Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications
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Acknowledgments
My knowledge of this subject has benefited from many others with whom I have
worked and talked over the years. These include colleagues from industry and
academia, and students, whether in short courses or longer-term degree courses and
research. I hope that the book does justice to all that they have taught me.
In writing this book,
I
have drawn on the information from many manufacturers,
and some have been particularly helpful in agreeing to the use of information and
diagrams. I have acknowledged these companies in the captions to the figures. Some
went out of their way to provide artwork, and I am particularly grateful to them.
Unfortunately, space, in the end, prevented me from using many of the excellent
diagrams and photographs with which I was provided.
In the middle of already busy lives, the following people kindly read through
sections of the book, of various lengths, and commented on them: Heinz Bernard
(Krohne Ltd.), Reg Cooper (Bailey-Fischer
&
Porter), Terry Cousins (T&SK Flow Con-
sultants), Chris Gimson (Endress & Hauser), Charles Griffiths [Flow Automation
(UK)
Ltd.],
John Hemp (Cranfield University), Yousif Hussain (Krohne Ltd.), Peter
lies-Smith (Yokagawa United Kingdom Limited), Alan Johnson (Fisher-Rosemount),
David Lomas (ABB Kent-Taylor), Graham Mason (GEC-Marconi Avionics), John
Napper (formerly with FMA Ltd.), Kyung-Am Park (Korea Research Institute of Stan-
dards and Science), Bob Peters (Daniel Europe Ltd.), Roger Porkess (University of
Plymouth), Phil Prestbury (Fisher-Rosemount), David Probert (Cambridge Univer-
sity),
Karl-Heinz Rackebrandt (Bailey-Fischer & Porter), Bill Pursley (NEL), Jane
Sattary (NEL), Colin Scott (Krohne Ltd.), John Salusbury (Endress & Hauser), Dave
Smith (NEL), Ian Sorbie (Meggitt Controls), Eddie Spearman (Daniel Europe Ltd.),
J. D. Summers-Smith (formerly with I.C.I), and Ben Weager (Danfoss Flowmetering
Ltd.).
I am extremely grateful to them for taking time to do this and for the con-
structive comments they
gave.
Of course,
I
bear full responsibility for the final script,
although their help and encouragement was greatly valued.
I am also grateful to Dr. Michael Reader-Harris for his advice on the orifice plate
discharge coefficient equation, and to
Prof.
Stan Hutton for his help and encourge-
ment when Chapter 10 was essentially in the form of technical papers.
I acknowledge with thanks the following organizations that have given permis-
sion to use their material:
ASME for agreeing to the reproduction of Figures 5.10(a),
10.11,
10.16, 17.1, 18.2,
and 18.4.
ACKNOWLEDGMENTS
Elsevier Science Ltd. for permission to use Figures 4.18, 5.5, 5.10(b), 5.12, 8.6,11.7,
11.11,11.13,11.14,11.17,13.9,
21.1,
21.2 and for agreement to honor my right
to use material from my own papers for Chapters 10 and 17.
National Engineering Laboratory (NEL) for permission to reproduce Figures 4.9,
4.14-4.16, 4.19, 5.11, 11.4-11.6, and 14.5.
Professional Engineering Publishing for permission to draw on material from the
Introductory Guide Series of which I am Editor, and to the Council of the Insti-
tution of Mechanical Engineers for permission to reproduce material identified
in the text as being from
Proceedings
Part C,
Journal
of
Mechanical Engineering
Science,
Vol. 205, pp. 217-229, 1991.
Extracts from BS EN ISO 5167-1:1997 are reproduced with the permission of BSI
under licence no. PD\ 19980886. Complete editions of the standards can be obtained
by mail from BSI Customer Services, 389 Chiswick High Road, London W4 4AL,
United Kingdom.
