Baker R.C. Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications
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6.6 APPLICATIONS, ADVANTAGES, AND DISADVANTAGES 137
where X is the Lockhart-Martinelli parameter and
The liquid and gas densities are given by
p
x
and p
g
and
n = 0.606(1 - e-°
746F
'g) for Fr
g
> 1.5 (6.4)
and
n = 0Al for 0.5 < Fr
g
< 1.5 (6.5)
where the superficial gas Froude number is given by
(6.6)
where V
sg
is the superficial gas velocity, and the Lockhart-Martinelli parameter is
given by
(6.7)
The relationship is valid for gas densities above 17 kg/m
3
and up to the liquid
density, for gas Froude numbers above 0.5, and for Xup to 0.3 (cf. Chisholm 1977).
6.6 APPLICATIONS, ADVANTAGES, AND DISADVANTAGES
Early work on the classical venturi by Herschel was aimed at use in the water indus-
try, and it has continued to be used in measurement of water flows. Where energy
conservation is an important consideration in large pumped flows, the low head loss
of the venturi becomes attractive.
Rivetti et al. (1994) have described a quite different application of a small size
venturi for liquid helium service, which was essentially a scaled down ISO 5167
design. Differential pressure must not be so high as to cause cavitation, and pressure
ducts should have a diameter between 1/10 and 1/5 of the diameter of the respective
venturi section and be four or more (an even number). Flow rates ranged from
0.5-4 g/s to 12.5-100 g/s for upstream/throat diameters of 6/2.6 and 16/12 mm,
respectively, and differential pressures of 34-2,190 and 33-2,140 Pa, respectively.
The classical venturi has also been used for slurry and two-phase flows for which
some data are available. Baker (1991a) has reviewed work on this application of the
venturi. There appears to be growing interest in the use of venturi meters in these
applications, although the behavior of the flow and the appropriate models of the
multiphase flow may not be straightforward. One such example is the performance
of the venturi meter in an oil-in-water emulsion in Figure 6.6 (Pal and Rhodes 1985).
The initial cost of the venturi is higher than the orifice because it is a much larger
device to make, and precision in manufacture may not be easy. Various manufactur-
ers offer ready-made designs (Figure 6.5).
138
VENTURI METER AND STANDARD NOZZLES
1
-5
ficient c
coet
ual
^d/act
Predict*
1.10
i n
1 .U
0.90
Symbol
O
A
D
X
-
_
-
1
Reynolds number
=
1
= 8
= 4
=
i
5
x
X
5
x
|
x 10
5
10
4
10
4
xlO
4
H
i
i
i
i i
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Oil fraction
Figure 6.6. Performance of
a
venturi meter in an oil-in-water emulsion (Pal and
Rhodes 1985; reproduced with permission from BHR Group).
The nozzles tend to be more stable than the orifice for high temperatures and
high velocities, experiencing
less
wear (due to the smooth inlet contour
as
opposed to
the sharp orifice upstream edge) and being less likely to distort. They are particularly
applicable to steam flows, where they have been widely used.
6.7 CHAPTER CONCLUSIONS
This chapter has been, necessarily, brief because there appears to be less significant
new material in the literature to add to the advice and guidance of the standards.
However, there does appear to be an increased interest by the oil and gas industry in
using venturi meters for both single and multiphase (especially wet gas) metering,
and
as
a result some new
work,
which appears to be yielding some unexpected results,
is taking place (Jamieson et al. 1996).
a. It should be noted that the coefficient for the ISA 1932 nozzle has been slightly
altered in the standard.
b.
Jamieson et al. (1996) observed discharge coefficients for a 150-mm (6-in.) diam-
eter venturi in high pressure air (70 bar) at Re up to 8 x 10
6
and throat velocities
up to 125 m/s, which were several percent higher than those obtained in water
calibration. The reason appears to lie in the tapping chambers and the possible
acoustic effects there.
c. Computational fluid dynamics (CFD) work is being developed (cf. Sattary and
Reader-Harris 1997) for venturi and nozzles. For venturi with
f$
= 0.4, 0.6, and
0.75,
CFD tended to underestimate relative to experimental and ISO by up to
about 1.2%. Sattary and Reader-Harris (1997) suggest that the difference is due to
the actual method of measuring static pressure compared with the CFD model.
6.7 CHAPTER CONCLUSIONS 139
They also looked at compressibility effects. There is likely to be growing interest
in developing the CFD work to explore the likely ultimate precision of these
instruments.
d. An inside-out nozzle or short venturi in the form of a bullet-shaped target meter
could yield a highly predictable meter with direct electrical output via a force
sensor. This could be combined with wall differential pressure measurements to
provide a condition-monitoring function.
