Fox J.A. Transient Flow In Pipes, Open Channels And Sewers
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ELLIS
HORWOOD SERIES
IN
CIVIL
ENGINEERING
Series
Editors
Structures: Professor
R. EVANS, Department
of
Civil
Engineering,
University College,
Cardiff
Hydraulic Engineering
and
Hydrology:
Dr R. SELLIN,
Department of
Civil Engineering,
University
of Bristol
Geotechnics:
Professor D.
WOOD, Department of Civil
Engineering,
University
of
Glasgow
Addis,
W.
Allwood,
R.
Bhatt,
P.
Blockley,
D.I.
Britto,
A.M. & Gunn, M.J.
Bljuger,
F.
Calladine,
C.R.
Carmichael,
D.G.
Carmichael,
D.G.
Carmichael,
D.G.
Cyras,
A.A.
Dowling,
A.P. &
Ffowcs-Williams,
J.E.
Edwards,
A.D. & Baker,
G.
Fox,
J.
Graves-Smith,
T.R.
Gronow,
J.R., Schofield,
A.N. &
Jain, R.K.
Hendry,
A.W., Sinha,
B.A. & Davies, S.R.
Heyman,
J.
Hoknes,
M. & Martin, L.
H.
Struc_tural
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Transient Flow in Pipes, Open Channels
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Introduction
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in Structural Analysis:
Vibrations,
Stability and Statics
TRAT{SIEF{T
FLOW
IN
PIPES,
OPEN
CHANNELS
AND
SEWERS
J.
A. FOX
Reader
in
Fluid
Mechanics,
Department
of Civil
Engineerin€
University
of
Leeds
ELLIS
HORWOOD
LIMITED
Publishers'
Chichester
Halsted
Press:
a
division
of
JOHN
WILEY
&
SONS
New
York
'
Chichester
'
Brisbane
'
Toronto
First
published
in 1989
ELLIS
HORWOOD
LIMITED
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Street,
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b
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Cetaloguing in
Publication Data
Fox, John
Adrian
Transient flow in
pipes,
open
channels
and sewers.
1. Fluid
dynamics. Data
processing
I. Title
537',.O51',A295
Library
of
Congress Card No. 89-15210
ISBN
LJ45H26ffi
(Ellis
Horwood Limited)
ISBN 0-47L2146T5
(Halsted
Press)
Printed
in Great Britain byThe Camelot Press, Southampton
COPYRIGHT
NOTICE
Ail
Riehts
Reserved.
No
part
of this
publication
may be reproduced, stored
in
a
retrieval
system,
or
transmitted,
in any form of by
any
meairs, electronic, rirechanical,
phot6copying,
recording or oiherwise,
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Sussex, Engfand.
Table
of
contents
Preface.
Nomenclature
chapter
1
The differential
equations
of
waterhammer
. . .
.19
Chapter
2
The
Allievi
interlocking
equations
and
graphical methods.
. . . . .
.45
Chapter
3
The
method
of characteristics
' ' '
'73
Chapter
4
Boundarycpnditions
"""85
Ttrrbines
. .
. .L26
Boundary
conditions
continud.
Junctions,
valves,
ftiction
and
wavespeed
-..130
Unsteadyflowinopenchannels.
.----t46
Finite difrerence
methods
.
. 189
Resonance
198
Threedimensional
free
surface
flow
. .
-
223
Transient
flow
in
gas
pipe
networks.
- .
-234
The
programming
of the
wave
equations
- . -
-242
References
Bibliography
.
.
7
9
Chapter
5
Chapter
6
Chapter
7
Chapter
E
Chapter
9
Chapter
l0
Chapter
l1
Chapter
12
259
261
275
Index
Preface
This
is the seconci
book
writien
'uy
author
on
ihe
subject
of
unsteady
flo'w.
The
first
was
published
in
1977
but
many
papers have
since
been
written so
it must
now
be
r"guid"d
as
being,
in
some
ways, a
little
dated.
These deficiencies
have,
at
least
in
part, been
rectified
in this
volume-
Unsteady
flow
occurs
far more
frequently
than
does steady
flow.
In open
channel
flow
it is
rari for
input
flows
to
a channel
to be steady
and
so
the flow
in
the
channel
itself
cannot
be coniidered
to be steady
either.
In
pipelines,
flow
may approximate
to
result
is that
the
pipeline spends
much
of
its time
in
an unsteady
condition
albeit
a
safe
one.
computer.
