Ellis,J. Pressure transients in water engineering, A guide to analysis and interpretation of behaviour
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such computations to be undertaken remains an area of particular
interest.
22.10 Cyclical oscillations
Kirby (1981) concluded that brittle failure of PVC pipes was the most
important operational factor caused by pressure transients. These
failures appeared to be particularly associated with pumped sewerage
installations. Unlike many water supply systems, sewage pumping can
be more intermittent with relatively frequent starting and stopping of
pumps and substantial pressure waves produced by closure of non-
return valves. This leads to a rapid accumulation of large-amplitude
oscillations. In laboratory tests, Gotham and Hitch (1975) found that
if maximum amplitude of these oscillations was 50% of the rated
pipe pressure, any need for reduction in hoop stress safety factor was
avoided.
Vinson (1981) found that repetition of large-amplitude pressure
surges in PVC pipelines could lead to failure even where the peak
transient pressure remained below the pipe burst pressure. Depending
upon the applied stress, the pipes could sustain millions of loading
472
Chainage
Invert lev.
Max. head
Min. head
Head (mAD)
30
25
20
15
10
5
0
–5
–10
–15
–20
Fig. 22.16. Composite system, envelope curves following pumping failure
Pressure transients in water engineering
cycles before failure. A log relationship was established between hoop
stress and the hydrostatic life of PVC.
For 150 mm diameter PVC pipes with SDR of 17 and 26, the number
of cycles C to failure was:
C ¼ 3:7 10
21
4:898
ð22:33Þ
where is peak hoop stress ¼ pD=ð2sÞ in psi.
The relationship above represents the lower 95% confidence limit of
the measured data.
Where the pipe wall has been scratched to a depth of 9% or more of
wall thickness, fatigue life is reduced and an effective wall thickness ¼ s
— scratch depth should be used when computing hoop stress. This
equation is a tool for use in large-amplitude conditions where peak
transient pressure > normal maximum working pressure.
Fedosoff and Szpak (1979) studied the effects of cyclic overpressures
in HDPE pipes. Their studies into ‘creep’ in this material showed that
when subject to cyclic overpressures of 2.5 times the normal design
stress, there was evidence of strain hardening in the material. They
concluded that if a ‘critical strain’ limit of 6—7% was not exceeded,
there was no permanent damage to the pipe.
473
Flexible pipe
23
Amplification of transient
pressures
The potential for actual transient pressures to exceed the expectations
predicted by a straightforward analysis has been demonstrated with
respect to the presence of modest pockets of air or gas within a pipeline
system. Where a network of pipelines occurs in which a range of pipe
sizes is present there also exists a potential for transient pressures to
increase in amplitude as pressure waves propagate through the
system. Some examples will be considered to illustrate circumstances in
which this can occur. Before considering actual systems within which
pressure transient amplification has been observed, three instances will
be examined which have the potential to produce transient pressure
wave increase.
23.1 Transmission of pressure waves through a branch
connection
These relatively abrupt rarefaction and compression waves developed
on the suction side of a booster pumping station, when pumps are
respectively started and stopped, propagate upstream along the trunk
suction main. Components of these pressure waves will travel into
any branch connections made from this main.
Considering Fig. 23.1 and assuming for simplicity that the trunk main
of area A
t
supplies only the pumping station with water at a velocity V
o
and that zero flow enters the distribution main of area A
t
. Further,
assume negligible pipeline resistance and a constant speed of pressure
waves. At each connection, the head at the connection point can be
calculated using the equation:
H ¼ 1=g
X
ðAJÞ=
X
ðA=aÞ
474
or
H ¼ a=g
X
ðAJÞ=
X
ðAÞð6:4Þ
where wave speed a is considered constant. Suppose a pump trip occurs
at the PS which reduces the initial velocity V
o
to zero. The resultant
invariant travelling upstream from the pumping station will have a
value V
o
. The invariant from the upstream conduit has value þV
o
and the value from the trunk main is zero. Head at the connection
point is then:
H ¼ a=gV
o
2A
t
=ð2A
t
þ A
d
Þ
where A
t
is the cross-sectional area of the trunk main and A
d
is the
cross-section of a distribution main.