CHAPTER
7
Critical Flow Venturi Nozzle
7.1 INTRODUCTION
This meter uses a fascinating effect that occurs when gas flows at very high velo-
city through a nozzle. As the gas is sucked through the nozzle, the velocity increases
as the cross-section of the nozzle passage decreases toward the throat. At the throat,
the maximum achievable velocity is sonic - the speed of sound. Downstream of the
throat, the velocity will either fall again returning to subsonic or will rise and become
supersonic. In normal operation of the nozzle, the supersonic region is likely to be
small and to be followed by a shock wave that stands across the divergent portion of
the nozzle and causes the gas to drop, very suddenly, from supersonic to subsonic.
The existence of a shock wave does not mean that one will hear a "supersonic bang"!
Such bangs are usually caused by moving shock waves carried forward by high speed
aircraft.
The fascinating effect of sonic conditions at the throat is that changes in the
flow downstream of the throat have no effect on conditions upstream. The sonic
or critical condition, as it is called, appears to block any information that is trying
to penetrate upstream. A simple picture of this is that the messengers carrying such
information travel at the speed of sound and so are unable to make any headway
over the fast flowing gas stream.
Two important effects of sonic conditions are that
i. the mass flow rate is a function of the gas properties, the upstream stagnation
temperature and pressure, and the area of the throat, provided that the nozzle
is actually running at critical conditions (which can be ascertained by checking
the downstream pressure);
ii.
the nozzle acts as a flow controller, creating steady conditions upstream even
though conditions downstream are unsteady.
Figure 2.6(a) shows a simple illustration of a convergent-divergent nozzle with
a throttle valve downstream. Figure 2.6(b) shows the variation of pressure through
the nozzle, and the critical flow conditions are those bounded by curves c and i for
which the flow becomes sonic at the throat.
For a fuller description of the fluid flow behavior, see Baker (1996). Two essential
equations apply when the velocity at the throat is equal to the speed of sound
(Mach number equals unity) and the conditions are critical. The first equation gives
140
7.2 DESIGN DETAILS OF A PRACTICAL FLOWMETER INSTALLATION 141
the pressure ratio for choked flow [Equation (2.18)]:
Po V 2 /
where
p*
is the pressure at the throat, p
0
is the stagnation pressure, and y is the ratio
of specific heats c
p
/c
w
.
The second equation gives the ideal mass flow q
m
for choked flow from Equation
(2.18),
making use of the relationships
(7
.
2)
where R is the universal gas constant (8.3143 kJ/kmolK), M is the molecular weight,
c
p
and c
w
are the specific heats at constant pressure and constant volume, A* is the
throat cross-sectional area, and 7o is the stagnation temperature.
Thus if we can measure upstream stagnation pressure and temperature po and
To,
and if we know the throat area
A*
and the gas properties, we appear to have a
mass flowmeter. The flow must be choked. Also this equation is limited to a perfect
gas and assumes an idealized and one-dimensional flow.
We are interested in the use of this device for real
gases.
For precise work, we have
to recognize that gases are NOT perfect. We also find that boundary layer growth
and curvature of the flow through the nozzle result in a flow that is not truly one-
dimensional.
Thus as with many flowmeters, there is a coefficient of discharge, to be discussed
later. First, we consider a practical flowmeter installation.
7.2 DESIGN DETAILS OF A PRACTICAL
FLOWMETER INSTALLATION
The critical flow venturi nozzle is fully specified by the most recent version of the
standard document BS EN ISO 9300: 1995. The reader is strongly advised to obtain
a copy of the standard for reference if planning to build a facility using this device
and to view this chapter as complementary to the standard.
Figure 7.1 shows a diagram of a critical flow venturi nozzle installation. Note
that the temperature tapping can be upstream of flow conditioners because the
temperature change is likely to be negligible. However, the pressure tapping must
be sensing a true inlet stagnation value. There may also be a downstream measure
of pressure to ensure that critical conditions are maintained.
142 CRITICAL FLOW VENTURI NOZZLE
2D±0.2D
6
flow
straightener
Figure 7.1. Details
of
venturi nozzle installation.
straight
length
>5D
D±0.1
D
i
throat
dia.
d
CUxl
CZXI
czxr
inlet
manifold
If
the
upstream pressure allows only limited variation, then
the
range
of q
m
will
be limited,
and
only
by
using
a
bank
of
nozzles
of
different areas
can we
achieve
a
satisfactory calibration range (Figure 7.2). Caron (1995) mentioned
a
test stand that
used
a
bank
of
sonic nozzles
in
binary sequence. Nozzle
2 has
twice
the
flow rate
of
nozzle
1, etc.
