Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications
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2.3 FLOW
IN A
CIRCULAR CROSS-SECTION PIPE 25
Table 2.1. Typical values
of
viscosity (approximate values
at 1
bar)
Viscosity
Water
Benzene
Oxygen
Nitrogen
co
2
Air
Temperature
(°C)
20
20
0
0
0
0
Density,
9 (kg/m
3
)
1,000
700
1.43
1.25
1.98
1.29
Dynamic,
McP)
1.002
0.647
0.019
0.017
0.014
0.017
Kinematic,
v (cSt)
1.002
0.92
13.3
13.6
7.1
13.2
Adapted from Kaye and Laby (1966).
If Re
is
less than about 2,000,
and the
fluid has
had a
sufficient length
of
pipe
to
reach
a
steady state,
it
all moves parallel
to the
axis
of
the pipe. At
the
pipe wall,
the
fluid
"sticks"
to the
pipe
in,
what
is
known as,
a
nonslip condition. The velocity of
the fluid increases, therefore, from zero
at
the pipe wall
to a
maximum
at
the center,
and
the
shape
of
the profile
is
parabolic.
In
this case,
the
flow
is
called laminar.
At about
Re
=
2,000,
a
major change takes place,
and the
smooth parallel nature
of the flow gives way
to
eddies
in the
flow.
These eddies mix the high velocity
at the
pipe axis with
the
lower velocity near
the
pipe wall.
The
resulting profile
is
flatter
(although
it
still goes
to
zero
at the
pipe wall). When fully developed, this
is
known
as
a
turbulent profile.
It is of a
well-defined shape
and
with
a
known range
of
eddy
sizes.
It
may, therefore,
be
misleading
to
use
the
term turbulent profile
for any
other
profiles created
by
upstream disturbances
due to
bends, etc. Between
the
turbulent
flow and
the
laminar flow, there
is a
transition when
the
flow alternates randomly,
in space
and
time, between laminar flow
and
turbulent flow.
Any flowmeter application
to
non-Newtonian fluids will need advice from
the
flowmeter manufacturer.
It is
probable that clear bore flowmeters,
if
applicable, will
be most suitable. A flowmeter with relatively low sensitivity
to
profile, such
as
elec-
tromagnetic
or
multibeam ultrasonic designs, will
be
worth consideration provided
the fluid
is
conducting
for the
first type
or
transmits
an
adequate level
of
sound
for
the
second.
It
is
apparent from
the
calculations
of
Reynolds number
for
water
and air
that,
in the majority
of
industrial applications,
the
flows
will be turbulent, and, therefore,
the behavior
of the
turbulent profile,
and of
flowmeters subject
to it, is of
primary
importance
to us.
For laminar flow,
we can
use
the
equation
V=V
0
1- [L.
(2.2)
where
V
o
is the
velocity
on the
axis
of
the pipe,
r is the
radial point
at
which we
are
measuring velocity,
and R is the
pipe radius
and
where
V is
zero
at the
wall
of the
pipe because
of
the nonslip condition
for a
fluid
at a
solid boundary.
26
FLUID MECHANICS ESSENTIALS
Table 2.2. Approximate turbulent velocity profiles from equation (2.4)
Re
n
V/V[air
=
V/V[a.tr
=
0.75R)
0.758/?)
4 x 10
3
6.0
1.003
0.998
2.3 x 10
4
6.6
1.004
0.999
1.1 x 10
5
7.0
1.004
1.000
1.1 x 10
6
8.8
1.005
1.002
2.0 x 10
6
to
3.2 x 10
6
10
1.005
1.002
The mean velocity in the pipe is then given by
" 2
For turbulent flow an approximate curve-fit is
V= V
0
(l-r/R)
l/n
(2.3)
(2.4)
where
Vo
is, again, the center line velocity, and we can relate n to Re from experi-
mental data (Table 2.2). Note that pipe roughness, if the pipe is not hydraulically
smooth, will also affect the profile.
This expression, therefore, provides not only a convenient, although approxi-
mate, representation of the boundary layer shape but also an index adjustment to
allow for the change in Reynolds number. The resulting profile shapes for the laminar
and the turbulent regimes, based on Equation (2.4), are shown in Figure 2.1.