Velocity in the trunk main upstream of the connection becomes:
V
t
¼ V
o
V
o
2A
t
=ð2A
t
þ A
d
Þ
and the resultant invariant propagating upstream of the connection
is:
J
t
¼V
o
ð1
1
2
A
d
=A
t
Þ=ð1 þ
1
2
A
d
=A
t
Þð23:1Þ
475
Suction trunk main
Rising main
J+ = +V
o
J+ = V
o
2/(1 +
1
/
2
A
d
/A
t
)
J– = V
o
(1 –
1
/
2
A
d
/A
t
)/(1 +
1
/
2
A
d
/A
t
)
+T
+X
+X
J– = –V
o
J– = 0
PS
Distribution mains
A
t
A
d
V
o
A
d
Fig. 23.1. Definition sketch for connection analysis
Amplification of transient pressures
Velocity at the distribution main is:
V
d
¼ V
o
2A
t
=ð2A
t
þ A
d
Þ
and the invariant travelling into the distribution main is:
J
d
þ¼V
o
2=ð1 þ
1
2
A
d
=A
t
Þð23:2Þ
when the ratio A
d
=A
t
< 1 the invariant travelling into the distribution
main will have a larger value than the invariant arriving at the branch
connection.
23.2 Pressure wave transmission through a change of
cross-section
As pressure waves travel into the distribution network there is a
tendency for the pipe sizes to become smaller as the more remote
parts of the network are encountered. If a pipe is connected to a
second pipe of small diameter (Fig. 23.2), then head H at the con-
nection will be given by:
H ¼ 1=gðA
1
J
1
A
2
J
2
Þ=ðA
1
=a
1
þ A
2
=a
2
Þ
Assuming for simplicity that a
1
¼ a
2
¼ a and that initial flow is at
rest V ¼ 0 under head H
o
¼ 0. Let a compression wave travel in pipe
476
J
1+
= V
1
+ g/aH
1
J
2+
= (J
1+
)2A
1
/(A
1
+ A
2
)
J
2–
= 0
V = 0
+a
V
1
H
1
+X
H
o
= 0
A
2
A
1
Fig. 23.2. Definition sketch for analysis of cross-section change
Pressure transients in water engineering
1 towards the connection with an invariant value ¼ J
1
. Then:
H ¼ a=gðA
1
J
1
A
2
J
2
Þ=ðA
1
þ A
2
Þ
¼ a=gðJ
1
A
2
=A
1
J
2
Þ=ð1 þ A
2
=A
1
Þ¼a=gJ
1
=ð1 þ A
2
=A
1
Þ
Velocity in the smaller pipeline 2 is given by:
V
2
¼ J
2
þ g=aH ¼þJ
1
=ð1 þ A
2
=A
1
Þ
The invariant value propagating into pipe 2 is then:
J
2
¼ 2J
1
=ð1 þ A
2
=A
1
Þ or J
2
=J
1
¼ 2=ð1 þ A
2
=A
1
Þð23:3Þ
The invariant in pipe 2 has a greater value than that in pipe 1 while
A
2
=A
1
< 1:0.
Suppose a compression wave propagates in the positive direction
towards the connection of two pipes producing a velocity change of
1 m/s and amplitude change 100 m behind the wave front. Further,
assume velocity and head in front of the wave to be zero. Let acoustic
wave speed a in both pipes be 1000 m/s and for simplicity let accelera-
tion due to gravity g be 10 m/s
2
J
1
¼ V
1
þ g=aH
1
¼ 1:0 þ 10:0=1000:0 100 ¼ 2:0m=s
and
J
2
¼ V
2
g=aH
2
¼ 0:0 10:0=1000:0 0 ¼ 0:0m=s
then
H ¼ 1000:0=10:0 ðA
1
2 A
2
0Þ=ðA
1
þ A
2
Þ
¼ 200=ð1 þ A
2
=A
1
Þ
If A
1
¼ A
2
then H remains at 100.0 m and V remains at 1 m/s in the
downstream pipe after passage of the compression wave. If A
2
< A
1
then H > 100:0. Substituting the value for H into the expression for
J
2
then:
J
2
¼ 0 ¼ V
2
10:0=1000 200=ð1 þ A
2
=A
1
Þ
giving
V
2
¼ 2:0=ð1 þ A
2
=A
1
Þ
and again if A
2
< A
1
then V
2
> 1:0 m/s. The pressure wave travelling
downstream in pipe No. 2 will have a quasi-invariant value:
J ¼ V
2
þ g=aH ¼ 2:0=ð1 þ A
2
=A
1
Þþ0:01 200=ð1 þ A
2
=A
1
Þ
¼ 4:0=ð1 þ A
2
=A
1
Þ
477
Amplification of transient pressures
When A
2
< A
1
then J > J
1
and so the pressure wave travelling into a
network containing pipes of reducing diameter may be amplified. For
example, if A
2
¼
1
2
A
1
then H ¼ 133:0 m/s and V
2
¼ 1:33 m/s, giving
a transmitted wave invariant value ¼2.67 m/s as compared with the
incident wave value of 2.0 m/s.
