hS

h

Second iteration: Assume hf= 146.47mS

fV

hf2

=146.51m

Recap


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Recap

Losses in pipes caused by friction

local head losses

Pipe friction theories large number of different pipe friction theories

most theories have limited application

Colebrook-White is widely applicable Moody diagram is widely applicable AND easy to

use

Recap


59/59

Recap

Head loss theories Darcy-Weisbach equation is generally applicable

to all types of flow


Hazen-Williams equation enables rapid

calculation of network wide conditions

Local head losses

concentrated losses occurring at pipe elements

expressed as some multiple of the flow velocity

head

Documents

Presentation - Hydraulics 2 - Pipe Flow Theory