One interesting feature of these turbulent profiles is that the ratio of the velocity
at about the 3/4 radius point to the mean velocity is approximately unity. The mean
velocity is given by
V
In
2
Hence
V/V =
2n
2
-a-r/R)
\/n
(2.5)
(2.6)
Table 2.2 provides the values of this ratio for various values of Re and two radial
positions. Thus if a single measurement of velocity in a pipe is to be used to obtain
the mean of the velocity in a turbulent flow, a point at 0.7S8R may be found best.
This figure will clearly be of importance when we consider where to position the
probes described in Chapter 18.
LAMINAR
Re < 2000
TURBULENT
Re =
2.3x10
Re
=1.1x10
Figure
2.1.
Laminar and turbulent pipe profiles.
Re = 3.2x10
2.4 FLOW STRAIGHTENERS AND CONDITIONERS 27
A final important point about the profiles in Figure 2.1 is that there is much
difference between the flows, even though they are drawn as smooth curves. In the
laminar profile, the fluid shears smoothly over
itself,
but it all moves essentially
parallel to the axis of the pipe. In the turbulent profiles, the curves represent a
mean profile and ignore the turbulent eddies, which can be as much as 10% of the
velocity of the mean flow and cause fluctuations in all directions. In the context of
flow measurement, this is particularly significant in that the meter must measure
accurately against this background fluctuation.
So far we have assumed that the pipe flow profile is fully developed. To achieve
such a profile requires an upstream straight length of pipe greater than may be
available in many industrial flowmeter installations. 60D is sometimes quoted as a
minimum requirement.
We,
therefore, have a problem when installing a flowmeter in that the calibration
conditions for a fully developed profile are unlikely to be achieved in the installation
and that the meter accuracy will be degraded. We have four options.
i. Find an installation point where fully developed flow occurs,
ii.
Calibrate the flowmeter with the upstream pipework (downstream has little effect
beyond a maximum of about 8D).
iii.
Take extensive measurements of the effects of bends, valves, T-pieces, multiple
bends,
expansions, and contractions on a particular flowmeter design and allow
for the change in precision,
iv. Attempt to reorder the flow to recreate a turbulent profile of the sort used for
calibration.
Outside a few laboratory situations, (i) is probably unlikely to be an option, but
(ii) may not be realistic or financially viable unless in situ calibration is possible. Op-
tion (iii) will be a recurring theme through the discussion of most types of flowme-
ters.
An alternative approach is to compute the profiles resulting from pipe fittings.
One example of such an approach is that of Langsholt and Thomassen (1991) who
used a commercially available computer program to obtain the flow downstream of
various piping fittings. Some indication that computer prediction and experiment
gave similar swirl decay downstream of a double bend in perpendicular planes was
given, and 41D reduced the angle to 3.3° for measured values and 2.4° for simulated
values. Mottram and Rawat (1986) suggested that the swirl from two 90° bends in
perpendicular planes is reduced by 50% in smooth pipes after L/D = 45 and in
rough pipes after an L/D value in the range 10 to 20 (cf. however Mattingly et al.
1987 who noted swirl effects on a turbine meter after 90D).
Langsholt and Thomassen (1991) also provided some flow contours at points in
a complex pipe configuration. In the context of this chapter, it is important to give
some attention to (iv).
2.4 FLOW STRAIGHTENERS AND CONDITIONERS
Flow straighteners and flow conditioners have been used, for many
years,
to attempt
to reorder
a
profile that has been disturbed by upstream fittings. The flow straightener
was designed to remove swirl from the flow. Swirl, the bodily rotation of the fluid in
28 FLUID MECHANICS ESSENTIALS
(c)
Figure 2.2. Approximate diagrams of flow straighteners: (a) Etoile; (b) Tube
bundle; (c) Box (a honeycomb layout may also be used).
a pipe, takes a long length to decay and can introduce severe metering errors. The
three types of straightener commonly used are shown in Figure 2.2.
It is essential that they are set straight or they will become the cause of swirling
flow rather than the correction for it. Examples of the effectiveness of etoile straight-
eners of various lengths were given by Kinghorn et al. (1991). Their conclusions
follow:
• An eight vaned (on four diameters) straightener of 0.5D length virtually removed
even the maximum swirl.
• The flat profiles downstream of the straighteners appear to have caused most of
the residual negative errors in the orifice coefficient, as opposed to the positive
errors due to swirling flow.
• To ensure an error of less than
±0.5%,
a straightener ID long should be placed
upstream of the orifice meter at least
6D for plates with a beta ratio of 0.5 and
14D for plates with a beta ratio of 0.8.
