Ellis,J. Pressure transients in water engineering, A guide to analysis and interpretation of behaviour
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20.11.6 Backflow check valve, 407
Chapter 21 Check valve characteristics 409
21.1 Check valve response, 409
21.2 Swing check valves, 411
21.2.1 Free-acting modifications, 414
21.2.2 Valve damping modifications, 415
21.3 ‘Recoil’ valves, 416
21.4 Tilting disk valve, 417
21.5 Rubber flap valve, 418
21.6 Split disk valve, 420
21.7 Butterfly valve used as a check valve, 421
21.8 Nozzle valves, 422
21.9 Moving ball, 424
21.10 Sleeve or duckbill valve, 425
21.11 Membrane valve, 434
21.12 Prediction of valve behaviour, 435
21.13 Use of the charts, 445
Chapter 22 Flexible pipe 446
22.1 Review of pipe materials and properties, 446
22.2 Pressure transient effects, 448
22.3 Strain and deflection, 449
22.4 Establishing the rate of ovalisation in the longer
term, 451
22.5 Long-term buckling pressures — unconstrained
surroundings, 452
22.6 Long-term buckling pressures — constrained
pipelines, 454
22.7 Deformation of a circular section and its effect on wave
speed, 458
22.8 Short-term elastic buckling under hydraulic transient
effects, 463
22.8.1 Unconstrained conditions, 463
22.8.2 Constrained conditions, 463
22.9 Application of a flexible pipe model, 465
22.9.1 Long horizontal pipeline, 465
22.9.2 Descending outfall, 467
22.9.3 Uniformly rising main, 468
22.9.4 Pipeline of differing properties, 470
22.10 Cyclical oscillations, 472
xi
Chapter 23 Amplification of transient pressures 474
23.1 Transmission of pressure waves through a branch
connection, 474
23.2 Pressure wave transmission through a change of cross-
section, 476
23.3 Meeting of opposing pressure waves, 478
23.4 Pressure waves in a suction main, 479
23.4.1 Protection of the rising main, 479
23.4.2 Conditions in the gravity main, 479
23.5 Amplification within a network, 481
23.5.1 Kirkleatham Lane Pumping Station, 481
23.5.2 System modelling, 484
23.5.3 Recorded pipe bursts and pressure extremes, 485
23.6 Wellfield transients, 488
23.6.1 Collector pipeline system, 488
23.6.2 Borehole and wellhouse configuration, 490
23.6.3 Wellfield operating conditions, 490
23.6.4 Pumpset inertia, 492
23.6.5 Sequenced pump operation, 493
23.6.6 Pumping failure, 494
23.6.7 Air valve operation, 496
23.7 Potential for amplification, 497
Chapter 24 Flow instabilities 499
24.1 Types of oscillation, 499
24.2 Pumping system — Glasgow East Main and Daer
network link, 501
24.2.1 Burnside booster pumping station, 503
24.2.2 Hydraulic transient computations, 503
24.2.3 Castlemilk Low pumping station, 503
24.2.4 Transient pressures, 504
24.2.5 Spread of unstable oscillations and consequences, 506
24.2.6 Possible remedies, 506
24.3 Gravity flow system, 507
24.3.1 The break pressure chamber, 508
24.3.2 Head losses, 508
24.3.3 Flow regulation, 508
24.3.4 H1 valve movement, 511
24.3.5 Remedial measures, 512
24.4 Small hydro station, 513
24.4.1 Observed behaviour, 513
xii
24.4.2 Comments on observed behaviour, 515
24.4.3 Modelling behaviour, 516
24.4.4 Remedial measures, 518
24.4.5 Final comments, 519
References 520
Further reading 525
Index 527
xiii
Acknowledgements
The author would like to express his appreciation for assistance received
from a number of sources. To Chris Bone of Morton & Bone Services,
UK agents for Charlatte bladder pressure vessels, who provided much
useful material on this form of surge protection. To Sam Gilbert of
Glenfield Valves Limited, Kilmarnock, Scotland, for many interesting
conversations regarding valve performance; also for providing infor-
mation on a wide variety of valve types. This book would not have
been possible without the work of engineers over many years who
laboured to improve understanding of this subject. From among now
retired engineers I would like to thank George A. Milne, formerly of
Crouch & Hogg, Glasgow, for taking time to discuss the subject at
length.