I am also grateful for the help and encouragement given to me by many in the
preparation of this book. It would be difficult to name them, but
I
am grateful for each
contribution. The support of my family must be mentioned. In various ways they all
contributed - by offering encouragement, by undertaking some literature searches,
by doing some early typing work, and by helping with some of the diagrams. I
particularly thank my wife whose encouragement and help at every stage, not to
mention putting up with a husband glued to the word processor through days,
evenings, holidays, etc., ensured that the book was completed.
My editor has been patient, first as I overran the agreed delivery date and then
as I overran the agreed length. I am grateful to Florence Padgett for being willing to
overlook this lack of precision!
Nomenclature
CHAPTER 1
Q
Sensitivity coefficient
f{x) Function
for
Normal distribution
K
K
factor
in
pulses
per
unit flow quantity
k Coverage factor
M Mean
of a
sample
of n
readings
m Index
N(/JL,
a
2
)
Normal curve
n Number
of
measurements, Exponent
p Probability, Index
q Mean
of n
measurements
qj,
Exponent
qj Test measurement
q
v
Volumetric flow rate
q
vo
Volumetric flow rate
at
calibration point
r Exponent
s Exponent
s(q) Experimental standard deviation
of
mean
of
group
qj
s(qj) Experimental standard deviation
of qj
t Student's
t
U Expanded uncertainty
u(xi) Standard uncertainty
for the zth
quantity
u
c
(y)
Combined standard uncertainty
x Coordinate
Xi Result
of a
meter measurement, Input
quantities
x Mean
of n
meter measurement
y
Output quantity
z Normalized coordinate
(x
—
/x)/a
l± Mean value
of
data
for
normal curve
v Degrees
of
freedom
o Standard deviation
(a
2
variance)
4>(z) Area under Normal curve [e.g., 4>(0.5)
is
the area from
z = -oo to z = 0.5]
</>(x)
Function
for
normalized Normal
distribution
CHAPTER 2
A
c
D
Cross-section
of
pipe
Local speed
of
sound
Specific heat
at
constant pressure
Specific heat
at
constant volume
Diameter
of
pipe
d Diameter
of
tube bundle straightener
tubes
g Acceleration
due to
gravity
H Hodgson's number
K Pressure loss coefficient
M Mach number
n Index
as in
Equation
(2.4)
p Pressure
po
Stagnation pressure
APioss Pressure loss across
a
pipe fitting
q
v
Volumetric flow rate
q
m
Mass flow rate
R Radius
of
pipe
Re Reynolds number
r Radial coordinate (distance from pipe
axis)
T Temperature
To Stagnation temperature
V Velocity
in
pipe, Volume
of
pipework
and other vessels between
the
source
of
the
pulsation
and the
flowmeter
position
Vb
Velocity
on
pipe axis
Kms Fluctuating component
of
velocity
V Mean velocity
in
pipe
z Elevation above datum
y Ratio
of
specific heats
li Dynamic viscosity
y Kinematic viscosity
p Density
SUBSCRIPTS
1,2
Pipe sections
CHAPTER 3
Cj
Sensitivity coefficient
for the zth
quantity
famin Lubrication film thickness
n Bearing rotational speed
p Bearing load
ui Standard uncertainty
for the /
th quantity
r)
Friction coefficient
k Specific film thickness
ix Fluid viscosity
NOMENCLATURE
a Combined roughness
of the two
contacting surfaces
of the
bearing
CHAPTER
4
Q Concentration of tracer in the main
stream at the downstream sampling
point
Cdmean Mean concentration of tracer measured
downstream during time t
Q Concentration of tracer in the injected
stream
C
u
Concentration of tracer in the main
stream upstream of injection point
(if the tracer material happens to be