• Even with 16D separation, virtually all tests suggested some effect from the
straightener.
• AID straightener is recommended for removing swirl and minimizing head loss.
Flow conditioners attempt to reorder the flow profile to recreate a fully developed
turbulent profile. However, this reordering must address both mean profile shape and
turbulent eddy pattern. Examples of conditioners are given in Figure 2.3.
Recent published work (e.g., that due to Laws) has thrown much helpful light
on the effectiveness of these devices. Bates (1991) reported on field use by Total
Oil Marine of a K-lab flow conditioner. Strong swirl was found to be present in the
gas metering system on a process platform in the Alwyn North field, which had
been installed to ISO 5167 in three 14-in. lines with beta ratio of 0.6. The meter was
separated from a pipe reduction of 0.86 by
30D,
but upstream of this there were other
fittings. Greasy deposits on the orifice plate gave an early indication of swirl being
present. This was rectified by installing K-lab flow conditioners. This conditioner is
machined from solid, with holes arranged approximately as in Figure 2.3(b). (See
Erdal et al. 1994 on the development of the K-Lab conditioner and Spearman et al.
1991 on LDV measurements downstream of a Mitsubishi flow conditioner and of
an orifice plate in a combined package.)
Karnik et al.
(1991,
1994) addressed the important question of how closely the
flow downstream of a flow conditioner represented fully developed flow as far as a
flowmeter was concerned. For undisturbed fully developed flow, they found that,
2.4 FLOW STRAIGHTENERS AND CONDITIONERS 29
(a)
(b)
THICKNESS-0.13D
(c)
flow
VANES
\
\\
IZ
77
ZZ
77
for Re of about 0.9 x 10
5
, the profile was ap-
proximated by n = 7A in Equation (2.4),
compared with n = 7 in Table 2.2. Mea-
surements of velocity and turbulence pro-
files and of orifice plate performance were
made to determine the effect of an el-
bow upstream of the tube bundle with
various spacings between the elbow, the
flow straightener, and the metering posi-
tion. They confirmed the point that, even
though the profile may be approximately
correct, the turbulence characteristics are
unlikely to be correct, and the orifice plate
appears to be influenced by both.
Laws and colleagues have done some
useful work on straighteners and condi-
tioners (Laws 1991, Laws and Ouazzane
1992,
1995b, 1995c). Her work initially,
and rather unexpectedly, suggested that
tube bundles of length recommended by
ISO 5167 (hexagonal pack of 19 tubes of
L > 20d), AGA(3), and ANSI/API 2530:1985
(circumferential pack with L > lOd) are
little more, and possibly less, effective than
one of only L =
1.25d.
She found that this
was as effective in swirl elimination and at-
tenuation of profile nonuniformities. She
also suggested that the AGA design was
more effective and had a lower Ap. This led
her to suggest that perforated plates may be
a better and more easily constructed option.
This observation was reinforced by work
on the Zanker flow conditioner with thicker
perforated plate, which performed, in some
cases,
better without downstream honey-
comb (Laws and Ouazzane 1992). A crite-
rion of ±6% of the fully developed pro-
file 100D downstream was used to assess
whether the resultant Zanker profiles at
8.5D were satisfactory. The reported experi-
ments suggested that, for a 12-mm plate in a 100-mm line, the Zanker flow con-
ditioner without downstream honeycomb performed better than with the honey-
comb.
Laws and Ouazzane (1995b) also suggested that a thick Zanker plate (10%
of the pipe diameter) with upstream honeycomb flow straightener is preferable to
the Zanker straightener with thinner plate and downstream straightener. Not sur-
prisingly, they suggested that the Laws plate is superior. In further tests, Laws and
Ouazzane (1995c) found that the etoile flow straightener was not a very good flow
= 0.13D
,PLATE
TABS
vortex tabs
Figure 2.3. Approximate diagrams of flow condition-
ers:
(a) Zanker; (b) K-Lab Mark 2; (c) Mitsubishi; (d)
Laws (after Laws 1990, Laws and Ouazzane 1995a);
(e) Vortab [reproduced with permission of Laaser
(UK)
Ltd.].
30 FLUID MECHANICS ESSENTIALS
conditioner, although it was effective as a straightener. They suggested that if the
etoile is made with the vanes stopping short of the pipe axis, the wake from the hub
of the etoile is removed and the performance is improved.