Instead
of
using
analytic
or
graphical
methods,
computers
are
now
being
progiur-ed
to
solve
the
basic
differential
equations
by
various finite
difference
methods
and the
early
methods
are
seldom
used.
The
author
still
teaches
them
Preface
because he feels that no subject
can
be fully
comprehended without an
understand-
ing
of its
foundations. The workers
who have contributed largely to this situation
include Abbot,
Chaudry, Courant, Cunge,
Ellis, Enever, Fanelli, L. Fox,
P. Fox,
Friedrichs,
Gray, Kranenburgh, Lewy,
Martin, Streeter, Swaffield,
Thorley,
Town-
.tt^-l--
It^^^:l:^-. lI/I^-^-+ IIr-.1:^ ^^A'7:^ll,^ l-.,+ +Li^ li^+:^:- -^..'^"
StlI,
ViarUlr
ValDDIllEV,
WfBBGlt,
YYylltrr.lllu
zJlslr\sr
lruL
tttlD
lt)t rJ
ru
II1J
w<rJ
exhaustive.
This work
has mostly occurred
in the last twenty-five
years.
The book
is,
in some ways, a textbook;
it is
hoped
that
students
will
be able to
use
it as
such
and learn
the subject
from
it.
Nomenclature
In this
book
there
are
too
many
variables
for
it to
be
possible to
give each
one
a
unique
name.
Whereas
some
variable
names
are used
throughout
the
book
others
are
duplicated
and
have
different
meanings
in
different
chapters
or
even
in the same
chapter.
This
should
produce
no
difficulties
as,
whenever
this
happens,
the
dupli-
cated
variables
are redefined
within
the text.
A
Throughout
the book
A may
represent
the
cross-sectional
area
of
flow.
A
In
Chapter
4, A is
a constant
in
the Stepanov
Pump
Equation.
A
In chapters
concerned
with open
channels,
A is
the
cross-sectional
area
of
flow.
In
Chapter
4,
A.,,
is the
area
of the
unfilted
portion
of
the
cross-section
of
the
horizontal
cylindrical
air
vessel.
In Chapter
4,46"1
is the
cross-sectional
area
of
the
pump's
delivery
pipe.
In Chapter
2,
A"is
the effective
area of
the
valve.
In Chapter
4, A"is
the
area of
exit of
the
impeller'
In
Chapte
r 6,
Aiis
the
cross-sectional
area
of the
fth
pipe
joining
at
a
junction.
In
Chapter
2,
Aois
the
pipe cross-sectional
area.
In Chapter
4, Apir,o'
is
the cross-sectional
area
of
the
piston of
a
reciprocating
pump.
In Chapte
t 4,
A""nis the
cross-sectional
area
of
the
pump's suction
A^i,
Aa.t
A"
A"
Ai
AP
Apiston
Ann
plpe.
Ar
inChapter
4, ATis
the
plan
area
of the
liquid
surface
in
an
air
vessel.
10
Nomenclature
In chapters
concerning open channel flow,
b
is
the mean breadth
of a channel.
In chapters
concerning open channel
flow, b" is the width of a
sluice
gate.
fn alronfarc ^n6^ar6i-- nno- alro--ol ffn.r, A ic alra ott-f-^o
rrr
vrrql/rvlJ wuwruurE vl/vu vrlqnlrvr uvw,
(/s
lit alrv Julrcw
breadth
of
a channel.
In
chapters concerning open
channel
flow, b* is the breadth
of a
weir.
In Chapter 4, C
is
a constant in
Stepanov's
pump
equation.
In
Chapter 9,
C is the capacitance/unit
length
of
an electrical
transmission
line.
In
ChapterT
,
Cois the coefficient
of contraction
of the
flow below
a sluice
gate.
In
Chapter
7,
Cois
the
coefficient of discharge
of the flow below a
sluice
gate, venturiflume, weir
or
valve.
In Chapte r Ll
,
Cois the specific heat
of
a
gas
at
constant
pressure.
In
Chapter
11.,
Cy
is the
specific heat of a
gas
at
constant
volume.
c is the
wavespeed
both in
pipelines
and in channels.
Wavespeeds at
A,B,C
and D in Chapter'l.
Wavespeed
at
the interpolated
point
on
the
ith
pipe
or channel
joining
at the
junction.
