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THE BEHAVIOUR OF ORIFICE AND VENTURI-NOZZLE METERS
IN, PULSATING FLOW
A Thesis Submitted for the Degree of Doctor of Philosophy
in the Faculty of Engineering at the University of Surrey
by
R. C. MOTTRAM,
/Lecturer, Department of Mechanical Engineering,
The University of Surrey.
0r: tober, 1971.
CONTAINS PULLOUTS
BEST COPY
AVAILABLE TEXT IN ORIGINAL IS CLOSE TO THE EDGE OF THE PAGE
SUMMARY
The conventional quasi-steady theory for the behaviour of meters
in pulsating flow is that at any instant the differential pressure is
only dependent on the acceleration of the flow due to the contraction
and is given by the steady flow relationship. The analysis presented
in this thesis is based on a quasi-steady theory modified to take
into account the additional instantaneous differential pressure due
to the temporal acceleration of the flow. Relationships are derived
for metering errors in terms of the r. m. s. amplitude of the differential
pressure pulsation and a Strouhal Number dependent on the waveform of
the velocity pulsation.
To test the validity of the derived theoretical relationships
the behaviour of square edge orifice plates with corner tappings and
of venturi nozzles were investigated in pulsating air flows. A
piston pulsator was built on which the stroke can be varied to obtain
a required pulsation amplitude while the machine is running at
frequencies up to 50 Hz.
The results of the tests showed that, although there were some
discrepancies, the theoretical relationships were basically sound.
It was found that it was possible to define when pulsations were
significant in terms of the r. m. s. amplitude of the differential
pressure fluctuation. It was also possible to determine an effective
Strouhal Number when temporal inertia effects became significant. No
basic differences in the behaviour of the two types of meter were
detected but certain predictable effects due to compressibility were
observed in tests on the venturi nozzles.
The techniques for reducing metering errors due to pulsations
are reviewed In the light of the analysis and experimental results.
Criteria by which the pulsation conditions can be properly assessed
and appropriate courses of action for reducing metering error are
4
Suggested.
ACKNOWLEDGEt4ENTS
I would like to thank my supervisor Professor J. M. Zarek for
introducing me to the field of pulsating flow measurement and for
his guidance and encouragement during the progress of the project.
I would like to thank technicians Mr. Terry Laws for his
considerable effort and skill in manufacturing the rig and Mr. Keith
Green for his cheerful assistance during the tests.
I would also like to thank research student Mr. Peter Downing
for his valuable contribution to the development of the rig and to
the conduct of the tests.
CONTENTS
PAGE
Chapter I
Notation
GENERAL INTRODUCTION 1-9
A THEORETICAL ANALYSIS OF
PULSATING FLOW MEASUREMENT 9- 69
SUMMARY 9
1.1 INTRODUCTION 9- 10
1.2 THE SIMPLE QUASI-STEADY THEORY 10 - 13
1.3 A QUASI-STEADY THEORY INCLUDING
TEMPORAL INERTIA EFFECTS 13 - 25
1.3.1 The Significance of Strouhal Number 13 - 14
1.3.2 Integration of the Temporal Inertia Term 14 - 18
1.3.3 An Expression for Metering Error
Independent of Temporal Inertia 18 - 19
1.3.4 The Relationship between Velocity and
Differential Pressure Pulsation Amplitudes 19 - 21
1.3.5 Waveform and the Effective Strouhal Number 22
1.3.6 The Relationship between Pulsation Error,
Temporal Inertia and Differential Pressure
Pulsation Amplitude 22 - 23
1.3.7 Residual Error due to Temporal Inertia
Effects 24
1.3.8 A Correction Factor in terms of a
Measurable Pulsation Amplitude 24 - 25
1.4 THE EFFECTS OF PULSATION ON THE
COEFFICIENT OF DISCHARGE 26 - 59
1.4.1 Meter Geometry and Pressure Tappings 26 - 27
PAGE
1.4.2 Orifice Discharge Characteristics 28 - 52
1.4.2/1 Kinetic Energy Distribution Effects 29 - 36
1.4.2/2 - Velocity Profile and Impact
Pressure Effects 36 - 41
1.4.2/3 Velocity Profile and the Contraction
Coefficient 42 - 45
1.4.2/4 Reynolds Number and the Contraction
Coefficient 45 - 50
1.4.2/5 The Combined Effect of Time-
Dependent Influences on Orifices 50 - 52
1.4.3 Venturi and Nozzle Discharge
Characteristics 52 - 59
1.4.3/1 Kinetic Energy Distribution Effects 52 - 53
1.4.3/2 Impact Pressure Effects on Nozzles
and Venturi Nozzles 53
1.4.3/3 Reynolds Number Effects 54 - 37
1.4.3/4 The Combined Effect of Time-
Dependent Influences on Venturi
Nozzles 57 - 59
1.5 COMPRESSIBILITY EFFECTS 59 - 69
1.5.1 The Expansibility Factor 59 - 63
1.5.2 The Effects of a Pulsating Upstream
Density, 63 - 67
1.5.3 The Acoustic Limitation of the
Quasi-Steady Theory 68 - 69
TABLES 1 -. 4 70 - 74
FIGURES 2- 24 75 - 96
0
" PAGE
Chapter 2 THE EXPERIMENTAL WORK 97 - 124
2.1 INTRODUCTION 97 - 100
2.2 THE DESIGN OF THE EXPERIMENTAL RIG
AND INSTRUMENTATION 100 - 109
2.2.1 The Air Flow Rig 101
2.2.2 The Pulsator 102 - 104
2.2.3 The Instrumentation 105 - 109
2.2.3/1 Pressure 105 - 106
2.2.3/2 Velocity 106 - 107
2.2.. 3/3 Instrument Frequency Response Tests 107 - 109
2.3 EXPERIMENTAL PROCEDURE 109 - 110
2.4 DISCUSSION OF EXPERIMENTAL RESULTS 110 - 122
2.4.1 The Quasi-Steady and Temporal Inertia
Effect Theory 111 - 116
2.4.1/1 Area Ratio 112
2.4.1/2 Reynolds Number 113 - 115
2.4.1/3 Effective length of the Restriction 115 - 116
2.4.2 Frequency Effects 116 - 122
2.4.2/1 Acoustic Characteristics 117 - 119
2.4.2/2 Velocity Profile Variations 119 - 120
2.4.2/3 Static Pressure Pulsations 120 - 122
2.4.3 Venturi Nozzles and Compressibility
. Effects 122
2.5 CONCLUSIONS ON EXPERIMENTAL WORK 123 - 124
TABLE 5 125
PLATES 1-5 126 - 130
FIGURES 25 - 57 131 - 163
Chapter 3
3.1
3.2
" 3.3
3.4
. 3.4.1
3.4.2
3.5
3.5.1
3.5.2
3.5.3
3.6
APPLICATION OF THE RESULTS TO THE
PROBLEMS OF METERING PULSATING FLOW
INTRODUCTION
THE CORRECT DESIGN OF THE SECONDARY
DEVICE FOR USE IN PULSATING FLOWS
THE PULSATION THRESHOLD
SQUARE ROOT ERROR ELIMINATION
Using an Electronic Square-Rooting
Circuit
Applying a Correction Factor
PULSATION DAMPING
A Review of Existing Techniques
An Analysis of the Hodgson Number and
Pulsation Error Relationship
Comparison of the New Damping Relationship
with'Existing Data
A PROCEDURE FOR REDUCING PULSATION ERRORS
IN ORIFICE METERS
TABLES 6- 10'
FIGURES 58 - 68
FINAL CONCLUSIONS
RECOMMENDATIONS FOR FURTHER WORK
REFERENCES
TABLES OF EXPERIMENTAL RESULTS - NOTES
TABLES 11 - 37
0
is
Notation
a amplitude of sine wave velo- city pulsation
ar amplitude of rth harmonic A' cross section area CD discharge coefficient CC -contraction coefficient CV velocity coefficient c speed of sound d meter throat diameter D pipe diameter E metering error F correction factor
(also pipe factor) f frequency H harmonic distortion factor J temporal inertia term
(equation 1/18) L+ pressure loss in Hodgson
number Le equivalent length of meter
restriction mass flow rate
m alternating component of mass flow rate
in throat to pipe area ratio NH Hodgson Number NS Strouhal Number p static pressure
impact pressure Q volumetric flow rate r, R radial distance Re Reynolds Number t, T time u local flow velocity U bulk flow velocity V volume w angular frequency, x, l axial distance a kinetic energy coefficient ß throat to pipe diameter
ratio Y isentropic index a boundary layer thickness 6' boundary layer displacement
thickness A differential pressure e expansibility factor A pipe friction coefficient p fluid density 0 reciprocal of Strouhal
number $(t) a cyclic function a phase angle
It except in section 1.4.2/1 both u and U are used to denote bulk velocity .
Suffixes
p pulsating conditions s steady conditions 1 upstream conditions 2 throat or vena contracta
conditions d refers to bore diameter
[also damped pulsation j
conditions (section 3.5) D refers to pipe diameter C, CLcentre-line conditions rms root mean square value R residual error T total error SR square root error obs observed th theoretical
i
GENERAL INTRODUCTION
Since the beginning of the century problems have been
encoubtered when flow meters have been used in pulsating flows as
opposed to steady flows. These problems have been particularly
severq with the differential pressure type of meter such as the orifice,
venturi and nozzle. These meters do not indicate the correct time-
mean flow rate when the flow is pulsating.
One can define a pulsating flow as one in which a regular cyclic
fluctuation is superimposed on a steady flow. The time-mean flow
rate may range from zero upwards while the cyclic fluctuation may
have any waveform, amplitude and frequency provided these do not vary
with time at a given flow condition. In very severe pulsating
conditions the amplitude of fluctuation may be so large that the flow
direction reverses during part of the cycle.
Pulsating flows arise whenever positive displacement machinery
of either the reciprocating or rotary type is used. They may also
be generated less obviously by oscillating valves and regulators or
by certain unstable flow conditions such as are sometimes found in
branched pipes (Appel and Yul).
In steady flow the flow rate is proportional to the square root
of the differential pressure measured across the primary device,
i. e. the meter restriction. If the steady flow relationship applies
at any instant in time during unsteady flow, which is the quasi-steady
assumption, it is possible to determine the correct value of the time
mean flow rate from the time-mean value of the instantaneous square-
rooted differential pressure.
2
I. e. Ma (Ap )
j where dt and Ap dt
00
Unfortunately most secondary devices, (devices for measuring the
differential pressure), are of the slow response type and at best can
only indicate the time-mean differential pressure. The square root of
the time-mean differential pressure, however, is not the same as the
required quantity
i. e. ( 4p )I #( np
This inequality leads to the well known square root error.
Researchers such as Hodgson32,33,86 and Judd and Pheley37
were well aware 50 years ago that the square root effect was responsible
for a large portion of observed pulsation errors in differential
pressure flow meters.
Hodgson also realised that a slow response secondary device did
not necessarily indicate the correct time-mean differential pressure.
He realised that any damping in the manometer limbs had to be of a
viscous nature and that there could be distortion of the pressure waves
transmitted along the connecting leads from the pressure tappings.
In short Hodgson was aware of the existence of secondary device errors 30,81 later identified and measured experimentally by Williams, 82,
Williams has made recommendations for the design of secondary devices
to be used in pulsating conditions. His design rules are applicable
to both slow and fast response devices.
Further sources of error lie in the limitations of the quasi-
steady theory. In this theory it is assumed that the differential
pressure required to accelerate the fluid with respect to time is
negligible compared with the differential pressure required to
accelerate the fluid through the meter restriction. Szebehely73
3
showed that the ratio of the temporal acceleration to the convective
acceleration is related to the Strouhal Number and that this should
be small compared with unity in quasi-steady flow.
Hence fd SU/St Ns -a ýUSU/6x ýý 1
Unfortunately it has not been easy to establish just how much
smaller than unity it is necessary for the Strouhal Number to be.
9 Analysis by workers such as Moseley-51 and McCloyý6 showed that the
"relevant length parameter in the Strouhal Number is not the meter
restriction diameter but its effective length. Again difficulty
arises in assigning a value to this effective length since it is not
simply the axial length of the restriction. This problem is discussed
in the relevant section of this thesis.
Metering errors which remain, when the square root effect and
secondary device errors have been eliminated are termed residual errors.
Residual errors due to neglect of the temporal acceleration (or
temporal inertia) term are discussed at length in this thesis. One
of the main objects of the experimental investigation was to
measure residual errors accurately. To do this both secondary device
and square root errors had to be eliminated from the measurements at
source. It was then possible to explore the relationships between
residual error and parameters such as Strouhal Number, Reynolds Number
and pulsation amplitude.
Residual errors are not necessarily solely due to temporal inertia
effects. There are good reasons for supposing that other assumptions
inherent in the quasi-steady theory are not strictly valid. One of
these assumptions is that the coefficient of-discharge determined for
steady flow. applies under pulsating conditions. In steady flow the
L `,
discharge coefficient is dependent on Reynolds Number and the upstream
velocity profile particularly for meters of large throat to pipe area
ratios (Johansen36, Engel 1s, Ferron23). ' Under pulsating conditions
there is a continuous variation of both Reynolds Number and upstream
velocity profile. There are thus grounds for suspecting that the
discharge coefficient would also undergo a cyclic variation with
subsequent effects on the measured differential pressure.
Unfortunately it is difficult to identify the sources of any
residual error measured experimentally. It might be possible to do
this by exploring the flow patterns in the immediate vicinity of the
meter restriction but such work was outside the scope of the investigation
described here. Priority was given to measuring and relating residual
errors to the flow and pulsation characteristics which could be
determined without making detailed local measurements in the flow
itself.
The square-edged orifice meter with corner tappings and the
venturi nozzle were selected for the experimental work because of the
marked dissimilarity in their geometry. There is a fundamental
difference in the flow through these two types of meter. In an
orifice plate the accelerated fluid is not guided by a solid wall and
the issuing jet forms the characteristic vena contracta. The extent
of the contraction is dependent on the throat to pipe area ratio and
the upstream velocity profile (Engel20, Ferron23). With a venturi
or venturi-nozzle meter, however, the fluid is guided by solid walls
and the behaviour of the boundary layer in the contraction and throat
plays an important part in determining the coefficient of discharge
(Engel16, Hall 28, Lindley43). It was felt that if there were
significant residual errors resulting from instantaneous variations in
the meter discharge coefficient the effects might well be different for
contrasting meter geometries and that it would be interesting, therefore,
to compare the behaviour of orifice plate and venturi-nozzle meters.
5
The immediate objective in this experimental investigation has
been to further the understanding of the behaviour of flow meters in
pulsating conditions. A better understanding, however, is only a
means to an end. The ultimate objective is to overcome the difficulties
associated with metering a pulsating flow.
The first and probably the most significant step in this direction
was taken by Hodgson32' 33,
who, 50 years ago, advocated inserting
capacitive and resistive damping elements into the flow system
between the pulsation source and the flow meter. The required capacity
and resistance to damp a given pulsation to an acceptable level was
calculated-from a dimensionless group since known as the Hodgson
Number. Much work has since been undertaken by workers such as Lutz 45,
Herning and Schmidt31, Kastner38 and Fortier25 on determining values
of Hodgson Number for pulsations of various waveforms and amplitudes
etc. The later work by Fortier took into account the effects of
temporal inertia and the possibility of acoustic resonance in the flow
system. The above work on damping is reviewed in this thesis and a
mathematical expression for calculating values of Hodgson Number is
developed which is applicable to pulsations of any waveform but of
known amplitude. Following Fortier, effects of temporal inertia and
the possibility of acoustic resonance of the system are taken into
account. ,
The theory on which the Hodgson criterion of adequate damping
is based is only applicable when the capacity chamber dimensions and
the pipe lengths connecting pulsation source, chamber, and flowmeter
are all small compared with the pulsation wavelength. When this is not
the case the amplitude of pulsation at the flowmeter and the meter
error can no longer be reliably predicted.