Nozzles
may
have
a
convergent portion only;
in
which case there will
be a
substantial pressure loss
at
exit,
and the
pressure ratio
of
outlet-to-inlet stagnation
must be less than 0.528
for
air.
On the
other hand,
the
convergent-divergent venturi
nozzle
in
Figure
7.3 may
achieve
a
pressure ratio (exit/inlet)
for
critical conditions
at
the
throat
of
pz/po
—
0.9. The one
shown
is the
toroidal throat venturi nozzle.
An alternative,
not
favored
by
some,
is the
cylindrical throat type where
a
constant
area throat
of
one diameter
in
length separates inlet contraction
and
outlet diffuser.
The diagram
in the
standard provides
the
details
of
design finishes
and
tolerances that differ slightly from,
but
cover, those
in
Figure
7.3. It is
also impor-
tant
to
note other features
of the
nozzle
in
Figure
7.3. The
inlet pipe diameter,
D,
must
be
greater than
4d, and
referring
to
Figure
7.1
with
a
pipe diameter
of 50 mm upstream, this would
put a
maximum
on the
throat diameter
of
12.5 mm. Also
the
preferred distance
to the
upstream pressure tapping
is
between
0.9D
and
1.1D.
Other spacings
are
allowed
pro-
vided that
it can be
shown that
the
tapping
gives
a
correct measure
of the
nozzle
in-
let stagnation pressure. The standard allows
for
a
larger space upstream with
its own
requirements.
The standard also sets
out
requirements
on temperature measurement that cover
the position
in
Figure 7.1
but
allow a greater
spacing
if it
gives, reliably,
the
stagnation
temperature
at
nozzle inlet.
The
standard
also warns that care needs
to be
taken
if
the surroundings
of
the installation differ
in
temperature by more than 5°C from the gas.
itxrzj
3XIZ=
3>C_
>o
>c=
critical
nozzles
outlet
manifold
Figure 7.2. Bank
of
critical flowmeters.
7.3 PRACTICAL EQUATIONS
143
2.5d
Figure 7.3. Details of toroidal throat venturi nozzle.
Where there is the possibility of condensate, drain holes are allowed provided
there is no flow through them when flow measurement is taking place, and they
are spaced upstream of the pressure tapping by at least D and not in the same axial
plane as the pressure tapping.
The standard requires that the nozzle be made from material capable of finish
to the required smoothness, resistant to corrosion in the intended application, and
dimensionally stable so that temperature corrections can be used for throat area.
7.3 PRACTICAL EQUATIONS
The practical use of critical flow venturi nozzles is set out in
BS
EN ISO 9300: 1995.
Apart from some nomenclature differences, this section gives the method set out
in the standard. Using the expression of Equation (7.2), we introduce a discharge
coefficient
C
to allow for the deviation from the ideal equation, and
we
also introduce
a coefficient for the gas Q, the critical flow function, obtainable from tables and
formulas, and for a perfect gas written as
(7.3)
With these coefficients, we can rewrite the equation as
A*C
C*po
T
0
(R/M)
Alternatively, the equation may be written in terms of p
0
and
q
m
= A
t
C
(7.4)
(7.5)
144
CRITICAL FLOW VENTURI NOZZLE
where
and
po
Po
ZQRTQ
M
(7.6)
(7.7)
In this case, we require values of
C*
and Zo, the compressibility factor at stagnation
conditions.
The design should, preferably, ensure that the values of p and T are obtained
where the Mach number is small enough to assume that they are equal to p
0
and T
o
.
Otherwise, the standard gives equations to obtain the corrected values.
In order to ensure that the nozzle operates critically, the standard requires that
the actual maximum outlet pressure
p
2
max,
when compared with the ideal value p
2
i,
for which critical conditions are achievable satisfies
- P*= 0.
(7.8)
0.9
0.85
0.8
0.75
Using tables such as those in Baker (1996), it is possible to obtain values of p
2i
for
various values of outlet area A
2
, assuming that the gas behaves sufficiently closely
to an ideal gas to use the ratio of the specific heats y. Figure 7.4 shows the values of
p2max/po
for
y = 1.4. The standard provides a plot for other values of the isentropic
exponent.
Note that y is the ratio of specific heats for an ideal gas, and
K
is referred to in the
standard as the isentropic exponent and applies to real gases. I have avoided using
the latter in this chapter. For further information, consult Bean (1971) who relates
the value (cf. Miller 1996, Sullivan 1979) of
K
to y via an expression (which he does
not derive) that recognizes that Z is constant for an ideal gas but varies for a real
gas.
Miller (1996) also gives plots of
K.
