Pumping Station Desing - Second Edition by Robert L. Sanks, George Tchobahoglous, Garr M. Jones

•

Advantages:

the

analysis

can be

applied

by a

person

with

little numerical analysis skill

and

with

limited

computational facilities.

•

Disadvantages:

the

solutions, which

are

always

approximations,

are

applicable only

to

simple pipe-

lines. Considerable experience

is

required

to

know

whether

the

results

are

applicable.

Elastic theory (water hammer).

The

elasticity

of

both

fluid and

pipe

affect

the

pressure changes. Pres-

sure

changes propagate with wave speed,

a,

which

varies

from

about

300 to

1400

m/s

(1000

to

4700

ft/s).

Flow conditions

are

described

by

nonlinear partial dif-

ferential

equations.

•

Solution: arithmetic, graphical method

or

method

of

characteristics using

finite

difference techniques.

•

Advantages:

the

theory accurately represents system

behavior

and is,

therefore, applicable

to a

wide range

of

problems. Pipe

friction,

minor losses,

and

varying

valve

closure procedures

can be

incorporated.

•

Disadvantages: applying

the

theory requires

a

sub-

stantial initial

effort

on the

part

of

user

to

learn

it, a

digital computer programmed

for the

method

of

characteristics,

and a

knowledge

of the

operational

characteristics

of

system components

to set up the

solution

for the

computer.

6-2.

Nomenclature

In

Chapters

6 and 7,

"velocity"

always means velocity

of

water

and

"speed"

means

the

velocity

of

pressure

waves.

The

symbols used

in

Chapters

6 and 7 are

defined

as

follows.

a

Elastic wave speed

in

water contained

in a

pipe

(in

meters

per

second [feet

per

second])

C

Coefficient

whose value depends

on

pipe

restraint

D

Inside diameter

of a

pipe

(in

meters [inches

or

feet])

e

Wall thickness

of a

pipe

(in

meters [inches

or

feet])

£j

Longitudinal joint

efficiency

in

welded

pipes

(dimensionless)

E

Modulus

of

elasticity

of

pipe material

(in

new-

tons

per

square meter [pounds

per

square inch

or

pounds

per

square

foot];

see

Table

6-1).

g

Acceleration

due to

gravity

(in

meters

per

second squared

[feet

per

second

squared])

h

Head

due

only

to

surge

(in

meters

[feet])

A/z

Change

of

head

due to

surge

(in

meters

[feet])

/z

a

Allowable head

due to

surge

(in

meters

[feet])

K

Bulk modulus

of

elasticity

of the

liquid

(in

newtons

per

square meter [pounds

per

square

inch

or

pounds

per

square

foot];

see

Table

A-8orA-9)

L

Length

of

pipeline

(in

meters

[feet])

AP

Change

of

pressure

due to

surge

(in

newtons

per

square meter [pounds

per

square

inch])

AP

a

Allowable pressure change

due to

surge

(in

newtons

per

square meter [pounds

per

square

inch])

P

atm

Atmospheric pressure

(in

newtons

per

square

meter (pounds

per

square inch);

see

Table

A-

6orA-7]

AP

C

Difference between external

and

internal

pressure

on a

pipe

(in

newtons

per

square

meter [pounds

per

square

inch])

P

v

Vapor pressure

of

water

(in

newtons

per

square meters [pounds

per

square

inch];

see

Table

A-8 or

A-9)

SF

Safety factor (dimensionless)

s

y

Yield stress

(in

newtons

per

square meter

[pounds

per

square

inch])

t

Time

(in

seconds)

t

c

Critical time

(2L/a\

in

seconds)

V

Volume

(in

cubic meters [cubic

feet])

v

Velocity

(in

meters

per

second [feet

per

sec-

ond];

in

this chapter,

"velocity"

is

average

velocity

of fluid flow)

Av

Change

in

velocity

(in

meters

per

second

[feet

per

second])

Av

a

Allowable change

in

velocity

(in

meters

per

second

[feet

per

second])

p

Density

(in

kilograms

per

cubic meters

[slugs

per

foot;

see

Table

A-8 or

A-9])

Ji

Poisson's

ratio (dimensionless;

see

Table

6-1)