In
the characteristic method, cw is the wavespeed
at
point
N.
is
the
wavespeed at
point
O in Chapter 7.
In the
characteristic
method, cp is
the
wavespeed
at the
point
P.
In
the characteristic
method,
cn and c5 are wavespeeds at the
points
R and S.
In
Chapter
7,
dt)
is a function describing the variation
of
wave-
speed with time.
In
Chapter
4, D is
the
diameter
of a
pump
impeller.
d is the diameter of a
pipe
or depth in a channel.
In Chapter 7
,
d"is the
critical depth.
In
Chapter
4, d6.1is the diameter
of the
delivery
pipe
of a
pump.
In Chapter'1, dsis the depth at the
point
O.
In
Chapter
4, dnn is the diameter
of the suction
pipe
of a
pump.
In Chapter 7, d(t) is a function describing the variation
of depth
with time.
In
Chapter
4, E is the efficiency
of a
pump.
In Chapter 7
,
E is
g(l
-
i)
the
net loss
of energy in a channel.
In
Chapter
I,
E is Young's Modulus
of the
pipe
wall
material.
In Chapter l, E.is Young's Modulus of concrete.
In
Chapter
L, E* is Young's Modulus of rock.
In
Chapter 7,
Euis the upstream specific energy.
In
Chapter
ll, elis a constant
in
the characteristic
equations
of
gas
flows.
b"
h
us
bn
C
C
c"
cd
CP
cv
c
C4
Cs
Cs
Cp
ci
CFI
Cg
-:
cP
cp and c,
c(t)
D
d
d"
do.,
do
d""n
d(t)
E
E
E
E"
ER
Eu
€1
€2
e3
F
Nomenclature
11
In
Chapte
r ll,
e2isa
constant
in the
characteristic
equations
of
gas
flows.
In
Chapterll,qis
a constant
in thecharacteristic
equations
of
gas
flows.
t- /rl^-r^-
r'|
D l--n+an d rtrorra f.orrallinc in
qn
rrnsfre.Arn
III
\-ffdPlVL
L1
I'
llsllt rllD
q
wsvv lrqrvruub
ur
srr
direction.
In
Chapter
7,
F1u
is
the Froude
Number
defined
as
u/c.
Terms
which define
friction
at
points R and
S
in Chapter
3-
In
Chapter
2,
/
denotes
a
wave
travelling
in
a downstream
direction.
Note:
subscripts
to
F and
/
above
denote
values
at
different
numbers
of
pipe
Periods.
The
Darcy-Weisbach
friction
coefficient
(the
Fanning
/
in the
usA).
In
Chapter
L,
f
denotes
stress,
/n
is the
hoop
stress
andfi
is
the
longitudinal
stress
in
a
pipe wall-
In
junctions,
"f,
is
the
Darcy-Weisbach
friction
value
at the
lnterpolation
point
on
the ith
pipe
joining
at the
junction.
Throughout
the
book,
g
is the
intensity
of
the earth's
gravity field.
I/ is
the
static
head
over
a
valve or reservoir
head
according
to
context.
In
chapterg,
H
is the
amplitude
of
the oscillating
component
of
head.
In Chaptet
4,
H is
the
head
across
a
pump
when
operating
at
a
speed
N.
The
amplitudes
of
the
oscillatory
head
at
the
receiving
and
sending
inds
of
a
resonating
pipe
respectively
in Chapter
9'
In
Chapter
4, H*
is
the
head
across
the
pump when
running
at
steady
speed.
Geneiatty,
ft denotes
either
pressure head
or
potential
head
according
to context.
In
Chaptetg,
his the
head
at a
point in a
resonating
pipe line.
In
chapterT,
his
the
depth
at
the
upstream
end
of
the
wedge of
water
running
over
a drY
bed.
In
Chapter
i,E
is
the steady
head
component
is
a
resonating
pipeline.
the
amplitude
of
the
oscillatory
head
component
in
a
resonating
pipeline.
in Ctr"pt"12,hpis
the
potential
head on
pipe no.L
adjacent
to
the
junction
at
node
D.
in Chapte
12,
hiis
the
potential
head on
pipe no.Z
adjacent
to the
junction
at node
D.
in
Chapter
2,
hr'
is
the
potential
head
on
pipe
no.
3
adiacent
to
the
junction
at node
D.
h,
denotes
the
friction
head.
