ISO/TR 3313:1992

SLOVENSKI STANDARD
SIST ISO/TR 3313:1996
01-november-1996
Measurement of pulsating fluid flow in a pipe by means of orifice plates, nozzles or
Venturi tubes
Measurement of pulsating fluid flow in a pipe by means of orifice plates, nozzles or
Venturi tubes
Mesurage du débit d'un écoulement pulsatoire de fluide dans une conduite au moyen de
diaphragmes, tuyères ou tubes de Venturi
Ta slovenski standard je istoveten z: ISO/TR 3313:1992
ICS:
17.120.10 Pretok v zaprtih vodih Flow in closed conduits
SIST ISO/TR 3313:1996 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST ISO/TR 3313:1996

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SIST ISO/TR 3313:1996
IS0
TECHNICAL
TR 3313
REPORT
Second edition
1992-07-1 5
Measurement of pulsating fluid flow in a pipe by
means of orifice plates, nozzles or Venturi tubes
Mesurage du débit dun écoulement pulsatoire de fluide dans une
conduite au moyen de diaphragmes, tuyères ou tubes de Venturi
Reference number
ISOiTR 331 3: 1992( E)

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SIST ISO/TR 3313:1996
Foreword
IS0 (the International Organization for Standardization) is a worldwide
federation of national standards bodies (IS0 member bodies). The work
of preparing International Standards is normally carried out through IS0
technical committees. Each member body interested in a subject for
which a technical committee has been established has the right to be
represented on that committee. International organizations, govern-
mental and non-governmental, in liaison with ISO, also take part in the
work. IS0 collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
The main task of technical committees is to prepare International Stan-
dards, but in exceptional circumstances a technical committee may
propose the publication of a Technical Report of one of the following
types:
- type 1, when the required support cannot be obtained for the publi-
cation of an International Standard, despite repeated efforts:
- type 2, when the subject is still
under technical development or
where for any other reason there is the future but not immediate
possibility of an agreement on an International Standard;
- type 3, when a technical committee has collected data of a different
kind from that which is normally published as an International Stan-
dard (“state of the art”, for example).
Technical Reports of types 1 and 2 are subject to review within three
years of publication, to decide whether they can be transformed into
International Standards. Technical Reports of type 3 do not necessarily
have to be reviewed until the data they provide are considered to be no
longer valid or useful.
ISO/TR 3313, which is a Technical Report of type 2, was prepared by
Technical Committee ISO/TC 30, Measurement of fluid flow in closed
conduits, Sub-committee SC 2. Pressure differential devices.
This second edition cancels and replaces the first edition
(ISO/TR 3313:1974), of which it constitutes a technical revision.
Annexes A, B and C of this Technical Report are for information only.
O is0 1992
No part of this publication may be reproduced or utilized in any f0rm
Ali rights reserved.
or by any means, electronic or mechanical, including photocopying and microfilm, without
permission in writing from the publisher.
International Organization for Standardization
Case Postale 56 CH-I211 Genève 20 Switzerland
Printed in Switzerland
ii

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SIST ISO/TR 3313:1996
ISO/TR 3313:1992(E)
Introduction
Methods of measuring fluid flow in a pipe by means of orifice plates,
nozzles or Venturi tubes are described in IS0 5167-1. However, it is
stipulated that the rate of flow shall be constant or, in practice, vary only
slightly and slowly with time (see IS0 5167-1:1991, 6.3.1). IS0 5167-1
does not provide for the measurement of pulsating flow.
The reasons for the publication of this document in the form of a Tech-
nical Report are the following:
- the restricted field of application of the document;
- the lack of available data on the relationship between measuring in-
stallation parameters and errors in measurement;
- if no appropriate action is taken, pulsation can cause very large er-
rors.
iii