6
In such a situation it would be necessary to measure the pulsation
amplitude at the meter and thus assess the likely error, if any, in
the indicated flow rate. It may not be necessary for pulsations to be
completely absent provided that their severity is so low as to cause
negligible error. The problem of defining a pulsation threshold
between steady and pulsating flow has intrigued workers in this field
for many years. Head30 proposed that the velocity pulsation amplitude
should be used as a measure of the severity. Unfortunately, velocity
pulsations can only be measured with delicate instruments such as the
hot wire anemometer.
The measurement of differential pressure pulsations is much
easier and nearly 40 years ago Lindahl6' 42 described a mechanical
pulsometer by which the peak-to-trough amplitude of the differential
pressure pulsations could be measured. In 1951 Hardway29 proposed
that the r. m. s. value of these fluctuations should be measured by
means of an electronic computer. * Unfortunately Lindahl's and
Hardway's techniques were not practical at the time they were proposed.
With present-day sophisticated electronic instruments, however, the
measurement of differential pressure pulsations is comparatively simple
and the proposition that their amplitudes be used as a pulsation
threshold criterion is therefore re-examined in this thesis.
The damping of pulsations has so far been the most reliable
method of overcoming the difficulties in metering but the use of
damping chambers and throttles involves bulk and expense which may be
quite unacceptable. A theoretical alternative is to modify the
design of the flowmeter (including the secondary device) so as to
eliminate at least part of the pulsation error. If the secondary device
is designed according to the rules laid down by Williams errors due to
ý', x :, aý
7
this element can be eliminated. Work described in this thesis shows
that if the secondary device incorporates an electronic square-rooting
circuit square root errors can also be eliminated.
Unfortunately there is no known method of eliminating residual
errors due to the effects of temporal inertia even if one ignores the
possibility of other kinds of residual error. Furthermore there is
the complication that whereas square root errors are always positive
(indicated flow rate greater than true flow rate) temporal inertia
effects always tend to produce negative errors. There is, therefore,
the strong possibility that under certain conditions the elimination
of the square root effect could leave a negative residual error that was
greater in magnitude than the original total pulsation error.
If it is not possible to correct a meter reading when temporal
inertia effects are significant it is essenkial to have criteria for
determining when these conditions exist. It is demonstrated in this
thesis that Strouhal Number should make an appropriate criterion
provided it is combined with a factor dependent on waveform.
Another possible technique for eliminating metering errors due
to pulsation is to apply a correction factor to the indicated flowrate
inferred from an orifice plate meter fitted with either a commercial
secondary device isolated from the pulsations or a simple U-tube
manometer. This is a very attractive possibility and in the past
strenuous efforts have been made notably by Zarek, Earles and Jeffery (12 -14,35,87,88) to correlate the required correction factors
with a number of different parameters expressing the flow and pulsation
characteristics. As with the pulsation threshold approach, however, it
seems that it is impossible to determine a correction factor without
making some kind of measurement of the pulsation amplitude and possibly
the waveform. Such measurements would require sophisticated electronic
8
devices and yet the use of such equipment would appear to defeat
the object of the exercise. As a laboratory technique, however, the
approach may well be useful and the state of the art is reviewed in
this thesis in the light of the experimental results.
This thesis is arranged in three major chapters.
The first chapter is a theoretical analysis of various aspects
of the behaviour of differential pressure flowmeters in pulsating
conditions;
The second chapter contains a description of the experimental work
and a discussion of the results;
The third chapter is a review of the various techniques for solving
the problem of metering pulsating flows in the light of the analysis
and experimental results.
The final conclusions and recommendations for further work
follow the third chapter.
I
9 p
Chapter 1.
A THEORETICAL ANALYSIS OF PULSATING FLOW MEASUREMENT.
SUMMARY
Relationships between metering errors and the pulsation character-
istics of amplitude, frequency and waveform are developed for the
quasi-steady theory with and without the temporal inertia term. The
conditions under which this term is significant are discussed.
The quasi-steady assumption of a constant coefficient of discharge
is subjected to critical analysis. The conclusion is that while
this assumption may not lead to large residual errors the amplitude of
the differential pressure pulsation may be underestimated for large area-
ratio meters subjected to a given amplitude of velocity pulsation.
Possible compressibility effects are analysed and predicted to
be significant under the following conditions: -
(a) when the equivalent steady flow expansibility factor is
less than 0.99,
(b) when the upstream static pressure pulsation amplitude,
(pus/pu), exceeds 0.035,
(c) when the meter bore is not small compared with the
pulsation quarter-wave-length.
1.1 INTRODUCTION
The equation used to express the flow rate in terms of the
pressure difference measured across a meter restriction is derived
from a one-dimensional-flow momentum equation written in differential
form. It contains a temporal and a convective acceleration term and
both terms make a contribution to the differential pressure. It can
be argued, however, that the convective acceleration of the fluid
10'
passing through an abrupt restriction is likely to be very much
larger than the temporal acceleration provided that flow fluctuations
are not rapid. If this is assumed the equation becomes the normal
steady flow relationship and the quasi-steady theory states that
this is valid when applied at an instant in time during pulsating
flow conditions. By including the temporal acceleration term it is
possible to assess the errors involved by making the quasi-steady
assumption and to relate these errors to the main pulsation character-
istics of waveform, frequency and amplitude.
It will be seen that it is possible to make a fairly complete
mathematical analysis of the quasi-steady theory modified to take
into account the temporal acceleration term (also called the temporal
inertia term). Unfortunately, however, the theory does not take
into account the possible effects pulsations may have on the coef-
ficient of discharge of the flow meter. In the quasi-steady theory
it is assumed to be constant and to have the value empirically
determined under steady flow conditions. In reality, however, it
must have a cyclic variation. The major factors influencing the value
of CD and the likely effect of pulsation will be discussed in this
chapter.
Further assumptions made in the quasi-steady theory are that
compressibility and acoustic effects are insignificant. The validity
of these assumptions will also be discussed.
1.2 THE SIMPLE QUASI-STEADY THEORY
The one-dimensional flow momentum equation is as follows: -
au/at + uau/ax + (1/P) Sp/ax =0..... 1/1
11
The one-dimensional flow continuity equation is written: -
Abp/St t d(puA)/dx =0 ..... 1/2
'In steady flow the time dependent terms Su/St in the momentum
equation, and Sp/St in the continuity equation, are zero. In
unsteady flow these terms can only be neglected if: -
(1) du/dt « udu/dx
i. e. if the temporal acceleration is very much less than the convective
acceleration.
(2) Sp/St « a(puA)/ax
or = zero
The latter condition will be true if the fluid can be treated
as incompressible (see section 1.5.3, page 68).
If the fluid is assumed to be incompressible and if the temporal
acceleration term is neglected then the momentum and continuity
equations can be integrated along the length of the restriction as
follows: -
Equation 1/1 becomes:
2+p1 (P2 PO =0..... 1/3 12
where suffixes 1 and 2 refer to the upstream conditicns and the
throat conditions respectively.
From equation 1/2 we have:
M2P Ul Al =p U2 A2 ..... 1/4
Combining equations 1/3 and 1/4 we obtain the relationship:
nd2 2p Ap
*pose 1/5
where Ap is the instantaneous differential pressure measured across
12
the restriction and m is the area ratio A2/A1.
Equation 1/5 does not take into account the differences
between ideal one-dimensional flow and the actual flow features
such as the non-uniform velocity profile in the upstream flow and
the jet contraction phenomenon in an orifice meter. These differences
are conventionally allowed for in the steady flow case by using an
empirical coeffieicnt of discharge, CD. Similarly minor changes in
fluid density during expansion are allowed for by using an expansibility
factor, c. Values for CD and c are given in the various codes of
practice3'8 for the standard meter designs when used under steady
flow conditions.
If it is assumed that these factors, CD and c can be treated
as constants (not time dependent) having the same values as deter-
mined for steady flow then the resulting flow equation is the
quasi-steady relationship, i. e.:
M= CD ( X42) e (12--M2 )ß ..... 1/6
The above equation gives the instantaneous mass flow rate in
terms of the instantaneous square-rooted differential pressure. The
time-mean mass flow rate is given by integrating with respect to
time, viz.
( 7rd2 ) 4 ..... 1/7
It should be noted that the time-mean of the instantaneous
square rooted differential pressure is required. If instead the time-
mean differential pressure is measured and the square root of this
quantity used in equation 1/7 the flow rate will be overestimated.
13
This particular pulsation error is known as the square root error.
It is due to the simple mathematical fact that
T Apdt To Apidt CfT_ 0 (. 1 ý The quasi-steady theory and the explanation of pulsation errors
in terms, of the square root effect date back 50 years to the time
of Hodgson. The main difficulty with the theory is in defining the
range of conditions for which it is applicable. To do this it is
necessary to make an analysis of each of the theory's assumptions.
1.3 A QUASI-STEADY THEORY INCLUDING
TEMPORAL INERTIA EFFECTS
1.3.1 The Significance of Strouhal Number
The temporal acceleration term is neglected on the grounds that
at low pulsation frequencies it is much smaller than the convective
acceleration term. Referring to equation 1/1 the condition is that
(du/dt)/(udu/dx) « I. ' As was shown by Szebehely73 and later discussed
by Oppenheim and Chilton54 the acceleration ratio (du/dt)/(udu/dx) is
closely related to the Strouhal Number, the dimensionless frequency
fd/Ü. The idea of using Strouhal Number to define the limit of
applicability of the quasi-steady theory was suggested by Schultz-
Grunowfi5 in 1941.
Schultz-Grunow argued that for dynamic effects (temporal inertia)
to be negligible the period of time for a fluid particle to travel a
characteristic path length had to be small compared with the pulsation
period. The characteristic length chosen by Schultz-Grunow was the
a
14 '
orifice diameter, d. The characteristic time is therefore (d/UUd).
According to his argument the condition for dynamic effects to
be negligible is, that:
1 ä >>
where 0 is the reciprocal of Strouhal Number. In his experiments the
value of 0 did not fall below 510 and in discussing his results
Schultz-Grunow suggested that the value could be as low as 10 before
dynamic effects would be significant. 4
It is possible, however, to approach the problem of determining a
limiting value of Strouhal Number in a more analytical manner.
1.3.2 Integration of the Temporal Inertia Term
If equations 1/1 and 1/2 are integrated with respect to x
along the length of the restriction, assuming that the fluid is
incompressible and including the temporal acceleration termwe can
obtain the following equation for the instantaneous differential
pressure:
ep = M2 (1 - m2)/[2p(nd'2/L+)2] + p12 i
du/dt dx ..... 1/8
In the following analysis the only non one-dimensional flow effect
that will be taken into account is the jet contraction phenomenon that
occurs in flow through an orifice. To do this the contraction coefficient,
CC, is introduced into equation 1/8 as follows:
2 Ap = M2 (1 - CC2m2)/ [2p(CC nd2/4)2] t pll Su/St dx
..... 1/9
where CC =1 for nozzles and venturis.
15
In order to develop the analysis it is necessary to evaluate the
last term in the equation. If we assume that the length of the
restriction is small compared with the wavelength of the pulsation the
variation of flow velocity along the restriction is given by the simple
continuity relationship that u= u2 . A2/A.
As an example let us consider that the restriction has a conical
shape as in a classical venturi (Fig. 1).
)
TIC. 1
A Venturi Contraction
Using the continuity relationship we have:
I221 pi ät )dxdt2ýi
Azdx-pddtfata2 ldx 0
Pdu (l. d/D)
12
16
Evidently the term is of the form [pät2 . Let, where Le is some
characteristic length depending on the geometry of the particular
restriction.
In the case of flow 2 through an orifice it is not possible to
dX evaluate the integral p11 () in the same way. By analogy with the
result for a venturi, however, we may assume that the term can 'similarly
be replaced by [p (du2/dt)Le] where Le is again a characteristic
axial length.
We will call the length Le the "effective length" of the
restriction and note that in acoustic terminology the quantity
[pLe(nd2/4)1 is its inertance.
02 If we replace the velocity u2 by [M/pCC '14 j. equation 1/9 now
becomes:
A= M2 (1 -C 2mz)/[2p (C rtd2/J+)2] + [Le/(CCird2/4)1 dt ..... 1/10 pCC
To determine the behaviour of equation 1/10 when applied to
pulsating flow we will suppose that the mass flow rate time relationship
is of the form:
M=M (1 + c(t) ) ..... 1/11
where ar sin (rwt + or) ..... 1/12 1
and represents a Fourier series.
where ar is the amplitude of the rth harmonic
w is the angular pulsation frequency
and a is the phase angle. Ir
The quantity M is the time-mean mass flow rate and we note that
if the flow was steady with the same flow rate the differential pressure
would be given by the equation:
17
As = M2 (1 - CC2m2)/ [2p (CCrd2/L. )2] ..... 1/13
Combining equations 1/10,1/11 and 1/13 we have
Ap = As (1 + $(t) )2 +5 e/(Ccnd2/4)1
M q'(t) ..... 1/14
where 0'(t) =w In arr cos (rwt + ar) ..... 1/15
Rearranging equation 1/14 we have
A= AS (1 + fi(t) )2 + 2(pU2/2)(wLe/U{)(ý'(t)/w) ..... 1/16
We see that the instantaneous differential pressure is made up
of two components. The first component is the pressure required for
the convective acceleration and is the steady flow differential
pressure modified by a time-dependent function of the pulsation
waveform and amplitude. The second component is the pressure
required for the temporal acceleration. It is the product of a
velocity head (p 22/2) and a Strouhal Number (WLe/UU2) modified by
another factor which ensures that the second pressure component is
n/2 out of phase with the first.
Note that U2 is the velocity at the vena-contracta and not the
bulk mean velocity through the orifice itself.
Rearranging equation 1/15 we have:
ap = AS [il + "(t) )2 + 2J($'(t)/w)I ..... 1/17
where j= (1/(1 - CC2m2) ) (wIAe%U2)
= (2trCC/(1 - CC2m2) ) (L8/d) (fd/Ud) ..... 1/18
where Üd - CCU2
18
In the above expression we have formed the more conventional
version of Strouhal Number based on the pulsation cycle frequency (f)
instead of the angular frequency (w), and on the orifice diameter (d)
instead of the effective length (La). It is more convenient to have
the ratio (Le/d) as a separate parameter as there is the difficulty
of assigning a value to it, particularly in the case of an orifice.
1.3.3 An Expression for Metering Error
Independent of Temporal Inertia
Expanding equation 1/17 using equations 1/12 and 1/15, we have:
n ep = os {1 +21 (arsin(rwt + ar) +ir ar cos (rwt + ar) )
r=1
nn +2Z [a sin(rwt +v) Z
r=1. rr q=1
n -I ar2sin2(rwt + ar)
r=1 ,
agsin(qwt + aq)]
0 .... 1/19
It can be seen from this equation that the temporal inertia
term (the one including J) is always ir/2 out of phase with the velocity
pulsation.
The time-mean differential pressure is given by
fT Gp =
Tl e dt and hence it can be shown that
10 p
a12 t a22 + a32 + ..... ap = AS (i +2) ..... 1/20
The only terms in equation 1/19 that make a contribution to the
19'
mean are terms like art sin2(rwt +a), where are sin2(rwt + ar) a2 2 (1 - cos 2 (rwt + ar) ).
On inspecting equations 1/11 and 1/12 we see that
MU 02(t) art rms)2 rms)2
..... 1/21 U M
where M and U are the root mean square of the alternating rMS rms components of the mass flow rate and bulk mean velocity respectively.
We have the interesting result that
U ep = ns (1 +( rms )2 ) -..... 1/22
U
The time-mean differential pressure depends only on the amplitude
of the velocity pulsations upstream of the meter. The error in the
meter reading in the absence of secondary device errors is, therefore: -
ET =( ap/IS )I -1= [I +( Urins/U )2 -1..... 1/23
Apparently the magnitude of the temporal inertia term cannot
affect the metering error if the velocity pulsation amplitude is fixed.