Y

Specific weight

of

water

(in

newtons

per

cubic meter [pounds

per

cubic

feet;

see

Table

A-8 or

A-9])

6-3. Methods

of

Analysis

Methods

of

analyzing

pipelines

for the

effects

of

hydraulic transients, with

or

without various means

of

controlling them

or

reducing

the

severity,

can be

sum-

marized

as

follows:

•

Graphical

[1]

•

Arithmetic

[2]

•

Algebraic

[3]

•

Method

of

characteristics

[3,4,5]

•

Finite element

•

Implicit differentiation

These

are

methods

of

analysis,

not

methods

of

design.

In

analysis,

the

system

is

described mathemat-

ically,

and the

behavior

of the

system

is

predicted

by

the

analysis.

In

design,

the

desired physical results

are

described

and

alternative means

for

attaining these

results

are

compared, which leads

to a

selection

of one

or

more control measures. Analysis should

be

per-

formed

as a

part

of the

design process.

All

of the

above methods involve

the

equations

of

motion

and

continuity used

to

describe

the

velocity

and

pressure variation

in the

pipeline.

For

computer

modeling,

the

most widely used

is the

method

of

char-

acteristics,

in

which

the

partial

differential

equations

of

motion

and

continuity

are

converted into

four

first-

order equations represented

in finite

difference

form

and

solved simultaneously with

a

computer.

The

method

provides

for

• the

inclusion

of

many possible pipeline

or

system

features,

such

as

junctions, pumping stations,

air

chambers,

air

release valves, reservoirs,

and

line

valves;

• the

inclusion

of fluid

friction;

• the

retention

of

small

or

secondary terms

in the

original equations

so

that accuracy

is

retained;

and

• the

computation

of

pressure

and

velocity

as a

func-

tion

of

time

at

various points throughout

the

entire

pipeline system.

6-4. Surge

Concepts

in

Frictionless

Flow

Water

hammer

can

occur

in a

pipeline

flowing

full

when

the flow is

increased

or

decreased, such

as

when

the

setting

on a

valve

in the

line

is

changed. When

a

valve

in a

pipeline

is

closed rapidly,

the

pressure

on

the

upstream side

of the

valve increases,

and the

pulse

of

increased pressure travels upstream

at the

elastic

wave

speed,

a.

This

pulse (called

an

"upsurge")

decreases

the

velocity

of flow.

Downstream

from

an

in-line valve,

the

pressure

is

reduced

and the

wave

of

decreased pressure travels downstream, also

at the

elastic wave speed,

a.

This pulse (called

a

"down-

surge")

decreases

the

velocity

of flow. If the

velocity

is

reduced

too

rapidly

and the

steady-state pressure

is

low

enough,

the

downstream pressure

can be

reduced

to

vapor pressure, which creates

a

vapor pocket.

A

large vapor pocket (called "column

separation")

can

collapse with

a

dangerous explosive force produced

by

the

impact between solid water columns

and can

cause

the

pipe

to

burst. This phenomenon

can

also

occur upstream

of the

valve when

the

reflected posi-

tive wave returns

to the

valve.

The

events following

a

sudden closure

of a

valve

located

a

distance

(L)

downstream

from

a

reservoir

is

described

in

Figure

6-1.

Friction

is

neglected,

and the

energy gradeline (EGL)

and

hydraulic gradeline

(HGL)

are

assumed

to

coincide because velocity

heads

are

small compared with water hammer pres-

sure heads.

The

steady-state

EGL is

called

//,

and the

added-pressure head pulse

is

called

h. The

velocity

of

the fluid

under steady-state conditions

is

V

0

just before

the

valve

is

closed

(at t = O).

The

sequence

of

events between

the

valve

and the

reservoir occurs

in a

four-phase cycle;

the

duration

of

each phase

is the

time

for the

pressure wave

to

travel

between

the

valve

and the

reservoir (the length

of the

Table

6-1.

Physical

Properties

of

Pipe

Materials

Modulus

of

elasticity

U.S.

customary units

Material

Poisson's

ratio

Sl

units (N/m

2

)

Ib/in.