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SIST ISO/TR 3313:1996

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SIST ISO/TR 3313:1996
TECHNICAL REPORT ISOlTR 331 3:1992( E)
Measurement of pulsating fluid flow in a pipe by means of
orifice plates, nozzles or Venturi tubes
2 Normative reference
O ’ Scope
The following standard contains provisions which,
through reference in this text, constitute provisions
of this Technical Report. At the time of publication,
This Technical Report defines pulsating flow, com-
the edition indicated was valid. All standards are
pares it with steady flow, and indicates the require-
subject to revision, and parties to agreements based
ments that allow accurate measurement of the mean
on this Technical Report are encouraged to investi-
rate of flow by means of orifice plates, nozzles and
gate the possibility of applying the most recent edi-
Venturi tubes in pipes where pulsations are present.
tion of the standard indicated below. Members of
IEC and IS0 maintain registers of currently valid
This Technical Report applies to flow in which the
International Standards.
pulsations are generated at a single source which
is situated either upstream or downstream of the
IS0 5167-1:1991, Measurement of fluid flow by means
flowmeter’s primary element. Its applicability is re-
of pressure differential devices - Part I: Orifice
stricted to conditions where the flow direction does
plates, nozzles and Venturi tubes inserted in circular
not reverse in the measuring section. It includes
cross-section conduits running full.
recommendations which apply to liquid flows and
gas flows in which the density changes in the
measuring section are small (expansibility factor
E > 0,99 and density fluctuation ratio prms/P Q 1/40).
3 Definitions
Critical flow Venturi nozzles are treated as special
cases.
For the purposes of this Technical Report, the fol-
lowing definitions apply.
There is no restriction on the wave-form of the flow
pulsation, but the pulsation wavelength must be long
3.1 steady flow: Flow in which parameters such as
compared with the dimensions of the flowmeter.
velocity, pressure, density and temperature do not
vary significantly enough with time to affect the re-
This Technical Report gives practical criteria for en-
quired uncertainty of measurement.
suring that flow pulsations are damped to such an
extent that systematic errors in the indicated time-
mean flow rate do not exceed a specified value.
3.2 pulsating flow: Flow in which the flow rate in a
Recommendations concerning the appropriate de-
measuring section is a function of time but has a
sign of the differential pressure secondary device
constant mean value when averaged over a suffi-
are also given.
ciently long period of time.
Finally, this Technical Report describes how uncer- Pulsating flow can be divided into:
tainty in the measurement of mean flow rate under
- periodic pulsating flow, and
pulsating flow conditions can be established.
- randomly fluctuating flow.
Annex A of this Technical Report defines the the-
oretical and experimental basis for the determi-
Unless otherwise stated in this Technical Report, the
nation of the total measuring error.
term “pulsating flow” is always used to describe
periodic pulsating flow.
Annex B gives the criteria for adequate damping.
1

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SIST ISO/TR 3313:1996
ISOlTR 331 3:1992(E)
Maximum permissible fluctuation in den-
4
4 Symbols and indices
sity
Maximum allowable percentage error
+, J/,
4.1 Symbols
w = 2nf Angular frequency
A Area
4.2 Indices
Area of the throat of a Venturi nozzle
A*
O Index for the pulsating source
Amplitude of the r'h harmonic component
crg hr
1,2 Index for measuring sections
in the damped or undamped pulsations
rms Root mean square
B Time-mean value of quantity B (see B.3)
B'
FluctAation of quantity B such that
5 General
B=B+B'
h The amplitude of pulsations of density
5.1 The nature of pipe flows
Contraction coefficient
cc
In practice, the flow observed in pipes is mostly a
statistically steady flow. When the pipe Reynolds
c Speed of sound
Number is sufficiently high, the flow is always tur-
n Internal diameter of the tube
bulent and is, therefore, a fluctuating flow as there
are irregular and random variations in quantities
d Internal diameter of orifice or throat bore
such as flow rate, pressure, density and temper-
ature. If the conditions are similar to those which are
Total error in the indicated time mean
-ET
typical of fully developed turbulent pipe flow and
flow-rate
there is no periodic pulsation, the provisions of
IS0 5167-1 apply.
Pulsation frequency (Hz)
f
The presence of pulsations is also very common in
Ii Harmonic distorsion factor
industrial pipe flows. Pulsating flows are generated
Effective axial lengths
L, LE!,,, 1, by both rotary and reciprocating positive displace-
ment engines, compressors, blowers and pumps.
m=P2
Orifice arealpipe area ratio
Hydrodynamic oscillations such as vortex shedding
can also be a source of pulsation.
Pressure (absol Ute)
P
It is possible that the damping effect of the flow
Mass flow-rate
4,
system between the pulsation source and the flow
meter is so great that flow pulsation in the metering
Volume flow-rate
4v
sections cannot be detected. In such a situation, the
t Time
flow is regarded as steady and the methods of
IS0 5167-1 are applicable. When flow pulsations in
U Axial velocity
the metering section have an amplitude above the
threshold value, however, IS0 5167-1 does not apply
V Volume
and the procedure outlined in this Technical Report
should be followed.
x Temporal inertia term for short pulsation
wavelengths
As a guideline, the threshold between steady and
pulsating flow can be defined in terms of the velocity
Orifice diameter/pipe diameter ratio
P
pulsation amplitude such that if
Ratio of specific thermal capacities
Y
U',,,/Ü < 0,05
Differential pressure
AP
and if the recommendations given in 6.4 are cor-
Aw Pressure loss
rectly followed,
Ap',,,/dl)o Q 0,IO
B Phase angle
K Isentropic index
U is the axial velocity component, such that
U= U+ U'
Fluid density
P
Ratio of pressure where U' is the velocity fluctuation,
= = P2lPt
2