Metering errors could, `therefore, be corrected if the r. m. s. ampli-
tude of the bulk velocity pulsation could be readily measured.
Unfortunately the measurement presents great practical difficulties.
Presumably if a suitable instrument existed it could replace the
differential pressure meter altogether when pulsation existed.
1.3.4 The Relationship between Velocity and
Differential Pressure Pulsation Amplitudes
Differential pressure pulsations are much easier to measure than
velocity pulsations and if ,a
relationship between the two can be found
20'
it should be possible to derive an expression for metering error
which would be practical to evaluate.
On rearranging equation 1/19 we have
n Ap = As {1 +2 Car sin(rwt + Qr) + Jrax, cos(rwt + ar)}
r=1.
n where r#q +2E araq sin(rwt + ar) sin (qwt +a)
1
a
-+ 22 (1 - cos 2(rwt + er) )
..... 1/24
Using equations 1/21 and 1/22 we have
ep =p {1 + (1 +1n a-2
) [2 11 (arsin(rwt + ar) + Jrarcos(rwt + ar)) ý1
r
f2n, r$q ara sin(rwt +a sin (qwt + aq)
a2 cos 2 (rwt +a) ..... 1/25
Equation 1/25 shows that the instantaneous differential pressure
has a 'dc' part, Ap, and an 'ac' part. The ac part can be arranged
in a Fourier series in which the coefficients of each harmonic
component are made up from terms containing, ar, ar2 and araq. If
we consider only flow fluctuations in which the flow never reverses
its direction the harmonic amplitudes, ar, are always less than unity.
Consequently it is reasonable to neglect the terms containing art and
ar q and the root mean square amplitude of the ac part of the
differential pressure is given by:
(MS )2= { ß+ d2 2
(1 + r2J2)} 1r
AP t in, ar2) 2 2
21
i. e. ( ems
)2= {4 (ýn ar2)
(1 t H2J2)} 1 ..... 1/26
p12 (1+rar2 )2
2
r2a 2a2
where H2 = Ln r (En r)..... 1/27 1212
We will call H the harmonic distortion factor.
Combining equations 1/26 and 1/21 we have
( Qrms
ý2 _ 4( Urms
)2-(1 + H2J2)(1 +( Urms
)2) -2 ..... 1/28
pÜÜ
The quantity (a s
p) is a ratio of two variables. By using
equation 1/20 we can also obtain:
v r ms )2 _ 4( Ems )2(1 + H2J2) ..... 1/29 su
Not only do we have a simpler expression but the ratio (ems/es)
is more convenient to use both in analysis and in experimental work.
Equation 1/29 shows the relationship between differential
pressure and velocity pulsation amplitudes.
When HJ «1 then (4S)2 (rms ) U
sU
This is the situation in quasi-steady conditions.
When HJ >1 then ( Ile s)a HJ s
for a constant velocity pulsation amplitude.
Alternatively ( vas
)a1
for a constant differential pressure pulsation amplitude.
22
1.3.5 Waveform and the Effective Strouhal Number
Before continuing the analysis it is worthwhile considering the
physical significance of the factor, H.
This was defined by equation 1/27, viz.
H2 =( ýi r2ar2 )/( ýi ar2 )
We see that when the velocity waveform is a pure sine wave of
the fundamental frequency the value of H is 1. When the waveform
is sinusoidal but the fluid system is resonating at the rth harmonic
frequency the value of H is r.
For any other waveform H>1.
We also see that J is multiplied by H. J. (defined in equation
1/18) contains the non-dimensional frequency, i. e. the Strouhal
Number. It is apparent that the function of H, the harmonic
distortion factor is to modify the Strouhal Number based on the
fundamental frequency. When the Strouhal Number and H are multiplied
together the product is an effective Strouhal Number adjusted to take
into account waveform and harmonic resonance effects. Effective
Strouhal Number henceforth refers to the quantity (H fd%ud).
1.3.6 The Relationship between Pulsation Error,
Temporal Inertia
and Differential Pressure Pulsation Amplitude
We have already derived equation 1/23 which defines metering
error in terms of the r. m. s. velocity pulsation amplitude. Eliminating
(U %U) by using equation 1/29 we have:
ET =( np/es )i -1=(1+a( Arms/As )2 / (1 t H2J2 ) )I -1..... 1/30
23'
The term H2J2 represents the contribution of the temporal
acceleration or temporal inertia to the metering error. When it
is zero the error is only due to the square root effect which is
always positive and dependent only on the r. m. s. amplitude of the
differential pressure.
It is interesting to note that as far as the square root effect
is concerned all waveform effects are accounted for if the r. m. s.
amplitude of the differential pressure pulsation is measured. If a
peak-to-trough amplitude had been used it would only have been possible
to make the analysis for a few simple waveforms such as a sire wave
and a rectangular wave. In his experimental work, Jeffery 35
measured peak-to-trough amplitudes and discovered that it was
necessary to modify these amplitudes by a waveform factor to obtain
a better correlation with metering error.
Fig. 2, (page 75) shows a graph on which the total pulsation
error is shown plotted against HJ for various constant values of the
pulsation amplitude (Arms/ps ). There would appear to be three
separate regimes.
(1) HJ < 0.1 Pulsation error solely due to the square root effect
(2) 0.1 < HJ < 10 Total pulsation error less that the square root
error alone.
(3) HJ > 10 Pulsation error tends to zero.
One should be cautious about predicting pulsation errors when
HJ exceeds 10 as there is a danger that acoustic compressibility
effects may be significant. The above theory has teen derived
assuming pulsation wavelengths are much greater than the meter
dimensions. (See section 1.5.3)
24
1.3.7 Residual Error due to Temporal Inertia Effects
The definition of a residual error is the error remaining in the
indicated flow rate when the square root effects have been eliminated.
From the preceding analysis it is evident that an expression for
residual error follows fron equation 1/30.
ER = (e /es )-1=C1t(e /s )2/(1 + H2J2) )
- (1 t4(e /As)2)1 ..... 1/31 rms
When ( Arms/es )<1 one can obtain the following approximate
expression:
ER S )2 H2J2/( 1t H2J2) s
..... 1/32
According to the above analysis it should not be possible to
have positive residual errors.
1.3.8 A Correction Factor in Terms of a
Measurable Pulsation Amplitude
Expressions such as equation 1/31 have only a limited application
since they involve the ratio ( Ams/As In most situations the value
of as would be unknown. It is possible, however, to obtain alternative
expressions in terms of ( Qrms%&p It would be useful to have an
expression for a factor which could be used to correct an indicated
flowrate-obtained under pulsating conditions.
From equation 1/23 we have:
FT =( A$/dp) =1+(u /U )21 - ..... 1/33
Equation 1/28 relates the ratios (4rmS% p) and (U S1U
) and
25
combining with 1/33 we have
(ems/ p)2 =4( UrmS/UU)2 (1+ H2J2 )( es/eP )2 ..... 1/34
Rearranging we have:
rC Urms /U )2 +1]_( ems/Ap)2/( 1+ H2J2 ) FT4 +1= 1/FT2
Solving the quadratic equation we obtain:
1t1-ý( Arms A)2/ (1 + H2J2 )] i
1/FT2 = j (Arms/ep)2 /(1 +H2J2 )
making the substitutions A2 = (A s/
)2 1t H2J2 )
and X2 =1- A2
we have:
_ A2_ 1-X2 Fx
L1±C1_A2J 1-X
=EI{1 +(1-A2) }1I 0
{1t(1-(A s/Ap)2/(1tH2J2))
} 11 ..... 1/35
This correction factor takes into account square root effects
and temporal inertia effects. The equivalent expression for the
pulsation error is:
ET =[{1+ (1 - (AMS/AP)2 /(1+ H2J2)) } ]- -1 ..... 1/36
26
1.4 THE EFFECTS OF PULSATION
ON THE COEFFICIENT OF DISCHARGE
It is always useful to reduce the complexity of a theory by
making simplifying assumptions. It is also important to determine
under what conditions the assumptions are no longer valid.
One of the main assumptions of the quasi-steady theory is that
the coefficient of discharge may be regarded as constant and having
the same value as under steady flow conditions. The coefficient
compensates for the discrepancy between the differential pressure
predicted from one-dimensional flow theory and that actually measured.
The value of the discharge coefficient for steady flow is
influenced by a large number of factors involving the meter geometry,
tapping position and flow conditions. In order to assess the validity
of the quasi-steady assumption it is necessary to study the likely
effects of pulsations on each of these factors.
l.. l Meter Geometry and Tapping Positions
It is impossible to discuss the discharge characteristics of
differential pressure flow meters without reference to the geometry
of particular designs. Not all designs are subject to the various
influences to the same extent.
The subsequent discussion on discharge characteristics will,
therefore, be divided into two main sections. The first section
will be concerned with orifice meters and the second with nozzles
and venturis.
Before leaving the topic of meter geometry it is interesting to
consider the relationship between pressure tapping position and the
axial pressure distribution through the constriction. Fig. 3, (page 76)
shows pressure distributions measured through a sharp edge orifice
27'
by Johansen36. Immediately below the distributions is a diagram
showing the positions of the three types of pressure tappings recommended
in BS 1042. It is clear that the stability of the pressure dis-
tribution is critical if the same discharge coefficient can be used
for steady and pulsating flow. The sharp rise in pressure immediately
upstream of the plate (known as impact pressure) is important for large
area-ratio orifices and influences the value of the discharge coef-
ficient for meters with corner tappings and flange tappings (large
pipes only). Impact pressure is also important for nozzles and
venturi-nozzles which have an upstream corner tapping. For all orifice
tapping positions but more particularly for D and D/2 tappings and
flange tappings on small pipes the downstream distribution is important.
Conventionally the downstream pressure minimum is associated with the
plane of the vena contracta and from Johansen's curves this appears
to move upstream towards the orifice and becomes more sharply defined
as the orifice area ratio increases. From the above it is clear
that whatever tapping position is chosen the larger area-ratio
orifices are likely tobe more susceptible to the effects of pulsation
or, indeed, any kind of flow disturbance.
Fig. 4, (page 77) shows the pressure distribution measured
by Lindley43 along the walls of a venturi tube. The upstream
tapping in this meter is well clear of any steep pressure gradients
but the throat tapping is situated between two regions of steep
pressure gradients associated with the intersections between the
cylindrical throat and the conical entrance and diffuser. The gradient
upstream of the throat tapping may well be significant when there is
pulsating flow as it marks a point at which there is a tendency for
flow separations to take place.
28 '
1.4.2 Orifice Discharge Characteristics
It is possible to derive a relationship for the discharge of
flow through an orifice taking into account some of the deviations
from one-dimensional behaviour normally covered by the discharge
coefficient. One of the features of pipe flow is that the velocity
profile is not uniform across the pipe section. The most significant
feature of the orifice is that it requires the flow to change
direction abruptly to pass through it with the resulting formation
of the vena contracta.
These two features have a number of consequences as follows: -
(1) the kinetic energy of the flow, and therefore the differential
pressure across the meter, depends on the velocity profile;
(2) the so-called 'impact pressure' which affects meters with
corner tappings depends on the velocity profile;
(3) the meter reading depends on the contraction coefficient
which not only varies with the velocity profile but also with
Reynolds Number.
These effects may be accounted for under steady flow conditions
by using the available empirical data. For pulsating flow an
attempt may be made to account for the effects by determining the
differential pressure step by step through a typical pulsation cycle
assuming appropriate steady flow conditions at the instantaneous
velocity profile and Reynolds Number.
29
1.4.2/1 Kinetic Enema Distribution Effects
The energy equation for flow through a restriction in a pipe
should be written:
R2 21Tr u2 3dr IR1 27rr U13 dr t1( P2 P1)=0..... 1/37
nR22 U2 0
irR12 U1 0
where U1 and U2 are local velocities dependent on the radial distance
from the pipe axis.
An alternative version of -this equation is
( P2 - P1 )=Q..... 1/38 22 U22 ' 21 U12 t1
where U2 and U1 are bulk mean velocities and a2 and al are kinetic
energy coefficients defined by the relationship:
I
a=21 (R)3R d(R) ..... 1/39 Jo
If the velocity profile is represented by:
i ü-
( 1- r) n ..... 1/40
Ct
It is possible to obtain an expression for a in terns of the index
n by integrating equation 1/39 hence:
a= (n +1 )3 ( 2n +1 )3 ++n nt3 2n +39....
1/41
For a perfectly uniform profile n tends to infinity and the
value of a is unity.
30
A convenient method of expressing the velocity profile is
by means of the pipe factor, F, which is defined by:
F=U..... 1/42 UCL
R Now u=Ä 2ffr u dr ..... 1/43
fo
and therefore
U- uCL n+ 12) (2n +1 """"" 1l44
using equation 1/35.
Hence: 2n2 F(1)(+1) """"" 1/45 n+ 2n
If equation 1/38 is used the following expression for flowrate
is obtained:
UA `�2e..... 1/46 G
a2 -a1 CC mp
alternatively this can be rewritten
2 aý ..... 1/47 U2 =CV 1-CCý4 -, m
where the velocity coefficient CV compensates for non-uniform
velocity profiles and is given by:
Cý =[1 Crým2m
..... 1/48 2 -'a1 C
31
Fig. 5, (page 78) shows CV plotted against the pipe factor,
F, for three sizes of orifice plate. The values of CV were cal-
culated from equation l/48 making the following assumptions: -
(1) a2 21i. e. the velocity profile after acceleration in
the contraction is uniform.
(2) The contraction coefficient could be assigned the theoretical
values tabulated by Engel and Stainsby18 after Von Mises48
(see Fig. 12 , page 84 ).
(3) At Reynolds Numbers above 60,000 the velocity profile conformed
to the Prandtl power law i. e. equation 1/40. This applies
for pipe factors above 0.83.
(4) A Reynolds Number below 60,000 the velocity profile could be
represented by the relationship attributed to Stanton 5
i. e. (ü) (1 -C rz ) ..... 1/49
CL
Corresponding expressions for a and F are: -
a=(I+(2 ý) 2)..... 1/50
F= (2 2C)..... 1/51
Stinton found that C=0.35 for ReD = 52,000.
For laminar flow C=1 and F
The most striking features of Fig. 5 are that CV only has,
significant values for large area ratio meters and that at pipe
factors below 0.8 the curve of CV begins to rise steeply. Although
the pipe factor normally exceeds 0.8 in fully developed turbulent
32
conditions it must be recognised that under pulsating conditions
the instantaneous pipe factor can fluctuate over a very wide range.
'Let us consider what happens when the flow has a sinusoidal
pulsation.
i. e. U=tJ (2tasin rwt )
Let us further suppose that the fluctuating component of
velocity is uniform across the pipe section,
i. e. u(r, t) =u(r)+aÜsinwt.
The instantaneous value of the pipe factor is given by:
F_Ü(1+asinwt )
üCL( itaÜ sin wt )
uCL
_(l+asinwt )
(1 t aTsin wt )
In a particular case F and the amplitude a might have the
values of 0.85 and 0.80 respectively.
Hence Fmax = 0.911
and Fmin = 0.531
The actual determination of the likely effect of a fluctuating
velocity profile on the differential pressure measured does depend
on the assumption made concerning the profile of the fluctuating
velocity component. Figs. 6 and 7, (pages 79,80 ) show some
examples of profiles measured during the experimental investigation.
. It would appear from these that the assumptions already made are
quite reasonable, i. e.
(1) that the time-velocity profile is the same as the steady
flow profile;
33
(2) that the fluctuating velocity component is uniform across
the pipe section.
An objection to the latter assumption is that it takes no
account of the fact that at the wall surface the flow velocity must
be zero at all times.
Fig. 8 shows the difference between the assumed instantaneous
profile and one that probably occurs in practice.
oft am ..
D _ _____
omw dw VLSI. Or
º`
lwt 1
IL air wo =
'FIG. 8
mom --- -- -- -- -- ý-" instantaneous profile - according to assumptions,
probable actual instantaneous profile.