2

Ib/ft

2

Aluminum

0.33

7.30

E+10

1.05

E + 7

1.51

E + 9

Asbestos-cement

0.30

2.30

E+10

3.4OE+

6

4.9OE

+8

Brass 0.34

1.03

E+11

1.5OE+

7

2.16E

+9

Copper

0.30

1.10

E+11

1.60

E

+7

2.3OE+

9

Ductile iron 0.28

1.66

E

+11

2.40

E + 7

3.46

E + 9

Gray

cast

iron 0.28

1.03

E+11

1.5OE+

7

2.16E+

9

HDPE 0.45

1.0 E +

9

a

1.5 E +

5

a

2.2 E +

7

a

PVC

0.45

2.7OE

+9

4.0OE+

5

5.76E

+ 7

Steel

0.30

2.07

E+11

3.00

E

+7

4.32

E

+O

Concrete

-

4

73

x

1Q

6^

b

57,000^

c

a

At

16

0

C

(6O

0

F).

Increases greatly

with

decreasing temperature

and

vice versa.

b

fc

is

ultimate

strength

in

newtons

per

square

meter.

c

/c

is

ultimate strength

in

pounds

per

square inch.

pipeline divided

by the

elastic wave speed,

L/a).

The

sequence occurs

as

follows:

1

.

O

<

t

<

L/a.

At t = O, the fluid

just upstream

from

the

valve

is

compressed

and

brought

to

rest. Part

of

the

pipe (section

BC) is

expanded

and

stretched,

as

shown

in

Figure

6-

Ia.

This process

is

repeated

for

each successive increment

of fluid as the

pressure

wave

travels upstream.

The fluid

upstream

from

the

wave

front

continues

to flow

downstream until

it is

stopped

by the

advancing pressure

wave

front.

When

the

pressure wave reaches

the

reservoir

at t =

L/a,

the

fluid

(at

rest

in the

pipe)

is

under

a

total pressure head

H

+

/z,

or h

greater than

the

static head

in the

reservoir.

2. L/a

<

t

<

2L/a.

The

pressure head

difference

at

t = L/a at the

reservoir causes

the fluid to flow

from

the

pipe back into

the

reservoir with

a

velocity

-V

0

.

The

pressure along

AB is

reduced

to the

original

steady-

state

level,

//,

and a

negative wave producing

normal

pressure propagates back

to the

valve,

as

shown

in

Figure

6-

Ib.

At

/

=

2LIa,

the

pressure

is

nor-

mal

(equal

to H)

along

the

pipe

but the

velocity

throughout

the

pipe

is

negative; that

is,

water

is flow-

ing

away

from

the

valve.

3.

2LIa

<

t

<

3LIa.

At t =

2LIa,

there

is no fluid

available

to

maintain

the

upstream

flow at the

valve,

and

the

normal pressure head,

//,

is

reduced

by h to

bring

the fluid in

section

BC to

rest (Figure

6-

Ic).

This

wave

of

reduced pressure propagates back toward

the

reservoir,

all fluid

comes

to

rest,

and the

reduced pres-

sure

allows

the fluid to

expand

and the

pipe walls

to

contract.

At t =

3LIa,

the

reduced pressure,

H -

h,

exists

all

along

the

pipe,

the

velocity

is

zero through-

out

the

pipe,

and the

static pressure head

in the

pipe

is

less than

the

pressure head

in the

reservoir.

4.

3L/a

<t<

4L/a.

Mt =

3L/a,

the

unbalanced pres-

sure head

at the

reservoir causes

the fluid to flow

back

into

the

pipe

at a

velocity

+V

0

.

A

wave

of

pressure

at the

original static pressure head level propagates down-

stream toward

the

valve,

but in

section

BC, the

pressure

is

still reduced

to H - h

(Figure

6-ld).

At t =

4L/a,

the

pressure throughout

the

pipe

is

normal (equal

to H),

and

the

velocity

is the

same

as the

original

+V

0

prior

to

the

valve closure.

But

when

the

velocity,

V

0

,

reaches

the

valve,

the

four-phase cycle repeats

and

continues

to

repeat periodically every 4L/a time period.