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SIST ISO/TR 3313:1996
ISOlTR 3313:1992(E)
damped pulsation can be measured, the discharge
Ap, is the differential pressure in the primary ele-
ment, such that coefficient may possibly be reduced to compensate
for the calculated square root error which is always
Ap, = Gp -+ Ap’
a positive systematic error. Methods for error cal-
culation are given in clause 8.
where Ap‘ is the differential pressure fluctuation.
-
The barred quantities, U and GP are time mean
6.2 Determination of pulsation amplitude
values.
When damping of the flow stream is required, it is
5.2 The detection of pulsations
necessary to determine the r.m.s. amplitude of the
undamped flow pulsation, q’vo,rms, at the pulsation
There is often no obvious indication of the presence
source, in order to calculate the required damping
of pulsations at the flowmeter. The secondary device
or throttling volume using the criteria described
used in industrial flowmeter installations is usually
in 6.3.
a slow response heavily damped instrument which
may not show any oscillation. The best method of There are a number of methods which can be used
detecting flow pulsations is to place a hot wire to obtain the amplitude. These are, in order of pref-
anemometer or similar device on the axis of the pipe e rence .
upstream of the measuring section. If this is not
0
possible, a fast response differential pressure
a) Direct measurement using a linearized hot wire
transducer may be connected across the primary anemometer or similar device and r.m.s.
element, provided that the recoinmendations given voltmeter. The r.m.s. value of the fluctuating
in 6.4 are strictly followed. component of the voltage output must be meas-
ured on a true r.m.s. meter. Mean sensing r.m.s.
meters must not be used as these will only read
6
Determination of the mean flow rate of
correctly for sinusoidal waveforms.
a pulsating flow
b) Use of a linearized hot wire anemometer or
similar device and computing q’vo,rms from a re-
6.1 Pulsation effects on the primary device
corded time trace.
The most important effect is due to the square root
c) Estimation of q’vo,rms from the action of the
relationship between flow and differential pressure.
pulsation generator (this method is possible
In pulsating flow, the time-mean flow rate should be
when a positive displacement compressor or
computed from the mean of the instantaneous val-
motor is situated near the meter location).
ues of the square root of differential pressure.
d) If the methods described in a), b) or c) cannot be
The practice of using slow, large displacement sec-
used, the maximum probable value of q’vo,rms can
ondary devices for pulsating flow, and averaging by
be approximately inferred from a measurement
damping before taking the square root, results in an
of using one of the following two in-
0 indicated flow rate greater than the true value. The
equalities:
amount greater depends on the ratio of the root-
mean-square amplitude of the flow rate fluctuation
q’V0,rms 1 AP’pû,rms
to the mean value (q’v,rms/4v).
< --
4V 2 APSS
Modern, fast, small displacement differential pres-
4’ VO, rm s
sure transmitters with square root circuits and sub-
<
sequent averaging exist. These can theoretically
4V
provice a more accurate output, but there is insuffi-
, 1/2
cient data to establish the magnitude of the im-
prove ment.
The recommended procedure for measuring the
mean flow rate of a pulsating flow involves the in- where
stallation of sufficient damping into the flow system
such that the pulsation amplitude of the flow rate in APSS is the differential pressure that
would be measured across the
the measuring section will be reduced to such a
level that the square root error is less than a given primary element under steady flow
conditions with the same time-
allowable value.
mean flow rate;
The flow rate is calculated in the same way as for
-
is the time-mean differential pres-
steady flow, using the discharge coefficients given
Ap,,
in IS0 5167-1. When the amplitude of the residual sure that would be measured
3

