ISO/TR 3313:2018
(Main)Measurement of fluid flow in closed conduits — Guidelines on the effects of flow pulsations on flow-measurement instruments
Measurement of fluid flow in closed conduits — Guidelines on the effects of flow pulsations on flow-measurement instruments
ISO/TR 3313:2018 defines pulsating flow, compares it with steady flow, indicates how it can be detected, and describes the effects it has on orifice plates, nozzles or Venturi tubes, turbine and vortex flowmeters when these devices are being used to measure fluid flow in a pipe. These particular flowmeter types feature in this document because they are amongst those types most susceptible to pulsation effects. Methods for correcting the flowmeter output signal for errors produced by these effects are described for those flowmeter types for which this is possible. When correction is not possible, measures to avoid or reduce the problem are indicated. Such measures include the installation of pulsation damping devices and/or choice of a flowmeter type which is less susceptible to pulsation effects. ISO/TR 3313:2018 applies to flow in which the pulsations are generated at a single source which is situated either upstream or downstream of the primary element of the flowmeter. Its applicability is restricted to conditions where the flow direction does not reverse in the measuring section but there is no restriction on the waveform of the flow pulsation. The recommendations within this document apply to both liquid and gas flows although with the latter the validity might be restricted to gas flows in which the density changes in the measuring section are small as indicated for the particular type of flowmeter under discussion.
Mesurage du débit des fluides dans les conduites fermées — Lignes directrices relatives aux effets des pulsations d'écoulement sur les instruments de mesure de débit
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Standards Content (Sample)
TECHNICAL ISO/TR
REPORT 3313
Fourth edition
2018-03
Measurement of fluid flow in closed
conduits — Guidelines on the effects of
flow pulsations on flow-measurement
instruments
Mesurage du débit des fluides dans les conduites fermées — Lignes
directrices relatives aux effets des pulsations d'écoulement sur les
instruments de mesure de débit
Reference number
©
ISO 2018
© ISO 2018
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ii © ISO 2018 – All rights reserved
Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and subscripts . 2
5 Description and detection of pulsating flow. 4
5.1 Nature of pipe flows . 4
5.2 Threshold between steady and pulsating flow . 4
5.2.1 General. 4
5.2.2 Differential pressure (DP) type flowmeters . 5
5.2.3 Turbine flowmeters . 5
5.2.4 Vortex flowmeters . 6
5.3 Causes of pulsation . 6
5.4 Occurrence of pulsating flow conditions in industrial and laboratory
flowmeter installations . 6
5.5 Detection of pulsation and determination of frequency, amplitude and waveform . 7
5.5.1 General. 7
5.5.2 Characteristics of the ideal pulsation sensor . 7
5.5.3 Non-intrusive techniques . 7
5.5.4 Insertion devices . 8
5.5.5 Signal analysis on existing flowmeter outputs: software tools. 8
6 Measurement of the mean flowrate of a pulsating flow.10
6.1 Orifice plate, nozzle, and Venturi tube .10
6.1.1 Description of pulsation effects and parameters .10
6.1.2 Flowmeters using slow-response DP sensors .12
6.1.3 Flowmeters using fast-response DP sensors .14
6.1.4 Pulsation damping.15
6.2 Turbine flowmeters .20
6.2.1 Description of pulsation effects and parameters .20
6.2.2 Estimation of pulsation correction factors and measurement uncertainties .23
6.3 Vortex flowmeters .24
6.3.1 Pulsation effects .24
6.3.2 Minimizing pulsation effects .25
6.3.3 Estimation of measurement uncertainties .25
Annex A (informative) Orifice plates, nozzles and Venturi tubes — Theoretical considerations .27
Annex B (informative) Orifice plates, nozzles and Venturi tubes — Pulsation damping criteria .34
Annex C (informative) Turbine flowmeters — Theoretical background and experimental data .40
Bibliography .44
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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electrotechnical standardization.
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described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
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URL: www .iso .org/ iso/ foreword .html
This document was prepared by Technical Committee ISO/TC 30, Measurement of fluid flow in closed
conduits, Subcommittee SC 2, Pressure differential devices.