present)
c
x
Sensitivity coefficient
M
n
Net
mass
of
liquid collected
in
calibration
p Pressure
q
v
Volumetric flow rate
in
the
line
q
V
i Volumetric flow rate
of
injected tracer
R Gas constant
for a
particular
gas
T Temperature
t
Collection time during calibration,
Integration period
for
tracer
measurement
V Amount injected
in the
sudden injection
(integration) method
v Specific volume
p Liquid density
CHAPTER
A
a\
a
€
b
b
€
C
C
Re
Claps
c
C\
C*
D
D
1
d
E
E
T
E*
e
F
5
Function
of p and Re
Expression
in
orifice plate bending
formula
Constant
Constant
Constant
Discharge coefficient
Part
of
discharge coefficient affected
by Re
Part
of
discharge coefficient which
allows
for
position
of
taps
Discharge coefficient
for
infinite
Reynolds number
Expression
in
orifice plate bending
formula
Constant
Pipe diameter
(ID)
Orifice plate support diameter
Orifice diameter
Velocity
of
approach factor (1
-
£
4
)~~
1/2:
,
Thickness
of the
orifice plate
Total error
in the
indicated flow rate
of a
flowmeter
in
pulsating flow
Elastic modulus
of
plate material
Thickness
of the
orifice
Correction factors used
to
obtain
the
mass flow
of a
(nearly)
dry
steam flow
f
H
h
K
Li
h
'2
M'
2
n
Pd
Pu
Ap
qm
qv
Re
r
t
V
V
vWs
X
a
P
y
Hm
€1
K
Pi
°y
4>
CHAPTER
C
c
Re
Frequency
of the
pulsation
Hodgson number
Thickness
of
orifice plate
Loss coefficient, Related
to the
criterion
for Hodgson's number
=
h/D
=
l'
2
/D (The
prime signifies that
the
measurement
is
from
the
downstream
face
of the
plate)
Distance
of the
upstream tapping
from
the
upstream face
of the
plate
Distance
of the
downstream tapping
from
the
downstream face
of the
plate
(The prime signifies that
the
measurement
is
from
the
downstream
face
of the
plate)
= 2L'
2
/(1
- p)
Index
Downstream pressure
Upstream pressure
Differential pressure, pressure drop
between pulsation source
and
meter
Mass flow rate
Volumetric flow rate
Reynolds number usually based
on the
pipe
ID
Radius
of
upstream edge
of
orifice plate
Time
Volume
of
pipework
and
other vessels
between
the
source
of the
pulsation
and
the flowmeter position
Mean velocity
in
pipe with pulsating
flow
Root-mean-square value
of
unsteady
velocity fluctuation
in
pipe with
pulsating flow
Dryness fraction
Flow coefficient,
CE
Diameter ratio,
d/D
Ratio
of
specific heats
Small changes
or
errors
in q
m
, etc.
Expansibility
(or
expansion) factor
Expansibility
(or
expansion) factor
for
orifice
Isentropic exponent
Density
at
the upstream pressure tapping
cross-section
Yield stress
for
plate material
Ratio
of
two-phase pressure drop
to
liquid flow pressure drop
Maximum allowable percentage error
in
pulsating flow
6
Coefficient
of
discharge
Part
of
coefficient
of
discharge affected
by Reynolds number
NOMENCLATURE
C
tp
Coefficient
for
wet gas
flow equation
Coo Discharge coefficient
for
infinite
Reynolds number
D
Pipe
ID
d Throat diameter
E Velocity
of
approach factor
(l-/*4)-l/2
Fr
g
Superficial
gas
Froude number
g Gravitational acceleration
k Roughness
n Index
Ap Differential pressure
q
g
Gas
volumetric flow rate
qi Liquid volumetric flow rate
q
m
Mass flow rate
q
tp
Apparent volumetric flow rate when
liquid
is
present
in the gas
stream
q
v
Volume flow rate
Re Reynolds number based
on D
Red Reynolds number based
on d
Vsg Superficial
gas
velocity
X
Lockhart-Martinelli parameter
£ diameter ratio
d/D
e Expansibility
(or
expansion) factor
p Density
p
g
Gas
density
p\