The

variation

of

pressure with time during

a

4L/a

interval

is

shown

at

several points along

the

pipeline

in

Figure 6-2. Friction

in a

real pipe system eventually

dampens

the

increased water hammer pressure head,

h.

The

principal

concepts

indicated

by the

events fol-

lowing

a

sudden valve closure

are as

follows:

• The

time

L/a is a

significant parameter

for

water

hammer analysis.

• The

time

2LIa

is

critical

because

the

pressure head

at

the

valve reaches

a

maximum

at

2LIa.

A

valve

closed

in any

shorter time produces

the

same maxi-

mum

pressure head

rise at the

valve, whereas

the

Figure

6-1.

Sequence

of

events

for one

cycle

after sudden valve closure.

pressure rise

is

reduced

if the

valve

is

closed

in a

longer time interval. Hence,

r

c

=

%

(6-1)

where

t

c

is the

critical time,

L is the

length

of the

pipe,

and

a is the

elastic wave speed.

Pressure Head

Change

A

momentum analysis

of the flow

conditions

for the

valve

closure shows

the

pressure head change,

A/z,

is a

function

of the

change

in flow, Av.

AA

-

^

AV

(6-2)

8

where

Ah is the

change

in

pressure head

in

meters

(feet),

a is

elastic wave speed

in

meters

per

second

(feet

per

second),

Av is the

change

in

velocity

(of

water) caused

by the

event

in

meters

per

second

(feet

per

second),

and g is the

acceleration

due to

gravity

in

meters

per

second squared

(feet

per

second

squared).

Use the

negative sign

for

waves traveling

upstream,

use the

positive sign

for

waves traveling

downstream,

and

note that

Av =

V

2

-

V

1

,

where

V

1

is

the

velocity prior

to the

change

in flow

rate

and

V

2

is

the

velocity following

the

change.

If the flow is

sud-

denly stopped,

Av =

V

1

and

A/z

=

0V

1

Tg.

Note, too, that

A/J

is

positive

if Av is

negative.

If the

valve

on the

downstream

end of a

pipe

is

closed incrementally,

Equation

6-2

becomes

ZA/2

=

--ZAv

(for

t <

t

c

)

(6-3)

O

Figure 6-2.

Head

fluctuations

at

selected

points

in the

pipeline

of

Figure

6-1

after valve closure.

Transient pressure heads

due to

valve closure

can

be

reduced

by

slowly closing

the

valve over

a

time

interval greater than

r

c

,

as

discussed

in

Section 6-5.

Elastic

Wave

Speed

The

pressure head change

due to a

change

in the flow

rate requires

the

calculation

of the

elastic wave speed

in

the

pipe.

The

wave speed depends

on

both

the fluid

properties

and the

pipe characteristics.

a=

I

K

'P

(6-4)

Ji

+4IY-I

y

(EJ(ej

where

K is the

bulk modulus

of

elasticity

of the

liquid

in

newtons

per

square meters (pounds

per

square

foot),

E is the

modulus

of

elasticity

of

pipe material

in

newtons

per

square meters (pounds

per

square foot),

D

is

inside pipe diameter

in

meters

(feet),

e is the

pipe

wall

thickness

in

meters (feet),

C is a

correction factor

for

type

of

pipe restraint,

and p is the

density

of the

fluid

in

kilograms

per

cubic meter (slugs

per

cubic

foot).

The

bulk modulus given

in

Table

A-8

must

be

changed

from

kilopascals

to

newtons

per

square

meter,

and in

Table

A-9 it

must

be

converted

from

pounds

per

square inch

to

pounds

per

square

foot.

For

thin-

walled

pipes (those with

D/e > 40) the

cor-

rection factor,

C,

varies according

to

pipe restraints.

Values

for

three cases

are

•

C

1

=

1.25

-

JLI

for

pipes anchored

at the

upstream

end

only

(Wylie

and

Streeter

[3] use 1.0 -

ju/2;

the

difference

has

little practical significance);

•

C

2

= 1.0 -

jLi

2

for

pipes anchored against axial

movement;

and

•

C

3

=

1

.0

for

pipes with expansion joints throughout.