This fourth edition of ISO/TR 3313:2018 is a technical revision of ISO/TR 3313:1998, which was
withdrawn in 2013.
iv © ISO 2018 – All rights reserved
TECHNICAL REPORT ISO/TR 3313:2018(E)
Measurement of fluid flow in closed conduits — Guidelines
on the effects of flow pulsations on flow-measurement
instruments
1 Scope
This document defines pulsating flow, compares it with steady flow, indicates how it can be detected,
and describes the effects it has on orifice plates, nozzles or Venturi tubes, turbine and vortex flowmeters
when these devices are being used to measure fluid flow in a pipe. These particular flowmeter types
feature in this document because they are amongst those types most susceptible to pulsation effects.
Methods for correcting the flowmeter output signal for errors produced by these effects are described
for those flowmeter types for which this is possible. When correction is not possible, measures to avoid
or reduce the problem are indicated. Such measures include the installation of pulsation damping
devices and/or choice of a flowmeter type which is less susceptible to pulsation effects.
This document applies to flow in which the pulsations are generated at a single source which is situated
either upstream or downstream of the primary element of the flowmeter. Its applicability is restricted
to conditions where the flow direction does not reverse in the measuring section but there is no
restriction on the waveform of the flow pulsation. The recommendations within this document apply
to both liquid and gas flows although with the latter the validity might be restricted to gas flows in
which the density changes in the measuring section are small as indicated for the particular type of
flowmeter under discussion.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
3.1
steady flow
flow in which parameters such as velocity, pressure, density and temperature do not vary significantly
enough with time to prevent measurement to within the required uncertainty of measurement
3.2
pulsating flow
flow in which the flowrate in a measuring section is a function of time but has a constant mean value
when averaged over a sufficiently long period of time, which depends on the regularity of the pulsation
Note 1 to entry: Pulsating flow can be divided into two categories:
— periodic pulsating flow;
— randomly fluctuating flow.
Note 2 to entry: For further amplification of what constitutes steady or pulsating flow see 5.1 and 5.2.
Note 3 to entry: Unless otherwise stated in this document the term “pulsating flow” is always used to describe
periodic pulsating flow.
4 Symbols and subscripts
4.1 Symbols
A area
A area of the throat of a Venturi nozzle
d
A turbine blade aspect ratio
R
th
a , b , c amplitude of the r harmonic component in the undamped or damped pulsation
r r r
B bf /q , dimensionless dynamic response parameter
p V
b turbine flowmeter dynamic response parameter
C turbine blade chord length
C contraction coefficient
c
C discharge coefficient
D
C velocity coefficient
v
c speed of sound
D internal diameter of the tube
d throat bore of orifice, nozzle or Venturi tube
E
R residual error in time-mean flowrate when calculated using the quantity Δp
E total error in the time-mean flowrate
T
f turbine flowmeter output signal, proportional to volumetric flowrate
f pulsation frequency
p
f resonant frequency
r
f vortex-shedding frequency
v
H harmonic distortion factor
Ho Hodgson number
I moment of inertia
I , I moments of inertia of turbine rotor and fluid contained in rotor envelope respectively
R F
k/D relative roughness of pipe wall
L turbine blade length
L effective axial length
e
l impulse line length for differential pressure (DP) measurement device
2 © ISO 2018 – All rights reserved
m = β orifice or nozzle throat to pipe area ratio
N number of blades on turbine rotor
p pressure (absolute)
q mass flowrate
m
q volume flowrate
V
R turbine blade mean radius
Re Reynolds number
r , r turbine blade hub and tip radii respectively
h t
Sr Strouhal number
Sr Strouhal number based on orifice diameter
d
t time
t turbine blade thickness
b
U axial bulk-mean velocity
U bulk-mean velocity based on orifice diameter
d
V volume
X temporal inertia term for short pulsation wavelengths
α
′
UU/
RMS
β orifice or nozzle throat to pipe diameter ratio
γ ratio of specific heat capacities (c /c )
p V
Δp differential pressure
Δϖ pressure loss
ε expansibility factor for steady flow conditions
ss
η blade “airfoil efficiency”
θ phase angle
κ isentropic exponent (= γ for a perfect gas)
μ damping response factor (see 6.1.4.1.3)
ρ fluid density
ρ turbine blade material density
b
τ = p /p pressure ratio
2 1
maximum allowable uncertainty in the indicated flowrate due to pulsation at the
φ
flowmeter
ψ maximum allowable relative error
ω = 2πf angular pulsation frequency
p
4.2 Subscripts and superscripts
o pulsation source
p measured under pulsating flow conditions, possibly damped
po measured under pulsating flow conditions before damping
RMS root mean square
ss measured under steady flow conditions
(over-bar) the time-mean value
1,2 measuring sections
'
fluctuating component about mean value, e.g. U′
5 Description and detection of pulsating flow
5.1 Nature of pipe flows
Truly steady pipe flow is only found in laminar flow conditions which can normally only exist when
the pipe Reynolds number, Re, is below about 2 000. Most industrial pipe flows have higher Reynolds
numbers and are turbulent which means that they are only statistically steady. Such flows contain
continual irregular and random fluctuations in quantities such as velocity, pressure and temperature.