Liquid density
CHAPTER
7
A2 Outlet cross-sectional area
A* Throat cross-sectional area
a Constant
a
c
Constant
a
z
Constant
b Constant
b
c
Constant
b
z
Constant
C Discharge coefficient
C
R
= C*^Z
C* Critical flow function
c Sound speed
c
p
Specific heat
at
constant pressure
c
v
Specific heat
at
constant volume
d Throat diameter
dt Diameter
of
tapping
d2 Outlet diameter
f Obtained from Equation (7.17)
M Mach number
Mi Mach number
at
inlet when stagnation
conditions cannot
be
assumed
M Molecular weight
n Exponent
in
Equation (7.12)
po Stagnation pressure
pi Pressure
at
inlet when stagnation
conditions cannot
be
assumed
p2i Ideal outlet pressure
p2max Actual maximum outlet pressure
p*
Throat pressure
in
choked conditions
R
Re
d
To
%
X
Z
ZQ
p
y
K
V
PO
Mass flow
Universal
gas
constant
Reynolds number based
on the
throat
diameter
Stagnation temperature
Throat temperature
in
choked conditions
Mole fraction
of
each component
of
a gas
mixture
Compressibility factor
Compressibility factor
at
stagnation
conditions
d/D
Ratio
of
specific heats
Error
Isentropic exponent
Kinematic viscosity
Density
at
stagnation conditions
CHAPTER
8
A Cross-sectional area of the pipe, Constant
A'
Constant
Af Cross-sectional area
of
float
A
x
Cross-sectional area
of
tapering tube
at
height
x
A2 Annular area around float, Annular area
around target
a Area
of
target
B Constant
C Coefficient
C
c
Contraction coefficient, Constants
in
curve
fit
for
target meter discharge
coefficient
D Pipe diameter
d Throat diameter
for
pipe inlet
E Full-scale
or
upper range value
of
flow
rate used
in
precision calculation
F Summation error
in
flow rate
g Gravity
K Loss coefficient, Precision class, Bend
or
elbow meter coefficient
L Length
of
laminar flow tube
M Actual flow rate used
in
precision
calculation
p Pressure
q
v
Volumetric flow rate
R Radius
of
bend
or
elbow
Re Reynolds number
V Velocity, Volume
of
float
V Mean velocity
in
tube
v Specific volume
of gas
x Height
of
float
in
tube
/x Viscosity
fig Viscosity
of
calibration
gas at
flowing conditions
/x
std
Viscosity
of
reference
gas at
standard conditions
p Density
Pi Density
of
float material
NOMENCLATURE
CHAPTER 9
E Young's modulus of elasticity
F Friction force
g
Acceleration
due to
gravity
L Length
of
clearance
gap in
direction
of flow
I Axial length
of
clearance
gap
£
ax
Axial length
of
measuring chamber
M Mass
of
rotor
N Rotational speed
of
rotor
N[ Rotational speed of cam disk of calibrator
(clutch system)
NIN
Rotational speed
of
shaft into calibrator
(epicyclic system)
N
m
ax
Maximum rotational speed
of
rotor
N
o
Rotational speed
of
outer ring
of
calibrator (clutch system)
NOUT
Rotational speed
of
shaft
out of
calibrator (epicyclic system)
N
s
Slippage
in
NIN
n Number
of
teeth
on
gear
pa Downstream pressure
p
u
Upstream pressure
<Zideai
As used in
Equation (9.A.6)
leakage Leakage flow rate
<?BULK
Bulk volumetric flow rate
<7siip
Volumetric flow rate through meter at no
rotation
q
v
Volumetric flow rate
R Radius of friction wheel
r Radius of point of friction wheel
contact
r
x
Inner radius of measuring chamber
r
o
Outer radius of measuring chamber
r
s
Shaft radius
T Temperature, Torque
TIN
Torque on input shaft to calibrator
TOUT
Torque out from calibrator
7b Constant drag torque
T\ Speed-dependent drag torque
t Clearance between stationary and
moving members
t
0
Thickness of outer casing
u Fluid velocity