2
34, '
Nevertheless, in order to obtain an idea of the likely effect
of a fluctuating profile we shall use equation 1/37 and the above
assumptions.
Hence, the instantaneous differential pressure across the
restriction, assuming quasi-steady flow is
R2 2Trr u2 3 dr RI 2nr U13 dr
ý 'pi P2 ) `2 p- fo 1/37 2f
irR22 U2 irR12 Ul 0
Assuming that the velocity profile in the contracted flow
stream is flat and that U2 = U2 (1+a sin wt )
and that U1 = U1 (1+a sin wt )
r( (üi +a1 sin wt) 3 (P1 '" P2) =2L Ü22 (1 +a sin wt)2 - 21 R)d(R
o U1 (1 +a sin wt)
..... 1/52
" Continuity gives ÜZ = CC M U2
Under steady flow conditions:
(P1 - P2)s, = 10 U22 (1 - al CC2m2)
Substituting into equation 1/52, we have:
(L )_(ltasinwt)2 C1 _C2M2
1(2(
. 21 )3t3(Ul )22asinwt
s (1 -XC 2m2 c 1c)c Ul Ul
.. t 3(-ul )2 a2sin2wt + 2a3sin3wt) 1+a sin wt)3(R)d(R ü
..... 1/53
If the time-mean velocity profile is represented by
ul )=(1- R) 1/n
uCL 1
00
35
it can be shown that
1 f ul 32 (- r) (n + 1)3 (2n. t 1)3 = a1 J C--) -=
o U1 R) d (R
4n4(n + 3)(2n + 3)
where al = the kinetic energy coefficient (equation 1/41)
I1 ( ui )22 (R) d (R) (n+ 1)2 (2n + 1)2 = ßl Jo.
U1 2n2 (n + 2)(2n + 2)
where ßl = the momentum coefficient (this coefficient is applied in
same way as a to determine the true momentum of flow at a pipe
section from that calculated using the bulk-mean velocity).
and J i-) 2 (R) d (R)
0 Ui
Equation 1/53 can now be re-written
(L )_ (i ta sin wt)2 Qs (1 -. a1CC2m2)
(a, + 3ß1 a sin wt + 3a2sin2wt t a3sin3wt)
..
[1_CC2m2
..... 1/54 (1 ta sin wt) 3
Note that if the velocity profile is flat al-and ß1 become
unity and the coefficient of (CC2m2) also becomes unity.
Having obtained an expression such as equation l/54 it is easy
to compute the effect of a fluctuating velocity profile on total
pulsation error, [(ßp/AS 1], residual error, C{ (e/4S} - 1]
aid the amplitude, (Q /As), by-evaluating the expression at 15° or rMs 300 intervals through the pulsation cycle.
As an example this was done for an orifice of area ratio 0.64
when subjected to sinusoidal pulsation of velocity amplitude a=0.8.
36,
The value of the index n for the equivalent steady flow was assumed
to be 7 giving ai = 1.06 and 01 = 1.02.
The calculated differential pressure cycle and the corresponding
square rooted pressure cycle are shown in Fig. 9 as the solid curves.
The broken line curves show the same quantities calculated from the
simple quasi-steady relationship,
()_ (1 +a sin wt)2 ..... 1/55 s
The differences between the two pairs of curves appear to be
very small. In fact they are not negligible as is shown in Table 1,
(page 70). The main effects are that the pulsation amplitude is
increased from 1.132 to 1.176, and there is a residual error of about
-0.8%. The total error is slightly larger than that predicted from
quasi-steady theory but not as large as would be inferred from the
same theory using the increased value of (AMs/as).
The accuracy. of the results obviously depends on how closely the
actual behaviour of the instantaneous velocity profile resembles
the assumed mathematical model. The fact that it is possible to
demonstrate some effects which are not negligible for large area
ratio orifices is itself significant.
1.4.2/2 Velocity Profile and Impact
Pressure Effects
The pressure distribution along the pipe wall in the vicinity
of an orifice plate shows a sharp rise immediately upstream of the
plate (Fig. 3). This rise in pressure is associated with the loss
of axial momentum of the outermost filaments of flow as they bend
inwards to pass through the orifice.
In their analysis of impact pressure and velocity profile
37 ,
effects Engel and Daviesis assured that the excess pressure on the
upstream face of the plate due to the reaction to the axial momentum
loss was uniform across the face. They further assumed that the
impact pressure was proportional to this excess uniform pressure.
Thus, if the excess pressure is pi
D/2
Pi (DZ- d2) I (pul) uZ 2nr dr ..... 1/56 1 d/2
If the impact pressure is P, and if the local velocity ul can
be represented by ÜI = f(R) then: U1
JI Pa pUI2 12 m) ýd
() ..... 1/57
m
The differential pressure required to accelerate the fluid
through the restriction is given by:
P2 =2 U12 (12- al ) ..... 1/58
C
The ratio. of impact pressure to this differential pressure is
p _K
(cc 2M2) I rf 2 --
rd pal _p2
j]()..... 1/59
(1 - al CC2m2) (1 - m) j �M
CR RR
where K is a constant.
There is no way of determining the value of K except by experiment.
In 1964 Engel and Stainsby20 were able to compare the results of the
earlier analysis26 with actual measurements of impact pressure by
Witte and Schrader on orifice plates. By assuming either the Bahkmeteff
or the Prandtl relationship for the velocity profile (equations 1/60
and 1/61 below respectively) and by selecting an appropriate value of
K they were able to match their predicted impact pressure/Reynolds'
38
Number curve with the experimental results for pipe Reynolds Numbers
above 60 000. Fig. 10 shows the graph published in the 1964 article.
There is discontinuity at Reynolds Number 60 000 clearly indicating
the limit of a separate flow regime. At Numbers below 60 000 Engel
and Stainsby extrapolated the curves through the experimental points
to meet at a common point at Reynolds Number 3 000.
The Bahkmeteff relationship as quoted by Engel and Davies15 is:
Ü=f (R) =k �8 {8 (k A+ Z) + In (1 - R)} ..... 1/60
where k is a constant and
A is the pipe friction coefficient.
The Prandtl relationship as stated previously is:
ÜCL = (l - R)l/n ..... 1/61
In laminar flow the velocity profile is parabolic and is
expressed by
u r2 = (1 R2) CL ..... 1/62
Equation 1/51 applies at Reynolds Numbers above 60 000 and
equation 1/62 applies at Reynolds Numbers below 3 000. in between
these limits the velocity profile can be expressed by the relationship
suggested by Stanton55, that is:
üCL - (1 "R2 ý.... 1/63
where C is a constant which depends on Reynolds Number, Stanton found
C to be equal to, r0.35 at Reynolds Number 52 000.
39
Fig. 11 shows curves of the impact pressure/differential
pressure ratio plotted against pipe factor. They were calculated
using equation 1/59, the factor K having been evaluated by matching
the equation against Witte and Schröder's results shown on Fig. 10
at a pipe Reynolds Number of 105. For Reynolds Numbers of 105 and
higher (pipe factors of 0.83 and higher) the Prandtl power law
(equation 1/61) was assumed for the velocity profile. At lower
Reynolds Numbers the Stanton relationship (equation 1/63) was
assumed. It should be remembered that the curves are only valid for
turbulent flow since equation 1/59 was derived assuming that the
differential pressure was proportional to the velocity squared as in
equation 1/58.
The most striking feature of both Fig. 10 and Fig. 11 are that
impact pressure is only significant for the larger area ratio meters
and that it falls off steadily as the pipe factor decreases.
Let us now consider what happens during pulsating flow by
carrying out an analysis similar to that concerning the kinetic
energy coefficient. As before we shall assume a sinusoidal pulsation
such that U=U (1 +a sin wt).
Again we shall assume that the time-mean velocity profile is
the same as the steady flow profile and that the fluctuating
velocity component is uniform across the pipe section.
Hence we can rewrite equation 1/59 as
I1 4 CC2m2 +a sin wt _, _-,
U12rr (Pi
P
P2) =K (1 - al CCZm2)(1 - m) �m( 1+a sin wt R) d (R)
..... 1/64
40 '
If the velocity profile is assumed to be expressed by the
Prandtl relationship
r)1/n R ü
CL
it can be shown that:
K2C 2m2 2+n pC Eßl(1-�m)n (1t(2nn)�I (Pi - P2) -
(1 - a2 CC2m2)(1 - At)
ltn
tz(1-�m)n (1t(lnn)�m)2asin wt
t (1 - m) F
a2sin2wt (1 t a1 sin wt)2 """"" 1/65
where ßl is the momentum coefficient, equal to (n t 1)2 (2n t 1)2
2n2(n + 2)(2n + 2)
and is the pipe factor, equal to 2n2
nt1)(2n--+--17
The differential pressure that should actually be measured on an
orifice fitted with corner tappings is:
(Pi - P2) (P1 " P2)(1 +P)..... 1/66 P1 - P2
where (pi P2) U22 (1 ta sin wt)2 (1 - a1 CC2m2) ..... 1/67
Ignoring all possible time dependent effects apart from the
impact pressure the ratio (ep/a is as follows:
41
2KCC2m2LA +B sin wt +C sin2wtj (_ )= (1+asinwt)2 { 1+ }
s (1 - al CC2m2)(1 - m) (1 ta sin'wt)2
2KC 2m2 {1+CA}
(1 - al CC2m2)(1 - m)
..... 1/68 2tn
where A=ßl (1 -mn (1 + 2+n
m)
1+n
B= (1/F)(1 - m) n (1 + lnn
mk) 2a.
and C= (1/F2)(1 - m) a2
The effect of the fluctuating impact pressure on the differential
pressure cycle can be determined as before by evaluating equation
1/68 at small angular intervals. The effects are only likely to be
significant at high values of m and a.
Again the example is taken of an orifice of 0.64 area ratio
subjected to sinusoidal pulsation of velocity amplitude a=O. B. The
results for the total and residual errors and the differential pressure
pulsation amplitude, (dam /es) are shown as item 2 in Table 1 (page 70 ).
As with the kinetic energy distribution effects the total error
and (dam/As) are both greater than one would expect from the quasi-
steady theory and the residual error is +0.6%.
Without claiming any great accuracy for the above analysis it is
perhaps significant that the effects are not negligible for a large
area ratio orifice subjected to severe pulsation.
42
1.4.2/3 Velocity Profile and the
Contraction Coefficient
As long ago as 1869 Kirchoff41 calculated that the width of a
jet issuing from a two-dimensional slit in a reservoir wall was
+) times the width of the slit. Later Rethy57, Voz Mises48 and
Hahnemann27 calculated the contraction of jets issuing from slits
and other apertures in the end walls of finite width channels.
Unfortunately the conformal transformation technique used for these
two-dimensional flows cannot be applied to three-dimensional problems.
Nevertheless, by using a reiterative process of computation and
by experiment, Southwell and Vaisey68 and Rouse and Abul Feton62
demonstrated that values of contraction coefficient predicted by
Von Mises for two dimensional slits in the end wall of channels also
applied to circular orifices at the end of a circular duct provided
the ratios of slit to channel width and orifice to duct diameter
were the same.
When the pipe continues downstream of the orifice, however, the
problem is even more difficult. The pressure cannot be assumed to
be constant everywhere downstream of the orifice as in the previous
case. In addition there is the difficulty of accounting for the
turbulent mixing of the jet with slower moving fluid around it.
There appears to be no method of predicting contraction coefficients
for orifices within pipes even for potential flow. The problem of
predicting the influence of upstream velocity profiles on contraction
coefficients by an analytical technique would, therefore, seem to be
intractable. It is possible, however, to make some predictions based
on the experimental work carried out by Ferron. 23
43 ,'
Ferron carried out tests on various meters including an 0.55
area ratio orifice fitted with both flange and D and D/2 taps.
The pipe Reynolds Number range was 105-to 106. Ferron varied the
upstream velocity profile by means of contractions and wire-mesh
pipe liners and measured the corresponding variations in the discharge
coefficient of the meters. Some of his results are shown tabulated
in Table 2, (page 72) with values of the contraction coefficients
calculated from equation 1/71.
The discharge coefficients for both flange and D and D/2 taps
increase by 2.7% when the pipe factor decreases from 0.96 to 0.76.
It is possible to make a calculation of an observed contraction
coefficient from the discharge coefficient by comparing the following
two equations.
/2(Pi_- P2) Q- CCobs (7r42 ..... 1/69
(a2 - al CCobs2m^)p
2(P1 P2)(1 t PAPI - P2) ) Q= CD (1! -4) ..... 1/70
(i - m2)P
where (pl - P2) is the maximum differential pressure when impact
pressure effects are absent.
P is the excess pressure due to the impact effect.
Combining equations 1/69 and 1/70, we have:
(1 + pi - P2
) a2
ýc ..... CCobs
D (1 - m2) + alm2CD2(1 + ills.
Pl--
When D and 0/2 taps are used the impact pressure P is zero. In
the case of flange taps P is, not zero if the pipe size is large.
44
Ferron's orifice was installed in a 10 inch diameter pipe line and
in the calculation of CCobs it is assumed that (PIP_ P2) has half
the value applicable to corner taps and is taken from FIG. 11.
The coefficient a2 is assumed to equal to 1.00.
It can be seen from Table 2 that the calculated values of
observed contraction coefficients, CCobs, for the two types of
tapping agree remarkably well. The increase in the contraction
coefficient for the decrease in pipe factors is about lj%. The
variation in the discharge coefficient was larger (2.7%) because
it takes into account the changes in the kinetic energy coefficient 01
The variation in contraction coefficient with pipe factor shown
in Table 2 and on Fig. 13, (page 85) is surprisingly small. The
ratio (CCobs/CCth) is the coefficient inferred from Ferron's discharge
coefficient divided by the theoretical value from Von Mises, (Fig. 12).
If the pipe Reynolds Number varied in the range from 107 to 104 the
corresponding normal pipe factor variation would be from 0.88 to
0.79. This is a much smaller range than in Ferron's experiments.
According to the ASME Power Test Codes Flow Measurement supplement3
the variation in discharge coefficient for a 0.55 area ratio meter
with flange taps in a 10 inch pipe over the Reynolds number range
would be from 0.600 to 0.772, i. e. an increase of 28.6%.
The obvious conclusion is that the contraction coefficient is
probably strongly dependent on Reynolds Number and is only slightly
dependent on upstream velocity profile.
Leaving aside the question of Reynolds Number dependency for
the moment let us consider the likely effects of pulsating flow on
the contraction coefficient in terms of the velocity profile changes.
During the accelerated part of the pulsation cycle the velocity
45 ,
profile becomes flatter and consequently the contraction coefficient
will decrease. The reason for this as Ferron pointed out, is that
for a flatter profile the outer flow filaments have more momentum and
as they bend inwards to pass through the orifice they have a greater
squeezing-in effect thus reducing the size of the vena contracta.
During the decelerated part of the pulsation cycle the reverse
happens; the outer flow filaments have less momentum and the vena
contracta has a larger diameter. In view of Ferron's results the
maximum cyclic variation in the contraction coefficient due to
velocity profile changes is unlikely to be more than about ± 1%.
These variations would cause a slight increase in the peak differential
pressure (about 0.02 "pmax) and a slight decrease in the minimum
differential pressure (again about 0.02 % Qpmin).. Both total
pulsation errors [(dp/Qs)' - 11, and residual pulsation errors
[(Ap/Os)I - 11 are likely to be negligible due to this cause.
1.4.2/4 Reynolds Number and the
Contraction Coefficient
In the previous section it was mentioned that the marked increase
in the discharge coefficient of a large area-ratio orifice as Reynolds
Number decreased could not be explained in terms of velocity profile
effects. Clearly the contraction coefficient is dependent on
Reynolds Number itself. This can be demonstrated by expressing the
variation in the contraction coefficient by the ratio (CCobs/CCth) as
suggested by Engel and Stainsby20. CCobs is calculated from
equation 1/71, and CCth is the theoretical contraction coefficient
originally derived by Von Mises48 for two dimensional slits and
shown in rig. 12.