Nevertheless, if the conditions are similar to those which are typical of fully developed turbulent pipe
flow and there is no periodic pulsation, the provisions of such standards as ISO 5167 (all parts) apply.
The magnitude of the turbulent fluctuations increases with pipe roughness, and this is one of the
reasons why ISO 5167 (all parts) stipulates a maximum allowable relative roughness, k/D, of the
upstream pipe for each type of primary device covered by ISO 5167 (all parts).
ISO 5167 (all parts), however, cannot be applied to flows which contain any periodic flow variation or
pulsation.
5.2 Threshold between steady and pulsating flow
5.2.1 General
If the amplitude of the periodic flowrate variations is sufficiently small there should not be any error
in the indicated flowrate greater than the normal measurement uncertainty. It is possible to define
amplitude thresholds for both differential pressure (DP) type flowmeters and turbine flowmeters
without reference to pulsation frequency. It is also possible to do this for vortex flowmeters but extreme
caution is necessary if even the smallest amplitude is known to be present in the flow.
For DP-type flowmeters, the threshold is relevant when slow-response DP cells are being used. In the
case of turbine flowmeters, the threshold value is relevant when there is any doubt about the ability
of the rotor to respond to the periodic velocity fluctuations. In the case of a vortex flowmeter the
4 © ISO 2018 – All rights reserved
pulsation frequency relative to the vortex-shedding frequency is a much more important parameter
than the velocity pulsation amplitude.
5.2.2 Differential pressure (DP) type flowmeters
The threshold can be defined in terms of the velocity pulsation amplitude such that the flow can be
treated as steady if
′
U
RMS
≤00, 5 (1)
U
where U is the instantaneous bulk-mean axial velocity such that
′
UU= +U (2)
where
is the periodic velocity fluctuation;
U′
is the time-mean value.
U
The threshold in terms of the equivalent DP pulsation amplitude is
Δp′
pR, MS
≤01, 0 (3)
Δp
p
where Δp is the instantaneous differential pressure across the tappings of the primary device such that
p
′
ΔΔpp=Δ + p (4)
p p p
where
is the time-mean value;
Δp
p
is the periodic differential pressure fluctuation.
′
Δp
p
To determine the velocity pulsation amplitude it is necessary to use one of the techniques described
in 5.5 such as laser Doppler or thermal anemometry. To determine the DP pulsation amplitude it is
necessary to use a fast-response DP sensor and to observe the rules governing the design of the
complete secondary instrumentation system as described in 6.1.3.
Theoretical considerations are covered in Annex A.
5.2.3 Turbine flowmeters
At a given velocity pulsation amplitude a turbine flowmeter tends to read high as the frequency of
pulsation increases and exceeds the frequency at which the turbine rotor can respond faithfully to the
velocity fluctuations. The positive systematic error reaches a plateau value depending on the amplitude
and thus the threshold amplitude can be defined such that the resulting maximum systematic error is
still within the general measurement uncertainty. For example, if the overall measurement uncertainty
is greater than or equal to 0,5 % then it can be assumed that a systematic error due to pulsation of
0,1 % or less has negligible effect on the overall measurement uncertainty.
′
The velocity amplitude of sinusoidal pulsation, UU/ , that produces a systematic error of 0,1 % in a
RMS
turbine flowmeter is 3,5 %. Thus the threshold for sinusoidal pulsation is given by
U′
RMS
≤0,035 (5)
U
Techniques such as laser Doppler and thermal anemometry can be used to determine the velocity
pulsation amplitude. If the flowmeter output is a pulse train at the blade passing frequency and if the
rotor inertia is known, then signal analysis can be used to determine the flow pulsation amplitude as
described in 6.2.