y Position coordinate across clearance gap
a
m
Coefficient of linear expansion of metal
«i Coefficient of volume expansion of
liquid
A Change in quantity, Distance between
center of rotation of arm and center of
discs in calibrator
8 Reduction in area of the measuring
chamber due to blade sections
9
Q
Angular position of center disk of
calibrator
0[
Angular position of cam disk of calibrator
0
o
Angular position of outer ring of
calibrator
Viscosity
Angular position of arm of calibrator
OTHER
/
1
m
1,2
1-8
SUBSCRIPTS
Dummy suffix for summation
Liquid
Metal
Different materials
Epicyclic gears
CHAPTER
10
A
a
do,
U\,
#2/
#3
B
b
C
c
D
c
D
c
t
c
h
C
L
c
D
D\
D
2
£>3
d\, d
2
e
F
f
I
i
i(r)
K
K(r)
Ki,K
2
K
h
L
N
n
P
poi
P02
Cross-sectional area of the effective
annular flow passage at the rotor blades
Turbine meter aerodynamic torque
coefficient
Constant coefficients
Axial length of rotor
Turbine meter aerodynamic torque
coefficient, Bearing length, Frequency
response coefficient unless used as an
exponent
Proportional to 1/K
Drag coefficient
Drag coefficient adjusted to allow
forC
h
Fluid drag coefficient
Constant for a particular design (Behaves
like an adjustment to the main
aerodynamic drag term coefficient)
Lift coefficient
Chord
Drag force, Pipe diameter, Impeller
diameter
Rotor response parameter
Fluid drag parameter
Nonfluid drag parameter
Additional geometric variables
Meter error
Nonfluid forces
Frequency of blade passing
Mass moment of inertia of rotor system
about rotor axis
Incidence angle
Incidence at radius r
Lattice effect coefficient, K factor
(pulses/unit volume)
Lattice coefficient at radius r
Turbine meter resisting torque
coefficients
Constant used in equation for helical
blade angle
Lift force, Helical pitch of blades
Number of blades
Rotational speed (frequency)
Index in error equation
Stagnation pressure at inlet
Stagnation pressure at outlet
NOMENCLATURE
pi Static pressure at inlet
p2 Static pressure at outlet
q Actual average flow rate, Index in error
equation
qo Time average flow rate over period T
q\
Initial flow rate
qi Final flow rate
q
b
Base flow rate
q\ Indicated average flow rate
<?max Maximum flow rate
for
which
the
meter
is designed
<7min
Minimum flow rate for which the meter
is designed
q
n
Normal flow rate
qt Flow rate at time t
^trans Flow rate at change of precision
q
v
Volumetric flow rate
R Pipe bend radius
Re Reynolds number
r Radial position, Index in error
equation
rj Journal bearing radius
r
h
Hub radius
r
o
Meter bore radius
r
t
Tip radius
S Slope of no-flow decay curve at
standstill
s Blade spacing
T Temperature, Fundamental period of
pulsating flow, Torque
TB
Bearing drag torque
T
c
Temperature
at
calibration
T
d
Driving torque
Tpo Mechanical friction torque
on
rotor
at
zero speed
T
h
Hub
fluid drag torque
T
n
Nonfluid drag torque
7^>
Temperature
at
operation
T
r
Retarding torque
T
t
Blade
tip
clearance drag torque
T
w
Hub
disk friction drag torque
t Time
£B
Blade thickness
£R
Relaxation time
V Nondimensional fluid velocity
Vo Time average value
of V
z
over period
T
V\
Inlet relative flow velocity
V2 Outlet relative flow velocity
^rnax Maximum value
of V
z
^rnin Minimum value
of V
z
V
z
Axial velocity (instantaneous inlet fluid
velocity
of
pulsating flow)
W Rotor blade velocity, Nondimensional
instantaneous rotor velocity under
pulsating flow
W
h
Relative velocity
at the hub
Y Tangential force