Where there is a divergence between
K
and
y,
the standard indicates which to use.
It is relatively simple to relate the area
~~~~ ratio between the throat and the outlet
based on design angles of the diffuser. If the
half-angle of the diffuser is 0, the radius of
toroid is r and the length of the diffuser is /
beyond the toroid, the increase in diameter
is (Figure 7.5) 2[r
(1
-
cos 0)+1
tan
0]
so that
the outlet diameter
0.7
0.65
d
2
= d+2[r(l-cos 6>)+/tan 0]
or
1.0
2.0
3.0 4.0
AJA*
5.0 6.0
2r(l-cos0) 2/ tan 0
(7.9)
Figure 7.4. Maximum values of back pressure for
y = 1.4 to ensure critical operation based on the cri-
terion in BS EN ISO 9300: 1995.
from which the expression in the standard
for
A
2
/A*
can be simply obtained.
7.4 DISCHARGE COEFFICIENT
C 145
/tane
- r(1 -cose)
'T
Figure 7.5. Geometrical calculation to obtain the outlet diameter from the
dif-
fuser geometry.
7.4 DISCHARGE COEFFICIENT
C
The data
on
which
the
discharge coefficient
is
based include work
by
Smith and
Matz (1962), Arnberg et al. (1973), Brain and MacDonald (1975), and Brain and Reid
(1978,
1980). Figure
7.6
shows
the
scatter
in the
data. The general equation that
gives
a
fit
to this data is
(7.10)
and the value of the constants, including the preferred ISO values, are given in Table
7.1.
Figure 7.6 shows that the values fall well within an uncertainty band
of
±0.5%
at 95% confidence level around the
BS
EN ISO 9300 preferred equation.
The values
of b
and
n
are affected
by
the nature
of
the boundary layer
in the
throat. Transition is likely to cause
a
change
in
these parameters. Miller (1996) also
suggested that the temperature gradient between gas and nozzle may affect
b.
0.99
0
98
——
— •—
'
1
+ 0.5%
£-.
a fio f
a
•
•* * • * \^
+ Arnberg
et al
a Smith & Matz
O
NBS
v Brain
&
MacDonald
• Brain
&
Reid
-ARNBERG ET AL
.
ISO 9300
a
BRAINS REID
(1978,
1980)
(1973)"
•
•
-
10
5
10
6
Throat Reynolds number-Re
d
10
7
Figure 7.6. Discharge coefficient for toroidal throat venturi nozzles (after Brain and Reid 1978
with permission from NEL).
146
CRITICAL FLOW VENTURI NOZZLE
Table 7.1. Values of constants in equation (7.10) for toroidal throat venturi
nozzles
0
(equations are plotted in Figure 7.6)
Range
n Reference
4 x 10
4
-3 x 10
6
0.99738 3.058 0.5 Arnberg et al. (1973)
3 x 10
5
-10
7
0.99103 0.0 — Brain and Reid (1978, 1980)
10
5
-10
7
0.9935
1.5250
0.5 BS EN ISO 9300:1995
a
Other values are available for cylindrical throat venturi nozzles but are not given here.
Nakao et al. (1996) gave variations of the discharge coefficient with Reynolds
number and came up with discharge coefficients for a venturi with throat diameter
of 0.301 mm of
(7.11)
(7.12)
C = 1.006-3./
and for a venturi with throat diameter of 0.501 mm of
C = 1.007 -3.195Re-°
5
with uncertainties of about
±0.1%.
These lie outside the ±0.5% limits in Figure 7.6.
Park (1995) provided experimental evidence in support of the standard but made
the point that, if the inlet radius is small, the discharge coefficient may decrease
because of separation in the inlet causing a divergence from the standard.
7.5 CRITICAL FLOW FUNCTION C*
Various gases have values of
C*
tabulated in the standard (cf. Miller 1996). Values
are obtainable for nitrogen, oxygen, argon, methane, carbon dioxide, air for —50
to 100°C, and steam for 100-600°C, and for 0-100 bar. A sample of these is given
in Table 7.2 for air. These should be compared with the value for a perfect gas with
y = 1.4 of
C*i
= 0.685.
For gas mixtures, a method (cf. Johnson 1970, 1971 and Miller 1996) is given in
the standard. This first uses an expression for the critical flow function for a real gas
(7.13)
Table 7.2. Sample values of
C*
for air (Miller 1996 I
and BS EN ISO 9300: 1995) I
Temp.
(°C)
0
50
100
0
0.6850
0.6849
0.6847
Pressure (bar)
10
0.6887
0.6870
0.6858
20
0.6925
0.6889
0.6869
50
0.7038
0.6949
0.6900
100
0.7224
0.7039
0.6946