Appendix A
Notation
ENGLISH
a pipe radius
a aspect ratio of capsule (capsule length divided by capsule diameter)
a1, a2, a3 arbitrary constants
A cross-sectional area of pipe
Ac cross-sectional area of capsule
Aj cross-sectional area of jet
Ao cross-sectional area of orifice
Ao cross-sectional area of electromagnetic pump
Ap cross-sectional area of piston
As surface area of reservoir
b blockage ratio
b pump impeller thickness
b depth of electrode penetration into ground in soil-resistivity measurement
b1, b2, b3 arbitrary constants
B Boltzmans constant
B a function of Mach number defined by Equation 3.46
B magnetic flux intensity
Bd effective width used for calculating earth load on conduits
B1, B2, B3 arbitrary constants
c specific heat capacity
cp specific heat capacity at constant pressure
cv specific heat capacity at constant volume
C celerity of water hammer waves in pipe
C cohesion coefficient of soil used to determine earth load on conduits
C1, C2, C3, C4 arbitrary constants
C1,C2,C3 arbitrary constants
CA volume concentration of solids in slurry at pipe axis (centerline)
Cc contraction coefficient
Cd discharge coefficient
Cd load coefficient in Marstons equation for buried conduits
CD drag coefficient
CH Hazen-Williams coefficient
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Pipeline Engineering
Cp compressibility coefficient
Ct celerity of water hammer waves in surge tank
CT volume concentration of solids in slurry near pipe top
Cv velocity coefficient for flow meters
Cv mean volume concentration of solids in a pipe
Cw mean weight concentration of solids in a pipe
ds size or diameter of solid particles in pipe
ds infinitesimal surface area
ds vector of infinitesimal surface area ds
D inner diameter of pipe
Db inner diameter (bore) of magnetic flowmeter
Dc capsule diameter
Dd capsule end disk diameter
Dm mean diameter of pipe
Dn nominal diameter of pipe
Do outer diameter of pipe
Dp pump impeller diameter
e absolute roughness
E bulk modulus of fluid
Eo voltage (used when V is needed to denote velocity)
Ep modulus of elasticity of pipe material
EGL energy grade line
EI energy intensiveness
f Darcy-Weisbach resistance (friction) factor
f line frequency
Df frequency shift due to Doppler effect
f Fannings resistance (friction) factor
fem electromagnetic force per unit volume
fm friction factor of solid-fluid mixture in pipe
F force
F force vector
FD drag force
Fe external force on piston
Ff contact friction force
FL densimetric Froude number for calculating limit-deposit velocity
FL lift force on a capsule in pipe
Fp piston force
Fx x-component of force
Fy y-component of force
g gravitational acceleration
G gas gravity
G seismic acceleration (i.e., number of gravitational acceleration g)
h total head
h head or height of liquid
ha atmospheric pressure head
hem head of electromagnetic pump
2003 by CRC Press LLC
Appendix A: Notation
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hj piezometric head at pipe junction
hL head loss
hp pump head
hs static head at pump location
ht turbine head
hv vapor pressure head
H water height (elevation) in a reservoir above pipe exit
H pump head
H Hedstrom number
Hm magnetic field intensity
HGL hydraulic grade line
DH pressure head rise in pipe caused by valve closure
DHs pressure head rise in pipe caused by slow closure of a valve
i specific internal energy
i pressure gradient of fluid in pipe (Dp/L)
im pressure gradient of solid-fluid mixture in pipe (Dpm/L)
I electrical current
I.D. inside diameter of pipe
J current density
k adiabatic exponent (cp/cv)
k capsule body diameter ratio (i.e., capsule diameter divided by pipe I.D.)
kd capsule disk diameter ratio (i.e., capsule disk diameter divided by pipe I.D.)
K local head loss coefficient
K consistency index of the power-law fluid
K bedding constant
variable distance along a pipe
n natural logarithm
log common logarithm
L pipe length
L spacing between electrodes used in soil resistivity measurement
L spacing between pipe supports
L length of pipe flow entrance region
Lc capsule length
Le equivalent pipe length for head loss calculation
Lem length of electromagnetic pump
Lp piston stroke length
m attenuation ratio of surge tank
m number of moles per unit weight
m mass
m mass flow rate (dm/dt)
M Mach number
Mi molecular weight of constituent i
Mo limiting Mach number
n Mannings coefficient
n number of moles per unit weight (n=1/m)
n power-law exponent for non-Newtonian fluid
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np number of poles in an electric motor
N angular speed (rpm, rad/s, etc.)