The

symbol

ji

is

Poisson's

ratio (see Table

6-1 for

properties

of

pipe materials). Expressions

for

C

3

for

thick-

walled

pipes (D/e

<

40), which

are

more com-

plex,

are

given

by

Wylie

and

Streeter

[3].

Buried pipes

are

best represented

by the

factor

C

2

.

Typical values

for

wave speed

for

water

in

pipes

are

given

in

Table

6-2. Because

the

elastic wave speed,

a,

for

steel

and

ductile iron

pipe

is

often

close

to 980 m/s

(3200

ft/s),

the

change

in

pressure head

from

Equation

6-2 is,

roughly,

±100v.

For

PVC,

h =

±34v.

The

speed

of the

pressure wave

in a

pipeline carry-

ing

liquids

is

greatly reduced

if

bubbles

of

free

gas are

entrained,

as

shown

in

Table 6-3.

A

detailed discus-

sion

of

this phenomenon

is

given

by

Wylie

and

Streeter

[3].

There seems

to be no way of

predicting

air

volume entrainment. Neglecting

the

effect

of air

entrainment provides

a

conservative analysis. How-

ever,

as the

wave speed decreases,

L/a

increases

and

the

time

for

closure

of

valves must

be

increased.

Table

6-2.

Typical

Wave

Velocities

in

Pipe

for

Water

Containing

Dissolved

Air

Wave

velocities

Pipe

material

m/s

ft/s

Asbestos

cement

820-1200

2700-3900

Copper

1000-1300

3400-4400

Ductile

iron

980-1400

3200-4500

HDPE

3

180-370

600-1200

PVC

300-600

1000-2000

Steel

600-1200

2000-4000

a

For

a

modulus

of

elasticity

of

1.03

E + 9

N/m

2

(150,000

lb/in.

2

)

Table

6-3.

Effect

of Air

Entrainment

on

Wave

Speed

3

Wave

speed

V/

ai

/V/

total

m/s

ft/s

O

1200 4000

0.001

610

2000

0.002

460

1500

0.004

300

1000

0.008

210 700

a

After

Wylie

and

Streeter

[3].

Example

6-1

Effect

of

Pipe

on

Wave

Speed

and

Pressure

Problem:

For

300-mm

(12-in.)

pipes

of

ductile iron, steel,

and

PVC,

find the

effects

of

pipe

material, bedding,

and

joint conditions

on

wave speed

and

water hammer pressure. Assume

the

pipes

are

3000

m

(9840

ft)

long

and

carry 0.15

m

3

/s

(5.3

ft

3

/s)

of

water

at a

temperature

of

15

0

C

(59

0

F).

Solution:

The

wave speeds

are

based

on

Equation 6-4.

The

calculation

of

wave speed

to the

nearest

30 m/s

(100

ft/s)

is

usually

sufficient,

because

the

accuracy

of the

data does

not

justify

greater precision.

The

critical

valve closure times,

f

c

,

are

based

on

Equation

6-1.

Valve closure

in

6-5. Slow Closure

of

Valves

To

limit

the

pressure

rise,

the

maximum deceleration

of

the

water during

the

critical time period,

t

c

,

must

be

limited.

The

maximum allowable deceleration

can be

calculated

by

using

the

ratio

A/>

*

AA

«

AV

"

(**•>

A^

=

A/T

=

A^

(6

-

5)

where

AP

is

pressure rise,

A/z

is the

head rise,

Av is the

change

in

velocity,

and the

subscript,

a,

means allow-

able.

The

terms

in the

denominator

are for

instanta-

neous

valve closure.

Manufacturers

can

provide either

(1)

curves

of the

valve

headloss

coefficient,

K,

in

Equation 3-15

wherein

h =

Kv

2

/2g

(shown

for a

butterfly

valve

in

Figure 6-3)

or (2)

values

of flow

coefficients,

C

v

,

wherein

C

v

equals water

flow in

gallons/minute

at

7O

0

F

and 1

lb/in.

2

pressure differential shown

for

ball

and

eccentric plug valves

in

Figure

6-4

(see Appendix

A

for

conversion

to SI

units.).

Both

K and

C

v

vary

greatly

with valve opening,

so

either

is a

convenient

aid for

solving valve problems.