Z Axial force
a
Pi
p
2
Pm
r
8
n
V
p
X
CHAPTER
A
a
B
D
f
H
h
K
L
A/7
Patmos
Pgmin
Pv
qv
Re
S
s
V
Vmax
W
8
Q
CO
Wave shape coefficient, Thermal
coefficient
of
expansion, Angle between
inlet flow direction
and far
field flow
Blade angle
at
radius
r
measured from
axial direction
of
meter
Relative inlet angle
of
flow
Relative outlet angle
of
flow
Mean
of the
inlet
and
outlet flow field
Full-flow amplitude relative
to
average
flow
Deflection
of
flow
at
blade outlet from
blade angle
Flow deviation factor
Dynamic viscosity
Kinematic viscosity
Fluid density
Nondimensional time
(t/T),
Period
of
modulation
Rotor coast time
to
standstill
Instantaneous rotor angular velocity
11
Pipe full flow area, Constant
of
value
about
3.0
used
in
equation
for
avoiding
cavitation
in
flows past vortex meters
Flow area past bluff body
Area when integrating vorticity
Constant
of
value about
1.3
used
in
equation
for
avoiding cavitation
in
flows
past vortex meters
Pipe
ID
Shedding frequency
Streamwise length
of
bluff body
Parallel flats
on
sides
of
bluff body
at
leading edge
Calculation factor
for
shedding
frequency (Zanker
and
Cousins 1975),
K factor
=
pulses/unit volume
Length
of
bluff body across pipe between
end fittings
Pressure drop across vortex meter
(about 1
bar at 10 m/s)
Atmospheric pressure
Minimum back pressure 5D downstream
of vortex meter
Saturated liquid vapor pressure
Volumetric flow rate
Reynolds number
Strouhal number
Length along curve when integrating
velocity around
a
vortex
Velocity
of
flow
Velocity
of
flow past bluff body
Diameter
or
width
of
bluff body
Shear layer thickness
Injected swirling flow rate/total flow
rate
Vorticity
NOMENCLATURE
CHAPTER 12
A Used for area of electrode leads forming
a loop causing quadrature signals in the
electromagnetic flowmeter
a Pipe radius
B Magnetic flux density in tesla
B Magnetic flux density vector
Bo Maximum value of magnetic flux
density
Be 6 component of magnetic flux density
b Inner radius of an annulus of conducting
fluid in two phase annular flow
D Diameter of pipe
E Electric field vector
f Frequency of magnetic field excitation
j Current density vector
/ Length of wire traversing magnetic field
Ap Pressure of the pipeline fluid above
atmospheric
q
v
Volumetric flow rate
r Radial coordinate
5 Electromagnetic flowmeter sensitivity
t Pipe wall thickness
U Electric potential
AI/EE
Voltage between electrodes
AC/p Voltage across a wire P moving through
a magnetic field (similarly for Q and R)
V Velocity
V
Velocity vector
Vm
Mean velocity in the pipe in meters per
second
W Weighting function
W Weight function vector
W Rectilinear flow weight function
W" Axisymmetric weight function
W
z
Axial component of W
a Void fraction
€ Strain
6 Cylindrical coordinate
\x Electric permeability
a Conductivity
<t>
Scalar potential
CHAPTER
A
a
c
D
f
/d
ft
fr
/u
13
Weighting factors for Gaussian
quadrature
Radius
of pipe
Velocity
of ultrasound in the medium,
Coefficients
of a polynomial
Pipe
diameter
Frequency
of stable quartz oscillator for
ultrasonic
measurement system
Frequency
for the downstream
sing-around
pulse train
Transmitted
doppler frequency
Reflected
doppler frequency
Frequency
for the upstream sing-around
pulse
train
y
z
z
(3
7
e
X
p
Pm
T
x
m
fit)
h
I
L
PR
PI
Re
Ryxir)
T
t
tm
At
V
V
m
V(X)
V
5
Vo
V
W
X
Difference between the sing-around
frequencies, Frequency shift in doppler
flowmeter
Function of t
