Fox J.A. Transient Flow In Pipes, Open Channels And Sewers
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t6
F
p
Nomenclature
In
Chapter 9, u is the real part
of
the propagation
constant
y.
In Chapter 4,
P
is used in
the anlalysis of
pumps.
In
Chapter 4,
p
is a constant used in
parabolic
interpolation to
obtain the
value
of the
required
wicket
gate
opening.
in
Chapter
9,
p
is the imaginary
part
of the
propagation
constant
v.
In
Chapter
4,
y
is the exit
angle
of a
pump's
blading.
In
chapters containing references to
gas
processes,
e.g.
4 and 11,,
y
is the ratio
of the specific
heats
of
a
gas.
In
Chapter
4,
y
is a constant used in parabolic
intelpolation to
obtain the
value
of
the required
wicket
gate
opening.
In
Chapter 9,
y
is the
propagation
constant.
In Chapter 7, 5 is used to define the ratio
of the
surface breadth of
a channel to its cross-sectional area, i.e.
it is the
mean depth based
upon
the
surface breadth.
In Chapter 6,
11
is
the
fractional
gas
content
in
bubble form in
the
fluid.
fn Chonfer r( c ic tha fronfinnql cqc.nnfanf rliccnktarl in thp flrrirl
L
v
)
v2
ru
lrrv
6su
vvrrlvrrt utJJvr l vu rrl rrlv rrurs.
(
denotes
\/hlho.
Subscripted
values
of
(
are
for times
which
are
multiples
of
the
pipe period
I.
In
Chapter
2,
\is
the
ratio
of the
valve
area
at time / to its area of
time zero.
In
Chapter
4,0
is the angle between the crank
and line
joining
the
centre of the crankshaft to
the centreline
of the cylinder.
In Chapter
4,0
is arctan
(N|N-/Q/Q.).
In Chapter 4,
0
is the angle
subtended by the
water
surface
at the
centre of
horizontal
cylindrical air
vessel.
In
Chapter
7,
0
is the angle between a
channel and the axis of
symmetry of
the
momentum being transported
towards
a
junction
of supercritical channels.
In Chapter
7
,
0 is the angle that the surge line
makes
with the r
ordinate.
In
Chapter
8,
0 is a
weighting
coefficient.
In
Chapter
1, l. is a constant in the
expression defining
the
magnitude of the wavespeed in a
pipe.
The
coefficient of
dynamic
viscosity.
Is the coefficient of kinematic
viscosity:
ptlp.
Takes
the
value
of 3.14159.
p
is the mass density of the fluid.
In
Chapter
2,
p
is
known as
the
pipe
characteristic
in Allievi's
interlocking equations and is
cul2gh".
In Chapter
4,
6 is
Poisson's
ratio of
the material
out of
which
the
pipe
is
made.
In
Chapter
4,
6 is the slip in a reciprocating
pump.
The
shear
stress
on
the boundary of a flow.
In Chapter 6, r is the
coefficient
of surface tension.
In
Chapter
3,
0
is
the
variable
used
to demonstrate
the chain
rule
Nomenclature
17
of
differentiation.
In Chapter
4,
0
is the ratio
Q/N.
In Chapter
9,
O
is
the angle
of the
propagation
constant
in the
expression
for it,
i.e.
in the expression
1:
r"€lo.
T-
/aL^-+^-1
.,,:-,,.^l +^ .|^.i^--ta flro hn norohnlqa
pmnlnrlerl
in
lll \.rldPtEl
Lt
Y
lJ uDtvu
t\,
u\.JrSrrotL llrw...4
ysrqvvrqv
vrrrHrvJ
valve analysis.
In Chapter
4,
V
is HlN2.
In
Chapter
4, Q is
the angular
velocity of
a
pump/motor
combination.
In
Chapter
4, f) is
the
angular acceleration
of
a
pump/motor
combination.
0
0
v
C2
C)
'l
f^
r
5
0
0
e
l"
It
v
tt
p
p
o
T
T
0
The differential
equations
of
waterhammer
I.I INTRODUCTION
Flows
in
pipes
and channels
are
usually
in unsteady state;
yet their
analyses
are,
almost
always, carried
out
using steady
state
theory.