Equation

6-5 can

lead either

to

simplified

equa-

tions

or to

computer solutions

for

determining

the

allowable rate

of

valve closure

so

that

a

given pressure

is

not

exceeded.

To be

valid,

a

method

of

solution

must include

• the

change

of

valve

coefficient,

K,

in the

formula

for

headloss (Equation

3-15),

or

• the

shape

of the

valve closure curve,

C

V

/C

VO

,

and the

nonlinear action

of

some valve stems,

and

• the

effect

of the

increasing pressure, which tends

to

sustain

the

original

flow

through

the

valve.

The

pressure,

for

example, depends

on the

shape

of an

associated pump curve

and the

location

of the

valve

as

well

as on

other characteristics

of the

system.

Because resistance

to flow is

relatively small

in the

initial stages

of

closure,

a

valve

can be

quickly closed

to

near

shut-off

if the final

closure

is

slow.

Of the

three

gate valve closure programs occurring

in the

same

interval,

3r

c

,

as

shown

in

Figure 6-5,

the

least pressure

is

generated

by a 95%

closure

in

0.5r

c

with

the

remainder

of the

closure occurring

in

2.5t

c

.

A

valve poorly selected with respect

to

C

v

closure

characteristics

can

often

be the

difference

between

a

problem situation

and no

problem

at

all. Ball valves

(with their worm

and

compound lever actuators) have

the

ideal characteristic

of

closing rapidly

at first

fol-

lowed

by

very slow closure without

any

complicated

control gear.

The

capacity

of the

valve

is

reduced

80%

any

time interval less than

t

c

subjects

the

pipe

to the

maximum pressure,

A/z,

given

by

Equation

6-2. Note that

K =

2.15

x

10

9

N/m

2

(4.49

x

10

7

lb/ft

2

)

for

water.

Sl

Units

U.S.

Customary

Units

Ductile

iron

Steel

PVC

Ductile

iron

Steel

PVC

OD(m[ft])

0.335 0.324 0.324 1.10 1.063 1.063

c(m[ft])

0.0094 0.0048

0.0149

0.0308

0.0157 0.0259

E

(N/m

2

[lb/ft

2

])

1.66XlO

11

2.OTxIO

11

2.7OxIO

9

3.46

XlO

9

4.32

x

10

9

5.76

x

10

7

\L

0.28 0.30 0.45 0.28 0.30 0.45

C

1

=

1.25-Ji

0.97 0.95 0.80 0.97 0.95 0.80

C

2

=I-JLi

2

0.92 0.91 0.80 0.92 0.91 0.80

C

3

=LO

1.00 1.00 1.00 1.00 1.00 1.00

Wave

speed,

a

(m/s

[ft/s]),

from

Equation 6-4:

C

1

=

1.25

-

ji,

no air

1220

1150

280

4000

3770

930

C

2

=I-Ji

2

,no

air

1210 1130

280

3970 3710

930

C

3

=

1.0,

no air

1190

1110

260

3920 3640

840

C

3

=1.0,0.2%

air 490 490 230

1620 1600

760

For

C

3

=

1.0

and no

air:

r

c

=

2LAz

(in

s) 5.0 5.4 23 5.0 5.4 23

Flow

(m/s

[ft/s])

1.8 1.8 1.9 5.9 6.0 6.2

h

(in

m

[ft])

219 206

49.4

718 676 162

P(inkPa[lb/in.

2

])

2150 2020

480 310 290 70

at

about

20%

closure (see Figure 5-2).

In

contrast,

the

eccentric plug with linear valve stem angle

(or

per-

centage

of

closure)

has a

near linear characteristic

and

would

require either

a

much longer time

for

closure

or

a

programmed closure

to

equal

the

ball

valve's

control

of

surge pressure.

Figure 6-4.

Valve

closure

functions,

(a)

Plug

valve

(DeZurik

Series 100),

the

plug

angle

is

linear

with

the

time

of

closure;

(b)

ball

valve

(Williamette

List

36),

actuator

travel

is

linear

with

the

time

of

closure.

Figure 6-3.

Headless

coefficient

for a

butterfly

valve.

As

a

rule, valves should

not be

closed

in

less than

2

to

10

times

t

c

.