In
junction
analysis
ft, is the
head
at
the
interpolation
point
on the
ith
pipe
joining
at
a
junction.
f
f
f,
G
H
H
F"
Fp
and
Fg
f
H
f/p
and
f/t
H*
h
h
h
n
h'
h;
h;'
hf
hi
1Z
ft,.,r*
h.,
hP
hpact
fto..r,
h*and
h,
ft*r,
hT
hT
hu
L
h*
Nomenclature
In
Chapter
1, ft,,.,-
is the maximum
head
at
the
valve
during
a
valve
closure.
In
Chapter 2,
h,, denotes the reservoir
head.
In
chapters concerned
with
characteristic
methods
/r" is the
potentiai
heaci at
point
P.
In
Chaptet
4,
hpd"r is the potential
head in the delivery
pipe
adjacent to the
pump.
In
Chapter
4, hp,,nis
the
potential
head
in the
suction
pipe at
point
P
adjacent
to the
pump.
In
chapters
concerned
with
characteristic
methods ft* and /r, are
potential
heads at the bases of the
two characteristics at points R
and
S.
In
Chapter
7,
y'r*.5
is the depth
of the reservoir
relative to
the bed
level at entry to the channel.
In
Chapter
4,
h, is the
depth
of liquid
in
the
air
vessel.
In
Chapter
4,
h, is the depth of liquid
in the air
vessel
initially.
In
Chapter
l, hu is the
potential
head
just
upstream
of
the valve.
l^
(-|^^^+^-'7
L i- +l'^ |.oi^k+ ^f +L^ .,,^:- -^+---^A +^ +L^ L^l ^$ +L-
l'rl
vrrdPlvr
,
,
taw
lJ
trr! rrvrElrL
\-rl f Irv
wLrl
I ulvr l L\.l LU LrrL r-rLu rJl f rlu
upstream
channel.
In
Chapter
4, h*is
the liquid head
equivalent
to the
gas pressure
in
the
air
vessel.
In
Chapter
4, h*i is the
value
of
h*
initially.
In
Chapter
4, I is the
Second Moment
of
Inertia
of the rotating
parts
of
the
pump
and motor.
In
Chapter
7,
i
is the bed slope
of a channel.
In
Chaptet 9,
i is the alternating
electrical current flowing in an
electrical transmission
line.
In
Chapter
4, i,is an integer taking the value plus
or
minus
one
so
defining a
characteristic
direction,
i.e. downstream
or upstream.
In
channel flows,
7
is the
energy loss
per
unit
weight per
unit
distance along the channel.
In
Chapter
I, K is the
bulk
modulus of the fluid.
In Chapter 6,
k is the multiplier
on
v2l2g
which
describes
the
energy
loss
across a
valve.
In
Chapter
L, K and k are constants describing
frictional losses.
In Chapters
4
and
11, k is
the constant in the
equation of state.
In Chapter 6, k is the mean height
of
roughnesses in the
pipe.
In
Chapter
4, k' is the coefficient
in the energy loss equation of the
impeller of
the
pump:
h:
kiu2lZg.
In Chapter 4, kuis
the
coefficient in the
energy
loss
equation of the
volute
of
the pump:
h:
k,uzl\g.
L is the
pipe
or channel length.
In
Chapter 9,
L is
the inductance
per
unit
length
at a
point
in an
electrical
transmission line.
In Chapter
4, L.a is the height
of a
vertical, cylindrical
air
vessel.
In Chapter
4, l,is the length
of the connecting
rod
of a reciprocat-
ing
pump.
h*i
I
T
K
K
Kand
k
k
k
ki
k,
L
L
LT
l.
Pi
m
N
A/
lY
I
N2
N*
N*
n
P
P,,..
Po
pwr
p
o
O
Q*
q
q
q"
Qo
Nomenclature
13
In chapters
concerning
open
channels,
rr is
the hydraulic
mean
radius.
In
Chapter
l, m
is the
reciprocal
of
Poisson's
ratio,
i-e.
o
:
llnt'
In
Chapter
4, N is
the
rotational
speed
of a
pump in
rev/min'
Tn
(-hqnrer
4 A/- is the rotational sneed in rev/min of
a
Oump
at
the
rrr vrrsHrvt
I
t.
!
I
.v
---- -r
beginning
of
a time
period
t.
In Chapter
4, N,
is the
rotationalspeed
in
rev/min
of
the
pump at
the end
of
the
time
r.