46
Fig. 14 shows the ratio (CCobs/CCth) plotted against pipe
Reynolds Number for an orifice with corner tappings. The discharge
3a coefficient data is taken from the AS14E Report on Fluid Meters.
The values for CCobs shown in Fig. 14 do not agree with those in
Engel and Stainsby's paper20 exactly as those authors did not account
for the impact pressure effects as is done in equation 1/71. The
variation of contraction coefficient with Reynolds Number is only
significant for the larger area-ratio orifices. For the 0.64 area
ratio orifice (CCobs /CCth) increases from 0.98*to 1.06 while the
Reynolds Number decreases from 107 to 6000. This variation is
considerably more than would be expected from Ferron's experiments
with velocity profile changes and one is forced to presume that the
viscous forces in the shearing and mixing regions around the issuing
jet are influencing the contraction effect. Clearly these effects
are'likely to be more pronounced for the large size orifices as
the radial distance from jet to pipe wall is smaller.
When considering what happens to the contraction effect during
pulsating flow it is necessary to decide whether an instantaneous
Reynolds Number has significance. For example supposing that a
0.64 m ratio orifice is being used to meter sinusoidal pulsating flow
when the time mean Reynolds Number is 105 and the velocity pulsation
amplitude, a, is 0.9. The maximum and minimum instantaneous Reynolds
Numbers are 1.9 x 105 and 104. Will the variation in contraction
coefficient be according to Fig. 14 (i. e. will (CCobs/CCth) vary
between about 0.99 and 1.04)? Presumably the answer depends on the
frequency.
The criterion by which we may judge whether or not the
instantaneous Reynolds Number is significant may be derived by the
47
following kind of analysis.
Consider the region of fluid between the jet, the orifice plate,
and the pipe walls. (See below. )
CL D/2 Id/2 FCC d /2
In steady flows the contraction coefficient varies with Reynolds
Number when the distance between the fluid jet and the pipe wall is
small (when d/D is large). Presumably this is because the shear force
between the jet and the pipe wall is having a significant effect on
the mechanism by which toroidal eddies are formed and which circulate
in the region between the jet, the pipe wall and the orifice plate.
The tangential acceleration of fluid becoming entrained in
this eddy is
tangential shear force mass of fluid entrained
where'the tangential shear force du (a cylindrical area)
U, "= µ 2(n�CCd)x
(D - �Ccd )
and the mass of fluid entrained is = p(n�CCd) xy
48
where y is a length in the radial direction and is of magnitude
T (D - �Ced) where n is a fraction.
The tangential acceleration of the entrained fluid is therefore
uä24
(1 - �Ccd)2pn
In pulsating flow the contraction coefficient will continue to
vary with the instantaneous Reynolds Number of the jet provided
the above tangential acceleration is greater than the maximum temporal
acceleration of the pulsating jet.
If the jet velocity is given by
u2 =u2 (l+asinwt).
the maximum acceleration is
ü2a w= 21Tü2a f.
The condition for the instantaneous Reynolds Number to be
meaningful is that _ u u2 t
2Tru2a f< (D - �CCd)2pn
that is (ß - �cc)(fd)(ü uzd) a<
-ff 2n
..... 1/72 u2
The product of Strouhal Number and Reynolds Number is sometimes
known as the Valensi Number and has been used by Sarpkaya63 and Combs
and Gilbrech9 to characterize pulsating viscous flow. Engel22
suggested that the pertinent parameters to characterize pulsation flow
conditions in the laminar/turbulent transition regime were Strouhal
and Reynolds Numbers.
If the distance between the pipe wall and the vena contracta is
y' the above criterion reads
()a<4 ..... 1/73
49
The criterion is still indeterminate unless a value can be
assigned to the fraction, n.
Clearly a more sophisticated analysis is required to determine
the criterion more exactly. It is felt, however, that as a demonstration
that Reynolds Number and Strouhal Number play a part in this criterion
the above simple analysis is valid.
If we assume that the fraction n is very small say, 1, then 1000
true quasi-steady flow would extend to frequencies when the Strouhal
Number was of the order 0.01.
Assuming that the latter condition applies the effects of
neglecting the Reynolds Number dependency-of the contraction
coefficient can be calculated as follows.
The differential pressure during pulsating flow is given by:
2,
Qp =pM (1 +a sin wt)2(l -OIC+Ö)..... 1/74
.; CC 2 ('ýd2 )2Pp P4
The differential pressure during steady flow is given by:
es =r (1 - a1CCs2º2)(1 +
..... 1/75
p
CCs (4 )s
Assuming that apart from the contraction coefficient the
normal quasi-steady assumptions apply we have:
L eC2 (lý - calCCT) m2) )=2 (1 +a sin wt)2 ... 1/76
s Cep (1 - a1Cc g2_.
_ )
If the example is taken of a 0.64 area ratio orifice subjected
to sinusoidal pulsations of velocity amplitude a=0.8 the effects of
the contraction coefficient varying with both instantaneous velocity
50
profile and Reynolds Number can be calculated from equation 1/76 using
data from Figs. 13 and 14. The results of such calculations are
shown'in item 3 of Table 1. As before the main effect is to increase
the amplitude of the differential pressure pulsation.
1.4.2/5 The Combined Effects of
Time-Dependent Influences on Orifices
The differential pressure under pulsating flow conditions but
in the absence of temporal inertia effects is given by:
P
pl M (1 +a sin wt)2(1 - a1PCC2M2)(1 + ý)
..... 1/77 C Z()2 P
where CCp, aip and Pp are all time dependent.
Under steady flow conditions the differential pressure is:
P
e= P1 ri (1 - a, scc
2M2)(1 + 5) ..... 1/78
CCS2(nd )2 s
Hence:
P
i -ý) =C CCS
)2 (1 +a sin wt)2 (1 - a12CCp2m2)
(1 tý p) ..... 1/79
-ý, As CC P (1 - a1SCCs2m2) Ps
tö ) S
Fig. 15 shows a differential pressure cycle calculated from the
above equation (the solid line curve) when the orifice area ratio
is 0.64 and the pulsation is sinusoidal and has a velocity amplitude
a=O. S. The broken line curve represents the simple quasi-steady
prediction. The square root error according to the quasi-steady theory
is (1 +12)l -1= 14.9%
51
The total pulsation error for the cycle calculated from
equation 1/71 is (ä -1=17.2%
s
The residual error for the above cycle is I=0.2% C1 s
In other words, despite the fact that the kinetic energy
coefficient, the impact pressure and the contraction coefficient are
all time dependent the error involved in assuming that they are
all constant is negligible.
The differential pressure amplitude (Arms/As assuming the
quasi-steady theory is given by equation 1/29
i. e. firms
_2` Ums
as Ü
In the case of sinusoidal pulsation
= 2r = 1.132 in this example AS
The value calculated for the cycle calculated from equation 1/79
using the definition
f( 4e s)-TT (a P-
ý) 2dt = 1.246
oss
and this is a 10.1% increase over the simple quasi-steady value.
The. pulsation error calculated from equation 1/30 (see below)
using the measured value of Arms/A8 - 1.246
is ET _ (1 +4(
Arms )2)-1- 17.8% As
52,
In the particular example considered it would appear that if the
pulsation error was calculated from the differential pressure fluc-
tuation amplitude it would be overestimated by t 0.6%.
If the error was calculated from the velocity fluctuation
amplitude it would be underestimated by 2.3%.
If an automatic square rooting device was used in conjunction
with a fast response differential pressure transducer the residual
error would be negligible.
It should be stressed that because of the assumptions the above
results may not be very accurate but they do indicate that arms is
likely to be higher than predicted by the simple quasi-steady theory
but that residual errors are not likely to be large.
1.4.3 Venturi and Nozzle Discharge Characteristics
Despite the obvious differences in geometry between venturi meters
and orifice plates most of the effects discussed already with respect
to the orifice plate are relevant to the venturi meter. Where the
effects are similar the comments set down below are very brief.
Apart from the remarks concerning impact pressure, which only apply
to nozzles and venturi-nozzles having upstream corner tappings,
there will be no differentiation between the various designs of nozzles
and venturi-meters.
I. 4.3/I Kinetic Energy Distribution Effects
These effects are similar whatever design of differential pressure
meter is used and have already been discussed at length with respect
to the orifice meter. In general one can say that at a given Reynolds
Number as the upstream velocity profile becomes flatter (pipe factor
increases) the coefficient of discharge decreases. Compared with
53, '
an orifice plate the variation in discharge coefficient is smaller.
One has to take care not to confuse effects due to Reynolds Number
with those due to upstream velocity profile. This point is well
illustrated by Ferron's experimental results (see rig. 16). For
each of the three different upstream profiles the discharge coefficient
of the venturi meter (m = 0.452) increases with Reynolds Number
despite the fact that presumably the velocity profiles would become _"
slightly flatter. Clearly the Reynolds Number effect is predominant.
1.4.3/2 Impact Pressure Effects on
Nozzles and Venturi-Nozzles
The upstream tapping will only be subject to the so called
impact pressure if it is in a corner position where the flow lines
have a positive curvature with respect to the pipe axis. The effect
is, therefore, relevant to the ISA 1932 nozzle, the venturi nozzle
and the Dall tube. It is not relevant to the classical vonturi tube
or the ASME flow nozzles.
Where they do apply the effects of a fluctuating velocity
profile will be very similar to those with an orifice plate. For
a given area ratio the. impact pressure for a venturi nozzle or
nozzle will be a greater percentage of the differential pressure
since the vena contracta effect is absent. One would expect that
a venturi nozzle of area ratio my to have the same ratio of
(P/Pl - P2) as an orifice of area ratio
CDv my
ECDo2 + v2 (CD V2 - CD02)]
This can be shown by equating differential pressures for the two
types of meter and assuming the same fiowrata and pipe size.
54'
1.4.3/3 Reynolds Number Effects
Typical CD - Red characteristics for smooth rounded entrance
nozzles and for classical venturi meters are shown on Figures 17
and 18. All the characteristics show that the discharge coefficient
increases with Reynolds Number but that the rate of increase decreases
with Reynolds Number. Indeed the characteristics taken from the
British Flow Standard for both nozzles and venturi tubes show that
at a sufficiently high Reynolds Number the discharge coefficient
can be assumed to be constant.
Various attempts have been made to predict the discharge
coefficient characteristics from an analysis of the likely
boundary layer behaviour in a smooth entrance nozzle. These predictions
have usually been confined to near-zero area-ratio meters in which
the influence of the upstream flow can be neglected. Fig. 17 shows
two such theoretical curves published by Hall28 and Rivas and sl Shapiro. In both these authors' analyses it was assumed that the
boundary layer thickness in the throat of the nozzle was the same
as that on an equivalent length of flat plate. It was assumed that
the. boundary layer grew from zero thickness at the nozzle entrance
and that in the throat the flow outside the boundary layer was
inviscid and had a uniform velocity distribution. These assumptions
allowed Hall to calculate the coefficient from the following
expression: ir(r -a max
)2 CD
..... 1/80 nr2
where 0 wax
is the maximum boundary, layer displacement thickness in
the nozzle tIroat.
55
These authors assumed that in the lower Reynolds Number range
the boundary layer was completely laminar but that at higher Reynolds
Numbers the boundary layer became turbulent. The turbulent boundary
layer is associated with the flatter part of the CD - Red characteristic.
Hall suggested that transition from laminar to turbulent boundary
layers was a gradual process and that over a range of Reynolds Numbers
the transition point moved upstream from the nozzle exit. The
transition regime is marked by the hump in Hall's predicted characteristic.
The experimentally determined characteristics obtained by Bean,
Johnson and Blakeslee5 and by Cotton and Westcott10 (also shown on
Fig. 17), however, do not show such a pronounced or well defined
hump as that predicted by Hall but it should be pointed out that these
experimental curves represent the average results of tests on a large
number of nozzles.
On examining the CD - Red characteristics for classical venturi-
tubes obtained experimentally by-Spencer and Thibessard71 and Lindley43
shown in Fig. 18 it is tempting to explain the hump on these curves
in terms of Hall's transition theory. It should be noted, however,
that the area-ratios of'the classical venturi meters of rig. 18 are
by no means close to zero as Hall's theory requires. A further
limitation to Hall's theory is that it only applies to smooth rounded
entrance constrictions in which no flow separations take place. In
the classical venturi meter design as standardised in BS 1042 the
conical entrance part may join the parallel throat section without
any blending radius. Lindley43 demonstrated quite clearly that
(a) the boundary layer in his 0.5 area ratio venturi meter was turbulent throughout the Reynolds Number range (Red 105 - 106) and
(b) there was a separation bubble just upstream of the throat tapping
at throat Reynolds Numbers below 3x 105.
56
Unfortunately the hump on Lindley's characteristic occurs at a
higher Reynolds Number. It would appear that neither the boundary
layer transition explanation or the separation bubble' explanation are
entirely satisfactory as far as the classical venturi meter is concerned.
In pulsating flow the question arises as to whether the coefficient
of discharge varies with the instantaneous Reynolds Number. Assuming
that the coefficient of discharge is determined entirely by the
boundary layer thickness,
i. e.
CD =1- 4SA/d
the criterion for true quasi-steady flow is that the characteristic
time of the boundary layer is small compared with the pulsation
period. Thus
t* = üW
<f..... 1/81
mr-
where u* = shear stress velocity =�; P
du u2 Now assuming that Tw =pd=pY
where u2 = the free stream velocity in the throat
0
hence u* d
U2 (u pu2d 701
The criterion for true quasi-steady flow reads
pu2d) 3/2 iü2) Cu C1 - CD) <8..... 1/82
Thus when Red = 105 true quasi-steady flow conditions exist at
Strouhal Numbers less than about 0.03.
Assuming that the coefficient of discharge does vary with the
instantaneous Reynolds Number the effect on the instantaneous
57
differential pressure cycle and the residual error if a square rooting
device is used will be extremely small. This is because the discharge
coefficient variations in the accelerated part of the cycle carry
far more weight than those in the decelerated part.
1.4.3/4 The Combined Effect of
Time Dependent Influences on Venturi Nozzles
The velocity profile variations affect the kinetic energy
coefficients and also the impact pressure. Both these effects are
significant for large area ratio nozzles only. On the other hand
Reynolds Number variations have a-much greater effect on the discharge
coefficient for small area ratio nozzles.
Te illustrate these effects three cases have been evaluated
for a given sinusoidal pulsation with velocity amplitude, a=0.8.
The cases are as follows:
1) m=0.5 Red= 5x105
2) m=0.1 Red =5x 105
3) m=0.1 Red=5x104
The following equation was used for the evaluation of the effects
on the instantaneous dif.
at) = (l ta sin wt)2
1 z s VP
Ferent:
1 Re
Lal pressure. P
(1 +a 11) P ..... 1/83
P (1 + S)
ZVP is the discharge coefficient correction factor for variations in
the kinetic energy coefficient and was evaluated from Ferron's
results (Fig. 16) for a venturi tube.
ZRe is the Reynolds Number correction coefficient for a venturi
nozzle taken from BS 1042. This is not quite accurate since the
58
published data inevitably embodies some effects due to velocity
profile changes.
From equation 1/65 we can see that the impact pressure is given
. by the following expression
2+n 2+r, p 2M2 PL-P2
K(1-xlm)(Z-m) X1(1-�m)n (1 +( n )�m)
1+n +l (1- �m) n (1+ lnn �m) 2a sin wt
t (1 - m)(1 } a2sin2wt 1 """"" 1/84
(1 ta sin wt)2
An idea of the magnitude of the impact pressure under steady
flow conditions can be gained by comparing the BS 1042 data for nozzles
and venturi tubes. At the same flow rate
} 2P-(p1 - P2)(1 + Pý P2 /2p (P1 - P2) CDv A2 (1 - m)
CDN A2 (1 - m2}
CD 2 i. e. (1 t
PI --- pp)=C=..... 1/85
nN
0
For an area ratio of 0.5 BS 1042 gives CDv = 0.986 and CDN = 0.993
for Rd t 300 000 and D, 12". Equation 1/85 gives the corresponding
value of P/(pl - P2) of 11.2% Using this result in equation 1/84
the constant K can be determined and the impact pressure can then be
evaluated during pulsating conditions.