5.2.4 Vortex flowmeters
A vortex flowmeter is subject to very large pulsation errors when the vortex-shedding process locks
in to the flow pulsation. There is a danger of this happening when the pulsation frequency is near the
vortex-shedding frequency. At a sufficiently low amplitude, locking-in does not occur and flow-metering
errors due to pulsation are negligible. This threshold amplitude, however, is only about 3 % of the mean
velocity and is comparable to the velocity turbulence amplitude. The consequences of not detecting the
pulsation or erroneously assuming the amplitude is below the threshold can be very serious. This issue
is discussed further in 6.3.
5.3 Causes of pulsation
Pulsation occurs commonly in industrial pipe flows. It might be generated by rotary or reciprocating
positive displacement engines, compressors, blowers and pumps. Rotodynamic machines might also
induce small pulsation at blade passing frequencies. Pulsation can also be produced by positive-
displacement flowmeters. Vibration, particularly at resonance, of pipe runs and flow control equipment
is also a potential source of flow pulsation, as are periodic actions of flow controllers, e.g. valve
“hunting” and governor oscillations. Pulsation might also be generated by flow separation within pipe
fittings, valves, or rotary machines (e.g. compressor surge).
Flow pulsation can also be due to hydrodynamic oscillations generated by geometrical features of
the flow system and multiphase flows (e.g. slugging). Vortex shedding from bluff bodies such as
thermometer wells, or trash grids, or vortex-shedding flowmeters fall into this category. Self-excited
flow oscillations at tee-branch connections are another example.
5.4 Occurrence of pulsating flow conditions in industrial and laboratory flowmeter
installations
In industrial flows, there is often no obvious indication of the presence of pulsation, and the associated
errors, because of the slow-response times and heavy damping of the pressure and flow instrumentation
commonly used. Whenever factors such as those indicated in 5.3 are present, there is the possibility of
flow pulsation occurring. It should also be appreciated that pulsation can travel upstream as well as
downstream and thus possible pulsation sources could be on either side of the flowmeter installation.
However, amplitudes might be small and, depending on the distance from pulsation source to
flowmeter, might be attenuated by compressibility effects (in both liquids and gases) to undetectable
levels at the flowmeter location. Pulsation frequencies range from fractions of a hertz to a few hundred
hertz; pulsation amplitudes relative to mean flow vary from a few percent to 100 % or larger. At low
percentage amplitudes the question arises of discrimination between pulsation and turbulence.
Flow pulsation can be expected to occur in various situations in petrochemical and process industries,
natural gas distribution flows at end-user locations and internal combustion engine flow systems. Flow-
metering calibration systems might also experience pulsation arising from, for example, rotodynamic
pump blade passing effects and the effects of rotary positive-displacement flowmeters.
6 © ISO 2018 – All rights reserved
5.5 Detection of pulsation and determination of frequency, amplitude and waveform
5.5.1 General
If the presence of pulsation is suspected then there are various techniques available to determine the
flow pulsation characteristics.
5.5.2 Characteristics of the ideal pulsation sensor
The ideal sensor would be non-intrusive, would measure mass flowrate, or bulk flow velocity, and
would have a bandwidth from decihertz to several kilohertz. The sensor would respond to both liquids
and gases and not require any supplementary flow seeding. The technique would not require optical
transparency or constant fluid temperature. The sensor would be uninfluenced by pipe wall material,
transparency or thickness. The device would have no moving parts, its response would be linear, its
calibration reliable and unaffected by changes in ambient temperature.
5.5.3 Non-intrusive techniques
5.5.3.1 Optical: laser Doppler anemometry (LDA)
This technology is readily available, but expensive. Measurement of point velocity on the tube axis allows
an estimate only of bulk flow pulsation amplitude and waveform but, for constant frequency pulsation,
accurate frequency measurements can be made. Optical access to an optically transparent fluid is
either by provision of a transparent tube section, or insertion of a probe with fibre-optic coupling. With
the exception of detecting low frequency pulsation, supplementary seeding of the flow would probably
be required to produce an adequate bandwidth. LDA characteristics are comprehensively described in
Reference [2].