Displacement of ultrasonic beam from
axis of pipe
Driving current in ultrasonic flowmeter
Adiabatic compressibility
Distance along path in transit-time
flowmeter, Distance between transducers
in ultrasonic correlation flowmeter
Counters for ultrasonic measuring
system
\/n is the index for turbulent profile
curve fit, Index
Ultrasound power reflected
Ultrasound power transmitted
Mass flow rate
Reynolds number
Correlation coefficient
Correlation time for integration
Pipe wall thickness
Downstream wave transit time
Mean wave transit time
Upstream wave transit time
Difference between these two wave
transit times
Beam positions for ultrasonic flowmeter
using Gaussian quadrature
Received voltages for ultrasonic
flowmeter
Flow velocity in pipe
Mean velocity in pipe
Flow velocity profile along pipe chord
Ultrasonic flowmeter created flow
Acoustic field for cases (1) and (2)
Relative flow velocity in the direction of
the acoustic beam
Vector weight function
Axial length between transducers in
ultrasonic flowmeter
Length of chord in the plane of the
ultrasonic path
Distance along chord used in ultrasonic
flowmeter
Acoustic impedance
Ultrasonic beam deflection distance on
opposite wall
Ultrasonic beam deflection angle
Ratio of specific heats for a particular gas
Ultrasound beam angle
Ultrasound wave length
Density of material
Mean density in ultrasonic meter
Measurement period for ultrasonic
system, Time period between
ultrasonic wave peaks
Mean ultrasonic correlation time
NOMENCLATURE
CHAPTER
14
A Area
of
duct
c Speed
of
sound
K Constant
k
s
Adiabatic compressibility
p Pressure
A/?A
Pressure drop
to
throat
of
venturi A
A/?B
Pressure drop
to
throat
of
venturi
B
ApAB etc. Differential pressure between
venturi throats, across limbs
of
the
hydraulic Wheatstone bridge,
or
across
diagonals
of the
hydraulic Wheatstone
bridge
q
m
Total mass flow
q
vp
Metering pump volumetric transfer flow
V Velocity
in
the meter
7 Ratio
of
specific heats
f>
Density
of
the fluid
CHAPTER
15
A
B
C,D
Area
of
the
duct,
Constant equal
to 1/n
Constant equal to
(2/TT)
0
-
5
Functions
of
temperature
c
p
Specific heat
at
constant pressure
c
v
Specific heat
at
constant volume
D
Pipe diameter
d
Heating element diameter
g Gravitational constant
/ Current through resistance
R
K Constants
k Thermal conductivity
of
the fluid
k
f
Constant allowing
for
heat transfer and
temperature difference
at
zero flow
I Finite difference dimension
Nu Nusselt number
n Index
Pe Peclet number
Pr Prandtl number
p Pressure
Q
h
Heat transfer
<j
a
Rate
of
heat addition per unit volume
q\y Heat flux
q
h
Heat flux vector
q
m
Mass flow rate
q
v
Volumetric flow rate
R Resistance
of
heating element
Re Reynolds number
5 Flow signal
So Flow signal
at
start
St Stanton number
T Absolute temperature
of
the fluid
Ti,Tz,T
c
Temperatures used
in
finite difference
approximation
AT Measured temperature difference
in
Kelvins
t Time from start
of
flow change
V Fluid velocity
V Vector velocity
W Weight vector
p Fluid density
[i Dynamic viscosity
of
the fluid
v Kinematic viscosity
T
Time constant
of
flowmeter
CHAPTER
16
q
m
Mass flow
R Radius
of
the annulus
of
angular
momentum meter
s Spring constant
X Angular momentum
0 Angular deflection
T
Time difference between markers
on
the angular momentum meter rotating
assembly
co
Angular velocity
of
the rotor
CHAPTER
A
d
F
h
/u
K
K
s
K
u
I
8m
qm
r
8r
8r'
T
t
V
V
v
(!)