Such
an
approach
can
deal
neither
with the starting
of
the flow
nor
with
what
happens
when
it stops.
Changing
a
flow causes
large
pressure or
depth fluctuations
which can
endanger
the
integrity
of
the
pipe
or
channel
and
are the most
dangerous
events to
which
the
pipe
or
channel
will
ever be exposed.
Such
pressure fluctuations
are called
pressure transients
and
the
state
of
flow in
which they
occur is
called transient
flow.
In solid
body mechanics,
one
can analyse
the situation of
a mass
when
being
acted
on
by a force by
considering
the
mass to be concentrated
at
its
centroid
and
then
applying
Newton's second
law to
it, taking into
account any
frictional
forces
present.
Of
course,
if the force
acts
in a line
which does
not
pass
through
the
centroid,
one
has
also to take
into account
any moments
acting
and then to
calculate
the
angular
accelerations
that
it causes.
This
technique
can be
applied to
a mass
of
water
with
equal
validity
but, if
the
mass of
water is
very long and
thin, as occurs
in
pipelines and
channels,
such an
approach
will
not
be realistic.
This is because,
when a
force
is
applied
to
a long mass of
water, the force
is
propagated through
the
water by
wave
action. Consequently
this
takes finite
time and
so the entire mass
involved
does
not
experience
the force
at
the same
time.
An analysis that
takes account
of this
effect
is
called an elastic
analysis
whereas the usual
method of
analysing
solid
bodies
is said
to
be
a
rigid
body
analysis.
Such
an
argument
applies
with
equal
force to
a
solid
body;
usually,
however,
solid
bodies are
not as extended
in
space
as
are
pipelines. The speed
at
which
waves travel
through solids
is very
much
higher than
that at
which waves
travel
through
liquids;
so
the conditions
of
waves travelling
in
pipelines are
very
different
indeed
from
those
that apply
to
solid
bodies.
From the above,
it
will be seen
that there
are
two basic methods
of
analysis
of
unsteady flows:
rigid column
theory
in
which the entire fluid
mass
is considered
to
20
The differential
equations of
waterhammer
[Ch.
1
experience
the force at
the sarne
time,
i.e.
the force
is transmitted
to every
particle of
fluid
by
a
wave action
which travels
with
infinite
velocity
and
elastic
theory
in
which
the
wave
speed
is finite. Obviously
the elastic
theory
is more accurate
than
the rigid
column theory
but the rigid
column
theory
can
be
very useful in many
circumstances.
|, -r - ----Ll^--- - ^tE ^---i--:-^- ^-l ^L^--^l^
'T'L^,,^-^
I
wO equauons
are useg
Io solve
Pfoutcllr5
ur
iluw
rll
PrPcb
auLl uualllrclb.
r trs)
crrs
hyperbolic
partial
differential
equations,
one
being a form of the continuity
equation
and
the other
being a
form of
the
momentum
equation.
The types
of
problem that
they
can
be used
to
solve
are
(1)
bounded
flows
(in pipelines)
and
(2)
free
surface flows
in channels.
Bounded flows include
compressible
fluid
flows and incompressible
fluid
flows.
Resonance
can occur
in
pipe networks
and
techniques of
analysis
have
been
developed, using the
above-mentioned
equations
and employing
an approach
originally used
to
solve
the
situation
in electrical
transmission
lines.
An
alternative
method has
recently appeared
which
uses
the
same
basic equations
in
conjunction
with
a
technique
originally
developed
to deal
with
vibrations in structures.
This
marlrnrl ic rqfhpr hpffpr than fhc irnnerlancc rnefhnrl menfionerl hefore nnd is drre fo
trrvlltvv ro roLrrvr
vvarvr
H.
Chaudhry.
Both
methods
are
included
in this
book.
However,
the author
has
another
technique
to
advance;it
was suggested
to
him by
Professsor
A. E.
Vardy
at
Dundee
University
and
is described
in Chapter
9.
r.2
THE
EQUATIONS
OF
WATERHAMMER
(The
equations
applicable
to channel
flow are
given in Chapter
7.)
1.2.1 The continuity
equation
Consider a short
section
of
a
pipe
of
length 6x.