5.5.3.2 Acoustic: Doppler shift; transit time
Non-intrusive acoustic techniques are suitable for liquid flows only, because for gas flows there is
poor acoustic-impedance match between the pipe wall and flowing gases. For the externally mounted
transmitter and receiver, usually close-coupled to the tube wall, an acoustically transparent signal path
is essential. The Doppler shift technique might require flow seeding to provide adequate scattering.
Instruments for point velocity measurements are available which, as for the LDA, provide only an
estimate of bulk flow pulsation amplitude and waveform. Moreover, Doppler-derived “instantaneous”
[3]
full-velocity profile instruments allow much closer estimates of bulk flow pulsation characteristics.
Transit-time instruments measure an average velocity, most commonly along a diagonal path across
the flow. All acoustic techniques are limited in bandwidth by the requirement that reflections from one
pulse of ultrasound should decay before transmission of the next pulse. Many commercial instruments
do not provide the signal processing required to resolve unsteady flow components. An investigation
[4]
by Hakansson on a transit time, intrusive-type ultrasonic flowmeter for gases subjected to pulsating
flows showed that only small shifts in the calibration took place and that these were attributable to the
changing velocity profile.
5.5.3.3 Electromagnetic flowmeters
When the existing flowmeter installation is an electromagnetic device, then, if it is of the pulsed d.c. field
type (likely maximum d.c. pulse frequency a few hundred hertz), there is the capability to resolve flow
pulsation up to frequencies approximately five times below the excitation frequency. This technique
is only suitable for liquids with an adequate electrical conductivity. It provides a measure of bulk flow
[5]
pulsation, although there is some dependence upon velocity profile shape .
5.5.4 Insertion devices
5.5.4.1 Thermal anemometry
The probes used measure point velocity, and relatively rugged (e.g. fibre-film) sensors are available for
industrial flows. These probes generally have an adequate bandwidth, but the amplitude response is
inherently non-linear. As with other point velocity techniques, pulsation amplitude and waveform can
only be estimated. Estimates of pulsation velocity amplitude relative to mean velocity may be made
without calibration. The RMS value of the fluctuating velocity component can be determined by using
a true RMS flowmeter to measure the fluctuating component of the linearized anemometer output
voltage. Mean-sensing RMS flowmeters should not be used as these only read correctly for sinusoidal
waveforms. Accurate frequency measurements from spectral analysis can be made for constant
frequency pulsation.
Applications are limited to clean, relatively cool, non-flammable and non-hostile fluids. Cleanness of flow
is very important; even nominally clean flows can result in rapid fouling of probes with a consequent
dramatic loss of response. A constant temperature flow is desirable although a slowly varying fluid
temperature can be accommodated.
5.5.4.2 Other techniques
Insertion versions of both acoustic and electromagnetic flowmeters are available. Transit-time acoustic
measurements can be made in gas flows when the transmitter and receiver are directly coupled to the
[6]
flow , although this might require a permanent insertion. Again there is the limitation of a lack of
commercially available instrumentation with the necessary signal processing to resolve time-varying
velocity components.
Insertion electromagnetic flowmeters are not widely available and are subject to the same bandwidth
limitations as the tube version, due to the maximum sampling frequency of the signal.
5.5.5 Signal analysis on existing flowmeter outputs: software tools
5.5.5.1 Orifice plate with fast-response DP sensor
A fast-response secondary measurement system is capable of correctly following the time-varying
pressure difference produced by the primary instrument provided the rules given in 6.1.3.2 can be
followed. In principle, a numerical solution of the pressure difference/flow relationship derived from
the quasi-steady temporal inertia model, Formula (A.11), would then provide an approximation to
the instantaneous flow. The square-root error would not be present, although other measurement
uncertainties (e.g. C variations, compressibility effects) produced by the pulsation would be. Successive
D
numerical solutions would then provide an approximation to the flow as a function of time and, hence,
amplitude and waveform information. Frequency information can be determined directly from the
measured pressure difference. At the time of publication of this document, there is no software tool
described for this implementation.