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SIST ISO/TR 3313:1996
ISOlTR 3313:1992(E)
6.3.2 Gas flow
across the primary element under
undamped pulsating flow condi-
tions;
A parameter which can be used as a criterion of
adequate damping in gas flow is the Hodgson num-
Ap’pO,rms is the r.m.s. value of the fluctuat-
ber, Ho, defined by
ing component of the differential
pressure across the primary ele- -
V Aa
ment measured using a fast re-
Ho = 7 x Y
sponse pressure transducer;
9vlf P
Appo is the instantaneous differential
where
pressure across the primary ele-
ment under undamped pulsating
V is the volume of the receiver and the
flow - conditions where
pipework between pulsation source and
APpo = APpo + AP’PO. flowmeter;
Note that reliable measurements of GPO and
qvlf is the time-mean volume flow per
Ap’pO,rms can only be obtained if the recommen- pulsation cycle;
dations given in 6.4 are strictly adhered to. _.
AW is the overall time-mean pressure loss
between the receiver and the source of
Note also that, if it is possible to determine Apss,
supply (or discharge) at constant pres-
equation (1) is to be preferred. EquaLon (2) will
sure:
only give reliable results if (AP’p0,rmçlAPpO) OS.
is the mean absolute static pressure in
p
the receiver.
6.3 Installation requirements
Damping will be adequate provided that the follow-
ing condition is fulfilled:
1 q’rn0,rms
Ho 0,056 3 -
K
6.3.1 General
&- x-4,
Pulsation in gases or vapours can be damped by a
where
combination of volumetric capacity and throttling
placed between the pulsation source and the
is the isentropic index of the gas or
IC
flowmeter (see figure 1). The volumetric capacity can
vapour (K =y is the ratio of the spe-
include the volume of any receivers and the pipeline
cific thermal capacities for an ideal
itself, provided the axial lengths involved are short
gas);
compared with the pulsation wavelength. The throt-
is the root-mean-square value of the
tling can be provided by frictional pressure losses
q’rno,rmç
in the pipeline can also contribute towards the fluctuating component of the flow rate
throttling effect. The straight pipe length between measured at the pulsation source:
the meter and any additional throttling device must
is the time-mean value of the flow
be in accordance with the installation requirements 4m
rate;
of IS0 5167-1. Care should be taken that the se-
lected throttling device does not create
is the maximum allowable percentage
$
hydrodynamic oscillations.
error in the indicated flow rate due to
pulsation at the flowmeter.
Figure1 shows the installation arrangement of a
pulsation source.
The pulsation amplitude ratio can be expressed in
terms of mass or volume flow rate or bulk mean
Pulsations in liquid flow can be similarly damped by
velocity.
sufficient capacity and throttling, the capacity being
provided by an air vessel. Alternatively, a surge
Thus
chamber can be used instead of the air vessel.
The equations in 6.3.2 and 6.3.3 are used for calcu-
lating whether damping is adequate to keep meter-
ing errors due to pulsation below a given level.
The derivation of the criteria for adequate damping
These formulae are insufficiently exact to be used to
implies that the dimensions of the damping chamber
predict actual values of pulsation errors in a given
and the lengths of pipe between the damping
system.
4

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SIST ISO/TR 3313:1996
ISOlTR 3313:1992(E)
chamber and the flowmeter are short compared with and pulsation source downstream. In this case, the
surge chamber or air vessel should be placed
the pulsation wavelength.
downstream of the primary element.
The following guidelines may be used.
6.3.3.1 Use of a surge chamber
a) The length of the damping tank, L,, should not
be greater than 1/10 of the pulsation wavelength.
The criterion for adequate damping when a surge
Thus the frequency limit is given by
chamber is used is that
C
f where c is the speed of sound.
where
b) The length of the pipe, L2, between damping tank
and flowmeter should not be greater than 1/5 of z is the time-mean value of the difference in
the pulsation wavelength. Thus the limiting fre- liquid level between the surge chamber and
quency must also fulfil the condition the constant head vessel;
C
A is the cross-sectional area of the surge
f chamber.
Another very important rule is that thg undamped
6.3.3.2 Use of an air vessel
velocity pulsation amplitude ( U’O,rms/UO) must be
determined very close to the inlet to the damping
The criterion for adequate damping when an air
tank.
vessel is used is that
6.3.3 Liquid flow
Pulsating liquid flow can be damped by placing ei-
ther a surge chamber or an air vessel between the
pulsating flow source and the primary element (see
figures 2 and 3).
In the diagrams, the flow is shown with the pulsation
where
source upstream. It is equally possible, however, to
is the volume of the air in the air vessel;
have a flow with a constant head source upstream
V,
Source of pulsntlons
Source of supply nt 4 Receiver
(for exnmple :
constnnt pressure
reciprocnting engine)
a) Downrtrenm of meter
‘qm
Qmo
___L I
I
Source of pulsation Receiver
ha
b) Upstream of meter
Figure 1 - Installation arrangement of a pulsation source
5