In Chapter
4, N
is the
rotational
speed
in
rev/min
of
the recipro-
cating
pump in
rev/min.
In Chapter
4, N*
is the
steady
running
speed
of
a
pump in
rev/min.
In Chapter
4, N. is
the specific
speed
of a
pump'
In Chapter
1
1, n
is the
polytropic
constant
in
the equation
of state.
In
chapters
concerning
open
channe
I P
is the
wetted
perimeter
of
the
flow.
In Chapter
4, P
is the
hydraulicpo\\'er,
which is the
energy
per
unit
wiight
given to
the
fluid by
the
pump
multiplied
by
the
number
of
unit
weights
of
water
flou'ing,
i.e-
V*ulg.wQH'
In Chapter
4,
Piis
the
power required
to
drive
the
pump,
i'e'
the
hydrauiic
power
above
plus the losses
such
as
friction
and bearing
losses.
In Chapter
4, P^is
the
power
generated
by
the
pump
motor'
In Chapter
4,
Po is
the
power
required
to
drive
the
pump
at
zero
flow.
In Chapter
4,
pwr is
the
power available
to
accelerate
the
pump.
p
ls
pressure.
In Chapter
2,
Q
is
the
flow
through
a
pump.
In
Chapte
r
g,
Q
is
the amplitude
of the
oscillatory
component
of
flow.
In Chapter
9,
the
flow
amplitudes
at the
sending
and
receiving
ends
of
a
pipe
in a
resonating
network-
In
Chapter
2,
Q*
is
the
flow
at the
pump's
steady
running
speed.
In Chapte
r
g,
q
is
the
flow
at
a
point in
the
resonating
pipe'
In
Chapter
9,8
is the
steady
component
of
q
above.
In Chapte
r 9,
q'
is
the
oscillatory
component
of
q
above'
In
Chapte,
2,
qo is
the
flow
in
the first
pipe transporting
flow
towards
junction
D.
In Chapter
2,
eo
is
the
flow
in the
second
pipe
transporting
flow
towards
junction
D.
In Chapiev
2,
e'
"o
is
the
flow
in the
third
pipe transporting
flow
away
from
junction
D.
In Chaptet
2,
et
is
the
flow through
a
valve
at time
f.
In Chaptet2,
e.,.,
is
the
flow
in
a
pipe at
a distance
x from
the
upstream
end
of
a
PiPe
at a
time
r.
In Chapter
2,
q*.7is
the
flow in a
pipe
at
a
location
X from
the
upstream
end
of
a
PiPe
at
a time 7..
In
Chapter
4, R
is
a constant
in the
pump
power equatton'
Q*
and
Q5
Q,,,O
Qt
Q...,
Qx.r
R
t4
R
Nomenclature
In Chapter
4, R is the
radius of
the
cross-sectional area
of an air
vessel.
In
Chapter 6,
R is the
radius
of
a bubble.
In
Chapter
9,
R
is the hydraulic
resistance in
the
impedance
equation.
In
Chapter
L1,
R
is
the universal
gas
constant.
In Chapter 6, R. is the
Reynolds'number.
In
Chapter 9,
R"1
is
the electrical resistance/unit length of
an
electrical transmission
line.
In
Chaptet
4,
r is the radius of the crank in a reciprocating
pump.
In
Chapter
9,
r is the modulus of the
propagation
constant.
In
Chapter
4,
S
is a constant in the
pump power
equation.
In
Chapter
7, s is
an integer
constant
which takes
the
value
of
*
1
or
-
1 and is used to specify the direction of the characteristic
lines.
*
1
indicates that the line is directed downstream in the
direction of x
increasing and
-
1 indicates that the line
is directed
upstream in the direction of x
decreasing.
Tn nhonfpr /. c rlcnnfpc +hc dicfonne hefrvacn fhe nenfrc nf the
llr vrrsl/ rvr
'I
t
u vvrrv!
crank
and the small end of the
connecting rod in
a
reciprocating
pump.
In Chapter
7,
when
a flow occurs over a dry bed
the
leading
wedge
of
water
has
a length
of s.
In Chapter
2, T
is
a selected
time associated
with
a
position
X.
In Chapter
2, T may also denote the
pipe
period
ZLlc.
In
Chapter
l, T is
the wall thickness
of
a
pipe.
In
Chaptet
11, T
is the
absolute
temperature.