′
However, the maximum probable value of q can be approximately inferred from a measurement
Vo,RMS
′
of Δp using one of the following two inequalities:
po,RMS
′
′ Δp
q
po,RMS
Vo,RMS
≤ (6)
2 Δp
q ss
V
8 © ISO 2018 – All rights reserved
12/
′
q
2
Vo,RMS
≤ −1 (7)
12/
′
qp11+−[(Δ /)Δp ]
V po,RMS po
where
is the r.m.s. value of the fluctuating component of the differential pressure across the
Δp′
po,RMS
primary element measured using a fast-response secondary measurement system;
is the differential pressure that would be measured across the primary element under
Δp
ss
steady flow conditions with the same time-mean flowrate;
is the time-mean differential pressure that would be measured across the primary
Δp
po
element under undamped pulsating flow conditions;
is the instantaneous differential pressure across the primary element under undamped
Δp
po
pulsating flow conditions where
Δ=ppΔ+Δp′ (8)
po po
po
′
NOTE 1 Reliable measurements of Δp and Δp can only be obtained if the recommendations given in
po po,RMS
6.1.2 and 6.1.3 are strictly adhered to.
NOTE 2 If it is possible to determine Δp Formula (6) is to be preferred. Formula (7) only gives reliable
ss
′
results if ΔΔpp/,<05 .
()
po,RMS po
5.5.5.2 Turbine flowmeter
The raw signal from a turbine flowmeter is in the form of an approximately sinusoidal voltage with a
level which varies with the flow but is usually in the range 10 mV to 1 V peak to peak. In most installations
this signal is amplified and converted to a stream of pulses. The extraction of information about the
amplitude and waveform of any flow pulsation from the variations in the frequency of this pulse train
depends on the value of the dynamic response parameter of the flowmeter. Flowmeter manufacturers
do not normally specify the response parameter for their flowmeters, and the measurements which
would be necessary to determine it are unlikely to be possible on an existing flowmeter installation.
However, the dependency of the parameter on the geometry of the turbine rotor and on the fluid
density is discussed in 6.2.1.4, and the range of values which have been found for typical flowmeters is
presented in 6.2.1.6, Table 1.
The response of a turbine flowmeter to flow pulsation is discussed in 6.2.1. It can range from the ability
to follow the pulsation almost perfectly (medium to large flowmeters in liquid flows) to an almost
total inability to follow the pulsation (small to medium flowmeters in gas flows with moderate to high
frequencies of pulsation). This latter condition is a worst case for a turbine flowmeter installation
because not only does the flowmeter output not show significant pulsation but if the flow pulsation
is of significant magnitude, the apparently steady flowmeter output is not a correct representation of
the mean flow. If this condition is suspected, other means of measuring the flow pulsation should be
employed.
In any particular installation, the first step in an attempt to interpret a turbine flowmeter output
should be to take the best available estimate of the flowmeter response parameter and using the results
summarized in 6.2.1, to estimate the general nature of the flowmeter response. In the interpretation
of any observable fluctuations in the turbine flowmeter output, unevenness in the spacing of the
turbine blades can give the appearance of flow pulsation at the rotor frequency. Unevenness in blade
spacing might be a result either of damage caused by the passing of a solid through the flowmeter or of
manufacturing tolerances. Unevenness of as much as 3 % or 4 % in the blade spacing has been observed
in a number of installations. A procedure for processing a turbine flowmeter output signal to remove
the effect of uneven blade spacing is given in Annex C.
If preliminary estimates of the frequency of any pulsation in the flowmeter output and the general
nature of the flowmeter response combine to suggest that the amplitude of pulsation in the flowmeter
output is being attenuated by limited flowmeter response, it might be possible to correct the output.
[7] [8]
Two possible methods of correction have been described by Cheesewright et al. and by Atkinson ;
both are summarized in Annex C. Within the constraints of the uncertainty about the value of the
flowmeter response parameter, this procedure can yield estimates of the amplitude and waveform of
the flow pulsation.