Insist

on a

computer solution wherever

more certainty than this

is

needed,

or

when

the

prob-

lem is

important because

of the

size

of the

pipe,

the

amount

of flow, or the

dynamic pressure involved.

Figure

6-5. Water hammer caused

by

various gate

valve

closure programs. After Waters

[4, p.

195J.

Example

6-2

Seating

a

Valve

Problem:

A

200-mm

(actually,

a

203-mm

or

8-in.)

butterfly

valve discharges

a flow of

12.6

L/s

(200

gal/min)

to

atmosphere

from

a

level pipe 4.83

km (3 mi)

long with

a

gauge pressure

of 690

kPa

(100

lb/in.

2

)

just upstream

of the

valve.

The

valve design

is

such that

it

will close

from

its

fully

open position

in 50 s.

Find

the

initial valve position (percentage

of

opening)

and

estimate

the

surge pressure when

the

valve closes.

Solution:

First,

find the

percentage

of

valve opening

from

Equation

3-15

and a

manufac-

turer's curve

of

K as

shown

in

Figure 6-3. Then

find the

approach velocity

in the

pipe.

Sl

Units U.S. Customary Units

G

0.0126

m

3

/s

n/im

,

Q

[2.00

gal/min]

[0.00223

ft

3

/s]

"

=

A

=

^^?

'

°-

401

^

"

"'A

=

[^^\[-&n*r\

=

1.28 ft/s

The

differential

head across

the

valve

is

(see Tables

A-8 and

A-9):

h

=

P

=

690,00ON

71

T

1

2

=

7Q5m

h

=

P_

=

100

lb/in

2

x

144Jn^

=

^

ft

V

9789

N/m

f

62.3

Ib/ft

ft

The

assumption

of a

steady pressure upstream

from

the

valve

is

often

erroneous.

The

pressure usu-

ally

changes with

a

change

in flow as the

system fol-

lows

the

characteristic pump

H-Q

curve.

For

pumps

with

type numbers greater than about

85

(specific

speeds greater than about 4400),

the

head developed

by

a

pump increases substantially

as the flow

decreases.

If the

type number

is

about

125

(specific

speeds about

6600)

or

more,

the

increase

in

head

is

dramatic,

as

shown

in

Figure

10-16.

If

the

time

of

closure

in

Example

6-2

were greater

than

t

c

,

the

resulting pressure surge would

be

less than

the 40

m

(130

ft)

calculated,

and a

computer analysis

would

be

used

to

reveal

the

pressure rise. Note that

the

shape

of the

valve closure curve would

be

very impor-

tant

in

such

an

analysis.

6-6. Surge Concepts

in

Flow

with

Friction

The

explanation

of

water hammer

in

Section

6-4 is

simplified

by

omitting

friction.

Such simplification

can be

justified

in

some systems, such

as

those with

short

force mains

in

which

friction

head

is a

small part

of

the

total design head.

In

most systems, however,

friction

contributes substantially

to the

total head,

and

it

is

important

to

understand

its

effect.

Figure

6-6 is

similar

to

Figure

6-

Ia

except

for the

hydraulic

grade line, which slopes during steady-state

flow.

Shortly

after

sudden closure

of the

valve

at

point

O, the

wave

front

would arrive

at

point

A.

Con-

sider

the

anomalies that would ensue

by

following

the

construction

shown

for

frictionless

flow in

Figure

6-

Ia

in

which

the

pressure

rise

for the

wave

front

at B

forms

rectangle

abed.

Its

counterpart

is

trapezoid

oabc

in

Figure

6-6,

but

because

the

hydraulic grade

line

be

is

sloping, water continues

to flow

past point

A

(corresponding

to

point

B in

Figure

6-1);

therefore,

v

(and, hence,

h)

would

be

less

in

Figure

6-6

than

in

Figure 6-

Ia

for the

same original velocity.

But flow

continues only until

the HGL

becomes level. Mean-

while

the

wave

front

progresses

to

point

B

(Figure

6-6)

and the

same phenomenon occurs again, with

more water

flowing

past point

A, the

pressure rising,

the

pipe expanding,

and the

water compressing.

Wylie

and

Streeter

[3]

call this occurrence

"packing."