v
(2)
w
0
p
X
Q
17
Cross-sectional area
of
the pipe
Width
of
U-tube
Force due
to
Coriolis acceleration
Inertia
in
plane
of
twisting
oscillation
Inertia
in
plane
of
normal oscillation
Constant, Allows
for
the fact that
the
twist
of
the tube will
not
form
a
straight
integration
Spring constant
of
the U-tube
in
twisting
oscillation
Spring constant
of
the U-tube
in
normal
oscillation
Length
of
the U-tube
Element
of
mass equal
to
pA8r'
Mass flow
Radius
Elementary length
of
tube
Elementary length
of
fluid
Torque
Time
Flow velocity
Vector velocity
Oscillatory velocity field set
up
in
the stationary fluid
by
the driving
oscillator
Oscillatory velocity field set
up
in
the
stationary fluid
by
the Coriolis forces
Vector weight function
Twist angle
of
U-tube
Amplitude
of
twist angle
of
U-tube when
in sinusoidal motion
Density
Difference
in
transit time
of
two halves
of twisted U-tube
Phase difference between the total
velocities
at
the two sensing points
Angular velocity
of
the pipe caused
by
the vibration
NOMENCLATURE
CO
co
s
COu
CHAPTER
Amplitude of the angular velocity of the
pipe caused by the vibration
Driving frequency
Natural frequency of U-tube in twisting
oscillation
Natural frequency of U-tube in normal
oscillation
18
A
At
Anin
A
x
A
2
A*
a
k Coefficient
M Mach number
p Pressure
p
0
Stagnation pressure
Ap Dynamic pressure
V
Velocity
y
Ratio
of
specific heats
p
Density
CHAPTER 20
C Cost
cp Component precision
D Pipe diameter
d Width of Coriolis flowmeter U-tube
eq Electronics quality
K
Constant
K
s
Spring constant
of the
U-tube
in
twisting
oscillation
ma
Manufacturing accuracy
me
Material consistency
Pi,
pi,
p3,
p4
Dimensional
and
other factors
q
m
Mass flow rate
q
w
Volumetric flow rate
5 Flow signal
At
Time difference between upstream
and
downstream waves
or
pulses
t
m
Mean time
of
transit used
to
obtain
sound speed
0
Ultrasound beam angle
r
Difference
in
transit time
of two
halves
of
twisted
U-tube
co
Driving frequency
co
s
Natural frequency of U-tube in
twisting oscillation
COMBINED ALL CHAPTERS
A Cross-section of pipe, Function of p and
Re,
Cross-sectional area of the pipe,
Constant, Cross-sectional area of the
effective annular flow passage at the
rotor blades, Constant of value about 3.0
used in equation for avoiding cavitation
in flows past vortex meters, Used for area
of electrode leads forming a loop
causing quadrature signals in the
electromagnetic flowmeter, Weighting
factors for Gaussian quadrature, Area of
the duct, Constant equal to 1 In
0, 01/
a
c
B
Bo
b
z
h
C
C
D
Qmean
Q
Q
C
L
CR
Constant
Cross-sectional area of float
Flow area past bluff body
Cross-sectional area of tapering tube
at height x
Outlet cross-sectional area, Annular area
around float, Annular area around target
Throat cross-sectional area
Constant, Area of target, Turbine
meter aerodynamic torque coefficient,
Area when integrating vorticity, Pipe
radius
Constant coefficients
Expression in orifice plate bending
formula
Constant
Constant
Constant
Constant, Axial length of rotor, Constant
of value about 1.3 used in equation for
avoiding cavitation in flows past vortex
meters,
Magnetic flux density in tesla,
Constant equal to
(2/TT)
0
-
5
Magnetic flux density vector
Maximum value of magnetic flux
density
0 component of magnetic flux density
Inner radius of an annulus of conducting
fluid in two-phase annular flow
Constant, Turbine meter aerodynamic
torque coefficient, Bearing length,
Frequency response coefficient unless
used as an exponent
Constant
Constant
Constant
Discharge coefficient, Coefficient,
Proportional to 1/K, Function of
temperature, Cost
Contraction coefficient, Constants in
curve fit for target meter discharge
coefficient
Drag coefficient
Drag coefficient adjusted to allow for C
h
Concentration of tracer in the main
stream at the downstream sampling
point
Mean concentration of tracer measured
downstream during time t
Fluid drag coefficient
Constant for a particular design (Behaves
like an adjustment to the main
aerodynamic drag term coefficient)
Concentration of tracer in the injected
stream
Lift coefficient