23.3 Meeting of opposing pressure waves
Pipe networks serving distribution areas can also contain looped pipe
arrangements. Individual pressure waves travelling into the system
along branch connections with the trunk main may eventually meet
within the distribution system. Consider a pair of opposing compression
waves of equal invariant value travelling within a pipe as illustrated in
Fig. 23.3. When the waves meet, the solution at the meeting point can
478
J+ = V + g/aH J– = –V – g/aH
aV/g
+a –a
H
+X
+V –VV = 0
+V –VV = 0
–a +a
H
H H
Fig. 23.3. Meeting of opposing pressure waves
Pressure transients in water engineering
be obtained from:
V ¼ðJþþJÞ=2 and H ¼ðJþJÞ=ð2g=aÞ
with velocity reduced to zero and head H ¼ a=gV. The head rise is
increased by the intersection of opposing compression waves. When a
pair of opposing rarefaction waves meet, the head drop at intersection
of the waves is likewise greater, as will be illustrated in section 23.5
concerning wellfield transients.
23.4 Pressure waves in a suction main
The first example of increased pressure wave amplitude comes from a
relatively simple treated water system in which a branch connection
is created halfway along an existing gravity main. A booster pumping
station is introduced on the new branch main a short distance down-
stream of the connection to the gravity main. Two duty pumps and a
standby unit were to be installed at the new pumping station. The
study was carried out to establish conditions along the cement-lined
ductile iron branch main which has a length of over 8 km and is ND
300. The gravity main is relatively short at 1700 m and is also of ductile
iron with diameter ND 600. The study was primarily concerned with
the new rising main and its protection, with information on the new
main and its profile, design flow rate and booster pump characteristics
readily available. When it came to the existing system, data were
more difficult to access and it would have been tempting to simplify
by assuming a range of constant heads on the suction side of the booster
station. This head range could have covered the normal operational
head range at the connection point.
23.4.1 Protection of the rising main
Studies showed that a pressure vessel was required to protect the rising
main against sub-atmospheric pressures following a station blackout
affecting all operational pumps. The resultant envelope curves for the
main were as shown in Fig. 23.4. The range in head just downstream
of the pumping station was around 80 m.
23.4.2 Conditions in the gravity main
Within the 17 900 m long upstream gravity main which was modelled in
detail, the corresponding envelope curves for the pump failure event
479
Amplification of transient pressures
were predicted to be as illustrated in Fig. 23.5 for the circumstance
when inlet valves at the downstream end of the main were shut. The
increased amplitude of oscillation over the lower parts of the main
can be seen with head range comparable to that in the rising main
and the appearance of sub-atmospheric pressures over parts of the
pipeline. Figure 23.6 shows time histories of head after pumping failure,
just upstream and downstream of the booster pumping station and at
the downstream closed valve at Rawyards over an interval of 1 min.
480
Elevation (mAD)
Chaina
g
e
350
300
250
200
150
100
50
0
Invert
Max. head
Min. head
Fig. 23.4. Envelope curves for rising main on pump start
Elevation (mAD)
Chaina
g
e
300
250
200
150
100
50
0
Invert
Max. head
Min. head
Fig. 23.5. Envelope curves for gravity main on pump start
Pressure transients in water engineering
The amplification of surge amplitude is quite evident at this down-
stream closed valve.
This simple case highlights the need to include all parts of a system
which may be significantly influenced by transient effects. The capacity
of pressure vessel necessary to protect the rising main would not be
greatly altered by simplifying conditions upstream — that is, by imposing
a constant upstream head at the pump suction. Adverse conditions
within the existing gravity main would not have been identified by
such a measure. It is very desirable to include, as far a possible, existing
elements of the system and the range of possible flow conditions
therein, even if full details cannot be obtained due to passage of time.
23.5 Amplification within a network
23.5.1 Kirkleatham Lane Pumping Station
Treated water is supplied to both residential and industrial areas on
Teesside through an extensive network of trunk mains as shown in
Fig. 23.7. Industrial users include a steelworks which receives a bulk
supply through a DN 250 steel main branching from conduit No. 62
(C62) which is 600 mm in diameter. A motorised valve is used to
control flow into the steelworks tank. As part of a plan to provide
treated water to areas of the North York Moors, a pumping station
was developed at Kirkleatham Lane (Fig. 23.7) at the downstream
481
Elevation (mAD)
Time
(
s
)
350
300
250
200
150
100
50
0
Rawyards
d/s pump
u/s pump
Fig. 23.6. Head variations in the gravity main after pump start
Amplification of transient pressures