xv
Notation
A cross-section area of pipe measured normal to axis (m
2
)
A
s
horizontal water surface area
a speed of pressure wave through composite liquid/pipe
medium (m/s)
a
H
, b
H
, c
H
coefficients of equation H
p
¼ a
H
V
2
þ b
H
V þ c
H
a
T
, b
T
, c
T
coefficients of equation T ¼ a
T
V
2
þ b
T
V þ c
T
C number of cycles till failure
C
p
specific heat of gas at constant pressure
C
v
specific heat of gas at constant volume
C
þ
characteristic with gradient dx=dt ¼ V þ a (m/s)
C
characteristic with gradient dx=dt ¼ V a (m/s)
c pipe circumference (m)
also pipe strength reduction factor
c
a
, c
g
pipe strength reduction factors
c
1
pipe contraction restraint factor
c
2
coefficient
D pipeline diameter (m or mm)
Df deformation factor
d handwheel diameter
dQ=dt rate of heat outflow from a pressure vessel
dr incremental change in radius
d incremental angle
E elastic (Young’s) modulus of pipe wall material (N/m
2
)
E
0
tangent modulus of soil
E
k
kinetic energy of rotating body ¼
1
2
I!
2
E
r
elastic modulus of rock (N/m
2
)
e kinetic energy of flow in a pipeline ¼ V
2
=ð2gÞ
F overall head loss coefficient ¼
P
K
L
þ fL=D
also flow factor for regulator
xvi
F
ro
¼ FV
2
o
=z
f friction factor in equation h
f
¼ fLV
2
=ð2gDÞ
f
c
ovality correction factor
f
e
equivalent overall friction factor ¼
P
K
L
D=L þ f
fos factor of safety
f
s
pipe support factor
G shear modulus of pipe material (N/m
2
)
g acceleration due to gravity (m/s
2
)
H
b
inertial head rise in a branch pipe
H
bs
base outlet head at a PRV
H
i
inertial head rise ¼ aV=g (m)
H
j
inertial head rise at a junction of pipes
H
o
initial head or piezometric level
H
p
head developed by pump
H
r
liquid level in reservoir or large tank
H
v
constant outlet head at a PRV or head rise at a valve
H
x
inertial head rise at a point ‘x’ along a pipeline
h pressure head ¼ p=ðgÞ at a point in the pipeline (m)
also absolute pressure head in vessel
h
abs
absolute pressure head
h
atm
atmospheric pressure head
h
f
head loss due to pipeline flow resistance (m)
h
vap
vapour pressure head (gauge)
I moment of inertia of a pumpset (kg/m
2
)
I
s
¼ s
3
=12
i pipe number
i.l. invert level of pipeline
Jþ invariant or quasi-invarient quantity propagating along
C
þ
path (m/s)
J invariant or quasi-invarient quantity propagating along
C
path (m/s)
J
o
initial value of invariant or quasi-invarient
K bulk modulus of fluid (N/m
2
)
K
g
bulk modulus of gas (N/m
2
)
K
s
bulk modulus of matter in sewage (N/m
2
)
k
i
þ1 if isolating valve open and 0 if valve closed
k
o
equivalent uniform sand grain roughness height of new
pipe (mm)
k
s
equivalent uniform sand grain roughness height (mm)
L pipe length measured along axis (m)
lnðxÞ natural logarithm (to base e)ofx
xvii
log
10
ðxÞ common logarithm (to base 10) of x
m number of waves in buckling calculation
m
g
mass of gas dissolved/unit volume of liquid
N rotation speed of a pump (rev/min)
n polytropic exponent of gas law
P
r
power output of a pump
p gauge pressure ¼ gh at a point in the pipeline (N/m
2
)
p
abs
absolute pressure (N/m
2
)
p
atm
atmospheric pressure (N/m
2
)
p
b
critical buckling pressure
p
ex
external pressure
p
H
percentage hydrogen ion concentration in water (%)
p
k
critical pressure
p
s
set pressure at which regulator starts to open
Q flow rate at a pipeline section at some time (m
3
/s)
Q
air
air inflow rate at air valve
Q
e
flow rate from an ejector system
Q
m
mixture discharge rate from a jet pump
Q
o
initial steady flow
Q
out
outflow to a gravity sewer
Q
p
flow rate of pumped medium through a jet pump
Q
r
discharge through relief valve or regulator
q ratio Q
p
=Q
m
for a jet pump
also vertical load on pipe/metre axial length
q
i
discharge at a demand point
R reactive force between sliding surfaces
R
c
collapse rate of flexible pipe ¼ minutes/% decrease in
diameter
R
e
Reynold’s number of flow at a section ¼ VD=
r
m
distance from pivot to centre of gravity of check valve
door
r
p
radius of pivot pin
S gradient of head loss due to pipeline resistance
S
o
initial steady flow gradient of head loss due to resistance
S
r
ring stiffness/unit length of pipe
s pipe wall thickness (m)
s
f
sliming factor
s
j
þ1 for C
þ
characteristic and 1 for C