Laws (1990) described her new flow conditioner, which benefited from these
findings. It has a central hole and two rings of holes on 0.4616D and 0.8436D. The
plate thickness is given as 0.123D and, for best performance, has an arrangement of
vanes and tabs as shown in Figure 2.3(d). The number of holes is
1:7:13
(or
1:6:12
in later versions), the ratio of open area outer to inner ring is
1.385,
and the ratio of
inner ring to central hole is 5.42. The total open area is 51.55%. Laws claims that the
unit will be an enhancement to most flow installations, enabling flow conditions
to be maintained close to fully developed irrespective of the operating conditions
and thus minimizing errors due to installation effects. The loss coefficient is about
1.4, and the claim appears to be made that with a flow distortion device 3D from
the conditioner and the conditioner 3D upstream of the upstream pressure tapping
of an orifice plate, the shift in performance is of order 0.2% (Laws and Ouazzane
1995a).
Smith et al. (1989) proposed a different approach to the recreation of turbulent
profiles using vortex-generating devices. Their results looked promising for the con-
ditioner 5D upstream of the flowmeter when they appear to claim less than 1% shift.
The device consists of four axial swirl tabs of about ID length, which project about
40%
of the radius into the flow from the wall and are essentially a straightening
device to remove swirl. Two sets are recommended; they should be set at 45° relative
to each other. These are followed by, preferably, two sections of ID length each,
with four trapezoidal tabs (vor-tabs) spaced around the circumference of the pipe
and projecting inward with an angle of about 30° to the pipe wall in the general
direction of the flow. Laws and Harris (1993) tested this device and considered that
it did quite well in recreating a turbulent profile but appeared uncertain of the sta-
bility of the resulting profile. The loss is claimed to be only 0.7 dynamic heads. A
commercially available device, the Vortab [Figure 2.3(e)], appears to exploit the same
idea and consists of a ID straightening section followed by a 2D section with three
sets of vor-tabs.
Gallagher et al. (1994) discussed the development of the Gallagher conditioner,
which also consists of an antiswirl device, settling chamber, and profile device moun-
ted sequentially in the pipe.
2.5 ESSENTIAL EQUATIONS
The continuity
equation
results from the classical physics concept that mass can nei-
ther be created nor destroyed and so the same mass must pass through each point
in a pipe
Mass flow
= q
m
= pq
v
= pVA (2.7)
and we can relate the velocity at different pipe sections 1 and 2 (Figure 2.4)
Pl
V
1
A
1
=
P2
V
2
A
2
(2.8)
2.5 ESSENTIAL EQUATIONS 31
•
Q
{
r
i
\
v
2
-
P2
—(f~
1
DATUM
(a)
(b)
Figure 2.4. Duct sections, (a) Varying height and duct section, (b) Varying area
duct to show positions referred to in equations.
Bernoulli's
equation can be derived from energy considerations (Baker 1996) and
relates the pressure change to the velocity change. Using Figure 2.4(a), we can relate
pressure, velocity, and height above datum in a compressible fluid without flow
losses as
2
d_p
+
-v
2
)
= 0
(2.9)
This equation is known as the compressible fluid form of Bernoulli's equation. If
p
is constant for an incompressible fluid with constant density, then we can rewrite
the Bernoulli equation as
(2.10)
Note from this, assuming that we can neglect changes in
z,
that, if the flow
is
brought
to rest, the resulting pressure, p
0
is the total or stagnation pressure, the pressure while
fluid is flowing is the static pressure, and the difference, Ap = \pV
2
, is the dynamic
pressure.
32 FLUID MECHANICS ESSENTIALS
If we neglect changes in height and combine Equations (2.8) and (2.10), we
obtain an equation that gives the relationship between pressure change and flow
rate through a duct such as in Figure 2.4(b)
(2
-
n)
Loss coefficients
result, in real flows, from viscosity in the fluid, and it is essential
to know these losses in a piping system to ensure that the pump, or whatever creates
the flow, will be sufficient to maintain the required flow. To understand how to
calculate losses, the reader is referred to the excellent book by Miller (1990).
Thus all flowmeters create a flow loss, although clear bore meters do not signifi-
cantly add to the loss due to an equal length of straight pipe. However, if the profile
is highly disturbed and flow conditioning is used, these devices cause an additional
pressure loss which is given by
(2.12)
K may be up to order 5 for the devices in Figures 2.2 and 2.3, but for some flow
conditioners it may be as much as 15 since the flow is effectively going through a
contraction with very poor downstream pressure recovery.