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SIST ISO/TR 3313:1996
ISO/TR 3313:1992(E)
is the pressure of the air in the air vessel; a) The bore of the pressure tapping must be uni-
po
form and not too small. Piezometer rings must
K
is the isentropic index for air:
not be used.
is the density of the liquid;
p
b) The tube connecting the pressure tapping to the
manometer must be as short as possible and of
is the acceleration due to gravity;
g
the same bore as the tappings. Lengths of head
near the pulsation quarter-wave length should
A
is the free surface area of the liquid in the
be avoided.
air vessel;
is the mean value of the difference in pres-
c) Volumes of gas must not be included in the con-
sure between the air vessel and the con-
necting tubes or sensing element.
stant head vessel.
d) Damping resistances in the connecting tubes and
sensing element must be linear. Throttle cocks
must not be used.
e) The natural frequency of the sensing element
6.4 Pulsation effects on the differential
must be much lower than the pulsation fre-
pressure secondary device
q u en cy.
Pulsations at the pressure tappings can cause seri-
9 When the above rules cannot be observed, the
ous errors in the indicated time-mean differential
secondary device may be effectively isolated
pressure. These errors are due to wave action in the
from pulsations by the insertion of identical
connecting leads and to non-linear damping both in
linear-resistance dumping plugs into both con-
the leads and in the differential pressure sensor it-
necting tubes, as close as possible to the pres-
self. The magnitude of the errors depends not only
sure tappings.
on the pulsation characteristics, but also on the ge-
ometry of the secondary device. At present, it is not
It should be understood that observance of the
possible to define a threshold level of negligible
rules listed in items a) to 9 for a slow response
pulsation applicable to all designs of secondary de-
device cannot eliminate square root error but
vices. However, it is possible to recommend a num-
merely reduces the error in the measurement of
ber of design rules.
the time-mean differential pressure.
For slow response secondary devices used to indi- For a fast response electronic differential-
cate the time-mean differential pressure in pulsating pressure transducer, rules a), b) and c) and the
fiow conditions, the rules are as follows. following apply.
Surge
chamber
‘h( I Constant
-
- - - ~ head
-
T
Pulsating -
-
flow
__L
4v i
Figure 2 - Surge chamber
n
1
Alr vessel
Constani
-
Pulsating
- ’ - head
-
-
flow
- -
qw) -
-Qv I
I”
Figure 3 - Air vessel

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SIST ISO/TR 3313:1996
ISOlTR 331 3:1992(E)
or
g) The mechanical and electronic frequency limits
of the transducer should be at least 10times
greater than the flow pulsation frequency.
h) The distance along the pressure passage from
or
tapping to sensing element must be small com-
pared with the pulsation quarter-wave length.
i) The device must be geometrically similar on both
upstream and downstream sides.
The following equations can be used to estimate the
actual error, ET, for the low amplitude pulsations.
7 Flow measurement
When the conditions specified in this Technical Re-
port concerning the determination of the mean rate
of flow and the installation have been satisfied, the
or
methods of measurement given in IS0 5167-1 can
then be used. It is, of course, necessary to respect
0
all the other requirements specified in this Technical
Report, apart from the need for steady flow through
the primary element.
or
8 Errors
The formulae given in 6.3 allow adequate damping
to be calculated for a given maximum allowable
relative error, tJ, in the indicated flow-rate due to the
residual damped pulsation. It is also possible to es-
The systematic error can be compensated for by
timate the actual error, ET, directly after measuring
reducing the discharge coefficient by a percentage
the pulsating amplitude at the orififce. The actual
equal to (1 - ET).
error, ET, should be less than 4.
There will be an additional uncertainty in the value
In theory, ET is always a positive systematic error,
of the discharge coefficient due to the pulsation,
but in practice there will also be an additional ran-
even after allowing for the systematic effect.
dom uncertainty mostly due to pulsation effects in
the secondary device. Calculations of the errors and
This percentage additional uncertainty is equal to
additional uncertainty are feasible provided that the
100 ET and should be added to the uncertainty cal-
0 pulsation amplitudes are not too large.
culated for steady flow.
Limiting pulsation amplitudes for error calculations
If the frequency response of the entire secondary
are
system, including the connecting tubes, can be
proved to be flat from O to lOf(wherefis the funda-
mental pulsation frequency), the additional percent-
age uncertainty may be reduced to 100 ET/2.
7