Assume
that
the fluid
in it is
under
a
pressure
that
increases
along
itslength
and that
its diameter
increases
in
the direction
of
x
increasing.
r
is measured
along
the
pipe
centreline
(Fig.
1.1).
This convention
is
the
normal
mathematical
convention
and
its adoption
here ensures
that no
difficul-
ties
will arise
with
signs.
The
mass
that enters
the
upstream
end of
the fluid
element
in a small
time
6r
is
pAu6r
and
the mass
that
leaves
the downstream
end
of the
fluid element
in the
time 6t
is
{pAu+[A(pAu)/ax/6x)
6r.
Thus
the mass
contained
in
the element
has increased
by
an amount
0(pAu)l0x6x6t
This
change
must equal
the
rate
of increase in
the
mass
contained
within the
element
with time multiplied
by
6r.
So
a(pAu)
=
_
0(p/)
0x
0t
Differentiating gives
10p,
lAA,
LOu,
1 0p
tar-Aa--;a.r-prat
*]P:o '
Av dt
(1.1)
(r.2)
Sec.
1.21
Rearranging
The
equations
of Waterhammer
Fig. 1.1.
2T
;(#.:#)
.)#-:#***:o
(1.3)
trd2
A--.
4
and so
aA
n
,0d
-:;u-
0-r Z Ox
AA
n
,0d
t:1" u
but
ctotum
22
The
differential
equations
of
waterhammer
Sec.
1.2]
The
equations
of Waterharnmer
Ah Ah czdu
0z
U;- r:
-r- -;-
-
U-: U
dx dt
8dx
ox
lch.
1
23
(1.4)
where/n is
the
hoop
stress
in the
pipe
(and
So.fr,
:
6p
dtZT),
E is Young's
mociuius
for
the
pipe wall
material
and
T is
the
wall
thickness
of
the
pipe. Note that
d is
the
pipe
internal
diameter. Therefore
Ed
:
6p
dzlzTE and
so 6dl0x:
(d2l2TE)(0pl0x)
and
OdlOt
--
(dzt2TE)@prct). Therefore
0Al0x
:
(nd3l4TE)
@plAx)
and
0Al0t
:
(nd3l4TE)
@prc|;
so
(1/.4)
@Alax)
=
(dlTE)
@prcx)
and
(1/,4)
@Alat):
(dlTE)
@p/a).
The bulk
modulus
of
the
liquid
is K and
K
-
-
6p/(6VlV)
where
V is
the
volume
of
a unit mass
of the
liquid.
P:
7lV
and so
V
:
Llp.6V
:
-
(llp')
6p;
hence
6VlV:
-
6p/p.
Therefore
6p
:
K0p/p;
thus
p(OplO-r)
:
K(ap/Ox)
and
so
10p
:
]
p(Aplax)
:t
0p
pOx
p
K
KOx
and
rop
uK 0t
Substituting
these
expressions
gives
The term u
(OzlOx)
is frequently neglected.
The maximum
value that1zlOxcan take
is
1.0
(i.e.
when the
pipe
is located
vertically);
so the term is small
compared with the
others
in
the
equation.
1.2.2
The
dynamic
equation
This equation
is a statement
of Newton's
second law. The net
force acting upon
the
element is equated to
the
rate
of
change
in momentum
occurring
across the
element.
This is an application
of Newton's
second
law.
Resolve all the forces
acting
upon the
element
along
the centreline
of
the
element.
These
forces
splitup
into three
groups (Fig.
1.2):
pressure
forces, frictional
forces and
weight forces.
D*
Fig. 1.2- The
pressure
diagram.
The axial
pressure
force
is
-
[A(p{/Ax]6x.
The force
produced
by
the
pressure
acting on the sides of
the element
equals
the
mean side
pressure
multiplied
by
the
projection
of
the area of
the sides on
a
plane
perpendicular
to
the
centreline
of
the
element.
It
equals
Ignoring second-order
small
quantities, this force equals
p(OAlOx)Ex.
The frictional forces
acting on
these sides of the element
equal tP6x
where
P is
the
perimeter
of
the element.
This force
acts in the
-
x
direction.