For a fuller derivation of these equations, the reader is referred to Baker (1996)
or other fluid mechanics books.
2.6 UNSTEADY FLOW AND PULSATION
In this section, we consider the effect of unsteadiness on flowmeters. The use of fluid
instability in flowmeters is covered in Chapter 11 where it is an essential part of the
meter designs.
Most flowmeters are affected by unsteady or pulsatile flows for the following
reasons.
• Pulsatile flow affects the velocity profile in the pipe, and the distorted profile may,
in turn, affect the flowmeter response. Hakansson and Delsing (1994) found that
flattening of the flow profile occurred due to the pulsating flows. This, in the case
of the ultrasonic flowmeter, causes incorrect averaging of the diametral beam.
• The characteristic of the meter may be nonlinear, and the resulting output aver-
age will not correspond to that for the flow. The differential pressure flowmeters
with their square law are examples.
• The inertia of parts of the flowmeter or of the fluid may not allow the meter
to track the pulsating flow correctly. The gas turbine meter is highly affected by
this.
The importance of this effect in the orifice plate has been the cause of some
discussion in the literature for which the reader is referred to Mottram's papers.
• The natural operating frequency of the flowmeter may be near the pulsation
frequency, and this may cause aliasing or other errors. Vortex meters are sensitive
to such pulsations.
2.6 UNSTEADY FLOW AND PULSATION
33
TOTAL VOLUME V
PRESSURE DROP Ap
CONSTRICTION
METER
x
ADDED
CAPACITY
ABSOLUTE
LINE
PRESSURE
P
SOURCE
OF
PULSATION
Figure 2.5. Diagram to indicate symbols used in the Hodgson number calcula-
tion.
• The secondary instrumentation may not be able to follow the pulsation, and, as
a consequence, errors may be introduced. The manometer is the most obvious
secondary device to be affected, but pressure transducers, connecting leads, and
other devices may also be affected.
One approach associated with Hodgson was developed to show how much damp-
ing was needed to reduce pulsation to acceptable levels.
Mottram (1989) reviews some of the work on Hodgson's number (cf. Mottram
1981) but makes the wise comment that "if you can't measure (the size of the
pulsation) damp it" (Mottram 1992). The Hodgson number is given by (Figure 2.5)
VfAp
P
H= -L-t-
(2.13)
where V is the total volume of pipework and other vessels between the source of
the pulsation and the flowmeter position, Ap is the pressure drop over the same
distance, f is the frequency of the pulsation, q
v
is the volumetric flow rate, and p
is the absolute line pressure. With this number, it is possible to plot curves to show
the likely error levels for various values of H (cf. Section 5.5 for orifice plates).
Mottram gives a simplified value for the damping based on the Hodgson number,
allowing for the worst case where resonance occurs, of
H
y
(V
rms
/V)
d
(2.14)
where (V
rms
/V)
U
d is the undamped velocity ratio, (V^
ms
/V)ci is the damped velocity
ratio,
and y is the isentropic exponent.
He cautions that the criterion resulting from using this with the Hodgson number
is untested. He also suggests that the Hodgson number should be doubled where the
pulsation amplitude is estimated and not measured.
34
FLUID MECHANICS ESSENTIALS
Some meters [e.g., differential pressure (cf. Section 5.5), turbine, or vortex] are se-
riously affected by pulsating flow, whereas others (e.g., electromagnetic or ultrasonic
time-of-flight) are probably little affected.
2.7 COMPRESSIBLE FLOW
When a gas flows at velocities comparable to the speed of sound, its behavior is
markedly different from the behavior of incompressible fluids. Such high velocity
flows occur in sonic or critical nozzles.
To appreciate the special features of compressible flow of a gas, we shall start
by imagining that we have the nozzle in Figure 2.6(a) set up in an experimental
rig with a compressor sucking air downstream of the control valve. We consider a
convergent-divergent nozzle that reduces in area from the inlet to a minimum at the
throat, and then increases in area to the exit, where it opens into the back pressure
chamber. As the flow increases, there would be a steadily increasing noise emitted
INLET
THROAT
EXIT
(b)
Figure 2.6. Convergent-divergent nozzle: (a) geometry; (b) p/po against distance through
convergent-divergent nozzle.