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SIST ISO/TR 3313:1996
ISO/TR 3313:1992(E)
Annex A
(informative)
Theoretical considerations and supporting experimental evidence
Hence
A.l General
The recommendations in this Technical Report are
based on the fact that, unless the amplitude of the
pulsations in a pulsating flow is reduced to an ac-
where
ceptable level, an accurate value of time-mean flow
-
rate will not be obtained by substituting the time-
App is the time-mean differential pressure
mean differential pressure in the ordinary steady
measured in pulsating conditions;
flow formula. At sufficiently small pulsation ampli-
tudes, it is possible to relate the error in the indi-
Apss is the differential pressure that would have
cated flow rate to pulsation amplitude and this has
been measured under steady conditions
been done in the derivation of the formulae for ade-
at a flow rate equal to the time-mean flow
quate damping in this Technical Report.
rate of the pulsating flow.
A.4 Quasi-steady temporal inertia theory
A.2 Derivation of equations relating error
The one-dimensional flow momentum equation is
in indicated flow rate to pulsation
amplitude
au uau I a~
. . (A.2)
-+- ax +FXZ=O
at
These equations are derived from the assumption
and the continuity equation is
that the fluid can be regarded as incompressible.
Experimental results obtained by placing orifice
meters in pulsating gas flows indicate that this as-
sumption is only valid if
If it is assumed that the flow is incompressible,
- the expansibility factor is very nearly unity, i.e. if
these two equations can be combined and inte-
E 2 0,99, and
grated with respect to x between the pressure
tappings to give an equation for the instantaneous
- the amplitude of the fluctuations in upstream
differential pressure, App:
density are very small compared with the mean
value, i.e. if p'r,,,s/F Q 1/40.
- cc P')
. . . (A.4)
APp = + pl*$ iJx
It is also assumed that the throat of the meter is 2p(cc n d*/4)*
small compared with the pulsation quarter-wave
length.
The first term on the right-hand side of the equation
is the differential presure associated with the
The special case of critical flow nozzles is covered
convective acceleration of the fluid through the re-
in 8.2.
striction. It is identical to the expression for the dif-
ferential pressure generated by a steady flow with
a mass flow rate equal to instantaneous mass flow
rate, qm, thus
A.3 Total error in the indicated flow rate
The definition of the total error, ET, is based on the
fact that the indicated flow rate during pulsating flow The second term is the differential pressure associ-
is calculated by using the measured time-mean dif- ated with the temporal acceleration of the fluid. Its
ferential pressure in the steady flow equation. It is magnitude increases with pulsation frequency and
assumed that there are no secondary-device errors. is zero for steady flow. The integral cannot be eval-
8

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SIST ISO/TR 3313:1996
ISOlTR 3313:1992(E)
uated exactly but can be replaced by an equivalent i.e
term:
(A. 12)
The ratio (U’r,,,s/Ü) is equal to (qm,r,,/~m) for
where Le is the effective axial length of the re-
incompressible flow and is the velocity pulsation
striction.
amplitude ratio.
Amplitudes of differential pressure pulsation may be
The instantaneous mass flow rate, qtn, can be ex-
easier to measure than those of velocity pulsations,
pressed by
however, and it is, therefore, useful to define:
. . . (A.5)
= 4m Cl + (b(t)l
. . . (A.13)
AP’p,rms = 4-
where
From equation (A.9) and neglecting terms of sec-
M
ondary importance we obtain:
. . . (A.6)
and
and introducing the Harmonic Distorsion Factor II:
-- dqm - 2 rw 4 cos(rwt + O,.)
. . . (A.7)
- 9m
dt
r= 1
For a steady flow:
&(I - c: P4)
. . . (A.8)
APS, =
2p(Cc n d2/4)2
where 4 is the amplitude of the rth harmonic in the
and hence for pulsating flow:
Fourier series defined in equation (A.6).
For a sine wave, H has a value of 1 and is greater
. . . (A.9)
than 1 for all other waveforms.
where
Thus
2 n C, Le fd
J= x-xT=- . . . (A.lO)
. . . (A.15)
O (1 - c; p4) d U,
Hence, from equation (A.ll) the following alternative
where C, is the contraction coeffic
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