5.5.5.3 Vortex flowmeter
The vortex flowmeter output can be used for instantaneous flow measurements, and hence amplitude
and waveform information, in a range restricted to pulsation frequencies less than 2,5 % of the lowest
mean-flow vortex-shedding frequency. Limited information can be obtained at higher pulsation
frequencies but, in order to avoid the substantial flowmeter errors which can arise from the shedding
frequency becoming locked-in to the pulsation frequency (see 6.3.1.3), the pulsation frequency should
be less than 25 % of the mean-flow shedding frequency. The detection of pulsation frequencies
substantially above the mean-flow shedding frequency can be achieved by spectral analysis. The
pulsation frequency is indicated by a local peak in the power spectrum.
6 Measurement of the mean flowrate of a pulsating flow
6.1 Orifice plate, nozzle, and Venturi tube
6.1.1 Description of pulsation effects and parameters
6.1.1.1 Square-root error
For steady flow, the flowrate through a restriction such as an orifice plate is proportional to the
square-root of the differential pressure measured between upstream and downstream tappings. The
relationship is given by
2ρΔp
d
ss
qC= π ε (9)
m Dss
1−β
If this relationship was assumed to apply instantaneously during pulsating flow, i.e. assuming quasi-
steady conditions, the time-mean flowrate would be inferred from a measurement of the time-mean
1/2
value of Δp .
Any attempt to infer the time-mean flowrate from the square-root of the time-mean value of Δp would
result in a square-root error, because
12/
12/
ΔΔpp≠ (10)
( )
In fact, the quasi-steady assumption is only valid for very low pulsation frequencies in incompressible
flow. For a more complete understanding of pulsating flow behaviour of DP flowmeters it is necessary
to consider temporal inertia effects, compressibility effects, and factors affecting the discharge
coefficient. A brief account of these is given in 6.1.1.2 and 6.1.1.3, and further details can be found in A.4
[9]
and in Gajan et al. .
10 © ISO 2018 – All rights reserved
6.1.1.2 Temporal inertia
When the flowrate is varying rapidly there is a component of differential pressure required to generate
the temporal acceleration in addition to that required for the convective acceleration of the fluid
through the restriction. The flowrate-differential pressure relationship is thus
dq
m 2
ΔpK=+Kq (11)
p 12 m
dt
On the right-hand side of Formula (11), the first term is the temporal inertia term and the second term
is the convective inertia term. The temporal inertia term is a function of the non-dimensional frequency
known as the Strouhal number, Sr , with respect to the throat bore, d, of the orifice, nozzle or Venturi
d
tube, where
fd
p
Sr = (12)
d
U
d
In the basic quasi-steady/temporal inertia theory the coefficients K and K are assumed to be constant
1 2
and are defined as
4L
e
K = (13)
πdC
c
1−C β 1
c
K = (14)
22 22
2ρ
CC (/πd 4)
v c
or alternatively
1−β 1
K = (15)
22 2
2ρ
Cd(/π 4)
D
where
C is the overall discharge coefficient;
D
C is the contraction coefficient;
c
C is a velocity coefficient.
v
The temporal inertia term is also a function of the geometry of the restriction and the axial distance
between the pressure tappings, and thus the coefficient K contains L , an effective axial length of the
1 e
primary device.
In pulsating flow the velocity profiles upstream and through the restriction are varying cyclically and,
thus, K and K are varying cyclically and even their time-mean values are not necessarily equal to the
1 2
steady flow values, except when pulsation amplitudes and frequencies are small.
6.1.1.3 Discharge coefficients
In steady flow, the discharge coefficients of all the different types of primary device are dependent
on the velocity profile of the approaching flow. The orifice plate tends to be particularly sensitive to
variations in velocity profile because of the jet contraction effect. A flatter than normal velocity profile
increases the contraction effect and consequently reduces the discharge coefficient. A velocity profile
which is more peaked than normal has the opposite effect.
In pulsating flow, the instantaneous velocity profile is varying throughout the pulsation cycle. The
degree of variation is dependent on the velocity pulsation amplitude, the waveform and the pulsation
Strouhal number. As a consequence, the instantaneous discharge coefficient also depends on the phase
angle in the pulsation cycle, the pulsation amplitude, the waveform and the Strouhal number. At the
time of publication of this Document it is not possible to relate mathematically the instantaneous
discharge coefficient to the pulsation parameters.