In Chapter
4, T-is the motor torque.
In
Chapter
4, T* is the torque
when
the
pump
is running at the
steady speed
N*.
t represents time.
In Chapter
4, u is
the
pump
impeller blade
tip velocity.
In
Chapter
L0, u is the component of
velocity in the x direction.
V may represent
volume
or
volume
per
unit
mass
of fluid.
In
Chapter
9,
V is alternating
voltage
at
a
point
on an electrical
transmission
line.
In Chapter
L
V
*
is
the
velocity
in
a
pipeline when time tends to
infinity.
In Chapter
7, u is the mean
velocity
in a
channel.
In
Chapter
4, u is the absolute
velocity
in
the
velocity
triangle of
the
pump's
impeller.
In Chapter
10, u
is
the component
of
velocity in the
y
direction.
In
Chapter
4, uris the
velocity
of
flow through a
pump
impeller.
In Chapter 6,
u,is the
velocity at the interpolation
point
on the
ith
pipe
joining
at a
junction.
In characteristic
solutions,
u* is the
velocity
at the
node
N.
In characteristic
solutions
u" is the
velocity
at the intersection
point
of the
characteristics
P.
R
R
R
R.
R"t
r
r
s
s
v*
u
t)
T
T
T
T
T^
T*
t
U
u
V
V
t)
Dg
Di
Dpr
Dp
Nunre
lrclature
In
Chapter
4,
Dpi,,o,,
is the
velocity
of the
piston
of
a reciprocating
pump.
in
Chapte
r
4,
t), is
the
relative
velocity in
a
centrifugal
pump.
In
Chapter
4, u*
is
the
velocity
of
whirl
in a
centrifugal
pump.
.F!=o
i*ra=nn!qt-od
.-relrrr-ities
ef fhe heses cf the characteriStie
lineS
I
IIE
Irrr9rP\rlaltvu
vwrvvrtrvD
st
frrv
in
an
xt space.
In Chapter
2,
u,.,is
the
velocity
at
a distance
x
from
the
upstream
end
at
a
time
f.
In
chapter
2,
uy.7
is the
velocity
at
a specified
point
X
at
a
specified
time
I.
In
Chapter
4,
Volun
is the
volume
of
air
in
the
horizontal,
cylindrical
air
vessel.
ln
Chapte
r
4,
W
ris
a
dimensionless
constant
which describes
the
mannei
of
variation
of the
head,
lLI,
in
four-quadrant
pump
operatton.
In
Chapter
4,Wris
a dimensionless
constant
which describes
the
n.lann"i
of
variation
of the
torque,
T,
in
four-quadrant
pump
operation.
, ir
th"
specific
weight
of
the
fluid,
i.e.
it is
the
weight
of the
fluid
per
unit
volume.
in
Chapter
10,
w is
the
velocity
component
in
the
vertical
direction.
15
Upiston
Dr
uw
up
anrJ
u5
u*,,
Dx,r
Vol^r,
wH
wr
w
w
X
x
y
ll8lxlfi"'lilI#'*T,'?:n",j:?'il'Jl:Ji;'Jtl.
in,he
relevant
direction,
In Chapte
r 2,
Xis
a selected
value
of
x-
x
is
thsdistance
along
the
centreline
of
a
pipeline
measured
from
the
upstream
end.
x
is
one
of the
two
coordinates
in
a horizontal
plane'
y
is one
of
two
coordinates
in
a
horizontal
plane'
in Chapter
9,
Z
and
its
subscripted
forms
is
the impedance
at
a
point
in
a network.
Zristhe
elevation
of the
base
of
an
air
vessel
above
the
common
datum.
z is
the
vertical
coordinate
out
of
three
coordinates.r'
y
and
z.
z
is
the elevation
of
a
point above
a
common
datum'
GREEK
CHARACTERS
o(
In
chapter
4, ais
used
as
a constant
in
the
pump analysis'
oc
In
Chapter
4, a
is
the
angle
between
the
connecting
rod
and
the
line
joining
centre
of the
crankshaft
and
the
centreline
of
the
cylinder.
o(
In
Chapter
4, u is
a constant
used
in
parabolic
interpolation
to
obtain
ihe
value
of
the
required
wicket
gate
opening'
In
Chapte
r
7
,
u
is
a constant
controlling
the
geometric
cross-
sectional
properties
of
a
channel.
x
v
z
zT
z
z