The component of the
weight force acting in the
-
x
direction
is
pgAlx
(dzldx).
l}p, |
0p, d
0p,l
d dP,t)r_n
ka,-- ,r ar
*
rEa*-;TE
u
-;a'
-'
Therefore
Uo:f#
rap:
pu
0r
,#
(i
.+,).*(+
.+)*#:o
(+.
+)(,x-#)
*#:o
So
Now, p
:
w(h- z)
where
lr is the
total
head and
z
is the elevation
of
the
pipe
centreline
above
the
datum
of the
head
h.
Therefore
0pl0x
-
w{(Ahlax)
-
@zrcx)}
andOp/Ot:
w(dhldt)
as z
does
not
vary with time.
Let
(1/K) +
@/Z:D:
1/pc2
(see
below
for an
explanation
p.
37).
Then
(o*o'#")ff''
The differential equations
of
waterhammer
lch.
1
The net force acting in the x
direction is
-
[A@filax]6x
+
p@NAx\6x
-
tP6x
-
pgA
6x
(dzldx). This force must equal
p.4
6x
(du/dr)
by Newton's
second
law.
So
and then
op
^
Az
,du
-#A
-
rP
-
psA;:
pAzdt
but
by
the
chain rule
Therefore
Oor0z0uOv
u*;+pgax+PDax*Par:Q
Dividing
through
by w
gives:
Ah 0z
r 0z
uOu 10u
ar
-
ar
*
***
ar*tar
*;
a,
:u
t
SO
Ah
uOu lOu
r
^
=-T--T-=T-:w
'
dx
8dx I
dt
wm
Here
z is the
hydraulic
mean
radius of the
pipe
and equals
AIP
where A is the
cross-sectional area of the
pipe
and P
is the
wetted
perimeter.
In this
book,
the form
of
the Darcy-Weisbach
equation
which
defines the head
loss due
to friction over
a length
L of.apipe
is hr:
fLuzl2gmwheremisthe
hydraulic
mean radius
and equals
AlP.
For
a circular
pipe,
m
=
dl4.
Now, American
practice
employs the form h,
:
fLuzlZgd
where d
is
the
diameter of
the
pipe.
American
readers
are
warned
of
this difference
between
British and
American
practice. The
effect
of this is that the equation
for
pipes
of circular
cross-section
in British
practice
is ft,
:
4fLuzlLgd
and in the
American form
it is
ft,
:
fLu2lZgd;
so
the American/is
four
times
the British
/.
Considering
a
urtit length
of
the duct,
/ A^
ez\ AA Az
drr
(r#
*
p;)sx
+rff
6x
-
tP6x
-
psAiu'
:
pA&;
du 0u
0u
-
-
rr- I-
a
-v
^
|
^
(lt
dx dt
Sec.
1.31
Rigid column theory
who4: P7.
So
r
,
zfulul
_:tif:_
wm
gd
Substituting
for
rlwm,
the equation
becomes
Ah u
Ou 1 Ou
Zfulul
J+:^
+:=++-0.
(1.5)
dx
gdx
I
dt
8a
Note the
use
of
ulul instead
of
u2 . This is introduced
to account for the
direction
of
the
frictional force.
The
equations
of
waterhammer
are thus
25
Ah Ah
c20u
0z
ti^
*
-
*-=-U;_:U
,
dx dt
8dI
dx
Oh
uOu 1Au
Zfulul
--L--r--J--
Ox'
glx' g}t
'
gd
(1.
1)
(
1.5)
1.3
RIGID COLUMN
THEORY
In this
theory the
assumption
is made that the
pipeline
is rigid and
does
not
distend
when
placed
under
pressure and
that the fluid is
totally incompressible.
This
means
that the
wave
speed
is
infinite.
Examining
the
continuity
equation,
if c approximates
to infinity,
then 0ul0t
approximatesto
zero.Thisisbecause
0u/0.risapproximatelyequalto
(1/c).0u/(0r) so
that
Ou/Ox tends
to zero
if c
tends to infinity.
By the same argument,
if c
is infinite,
Ohldx
must become
zero
also; substituting
these
values
into the continuity
equation
gives a
value
for 0hl0t
of
u(dzldx)
which is relatively small. No
useful information
can
therefore be obtained
from
this equation.