As

the

wave

front

approaches

the

reservoir,

friction

decreases

v and

/z,

a

phenomenon called "attenua-

tion."

Eventually,

the

pressure builds

to a

maximum

above

the

level

of the

reservoir

and finally

dies

to the

reservoir level.

Depending

on the

length

of the

pipeline,

the

pres-

sure

surges

at the

valve might appear,

as

shown

in

Fig-

ure

6-7.

With each reversal

of the

pressure wave,

Sl

Units

U.S.

Customary

Units

From Equation

3-16

R

=

2gh

=

2X9.81X70.5

=

^

R

=

2_gh

=

2x32.2x231

=

^

v

2

(0.401)

2

v

2

(1.28)

2

The

discrepancy between

SI and

U.S. units

is due

only

to

rounding

off

the

difference

between

200 mm and 8 in.

From Figure

6-3,

the

valve

is

open about

7°

(nearly closed).

The

wave speed

can be

calculated

from

Equation

6-4 or

simply estimated

as

about

100 g. For

linear valve closure,

the

last

7° of

valve movement would require

(7/90)50

= 3.9 s.

a

=

100

x

9.81

«(980)

a =

10Ox

32.2-(3200)

2L

2x4830

in

,

2L

2x3x5280

in

'c

=

T

=

-98(r~

1()S

?C

=

T

=

3200

~

1QS

Because

the

valve will close

in 3.9 s

(less than

the

critical time,

10 s), it

closes quickly enough

to

generate

the

full

transient pressure given

by

Equation

6-2.

Al

0Av

980x0.401

Ar

.

A7

0Av

3200x1.28

ian

.

Ak

=

V

=

9.81

~

4

°

m

M

=

T

=

32.2

~

13

°

ft

And

the

total pressure

in the

pipeline

is

P

= 690

kPa

+ 40 x

9.81

=

1080

kPa

P=

100

lb/in.

2

+ 130 x

Q.433

1

-^^-

=

156

lb/in.

2

it

friction

decreases

the

pressure changes,

so

pressure

eventually

coincides with

the

reservoir level.

6-7.

Column

Separation

In

the

preceding sections,

the

pipelines

are

assumed

to

be

level,

but in

real systems pipes

may

slope

and

downsurges

result when power fails

or

when valves

at

the

upstream

end

(the usual configuration) close

quickly.

Under some conditions,

the

downsurges

can

cause column

separation

—

a

condition

to be

avoided

at

any

cost

by a

proper control strategy.

As

shown

in

Figure 6-8,

a

knee

(a

reduction

in

gradient

or a

change

from

a

positive

to a

negative gradient) makes

a

pipe-

line especially vulnerable.

If the

power

fails,

the

pumps

stop quickly with

an

effect

like closing

a

valve.

The

upstream

flow

(near

the

pumping station) stops

whereas,

due to

inertia,

flow

continues

at the

down-

stream

end

(near

the

discharge).

The

static hydraulic

gradeline begins

to

decay

as

shown

by

successive

curves labeled

t = 1 s, t = 2 s, t = 3 s,

until,

at t = 4 s, a

slight negative pressure occurs between

the

pump

and

the

knee.

At t = 4.5 s,

vapor pressure exists over

a

con-

siderable length

of the

pipeline,

and the

water

is

boil-

ing

and

forming large pockets

of

vapor. Column

separation

has

occurred.

On the

upsurge

of

pressure

that

follows,

the

vapor pockets collapse

and the two

liquid columns

can

come together

at

literally express-

train speed. Since

the

water

is

almost incompressible,

the

forces

at

impact

can be

enormous.

The

knee makes

the

situation

in

Figure

6-8

worse,

but

note that column separation would occur with

or

without

the

knee

and

would occur even

if the

pipeline

had a

uniform gradient. However,

if the

pipeline

pro-

file

were

flat

near

the

pump

and the

steep gradient

occurred near

the

reservoir, column separation might

be

avoided

—

a

useful

control strategy that

is

mainte-

nance

free.

A

computer analysis

of a

Pumping Station Desing - Second Edition by Robert L. Sanks, George Tchobahoglous, Garr M. Jones

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