The practical approach to the calculation of a flowrate in pulsating flow conditions is to use a constant
value of the discharge coefficient, preferably the value used in steady flow conditions. This approach
gives accurate results for low amplitude and low frequency pulsation in incompressible flow and
limiting values of the relevant parameters defined in 6.1.2.3. Residual errors due to temporal inertia
effects and variations in discharge coefficient increase with pulsation amplitude and frequency as
[9]
shown by Gajan et al. .
6.1.2 Flowmeters using slow-response DP sensors
6.1.2.1 Limiting conditions of applicability
For a slow-response DP sensor (upper frequency limit of about 1 Hz) then, at best, the time-mean
differential pressure Δp is indicated. The corresponding indicated mean flowrates derived from
p
12/
Δp include both the square-root and temporal-inertial effect errors. Conditions limiting
( p)
applicability are those which prevent the secondary measurement system producing a correct time-
mean pressure signal. These include distortion of the pressure waveform or phase relationship in either
of the two lines connecting tappings to the sensor. These effects arise from boundary friction, finite gas
volumes and non-linear damping. In addition, the connecting line length should be restricted to prevent
resonance due to this length being equal to the pulsation quarter-wavelength. This resonance occurs at a
frequency, f , given by f = c/(4l). In practice minimum connecting line length is restricted by physical
r r
size of primary and secondary instrumentation and associated valve assemblies. At the time of
publication of this Document, it is not possible to define a threshold level of negligible pulsation applicable
to all designs of secondary device. However, it is possible to recommend a number of design rules.
6.1.2.2 Design of flowmeter secondary instrumentation
For devices used to indicate the time-mean differential pressures in pulsating flow conditions, the
design rules are as follows.
a) The bore of the pressure tapping should be uniform and not too small, i.e. ≥3 mm. Piezometer rings
should not be used.
b) Distance between pressure tappings should be small compared with the pulsation wavelength.
c) The tube connecting the pressure tappings to the manometer should be as short as possible and of the
same bore as the tappings. A tube length near the pulsation quarter-wavelength should not be used.
d) For gas-filled secondary systems, sensor cavities or other discrete volumes should be as small as
possible.
e) For liquid-filled secondary systems, gas bubbles should not be trapped in the connecting tube or
sensing device; thus vent points are required.
f) Damping resistances in the connecting tubes and sensing element should be linear. Throttle cocks
should not be used.
g) The device time constant should be about 10 times the period of the pulsation cycle.
h) When the above rules cannot be observed, the secondary device might be effectively isolated from
pulsation by the insertion of identical linear-resistance damping plugs into both connecting tubes,
as close as possible to the pressure tappings.
Observance of the rules listed in items a) to h) for a slow-response device cannot eliminate the square-
root error but merely reduces the error in the measurement of the time-mean differential pressure.
12 © ISO 2018 – All rights reserved
6.1.2.3 Estimation of correction factors and measurement uncertainties due to pulsation
The formulae given in 6.1.4 allow adequate damping to be calculated for a given maximum allowable
relative error, ψ, in the indicated flowrate due to the residual damped pulsation. It is also possible to
estimate the total error, E , directly, after measuring the pulsation amplitude at the orifice. The total
T
error, E , should be less than ψ.
T
In theory, E is always a positive systematic error, but in practice there is an additional random
T
uncertainty mostly due to pulsation effects in the secondary device. Calculations of the errors and
additional uncertainty are feasible provided that the pulsation amplitudes are not too large.
Limiting pulsation amplitudes for error calculations are
′
q ′
U
V,RMS
RMS
= ≤03, 2 (16)
q U
V
or
′
Δp
p,RMS
≤06, 4 (17)
Δp
ss
or
′
Δp
p,RMS
≤05, 8 (18)
Δp
p
The following equations can be used to estimate the total error, E , for the low amplitude pulsation:
T
12/
U′
RMS
E =+11− (19)
T
U
or
′
Δp
p,RMS
E =+1 −1 (20)
T
4 Δp
ss
or
−12/
12/
Δp′
1 p,RMS
E =+11− −1 (21)
T
2
Δp
p
The practicality of using Formula (19) is limited by the requirement of an independent measurement
of pulsation bulk velocity. Similarly, use of Formula (20) is restricted in practice because of the
requirement to know the steady flow differential pressure. Neither Formula (19) nor Formula (20)
contains terms for temporal inertia effects and they thus tend to give slightly overestimated values of
E when the Strouhal number
...
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