Examining
the dynamic
equation,
zfulul:9_h,
gd
0x
Then
Ah ldu
,0h, ^
:-T--?-:r'
Ox
'gdr
'
O-r
The differential
equations
of waterhammer
lch.
I
Integrating
with
respect
to x;
,-:#l'*
*
h,:o
So
-:H:h*h'l'
1 du
[*,
ffiJo*:
-(h+hr)
'
Integrating with respect
to x,
(1.6)
This integration
is legitimate
because,
in
the case of an incompressible
fluid,
Ou/Ox:0
(see
above);
so
du/Ot
and
therefore
du/dt
is
independent
of
x so
that the
integration can be
performed
with complete
validity.
The use
of this
equation
is
illustrated
by its
application
to some
problems of
interest.
l3.f Sudden
valve opening
Consider
a simple
pipeline
of
length
L
and diameter
d
(Fig.
1-3).
Fig. 1.3.
length L
diomejer
d
Sec. 1.3J
Rigid
column
theory
where
C
is a constant
of
integration.
When r
:
0, h,:
h,,
0du
;;:
-(h"+o)+c
andso C:hr.
When x: L,
Ldu
4fLtt2
iA:tt'-h1-A
where
lrr
is
the head
just
upstream
of
the
valve.
This
is obviously
correct.
Now
a''l: Coa,1/(Zght)
where a" is the crosS:sectional
area of
the
pipe,
C6 is
the coefficient
of discharge
of
the
valve and a., is
the flow
area
through
the
valve.
So
,
_(or\'r,
_ku,
":
\coo")
zg-
zg
Thus
'a|#,:^-(ry.r)*
Denote
(4fLld)
+ &
by K. So
Ldu
,
Kn'
Ea:n'-E
Then
28
The differential equations
of
waterhammer
.Ldu
o.t:E
h_1Kuztzg)
lch.
1
du
Kl2g)u
\/h,+
t/(K/zg)u
Integrating
gives
D
g!(2sh,K)tttl
-
|
,-:@T
where
u*
is the steady
velocity
which the
velocity approaches
asymptotically
as
/
becomes
large
(Fig.
1.a).
Fig. 1.4
-
The
plot
of u against
t.
f .3.3 Slow
valve opening
Equation
(1.6)
developed
above
can
be used equally
easily
to deal
with the
circumstance of
a slowly
closing
valve.
However,
the
resulting
equation
will most
probably
not
be integrable.
The
difficulty
arises
because
the
relationship
between
the
acceleration term
duldt
and
the associated
value
of
the time
t may be of a complicated
type
and
this, in
conjunction
with
the
friction
term
which is u2
dependent, can
make
the
integration
of equation
(1.6)
either
extremely
difficult
or impossible.
In
this
circumstance
it is
always
possible
to
perform
a simple
finite difference
integration.
N-
octuol
velocily
history
Z
\
theoreticol velocity
hisfory
(Zqnrx)o'5
tL
Sec.
1.31
Rigid column
theory
29
on a
hand-held
This
is usually
a
very
straightforward
process and can
be
done
programmable calculator.
The
basis of
such
a
finite difference
integration
is
given below.
1.3.4
Slow
valve
closure
A simple
pipeline is
one in
which none of
its
properties
changes
within
its
length.
In
other
words it
is made
of
the same
material
throughout
its
length, has a constant
wall
thickness,
diameter
and roughness
and connects
two
hydraulic
devices such
as
a
reservoir at
the
upstream
end
and a
valve at
its downstream
end.
Of
course
it
could
connect
any
fwo other
devices,
e.g.
a
pump
and
a sewage
works. Provided
that
both
of
these devices
can be
given
a
defined
mode of operation
which can
be specified
mathematically,
it is
possible
to
solve
the
system
using
rigid
column
theory,
so
obtaining
a detailed
description
of the
way in
which
the
velocity
and head
at the
valve
varies
with time.
To illustrate
how
this
is done the analysis
of
an
upstream
reservoir
and a
slowly
closing
valve is
given
below.
The
area of
the
valve
is defined
by the equation
a":
a;flt)
where
ao is the
futly open
area of
the
valve, a.
is the
effective
area of
the
valve at time
t
and/(r)
is a continuous
function
of t.
The flow through
the
valve is
given
by
applying Bernoulli's
equation
q,=
a"\t/(Zgh,)
.