Nc number of capsules in pipe
Nlr loading ratio (weight of solids in pipe divided by weight of fluid in pipe)
Ns specific speed of pump
Nsp specific pressure ratio (pressure drop of mixture divided by pressure drop of
fluid)
Nt number of capsules in a capsule train
NPSH net positive suction head
p pressure of fluid at a given point
p pump pressure (discharge pressure minus suction pressure)
pb pipe buckling pressure (external pressure that causes pipe to buckle)
pc critical pressure
pc pseudocritical pressure
pe external pressure
pi internal pressure
po limiting pressure
pp piston pressure
ps static pressure in pipe
Dp pressure rise due to water hammer in pipe
Dp pressure drop along pipe over distance L
Dpc pressure drop across a capsule
Dps water hammer pressure due to slow closure of valve
Dpt pressure rise due to water hammer in a pipe protected by a surge tank
P power (brake horsepower)
Pe power delivered to piston by an external force
PL power loss
Pi power input of pump
Po power output of pump
Pr reduced pressure (p/pc)
Q rate of heat loss through unit length of pipe
Q volumetric discharge of fluid through pipe
Qc rate of heat loss through pipe of length L
Qm volumetric discharge of mixture through pipe
Qs volumetric discharge of solids through pipe
DQ discharge correction factor used in the Hardy Cross method
r radial distance from pipe centerline
r compression ratio (pressure after compression divided by pressure before
compression)
ri impeller radius
ro plug flow radius for Bingham plastic fluid in pipe
ro rotor radius
Reynolds number
R critical Reynolds number
R engineering gas constant
R electrical resistance
2003 by CRC Press LLC
Appendix A: Notation
Rb bend radius
RH hydraulic radius
Rx x-component of force by pipe on fluid
Ry y-component of force by pipe on fluid
s surface area
S surge height in a surge tank
S density ratio (solid density divided by fluid density)
S slipan electrical quantity defined as (Vs-Vm)/Vs
Se energy slope (Se=hL/L)
So maximum surge height in a surge tank
Sp linear speed of piston
t time (variable)
to time to drain a reservoir
tp time when pump is on
Dt cycle time of pump
T temperature
T torque
T tensile force
Tc valve closure time
Tc critical temperature
Tc pseudocritical temperature
Tm torque of motor
Tp torque of pump
Tr reduced temperature (T/TC)
DT temperature change (T2-T1)
u local or point velocity at time t and at a distance y from wall
u shorthand for u starting Section 2.3.1
u turbulent (fluctuating) component of u
u temporal mean of local velocity u
u* shear velocity
u+ dimensionless local mean velocity
U tangential velocity at pump impeller tip
V mean fluid velocity across pipe (V=Q/A)
V voltage
Va mean fluid velocity in capsule-pipe annulus
Vc pipe centerline velocity
Vc capsule velocity in pipe
Vd differential velocity between fluid and capsule (or pig)
Vi incipient velocity
VL limit-deposit velocity
VL lift-off velocity of capsule in pipe
Vm motor linear speed
Vm meridian (radial) velocity component of centrifugal pump blades
Vo steady-state mean flow velocity in pipe
Vo critical velocity of capsule flow
Vp mean velocity of particles moving through pipe
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Pipeline Engineering
Vp average piston velocity
Vp velocity of pig in pipe
Vr velocity relative to blade tip of a centrifugal pump
Vs speed of water surface decrease in a reservoir or tank (Vs=-dH/dt)
Vs settling velocity of solids in fluid
Vs synchronous speed of electric motor
Vt tangential velocity component of blade tip of a centrifugal pump
w molecular weight
w work per unit mass (specific work)
w weight flow rate (weight per unit time)
W weight
Wc capsule weight
Wc earth load (force) per unit length on a buried conduit or pipe
x longitudinal distance along pipe in flow direction
xi mole fraction of component i of a gas mixture
xo distance along a pipe to produce limiting condition
X distance from valve subjected to maximum water hammer pressure
DX horizontal deflection of pipe cross section under vertical load
y distance from pipe wall perpendicular to wall (y=0 at wall)
y mole fraction
yi mole fraction of component i in a gas mixture
y+ dimensionless distance from wall (y+=ru*y/)
Y yield number of Bingham plastic fluid through pipe
DY vertical deflection of pipe cross section under vertical load
z elevation
z supercompressibility factor (also called compressibility factor)
GREEK
a energy correction factor
a angle of pipe incline (relative to a horizontal plane)
a thermal expansion coefficient of solid material
momentum correction factor
angle between Vr and U
clearance ratio, (A-Ac)/A
g specific weight of fluid (weight per unit volume)
gs specific weight of solid particle (weight per unit volume)
d pipe thickness
D differential (e.g., DT=T2-T1, or Dp=p1-p2)
e void ratio of solids in pipe
e volume reduction ratio in mixing two liquids
e dimensionless factor in Equation 2.80
emmagnetic permeability of fluid
h contact friction coefficient
h efficiency of pump and other machines
hm motor efficiency
2003 by CRC Press LLC
Appendix A: Notation
hp pump efficiency
q angle of pipe bend
q angle between electric current I and voltage V (i.e., phase angle)
k von Karman constant (k=0.40 for pipe flow of Newtonian fluids)
l linefill rate of capsules
dynamic viscosity of fluid
1 viscosity of gas at one atmospheric pressure
p Poissons ratio of pipe material
n kinematic viscosity of fluid (n=/r)
p 3.1416
r density of fluid in pipe
r electric resistivity of material
ra density of gas at standard atmospheric condition
rm density of solid-fluid mixture
rs density of solids
s stress
st tensile stress in pipe
sT thermal stress in pipe
S summation sign
t shear in flow at radius r
to shear in flow at pipe wall
ty yield stress of non-Newtonian fluids with yield
w angular velocity (rad/s)
w m angular velocity of motor (rad/s)
w p angular velocity of pump (rad/s)
OTHERS
arrow sign placed above any vector quantity
proportionality sign
volume sign
derivative with respect to time
2003 by CRC Press LLC
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