The calculated results of the evaluation of the effects of
pulsating flow for the three cases are summarized in Table 3 on
page
It will he seen that for the m=0.5 case there is a small
positive residual error (0.45%), whereas for the m=0.1 case the
59
residual error is negligible although tending to be negative at the
lower Reynolds Numbers. As with the orifice the differential
pressure fluctuation amplitude is greater than would be expected
from the simple quasi-steady theory.
1.5 COMPRESSIBILITY EFFECTS
Gases can often be treated as if they were incompressible as far
as metering is concerned. The density which appears in the flow
equation is the upstream density and the expansion through the
restriction is allowed for by an expansibility factor.
In the simple quasi-steady theory the differential pressure is
assumed to be the only significant time dependent variable and quantities
such as the upstream density and expansibility factor are treated as
constant at their time mean values. The validity of these assumptions
are investigated below.
The other aspect which is considered is the assumption that as
far as continuity through the meter is concerned density does not
vary with time. This assumption is made even when temporal inertia
effects are allowed for.
1.5.1 The Expansibility Factor
Even when the fluid being metered is a gas the ordinary incompressible
relationships can normally be used provided the density drop through
the restriction is accounted for by means of an expansibility factor, e.
When the expansion can be regarded as isentropic as in a venturi or
a nozzle it can be shown that e is given by the following relationship;
E={ý. __ 2/y 1-m2 1-r} 1
Y -. l 2r2/Y 1-r..... 1/86 m I-
where r is the pressure ratio Pz/pl.
60 1
When the expansion cannot be regarded as isentropic empirical
data is used which is given in the various codes of practice.
The ASTE code3 gives the following relationship for c to be used
with square edge orifices.
E1- (0.41 + 0.35m2) ( P1
y-
P2 -) ..... 1/87
When the flow is pulsating it is clear from equation 1/73 that
the expansibility factor will be time dependant.
If the differential pressure (pi - P2) is represented by
(pi - p2) = As (l ta sin wt)2 we can evaluate the effects of a
fluctuating expansibility factor from the following relationship.
A [1- (0.41t0.35m2) ]2
=(1tasinwt)2 AS E1 - (0.41 + 0.35m2)
,1 (l +a sin wt)2J 2
- (1 +a sin wt) 2-- -------- ..... 1/88 rl-A(l+asinwt)2]2
where A= (0.41 + 0.35m2) e1=1-c.....
1/89
Equation 1/88 has been developed using the expansibility factor
expression 1/87 for square edge orifices.
For a venturi the expansibility factor is given by the isentropic
relationship of equation 1/86 which appears to be completely different
from the empirical equation 1/87.
The following similar equation, however, does give approximate
values of the expansibility factor for a venturi as can be seen from
Table 4 (page 74)
P2 e=1- (0.79 + 1.27m2).
P1
Y
`' p ..... 1/90 aPProx
61
Since this expression is similar to equation 1/87 it follows
that equation 1/88 and those following are valid.
If A<1 we can use the binomial theorem to obtain
_ (1 +a sin wt)2 [1 - A]2 [l + 2A(1 +a sin wt)2+ 3A2(1 +a sin wt)4] A P-
s where terms of higher orders have been neglected.
By integrating this expression over a complete pulsation cycle
it can be shown that
(ý) (1- A)211 +a2+ 2A(1+3a2+3a`') s 2 g
+ 3A2(1 + 12 a2 +
48 a+ 16
a6)ý ..... 1/91 5
and that
C(ä )9 _ (1 - A) rl + A(1 +2 a2) + A2(1 +5 a2 +
15 a4)] ..... 1/92
s
Equations 1/91 and 1/92 enable us to evaluate total pulsation
and residual errors.
Equally important is the effect on the amplitude of the differential
pressure fluctuations.,
From equations 1/88 and 1/91 the following expression for the
alternating component of the differential pressure can be written as
( as _ (1 - A)2[(1 +a sin wt)2{1 - A(1 +a sin wt)2}'?
- (H1 + 2AB2 + 3A2B3)] ..... 1/93
where B1 = (1 + 22 )
B2 = (1 + 3a2 +8 a4
ý B3 = (1 + 15 a2 f
45 a`' + 5a6
2a is
62
Again if A«1 we can use the binomial expansion to obtain
e -e (-ý-ý )2 = (1 - A)4[(1 +a sin wt)4 + 4A(1 +a sin wt)6
+ 1OA2(1 +a sin wt)8 - 2C{(1 +a sin Wt)2 + 2A(1 +a sin wt)4
t 3A2(1 +a sin wt)6} + C2 3 ..... 1/94
neglecting higher order terms of A.
and where C= (B1 + 2AB2 + 3A2B3)
Hence it can be shown that
A
rms )_ (1 - A)2[(B2 + 4AB3 + 1OA2B4) - (B1 + 2AB2 + 3A2B3) (
..... 1/95
8
445 a4 +
35 a6 t 135 a
where By = (1 + 14a2 t 10
By combining equations 1/91 and 1/95 we also have
e (a2 + 4AB3 + 10A2a4)
ap (81 + 2Aß2 +8a *)2
Fig. 19 shows the residual error, {. [(ep/es) - 1, plotted
against the velocity pulsation amplitude, a, for various steady flow
values of the expansibility factor es. It is clear that positive
residual errors result when the fluid is compressible and that these
errors become significant when the expansibility factor is less that
about 0.99.
Fig. 20 shows the total error {(Äp/As) - 1) plotted against a.
These errors also increase as the expansibility factor decreases.
Fig. 21 shows that there is a marked variation in (AMs/4s) with
the value es at a given value of velocity pulsation amplitude a.
The effects of expansibility on the relationship between residual
and total errors and the pulsation amplitudes Q. s/äp and aS/QS
63 '
can be derived from Figs. 19 - 21 provided it is remembered that it
has to be assumed that the velocity waveform is sinusoidal.
'0n inspecting equation 1/88 it is clear that the amplitude of
the differential pressure fluctuation increases as the expansibility
factor decreases.
For the above effects to be small enough for the quasi-steady
theory to be still valid the value of the expansibility factor should
be greater than 0.99.
1.5.2 The Effects of Pulsating Upstream Density
In the quasi-steady theory it is assumed that the measured time-
mean static pressure will enable the correct density to be calculated.
The fact that the density term is time dependent,. however, indicates
that there is a possibility of interaction with the other time
dependent terms in the equation.
Let us assume that the fluid velocity in the pipe is fluctuating
sinusoidally and is represented by:
-u=Ü (I+asinwt)
The static pressure will also fluctuate sinusoidally and may be
represented by
p=p (1tbsin(wtta) )
where a is the phase angle between pressure and velocity fluctuations.
Assuming that the fluid temperature is sensibly constant then
for a gas the density is given by
P= p(1+bsin(wt+a))
64
The differential pressure across the flow meter can now be
written as
ep =I Ü22(1 ta sin wt)2 7 (1 +b sin (wt + a))(1 - a1CC2m2) ..... 1/97
and hence
A A= (1 +a sin wt)2 (1 tb sin (wt + a)
Clearly the density fluctuations may be significant when the
fluctuation amplitude b is not negligible compared with unity. In
this case the phase angle a is obviously of importance.
In a pipe of infinite length with no obstructions in it the
pressure and velocity waves must travel in phase with each other. if
this is the case then it can be shown that
a2 + 2ab =1t2..... 1/99
s
and D L(L)ý =1+ab ..... 1/100
.s
It can also be shown that
Arms z ()={(A}= (2a2+äa4 +2ab+14a2b2+16akb2-aab)
s' s
.... " 1/101
If we had assumed that the expansions and compressions take
place isentropicaily then 1/Y
(P/P) = (1 +b sin (wt + a) ) .... 1/102
If b«1, then
( E_) 1,1 +Y sin (wt + a) ) ..... 1/103
P
65
Equations 1/80,1/81 and 1/82 would become
a2 + 2ab/y ._t2..... 1/104
s
[(. )] =1+..... 1/105 s
and (A rms (2a2 t1 a4 + 2ab/Y +
17 a2b2 + 165 a`'b2
- ab)
7 7- -j
.... 1/106
respectively.
We see that for a given amplitude of pressure fluctuation b
the effects on the meter differential pressure are greater if the
processes are isothermal rather than isentropic.. Henceforth in the
analysis we will assume the isothermal case in order to account for
the worst situation.
Fig. 22 shows the residual error and Fig. 23 the total pulsation
error that would be incurred if the effects of the fluctuating
density were ignored in a situation in which these fluctuations were
in phase with the velocity. These errors are significant when the
value of b exceeds 0.025. Fig. 24 shows that a pulsating upstream
density tends to increase differential pressure pulsation amplitude
and the effects are most marked at low velocity pulsation amplitudes.
In most pipe work systems, however, wave reflection at
obstructions and at closed and open pipe ends complicate the behaviour
of the pressure and velocity waves. At resonance frequencies a
standing wave pattern is formed in which the pulsation amplitudes vary
sinusoidally with distance along the pipe. When there is a mechanical
device forcing the fluid to pulsate at non-resonance frequencies a
66,
somewhat less well defined standing wave pattern tends to form as
incident and reflected waves combine. In a standing wave system
the fluctuating component of velocity at any time and at any point
along the pipe length can be expressed by
M=M (1 ta sin wt sin 27rX)
..... 1/107
where A is the pulsation wavelength.
The continuity equation states that
aM nap ..... 1/108 ax et
For small pressure fluctuations about a mean value we may
assume an isentropic relationship between pressure and density,
hence
C2 = (6P) = ..... 1/109 dp s dp
From equations 1/108 and 1/109 we may write
p- _- C2 dM
0.0060 1/110 öt A dx
hence
and
C2 , -; =ÄMa2 sin wt cos 2A"
..... 1/111 6t X
p- po =CÄa cos wt cos r ..,. 1/112
It is clear from equations 1/107 and 1/112 that the fluctuating
components of the mass flow rate (and therefore the velocity) and
the static pressure are out of phase by n/2 when there is a standing
wave pattern in the pipe.
67 -
Assuming a phase difference of 7/2 the differential pressure
is given by
Ö= (1 +a sin wt)2 (1 +b sin (wt - 1/2) ) s
Hence it can be shown that
()= (1 +az for any value of b.
Similarly
)1=1 indicating zero residual error. J As
It can also be shown that
..... 1/113
a-{()} 2a2t1a4b23a2b2a4b2 ( rms
. QS [D]8t2tt
16 s
..... 1/114
The above expression indicates that unless the amplitude b A
exceeds a value of 0.1 the value of äS is virtually unaffected by s
the static pressure pulsations.
Comparing the above effects on the differential pressure with
those in the in-phase case density pulsation effects are less when
there is a n/2 phase difference between pressure and velocity.
Although the situations in which the in-phase relationship
exists are probably rare it would be advisable to check this in any
given system. If the static pressure and velocity are in phase with
each other the amplitude of the static pressure fluctuation should be
measured.
The value of b should not exceed 0.025 or the value of
(erns/p) should not exceed �2 0.025, if residual errors due to
static pressure effects are to be avoided.
68
1.5.3 The Acoustic Limitation of the
Quasi-Steady Theory
To a certain extent the acoustic behaviour of flow meters such
as orifices and venturi meters has already been discussed in the
section concerning the temporal inertia effects. These effects are
significant when the acoustic inertahce is a substantial part of the
acoustic impedance of the meter. Acoustic inertance is equivalent
to electrical inductance.
There is, however, another aspect of the acoustic characteristics
of the metering installation which has to be considered before the
quasi-steady and temporal inertia theory can be applied. This aspect
concerns the wave length of the pulsation in relation to the
dimensions of the flow meter.
If these are comparable then it is no longer possible to assume
that the continuity equation can be applied. as if the fluid is
incompressible. The continuity equation 1/2 is as follows
A Sp/St + d(puA)/ax =0
The quasi-steady theory including temporal inertia effects has
been derived assuming that
iA ap/6t1 « +a(PuA)/axl
A second consequence of acoustic wave length concerns the standing
wave pattern.
If a standing wave farms due to wave action in the pipework
system there will be a large variation in the pulsation amplitude
along the length of the pipe. The largest variation will be between
the nodal and anti-nodal points which are a l/4 wave length apart.
69
The condition for the quasi-steady and inertia term theory to
be valid is that the meter dimensions are small compared with the
quarter wave length, i. e.
d«c 4f
f
70
14
E-
0 4J b a) 4J U O
.n
N
to
O
t0 II
0A
II p4
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11 rd ý ro ro ý w C) o N bO
fd "rýi iJ cn
cý ro FOI
r-1 C) v
r
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N
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N
0
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UU
N
N
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ca F3 UN
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ý' P-4 aN as ö 0V QýGN 11 ++ 11 11
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71
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72
N
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r
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t9 cD U
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to n)
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Pt N 0 0 0 P., to pi p1 pi
ri 1
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10 p N
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0 H ri pi N ro
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Co 0
Ln N Ä
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(0
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"
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V f w
un .
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to
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73'
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F
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z 0 U
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74 '
: ti
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p, rn co N 04 b ý1 0 Q1 W
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is 0 O tO H to N C4 Lo t'- N y 0
0 rn co
(a 4 ö ö ö ö
p rn `g' N 0 co a o pq . . . .
ro .ý W 0 ö Ir
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75
FIG. 2' TEMPORAL INERTIA & TOTAL ERROR
ý_ 3.. - r ..
-T 7- - 77 17-- 7
t -z
ýf t
007
- 4 7
1 ;' 0
l 1
} --- } r
.. 3 ::
: .. .,. . . .. ... . . :N .
t -ý .. ..
ý.. (r) . fi . , .. /1 - f i. .....
'rr" .. .. I i ý. .. ý
: [ty 1
f}C" -: °«S :: ýä' 1
: S"
:1 1
x.:
7- i ý'..
t `iii rL. . T;
t Ito
,
l
ýi H.. h h« 'ý ..:
M + . +f . .- »r . -" h y. ýl" :.
' r.. '
W . M. . N+ 4: ,
"{}f r" . ý l r- r , _ ~ r. "
ý iý T" «
.4 3 "+ý ; '. T H + r} ß. T^3 . yý.. . 1br. T. r.
. .. " - t" - r. ý. ' r 1yr t. t -.. r : T.. Y fr+ 1-«. 1i+ . _.. rtý. l r4.. «. t 41: ... +"i. r ri . ri. ,
. ýý
ý i+r. rý r .«.. :
"ý "',
«,: ý
.. . .. . - "
rt
..... .
Ft
..
«+ ý I, i
irr `
k"
t
i..
Li+ ..
`
1 T
.". Lr i
..
ýaý
. .H t+, i ,
.1 .a
«hat
. t..
3,. ýý ý
r .., . T r. . +il }. .... a .. f ý i,:.. '. ý p y Tf. x. " i4T :
.. r
T ! 4 4ý.. f 1t ,.,
. ý, "ýt} .: 'ý +" ý r .. ý
! ý .. +, ti
.
. t ý ý..
L. ^r: i : ýY
, "ý ýý ýt ý 1 i r F:
ý
- ' Fj
ý ýý}Iý +- ? "rT "i ý «,. � t: ht .ý T..
ý aa. h ý: 1y rýt" ; ý" :t .,.. : ., : . r.