/l" is the head
just
upstream
of the
valve.
where,4
is the
cross-sectional
area
of
the
pipe.
So
',:kA\/Qgh')
Therefore
,r:fue
Also
#:*ry+,s#x*
30
Substituting
for
a",
%
:nt/Qsn)
(
.
_
o
_4fLu2\ _Z4ortOt
dt- Lf(t)
\"o-""-
28'l
J-f,-a'
where n: Alao.
Integrate this expression
by a finite
difference method.
Again, this can
be
done
on
a hand-held
programmable
calculator. However, the
maximum head
can
be
obtained
analytically,
as follows.
First multiply through by/(r).
The differential
equations
of
waterhammer
!1':
Hn- h..-4fLu2
g
dt
^-('
"v
Zgd
f@*:NP(""-
o"-W)-h"Y
o
:
NP
(Ho-ft,.u*)
-zn^^-!r6g11
Srro^
*(Ho-
h^o)':
qh^*'#fv,4l'
)E*1=o
lch.
1
can be neglected,
as it
be obtained:
It
f(il
or
dhldt or
/(r)
dhldt
equals zero
and friction
frequently can, an
analytic solution
is
possible.
Then, when
dhldt is
zero or/(r)(dftldr
is
zero, ft-.* can
So
then
2 + 2{L
d[f(t)ltdt]
gnzHo
Sec.
1.41
Simple
elastic theory
Of course
O[(f(r)]/dt
must not itself
be a function of r. If it is or if friction cannot
be
neglected,
the
finite
difference
integration
method referred to before
must be
employed.
These
two examples show
how rigid column theory can be
used
by
relatively
simple methods
and
demonstrates
how it can also
give
analytic results.
Many
problems
can be solved
by such
methods and, even though
the results naturally
cannot include
any
consideration
of
the
wave
mechanisms that
are actually
present,
they are frequently of adequate
accuracy.
I.4 SIMPLE
ELASTIC
THEORY
1.4.1
Instantaneous valve closure
By consideration
of
what
happens
when a
valve
at the downstream end of
a
pipeline
closes,
much can
be understood
about
the behaviour
of
wave action in transient
flows.
Assume
that the
valve closes instantaneously,
and that
pipe
friction
can
be
neglected.
This
is obviously
impossible
but
by
consideration of this hypothetical
circumstance
it is
possible
to
deduce
what
happens
when a valve closes rapidly
but
not
in
zero time, as
will be seen
from
what
follows.
The
velocity in the
pipeline
will
be u everywhere
initially.
When the
valve closes,
the fluid layer
immediately
upstream is suddenly
brought to rest
and consequently
its
pressure
rises. Just behind
this
layer, the next layer
is brought to
rest
and experiences
the same
pressure
rise.
This
pressure
rise maintains
the
pressure
of the first
layer.
This
process
continues
in an
upstream
direction, layer after
layer being brought
to
rest
successively,
their
pressures rising correspondingly.
However, as each
layer
is
brought to
rest, because
its
pressure rises, it must also compress and
the
pipe around
it must expand; consequently,
fluid must
flow into the layer to occupy the
additional
volume
so
provided.
When the layer
under consideration experiences
this compres-
sion
and
the
pipe
expands,
just
upstrearn
of it, the flow is travelling at
the
initial
velocity
u; so it
will take
a little time
to fill the extra
volume
that the
layer's
compression
has now
provided.
Consequently,
the sequence
of
successive
layers
being brought to rest
does not
occur instantly. Instead, each
layer
takes
a definite
time to stop
and so this
process
of stopping
all the
fluid
layers
in the
pipe
length
takes
a finite
time.
In a
normal
pipe,
made of
steel, and
of the thicknesses that
are commonly
used
in
pipes,
the velocity of a
wave is about
1300 m/s. This
wave
speed
can be
calculated
and
later this
calculation
will
be
demonstrated.
The
process
of successive
fluid
layer
compression is
shown
in Fig.
1.5(a) below.
As the
wave passes
up the
pipe,
the
pressure
rise associated
with it distends
the
pipe;
this is illustrated in Fig.
1.5(a).
In Fig.