characteristic
s
v
V
o
=jV
o
j provided jV
o
j > 0
T torque
T
abs
absolute temperature
xviii
T
eff
effective valve closure time
T
r
resultant thrust on valve hinge
t time (s)
V mean velocity of flow at a cross-section at some time
(m/s)
V
o
initial velocity
Vol volume of liquid þ volume of gas ¼ Vol
l
þ Vol
g
(m
3
)
Vol
f
volume of free or undissolved gas (m
3
)
Vol
g
volume of gas dissolved in liquid (m
3
)
Vol
in
inlet volume of sleeve valve
Vol
L
volume of liquid (m
3
)
Vol
p
volume of a pipeline
Vol
s
volume of matter in sewage (m
3
)
Vol
t
total volume (m
3
)
W
s
buoyant weight of check valve door
x axial distance along pipeline from some datum (m
3
)
Z head relationship for jet pump
z pipe centreline level measured from a common
horizontal datum (m
3
)
also d/s reservoir level pressure vessel level þ h
abs
z
¼ V
o
=
p
½LA=ðgA
s
Þ
z
w
weir crest level
¼ y=D
m
k rate of increase in roughness height with time
x distance increment in numerical scheme
x
p
small distance increment
t time increment in numerical scheme
t
p
small time increment
y vertical change in pipe diameter
! rotation speed change over a time increment
relief valve diameter
void fraction ¼ð
1
þ
f
Þðp
atm
=p
abs
Þ
ð1=Þ
¼ D
min
=D
m
¼ 1
f
void fraction of free gas ¼ Vol
f
=ðVol
f
þ Vol
l
Þ
s
sewage matter void fraction ¼ Vol
s
=Vol
t
1
void fraction of free gas
2
void fraction of dissolved gas ¼ Vol
g
=ðVol
g
þ Vol
l
Þ
ratio of specific heats for gas ¼ C
p
=C
v
h
horizontal radial deformation of pipe
i
initial pipe deflection
xix
y
vertical radial deformation of pipe
" maximum strain
"
cr
critical strain for a perfectly round pipe
"
f
effective maximum strain
pump efficiency
angle between flats of a polygonal duct (rad)
also angle of pipeline with the horizontal
also angle of check valve door from the vertical
Poisson’s ration for pipe wall material
also coefficient of sliding friction
kinematic viscosity of fluid (m
2
/s)
ratio h
m
=h
liquid density (kg/m
3
)
d
density of disk material
g
density of gas (kg/m
3
)
m
mean density of mixture ¼
g
þð1 Þ (kg/m
3
)
also mean density of mixture exiting from a jet pump
p
density of pumped medium through a jet pump
s
density of sewage matter (kg/m
3
)
circumferential hoop stress in pipe wall (N/m
2
)
also axial spring force
! rotation speed of a pump (rad/s)
!
o
rotation speed at the start of a time increment
!
s
specific speed of a pump
xx
Introduction
A number of terms have been associated with the phenomenon of
transient motion within pipeline and tunnel networks and it may be
worthwhile to attempt a definition of some of these expressions. As
far as the writer can discover, there are no universally agreed definitions
for some of these terms.
Unsteady flow is a general term to describe any motion which
embodies a time-varying component.
Waterhammer has been used by Streeter and Wylie (1967) as synon-
ymous with unsteady flow but embodying the word ‘hammer’ suggests
noise and can be specifically associated with a fast-changing pressure
which imposes a force on pipework and valves sufficient to generate
sound. Joukovsky (1898) was one of the first to analyse ‘waterhammer’
waves in which flow varies spatially as well as over time. Jaeger (1977)
defined waterhammer as a rapid oscillation.
The expression transient flow was used by Streeter and Wylie (1967)
to indicate a change from one state of steady motion to another.
Steady oscillatory flow, periodic flow or pulsatile flow all define condi-
tions of flow which are repeated at intervals of time called the period
of oscillation.
Resonance describes motion which can progressively develop into a
steady oscillatory flow.
The term surge has been associated with the concept of mass oscilla-
tions in which although flow is changing over time is essentially constant
along a pipeline. This expression often has been used in connection
with longer-period transients in a low-pressure tunnel system for a
hydropower plant whereas waterhammer has been used in relation to
faster transients in the high-pressure parts of hydroelectric plant.
Jaeger (1977) defined mass surges or mass oscillations as slow transients.
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