. . 7 + . + 7T. h +} ýa
Y .7r f ý 1 ý3 . l . r
M ci ýN0 C0 C\l
N 00 T-
Q,
Wt
V
Lt. zc a0 ý. z to 4 L IL 1- o
O 08
Wz octx
wS 0. W
76
1 N ýn c fj
I j
ICL A L" I i
"+
1r OU 0 1I L
i ä" v -il 0
c
y - 1 .+ O
-I Z
L. 1I I
0 U
.ý
rn
11 ,
a ~ N II rn tmc p 10
11 CL
/ C .+ ,p I1
0 I I.
I 1
00 C1 I
1LL. C36 0 3c
a. o"
u
M° O
n. "
0 0 U)
a 9 .2 a
Q
Qm w
G Al
"ý L v
u s
N gt
u CO
E10
,ýO
M v: 4) L
Ii)
.r 0
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H-
v u U-
l0
6
U-
a:; jwj) G: oJ p': 1 yo auýo rao ýº, ".. a. a
1ý
77
+I0 Throat ap ; ý
0 %
Vintun profila .-vn un f % Throat Vý Itroth
W 2.0 Flow o I
-70
(Symbol Throat Ra No.
. --- 1- P 1.136 r 105 :: 7" 5690 x Ids ic 0 ( 10 IS 20 25 30 3 40 45 0
DISTANCE (in) ALONG VENTURIMETE4 00WNSTREAM FROM INLET TAP Wall pressure distribution along 10 in diamdter 0"6 area ratio, bronze
ve : turimecar (air as workinj fluid)
from Lindley (ref. 43)
I
FIG. 4 PRESSURE DISTRIBUTION
IN A VENTURI
78
0 FIG. 5 VELOCITY COEFFICIENT &
PIPE FACTOR
1.16
1.14
Ü 1.12
z 1.10 W, U L- 1.08 LL w 0 U
1-"06
U 0 1.04
1.02
1.00 0.5
1
1 1
1m =0.64
1
1
\\m=0. 36
m=0.09 r aft- 0.6 0.7 0.8
PIPE FACTOR
0.9 1.0
F= U/u,:
D dý N
V
LO Lo
o (; ) LO o d'
o® aooD
ra
00
o1
a L
4
o Co (0 qr :m , ý- ooo"o : )n/[j O Il`db All 00"13A--,,
O r
0)
6
co
0
N
o
o
0
LO X
6
J N
o Q -- o U 6 0 ý M
2 J " p W O >
a. N (D N
c'i W z 0
z F. a :' 4)
0 Cl a- >
Jp 0
V
IL
--- - t3 V
0 e mu N
T
Oc ýCj c3ý f CV
" ® iff a C7. O t>
c R: D
M
0 -112 ea cv 0 r" r
swi 9. /I
WW
000
OI. LVd 3afli n"ldWV
O F- z N
O Z O ä
co -lb
L <1
< o x o, ob i (D J W N
- 0 < . T
LL 4) O It
CL
O
a N
6 W U O V p Z J E
w b u) N
N
z a N 0
J
ý-- Q. > 0
N ö
V
L
i
-J
81
FIG. 9 KINETIC ENERGY DISTRIBUTION EFFEC1 orifice m 0.64
, amplitude a" 0.8
4.0
3.0 -ý`
'ýS 2.0
f
1.0 I.
0.
2.0 1/2
P 41
.5
1.0
0.5
--cc1Ct) "oc1 s
60
Qýw V '-ats
o__ 0
120 180 240 300 360 wt
60 120 180 240 300 360 wt
82
p
JI Calculated impact pressure
based on $aksmete(f relation shown
Prandfl power law shown -------
Y -4 zi
-Impact pressure In per cent of the differential pressure measured between both upstream tapping
t! points A and a- z
"O -y
" os, ýl 2
'.. ýýesepö ei
' M-04 BMG , 0-
" TffºI£füu SV.
"L M7£1[ AND A. KWÖD£A t*3MM.
tot PIPI DW4 JJMiM. 0 . .. IDRISH014IXMM 0
MA wtmJw D
2m P1
esL -P,, "-
0'
ms06 0fk: --
ap-pO o
m "O"S
IT /ýý O O"
m: 02
"i4 fo' 21f ýýs !11 ý0' 2i
POE REYNOLDS NUMUE*
from Engel & Stainsby (ref. 20)
i
i i i
E i
.I t
r
FIG. 10 IMPACT PRESSURE
-Schematic dig; ram _howing static pressure distribition along a ;i liac in we vicinity of an orifice plate. A, L, ºcn: a conts cta tappiausi a, b, corner
tappin, ̂ , s
FIG. 11
18
16
ý 14
0 12 I
1 0 W .
(1)
W w
a.
I- -6 V
4
2
01 1
0.5 0.6 0.7 O. 8 0.9
PIPE FACTOR U uc
M =0.8
0.7
0.64
1 / 0.6
0.5
IIý, ý, /,
9!:: 1
- 0 op,
f/I ' 0.0 --oo
r
0.2
83
IMPACT PRESSURE & PIPE FACTOR,
orifice with corner tappings
1'0 i l r
84
I=IG. 12 THEORETICAL CONTRACTION COEFFICIENTS (after Von Mises)
1.0
0.9 CC th
0.8 0
0.7 40
t
0.6 E----------; ---- II 0
t
0.2 0.4 0.6 0.8 1.0 d/D
85
O-
FIG. 13 VELOCITY PROFILE EFFECTS
ON CONTRACTION COEFFICIENTS
inferred from Ferron's results on
a square edge orifice m=0.551
1.00
" Cc
obs Cc th
0.99
0.98
0.97
0.96
0,951 _ 0.5 0.6
1
r
M
0.7 0.8 0.9 1.0
PIPE FACTOR F
0
', FIG. '14
1.08
1.04 Ccobs Cc th
1.00
0 0.96
o"s2 L. 103
REYNOLDS NUMBER EFFECTS
ON CONTRACTION COEFFICIENTS
for square edge orifices with
corner tappings
M= 0.64 0.5
0.2 0.1
104 105 -106 107
Rep
(4
ý
J
's
87
FI G. 15 TIME DEPENDENT DISCHARGE EFFECTS
-orifice m. 0.64, amplitude a 0.8 4.0
Ivi-
R
30 Op
s 2.0
1.0
OL 0
2.0 1Q 1/2 P
as 1.5
0
1.0
0.5
OL 0
-- 10 Nk
-Cp(t) ý --Cps
60 120 180 240
wt
'C 300 360
out c
Ds
a(t)
*aft .0 0#0
60 120 180 wt
f
Ig 3
1
i tl
240 300 36Q,
88
aP too V!
0.99
O-VELOCITY PROFILE I UPSTREAM X-VELOCITY PROFILE II UPSTREAM "-VELOCITY PROFILE UPSTREAM
° o x x x
8 I
0.98
ý, ox466, to PIPE REYNOLDS NO R 10'1
Pit. 6 Calibration of 12-In. X 8.0794n. vonlurl color. -on'odc of Velocity distribution
I00
y 90 W u
a 80
a
>I. 4''70
co
0 20 40 60 60 too PER CENT OF PIPE DIAMETER
Fill. 7 Velocity trevors, f upstream of 0.667 ß rails vontvrl motor and 0.743 ß ratio oriflce
N: ýo ö3 iý1
W0.99 o' u WO.
98
0
from Ferron (ref. 23)
FIG. 16 VELOCITY PROFILE EFFECTS ON
VENTURIS i
0.6 O. T 0.8 0.9 1.0
PIPE FACTOR VT-
Fig. I [Sods of volocIty dtshibutlon en 0.667 S. -onlvri m-tw
6-
FIG. 17 CD DATA, SMOOTH ENTRANCE NOZZLES
" small m ratios 0
I
0
i ö ý. ýý I E E
ýo ý
o
ý. 4
r fý ýý
`ý ` ``ý
ý`
ýý ý ý`ý ý`
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iC c
CL
c
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.u rn
v)rn aE
cý ., ( z L: ý o C6 LO 06 L Of
: tj u>- T- orir d tn c oa. c =Co
") v) °' rn 12) cn v o+
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o rn o rn 'ý o cý
ýZ3 .
S
9
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4
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i
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F'IG. 18 CD DATA, CLASSICAL VENTURI TUBES
P1
0)
p
o CD o CD V
T"`
v, 0) 0)
a o a ö ö
91
FIG. 19 EXPANSIBILITY & RESIDUAL ERROR
1R
16
14
12
ER °/o
10
8
9
6
4
2
0
ES= 0.90
0.925
/0.95
0.975
099
0 0.2 0.4.0.6 0.8 1.0 a
%V
FIG. 20
36
32
28
24
ET°/o
20
16
12
0
8
4
OL 0
92
EXPANSIBILITY & TOTAL ERROR
0
0.2 0.4 0.6 a
0.8
5
1.950
1.975
. 990
. 000
1.0
a
FIG. 21
1. 8
1. 6
1.4
1.2 Arms AS 1.0
0.8
rms O P 0.6
0.4
0.2
oý 0
93 EXPANSIBILITY & DIFFERENTIAL
PRESSURE AMPLITUDE
VIA
/ ýý
loop
OOF
0.2 0.4 .
0.6 a
`'S-
0.950
0.975
1.000
0.950 D. 975 1.0001
0.8 1.0
i{
., _ ..... ... 3
94
FIG. 22 UPSTREAM PRESSURE PULSATION
& RESIDUAL ERROR
pressure in phase with velocity
5.0
4.0
c/o R 3.0
2.0
1.0
OL 0
i
7
J
0.2 0.4 0"b o"b
d
b= D. 2
0.1
0.05
0.025
1.0
9b
FIG. 23 UPSTREAM PRESSURE PULSATION
& TOTAL ERROR
pressure in phase with velocity 32 .
28
24
20
ET °/o
16
P
12
8
4
0
b= 0.2
0.1 0.05 0.025
0
16
0 0.2 0.4 0.6 0.8 1.0
a
96
FIG. 24 UPSTREAM. PRESSURE PULSATION
& DIFFERENTIAL AMPLITUDE
pressure in phase with velocity
b I-'.,
_ 1.4
II
1.2
1.0 Arms O s 0.8
0
I
0.6 S
0.4
0.2
0 0.2 0.4 0.6
a
0.2 0.1 0
0.8 1.0
97
Chapter 2.
THE EXPERIMENTAL WORK
2.1 INTRODUCTION
The main objective of the experimental work was to measure
accurately metering errors incurred by orifices and venturi-nozzles
when subjected to sinusoidal pulsations of varying amplitude and frequency.
It was hoped that it would be possible to determine to what extent
the quasi-steady theory was valid, and, if residual errors were
found, to see if these could be attributed to temporal inertia
effects.
At the time this investigation was planned others had been
14, recently completed in the U. K. and U. S. A. 35,50,69
which
had had similar objectives. These previous workers had experienced
considerable difficulty, however, in obtaining conclusive results.
Indeed the American investigator, Sparks69 went so far as to say
that the results for air could not be explained by the quasi-steady
and temporal inertia term theory. Nevertheless the writer thought
that it was possible that the design of their rigs and their experiments
could explain the unsatisfactory results and that important lessons
could be learnt from their experience. Indeed the author was
fortunate in receiving the benefit of Professor Zarek's experience
at first hand.
One of the most important pieces of advice passed on by
Professor Zarek concerned the design of the pulsator. He stressed
the necessity of being able to investigate the effects of pulsation
amplitude and frequency independently. He felt that it was essentihl "
to be able to adjust the amplitude of pulsation while the machine
was running at a previously selected frequency.
98 1
Another important aspect of design concerned the possible effect
of the pulsator in producing swirl and irregular velocity profiles.
Clearly it is impossible to investigate the effect of pulsations
with any precision if these other undesirable flow features are
present. There is considerable evidence from Lugt 44, Zanker8 ' 85
and Sprenkle 72 that swirl and asymmetric profiles produce significant
errors in orifice meters.
Both Earles and Zarek14 and Jeffery 35 placed their pulsators*
about 20 diameters upstream of the orifice. The Earles and Zarek
pulsator was a piston device in a tee-branch at right angles to the
test pipe whereas Jeffery's was an oscillating butterfly valve
within the pipe. There is no direct evidence that these devices
did produce swirl or asymmetric velocity profiles but their results
did show effects which could not be explained by the quasi-steady
and temporal inertia term theory. In particular, Jeffery found
positive residual errors with orifices of area ratio 0.55 and 0.68.
In view of these results it was decided to place the pulsator
some 56 pipe diameters upstream of the test meter and to design a
piston displacement device that was axially symmetric with the pipe.
The rig that Sparks69 used had about 120 pipe diameters between
the meter and pulsator ; ºhich was a siren divice. It seems unlikely
that the supposed deviations from quasi-steady behaviour were due to
swirl or. irregular velocity profiles. There are a number of points
about their rig and experimental technique, however, where the writer felt a different approach might have produced more satisfactory
results. These points are as follows: -
(1) The connecting tubes to the differential pressure transducer 9
contained three way valves and a dead leg (part of an equalizing bypass).
99
Such discontinuities could have produced distorting wave action.
(2) The only measure of pulsation amplitude was of the velocity
pulsation (Urms /Ü). This measurement was made with a hot wire
anemdmeter probe placed immediately downstream of the orifice in
the issuing jet. Since a hot wire probe can only give a local
velocity measurement assumptions must be made about the profile of
the dc and ac components of flow in the issuing jet. Differential
pressure pulsation amplitude was not measured.
(3) Tests appear to have been conducted at randomly selected
pulsation amplitudes and frequencies. It is not possible, for
example, to see the effect of increasing frequency while maintaining
pulsation amplitude constant.
(4) In their theoretical analysis Strouhal Number was not
coupled with a waveform factor. In the report and paper69' 70 it
is mentioned that there was severe harmonic distortion at times and
yet it was not recognised that this greatly affects the effective
Strouhal Number.
While the author's investigation has been in progress several
related investigations have been reported. Benson and Shafie7 described
the behaviour of a square edge orifice metering flows similar to
those found in I. C. engine intakes. They explained their results
entirely in terms of the square root effect but since their frequencies
were very low temporal inertia effects were unlikely to be significant.
Kastner, Tindall and Williams40 described another investigation
concerned with metering air flows to I. C. engines. A particularly
interesting feature of their equipment was the electronic square
rooting device used with two separate miniature pressure transducers
which eliminated both the square root error and the need for
connecting pressure leads.
100
Yazici83 working under the supervision of Professor Fortier
related the metering errors to the resonance frequencies of the
flow-system. No attempt was made to separate square root and temporal
inertia effects.
Crimson and Hay26 have published work in which the behaviour
of orifice meters was correlated with differential pressure peak-to-
trough amplitude and Strouhal Number. Unfortunately, although
waveform was maintained constant in a series of tests on a given
meter no attempt was made to account for the effect of waveform on
their results. For this reason it is not possible to compare their
results with the quasi-steady and temporal inertia theory.
The author is unaware of any results which can be compared
I
directly with his own. The main reason for this is that in the past
the effects of waveform have not been properly accounted for. In
the present investigation the pulsator produces basically sinusoidal
waveforms although with slight distortion. As far as the square root
error theory is concerned the distortions do not matter provided
the r. m. s. amplitude of the differential pressure pulsation is
measured. Harmonic distortion does affect the effective Strouhal
Number, however, and it has to be admitted that so far no attempt
has been made to compensate for it. It is thought unlikely that
- the harmonic distortion factor exceeded 1.5 at any time and that
during most tests it was less than 1.1.
2.2 THE DESIGN OF THE, EXPERIMENTAL
RIG AND INSTRUMENTATION
The apparatus can be considered to consist of three main parts
namely, the basic air flow measurement rig, the machinery for generating
pulsations, and the instrumentation. These are described separately
below.
2.2.1
101
The Air Flow Rig
Air was chosen as the working fluid so that it would be possible,
when required, to study the effects of compressibility. Provided that
differential pressures across the flow meter are sufficiently low,
however, the fluid can be regarded as incompressible and thus the less
complicated type of flow can be studied in the first instance.