1.5(b)
the situation
when the
wave reaches the reservoir
(at
timeZLlc)
is
illustrated.
31
E:r+kz.-,,/(i.*1)
32
The
differential equations
of
waterhammer
lch.
1
t.+
h.+hs
pressure
heod
plol
Fig.
1.5(a)
-
The
pressure wave
starting
off up the
pipe.
Note
that,
in this
argument,
friction
is completely
neglected.
Once the
wave
has traversed
the
pipe,
the
entire
pipe
is at
a
pressure of
ft, * htand
the velocity
is everywhere
zero.
When the
wave arrives
at the
reservoir,
the
pressure
in it
is higher
than that
in the
reservoir
by
the amount
h, and
the
velocity everywhere
will be
zero. This
is an
unstable
circumstance;
so the
water
starts
to flow
backwards
out
of the
pipe
into
the
reservoir.
As each
layer
of fluid
moves
backward,
its
pressure is relieved,
dropping
back
to its original
value, i.e back
to
the reservoir
head.
In effect
the
pressure
energy
it
possessed
is
converted
into
kinetic
energy.
The
pressure head
drops by an amount
of ft;
and
the
velocity becomes
-
D.
The
situation
then
develops
as
illustrated
in Fig.
1.5(c)
As
the
first layer
attempts
to move
away from
the
valve it
will be unable to
do so
t=
L
'C
dislended
prpe
Sec.
1.41
4>r
>*
Simple
elastic
theorY
pressure heod
plol
Fig.
1.s(b).
33
.?L
l=
z
and
the
pressure
will
drop
by
the same
amount
as it
rose
when
the
original
flow
towards
the
valve
*u,
uri"rted.
This
drop
in
pressure
will be
maintained
by
the
attempt
of the
second
layer
to move
away
from
the
first
layer,
and
the
second
layer's
pr"rrur.
will also
drop
by
the
same
ft,
value.
The
velocities
of these
layers
will all
be
ieduced
to zeroatso.
'itre
rarefaction
wave that
this
phenomenon
represents
will
pass
up
the
pipe at
wave
speed
taking
a
time
of.
Llc to
do
so.
At
the end
of
a
total
time
of
ntc
(iii.1.5(d))
there
will
a mass
of
stationery
water
in the
pipe all
at
a
pressure
head
/r, below
reservoir
pressure
head
ft,.
This
situation
is
again
unstable
and
flow
will immediately
start
fromthe
reservoir
end
at a
velocity
u towards
the
valve.
This
will
raise the
pressure
of each
layer
of
fluid,
starting
from
the
upstream
end,
back
to the
reservoir's
pressure
head
and
raising
its
velocit! back
to
the
original
value u.
When
this'wave'
reaches
the
valve
the
situation
Dressure
heod
plol
tts
34
The
differential
equations of
waterhammer
lch.
I
hrs-h;
I
pressure
heod
olol
3L
'C
Fig.
1.5(c)
-
The
fluid is now moving
away from
the
valve
end of
the
pipeline.
is
exactly the
same
as
when
the
valve
first closed; so the entire
process
repeats and
continues
to do so
for
ever.
In actuality
it does not repeat endlessly but is
gradually
attenuated
away to
nothing by the effects
of friction
which were
neglected earlier.
How
this happens
will be explained
later
(see p.
38, 39)
To calculate the
value
of
LH,
we
proceed
as follows.
In
the absence
of friction,
the
value of AH is
given
by
Ldu L
Lu
L,H:
gdt g&t
pressure
heod
plol
L,
t>4
reduced
oioe
diomeler
The
Au value
in this circumstance
is u.
It remains to find the time required to bring
all
!*orio,t.o#
Sec.
1.41
Simple
elastic
theory
Fig.
1.s(d).
Note
thatfi,
is the
hoop
stress
in the
pipe
wall.
Now/5
=
Lpdl2T
where
Zis the
pipe
wall
thickness.
So
the
volume indicated
by the shaded
area
in
Fig. 1.6 equals
Lpd3nLlZTE.
Thus
the
total additional
volume of
liquid
admitted
to
the
pipe
due
to its exposure
to
the
pressure Ap is
35
4L
oressure
heod
Ptol
t=-
c
originol
pipe
diorneter