fluid flow in pipes - lecture 4

CIVE2400: Pipeflow - Lecture 4 09/04/2009

1

School of Civil EngineeringFACULTY OF ENGINEERING

Fluid Flow in Pipes: Lecture 4

Dr Andrew Sleigh

Dr Ian Goodwill

CIVE2400: Fluid Mechanics

www.efm.leeds.ac.uk/CIVE/FluidsLevel2Fluid Mechanics: Pipe Flow – Lecture 4 2

Local Head Losses

• Local head losses are the “loss” of energy at point where

the pipe changes dimension (and/or direction).

Pipe Expansion

Pipe Contraction

Entry to a pipe from a reservoir

Exit from a pipe to a reservoir

Valve (may change with time)

Orifice plate

Tight bends

• They are “velocity head losses” and are represented by

g

ukh LL

2

2

Fluid Mechanics: Pipe Flow – Lecture 4 3

Value of kL

• For junctions and bends we need

experimental measurements

• kL may be calculated analytically for

Expansion

Contraction

• By considering continuity and momentum

exchange and Bernoulli


Losses at an Expansion

• As the velocity reduces (continuity)

• Then the pressure must increase

(Bernoulli)

• So turbulence is induced and head losses

occur

Turbulence and losses


Value of kL for Expansion

• Apply the momentum equation from 1 to 2

• Using the continuity equation we can

eliminate Q

• From Bernoulli1 2

122211 uuQApAp

21212 uu

g

u

g

pp

g

pp

g

uuhL

12

2

2

2

1

2 Fluid Mechanics: Pipe Flow – Lecture 4 6

Value of kL for Expansion

• Combine and

• Using the continuity equation again

u1A1 = u2A2 u2=u1A1/A2

• A1 >> A2, kL = 1 exit loss

21212 uu

g

u

g

pp

g

pp

g

uuhL

12

2

2

2

1

2

g

uuhL

2

2

21

g

u

A

AhL

21

2

1

2

2

1

2

2

11A

AkL


2

Fluid Mechanics: Pipe Flow – Lecture 4

• Flow converges as the pipe contracts

• Convergence is narrower than the pipe

Due to vena contractor

• Experiments show for common pipes

A1’ = 0.6A2

• Can ignore losses

bewteen 1 and 1’

As Convergent flow is very stable

7

Losses at an Contraction

1 1’ 2


• Apply the general local head loss equation

between 1’ and 2

Using A1’ = 0.6A2

And Continuity

8

Losses at an Contraction

1 1’ 2

g

u

A

AhL

21

2

'1

2

2

'1

6.06.0

2

2

22

'1

22'1

u

A

uA

A

uAu

g

uhL

244.0

2

2


• Whenever there is expansion

• Pressure increases down stream

• Danger of boundary layer separation as the fluid

near the walls had little momentum

9

Other Losses


Losses: Junctions

Reduced velocity

Increased pressureReduced velocity

Increased pressure


Losses: Sharp bends

Reduced velocity

Increased pressure


kL values

kL

valu

e

Bellmouth entry 0.10

Sharp entry 0.5

Sharp exit 0.5

90 bend 0.4

90 tees

In-line

flow

0.4

Branch

to line

1.5

Gate value (open) 0.25

kL = 0.1

Bell mouth Entry

kL = 0.5

Sharp Entry/Exit

T-inline

kL = 0.4

T-branch

kL = 1.5


3


Pipeline Analysis

• Bernoulli Equation

equal to a constant: Total Head, H

• Applied from one point to another (A to B)

With head losses

Hzg

u

g

pA

AA

2

2

fLBBB

AAA hhz

g

u

g

pHz

g

u

g

p

22

22


• Reservoir

• Pipe of Constant diameter

• No Flow

Bernoulli Graphically

Datum line

zA= HH

z

p/ g

Pressure head

Elevation

p/ g

z

z

p/ g

Hzg

u

g

pA

AA

2

2

Total Head Line


• Constant Flow

• Constant Velocity

• No Friction


Datum line

zA= HH

z

p/ g

Pressure head

Elevation

Hzg

u

g

pA

AA

2

2

Total Head Line

Hydraulic Grade

Line

u2/ g

Velocity head


• Constant Flow


• No Friction


Datum line

zA= HH

z

p/ g

Pressure head

Elevation

Hzg

u

g

pA

AA

2

2

Total Head Line

Hydraulic Grade

Line

u2/ g

Velocity head

u22/ g

Wider Pipe

Change of Pipe

Diameter


• Constant Flow


• With Friction


Datum line

zA= HH-hf

z

p/ g

Total Head Line

Hydraulic Grade

Line

u2/ g

fLBBB

AAA hhz

g

u

g

pHz

g

u

g

p

22

22

u2/ g u2/ g


Reservoir Feeding Pipe Example

• d = 0.1m

• Length A-C = L = 15m

• Length A-B = L = 1.5m

• f = 0.08

• kL entry = 0.5 Sharp

• kL exit = 0 Opens to atmosphereB

A

CzA zB

zCpc = Atmospheric

zB-zA = 1.5m

zA-zC = 4m

Find

a) Velocity in pipe

b) Pressure at B


4



• Apply Bernoulli with head losses

fLCCC

AAA hhz

g

u

g

pz

g

u

g

p

22

22

B

A

CzA zB

zCpc = Atmospheric

d

fL

g

u

gd

fLu

g

u

g

uzz C

CA

45.00.1

22

4

25.0

2

2222

1.0

1508.045.1

81.924

2usmu /26.1

pA= pc = Atmospheric

uA= negligible



• Find pressure at B: Apply Bernoulli A-B

fLBBB

AAA hhz

g

u

g

pz

g

u

g

p

22

22

B

A

CzA zB

zCpc = Atmospheric

smuu B /26.1

pA= Atmospheric = treat as 0

uA= negligible

ABABBB zzgd

ufL

g

u

g

u

g

p

2

4

25.0

2

222

5.11.0

0.508.045.1

81.92

26.1

81.91000

2

Bp 23 / 1058.28 mNpB

Negative

i.e. less than Atmospheric pressure

21

Today’s lecture:

• Local head losses

Expansion loss

Contraction loss

Junction

+ other minor losses

• Graphical representation of Bernoulli

Total Head Line

Hydraulic Grade Line

• Analysis of pipeline, including losses

g

ukh LL

2

2

g

u

A

AhL

21

2

1

2

2

1

fluid flow in pipes - lecture 4

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