It was decided to supply the rig with air through a critical
nozzle-and to control the flow rate by regulating the pressure
upstream of the nozzle. The layout of the rig is shown in Fig. 25(page 131)
The pressure regulator is of the type in which the delivery pressure
is controlled by a pressure applied externally to a diaphragm. This
applied pressure is set manually with a control valve and by continuously
adjusting it the pressure upstream of the critical nozzle as indicated
on a precision two-revolution bourdon-tube gauge, can be maintained
to within ± 1/10% during both steady and pulsating flows. The fact
that the pulsator is downstream of the critical nozzle ensures that
the control of the flow is independent of whether the pulsator is
on or off.
A further safeguard against severe pulsations affecting the
critical nozzle when operated near the critical pressure ratio is
provided by a settling chamber of volume 10 ft3 and a flow resistance
within the pulsator of the order of 2 lbf/m2. This damping system
also ensures that the noise from the critical nozzle does not travel
downstream to the test section.
The test section consists of a straight length of 3.2 inch
internal-diameter cold drawn steel pipe. There is a length of 56 pipe
diameters upstream of the test flow meter and a similar length down-
stream (see Plate 1, page 126). As the. pipc discharges into the laboratory
the pressure in the test section is approximately atmospheric.
102
2.2.2 The Pulsator
- From the experience of previous experimenters such as Zarek87,
Earles14, Jeffery 35, Moseley 51, Sparks 70 it was evident that it was
essential to be able to control the pulsation amplitude independently
of frequency. It was also evident that in most of the experimental
rigs previously designed insufficient precautions had been taken to
ensure that other flow irregularities such as asymmetric velocity
profiles and swirl were not introduced by the action of-the pulsator.
The choice of pulsator was also influenced by the desire to
generate a regular and reproducible wave form (preferably sinusoidal)
for the initial stages of the investigation. Both reciprocating piston
and rotating siren type devices were considered and it was decided that
the piston type would be more capable of generating large amplitude
pulsations in the frequency range of most interest to industry.
Fig. 26 shows a section through the pulsator designed and built
for the investigation. The 6.5 inch diameter piston reciprocates
in a cylinder coaxial with the test section. The air first passes
through an annular flow straightener on the outside of the cylinder,
then through a flow resistance element composed of latex foam
material, and finally enters the converging inlet to the pipe through
an annular slit J" wide. Plate 2 (page 127) shows a view of the
pulsator with the converging inlet piece removed.
The air trapped behind the piston has access to a sealed
expansion chamber external to the pulsator. and is thus maintained
at almost constant pressure throughout the pulsation cycle (see
Plate 3, page 128). This air could not be vented to the atmosphere
since there would then have been a possibility of leakage of air past
the piston.
103
The piston was driven by a crank with a stroke changing mechanism
shown in rig. 27 and Plate 3. The crank shaft (C) is set eccentrically
in a hub which is the casing of a rotating gear-box. The gear-box
is a double-worm speed-reduction type the output shaft of which is
the crank shaft. To change the stroke the crank shaft must be
rotated with respect to the gear-box casing and this is done by
rotating the input shaft (B) which is coaxial with the gear-box
hub. The speed ratio between input and output shafts is 729: 1 and the
worm gears are such that they are self-locking. The input shaft B
is rotated with respect to the hub by means of a differential gear-box
(see Plate 4). The relationship between the number of turns of the
handle on the input shaft to the differential gear-box and the stroke
of the pulsator is shown on Fig. 28. The relationship is very nearly
linear and an j turn of the handle produces a stroke change of less
than 0.008". Unfortunately the back lash in the system makes the
resolution of the system less fine than it would appear but, nevertheless,
if care is taken to approach the required setting from the same
direction each time the device is very precise.
The rotating gear-box is driven by a7h. p. induction motor
through an infinitely variable speed hydraulic gear-box (see Plate 4).
There is a 3: 1 step-up gear ratio between the drive and the rotating
gear-box hub which gives a theoretical maximum speed in excess of
4000 rev/min. The rotating gearbox, however, was not intended to be run
faster than 3000 rev/min.
The stroke-changing mechanism did require a certain amount of
development before it achieved the performance required of it. The
two main problems were balancing of the rotor and the overheating of
its bearings. The writer here acknowledges the assistance of
. 104
Mr. P. M. Downing who was given the task of overcoming these problems
as his contribution to the development of the rig which he was also
using for his own project.
The balancing problem was solved by installing a dummy crank
diametrically opposite to the working crank and with the same
eccentricity. '
The dummy crank greatly helped to compensate for the out of
balance forces which changed as the stroke was altered. It was found,
however, that additional weights had to be bolted on to tha periphery
of the hub to reduce vibration to an acceptable level.
Plain leaded phosphor bronze bearings originally designed for
the rotating gearbox hub were found to overheat and to absorb too
much power. The result was that high speeds could not be achieved
and time was wasted waiting for the bearing to cool between runs.
The plain bearings were eventually replaced with roller bearings
and these were found to completely cure the problem.
The pulsator and stroke changing mechanism was first run in
the summer of 1969 and by December 1969 the speed was limited to
about 17 Hz. By December 1970 with the installation of the roller
bearings the design speed of 50 Hz was achieved.
It should be understood that it is not possible to run the
pulsator at 50 Hz with large strokes-because of the limitation of
power and vibrations.
Currently the maximum stroke at 50 Hz is about 4"
the maximum stroke at 30 Hz is about 11"
and the maximum stroke up to 18 Hz is 5".
In the tests completed so far these strokes have been sufficient
to generate the required pulsation amplitudes over most of the
frequency range.
log
2.2.3 The Instrumentation
An instrument block diagram is shown on rig. 29 and the
arrangement of instruments and test section can be seen in Plate S.
2.2.3/1 Pressures
The most important quantity to be measured is the differential
pressure across the meter. The instruments used were inductive
type differential pressure transducers.? ` In these transducers
the diaphragm is housed centrally and the upstream and downstream
pressures are connected through symmetrical i" bore passages to
each side. The dead space at the diaphragm is negligible and the
leads connecting the transducer are i" bore nylon tubes and are
about 6 inches long.
Similar pressure transducers were used to measure the upstream
and downstream static pressures at points about 3 pipe diameters
on either side of the flow meter.
In addition to the electronic transducers U-tube water manometers
were also used to indicate the time-mean differentail and upstream
static pressures to prövide a rough check on the behaviour of the
transducers. The U-tube readings were not otherwise used.
The signals from the pressure transducers were processed in
ways indicated on the block diagram shown in Figs 29,32,33 and
as listed below: -
(1) The signals were displayed at all times on an oscilloscope
screen.
(2) The amplitude of any harmonic could be checked on a wave-
analyser. The amplitude of the second harmonic was always checked.
* S. E. Laboratories Ltd. Type SE 150/5947/50" WG and 5 p. s. i.
106 ,
(3) The root mean square of the alternating component of the
signals were measured on a true r. m. s. meter - i. e. (perms' Pdrms' Arms)
<4) The signal was averaged using an R. C. filter and measured
on a Solartron digital voltmeter (DVM) - i. e. (Fu-' pd' X)
(5) In the case of the differentail pressure the signal was
processed by the square rooting circuit, averaged with an R. C. filter,
and measured on the DVM - i. e. ( () )
The square rooting circuit was made up from solid state integrated
components supplied by Ancom Ltd. The circuit is shown in Fig. 30
and it comprises a solid state amplifier with squaring modules
placed in the-feed back loop. Its calibration (voltage output versus
(voltage input) ) was found to be linear and its accuracy was
within ± 0.5% of the output. The device is capable of processing
both positive and negative signals.
The linearity and drift characteristics of the transducers were
of great importance. Calibration of the transducers was carried out
every day and their linearity checked. The effects of drift were
eliminated by carrying out pulsating and the corresponding steady
flow runs within minutes of each other and checking the zero drift
before and after the runs.
2.2.3/2 Velocity
A hot wire anemometer probe''twas used to monitor the centre-
line velocity and to measure velocity profiles. The probe was
situated about 3 pipe diameters upstream of the flow meter and could
be traversed across the pipe section by means of a micrometer-screw
head.
A diagram showing the L-shaped probe mounted in the pipe with
the traversing arrangement is shown in Fig. 31. The anemometer
Disa Electronik, Miniature "L" Probe Type 55A36
107.
equipment was supplied by DISA and comprised of a constant temperature
anemometer, an auxiliary unit confining high and low pass filters,
a linearizer, a true r. m. s. meter and a digital voltmeter.
The output of voltage of a constant temperature anemometer is
proportional to approximately the 4th root of the local mass flow
rate past the wire. It is thus necessary to linearize the output
if large velocity fluctuations are to be measured. The linearized
signal is then processed in the following ways (see also Figs. 29,34): -
(1) It is displayed on an oscilloscope screen (the centre
line velocity and meter differential pressure were normally displayed
together on a double beam oscilloscope).
(2) The harmonic content (particularly the second harmonic)
was checked on a wave analyser.
(3) The root mean square of the alternating component was
measured on the true r. m. s. meter. (The'DISA r. m. s. meter-was
used for both the velocity and pressure measurements. )
(4) The signal was averaged with a low pass filter and measured
on the DISA digital voltmeter.
The linearized hot wire output was calibrated every day against
the centre line velocity established by means of a pitot static
tube.
2.2.3/3 Instrument Frequency Response Tests
Tests were carried out to check that each electronic measuring
system was giving the same response to a given signal over a sufficiently
wide frequency range (i. e. that the frequency response characteristic
was flat for a frequency range of, say 0- 500 Hz). Although the
frequency response was in most cases known for individual instruments
the fact that many were connected together with a number of resistance-
109
capacitance filters made it necessary to check the complete signal
processing systems. It would have been desirable to check the
response of either the pressure transducers to actual pressure
fluctuations or the hot wire to actual velocity fluctuations. This
was Dot possible, however, as there were means available by which
fluctuations of constant amplitude and variable frequency could be
generated. It was only possible to feed into the processing systems
a constant amplitude voltage fluctuation from a signal generator and
to vary the frequency of this signal.
The arrangement of the signal processing system for the differential
pressure is shown in Fig. 32. The test signal from the signal
generator was applied to the input to the square-rooting circuit for
the frequency response tests.
A digital voltmeter was used to monitor the a. c. amplitude
and the d. c. bias of the sine wave test signal to check that these
did not alter as the frequency was varied. It was essential that
the test signal had a substantial d. c. bias so that the response
of the square rooting device could be adequately tested.
The arrangements of the signal processing equipment for the
static pressures and the hot wire signal are shown in Figs. 33
and 34k.
The results of frequency response tests carried out with the
electronic systems arranged as in Figs. 27,28,29 are shown in
Table 8. The frequency response of all systems is substantially
flat up to a frequency of 500 Hz.
A frequency test such as described above revealed that a filter
in the hot wire signal processing circuit was attenuating the
fluctuations in the working frequency range. When the fault had been
109 ,
rectified it was possible to retrospectively correct all the
velocity r. m. s. readings previously taken.
2.3 EXPERIMENTAL PROCEDURE
Before any test runs were made on a given day all the pressure
transducers and the hot wire anemometer were calibrated and checked
for linearity. To calibrate the pressure transducers the end of
the test pipe was closed off with a steel plate and compressed air
was fed into the pipe, the pressure being controlled by a system
of needle valves for feeding and bleeding. The pressure in the pipe
was measured with either a liquid filled single limb manometer for
pressures above 40 in WG or with a Betz-type projection micromanometer
for lower pressures.
The linearized signal from the hot wire anemometer was
calibrated against the know centre line velocities at given flow
rates. The flow rates were determined by means of the critical
nozzle and the centre line velocities by means of a pitot static
tube.
When all the equipment checks and calibrations had been completed
the following procedure for carrying out a test was adopted:
(1) Immediately before the test all the instrument outputs
were zeroed.
(2) The air was turned on and controlled at the required flow
rate as indicated on the precision pressure gauge.
(3) The differential pressure as measured by the transducer
and indicated on the DVM was read so that the required value of
Arms could be calculated.
(4) The pulsator was switched on and adjusted to the required
frequency. The stroke was also adjusted to give the required
pulsation amplitude (a. s)
iia
(5) While pulsation conditions and the flow rate were .
accurately maintained constant (by a technician) readings of all
the instruments were made with particular attention being paid to
the accuracy of the time-mean differential pressure and the time-mean
square-rooted differential pressure.
(6) The pulsator was switched off and the readings of the
steady flow differential pressure and square-rooted differential and
other quantities were immediately made.
(7) The flow was switched off and the zero readings of all
instruments were checked.
2.4 DISCUSSION OF EXPERIMENTAL RESULTS
The majority of the test series were designed to test the
validity of the quasi-steady and temporal inertia effect theories.
It was recognised that there were likely to be other effects which
varied with frequency rather than Strouhal Number. For this reason
it was decided to conduct series of tests at a fixed pulsation
amplitude as measured by (Arms /As) with frequency as the controlled
variable. The frequency range was 0- 50 Hz and the maximum
amplitude of Arms/As used was 1.5. This was the amplitude which
marked the threshold when flow reversal started to occur at low
Strouhal Numbers. Three sizes of orifice plate (m = 0.104,0.393,
0.612) and two sizes of venturi-nozzle (m = 0.063 and 0.393) were
tested. The constant amplitude test series were repeated at
different Reynolds Numbers (based on the bore diameter) in the range
3-23x10`'.
111
2.4.1 The Quasi-Steady and
Temporal Inertia Effect Theory
Figs. 35 and 36 show graphs of total pulsation error plotted
against the Effective Strouhal Number, (H fd). H, the harmonic
ü distortion factor, was assumed to be unity
d for all the tests.
Fig. 35 shows the results for the two larger orifices and demonstrates
that the error does decrease with increasing Strouhal Number as the
theory predicts. Comparison with the theoretical curves shown in
Fig. 2 (page 75) shows that the maximum frequencies would have to
be increased by a factor of 10 before one could expect the pulsation
error to decrease to zero.
The results for the smallest orifice are shown on Fig. 36.
Since it was not possible to attain such high Strouhal Numbers with
the small orifice the temporal inertia effects are not so evident.
The considerable amount of scatter among the experimental
points indicates the extent of the discrepancy between the actual
behaviour and that predicted by the simple quasi-steady and temporal
inertia term theory. The scatter is particularly pronounced for
the high pulsation amplitude tests and for the tests on the smallest
orifice.
The temporal inertia effects are clearly shown on Figs. 37 - 40
where the pulsation residual errors are plotted against the Effective
Strouhal Number. The solid line curves represent the theoretical
relationship expressed by equation 1/31. The quantity J (equation 1/18)
was evaluated for the 2" diameter orifice and nozzle assuming two
values of the effective Le. It was arbitrarily assumed that this
length was
a) twice the orifice or nozzle diameter
b) equal to the diameter.
112, ,
Despite the fact that the results for the smallest orifice are
plotted with those for the two larger sizes the scatter of the
experimental points is very much reduced and the disposition of the
points clearly follows the shape of the theoretical curves. It
would be fair to conclude that for orifices at values of (H rd)
Ud below 0.02, the effects of pulsation can mostly
be explained in terms of the quasi-steady theory whereas at higher
values temporal inertia term effects are significant. For venturi
nozzles temporal inettia effects appear to become significant at
values of H fd%ud above 0.01.
Clearly other influences are present besides those of the square
root and inertia term effects and these require further discussion.
2.4.1/1 Area Ratio
At first sight it appears from the results that the behaviour
of the smallest size orifice does not conform so well to the theory.
The scatter of points is greater than that for the two larger
orifices. If this observation indicates that quasi-steady assumptions
are less valid for small area ratio orifices this would be a most
surprising finding. The analysis contained in Chapter 1 demonstrates
clearly that changes in instantaneous velocity profile and Reynolds
number should only be significant for the largest values of area
ratio. Indeed it is difficult to provide any explanation why a
small area-ratio orifice should not behave in accordance with the
quasi-steady theory. Without, ruling out the possibility of such
an explanation, 'however, it should be noted that the time-mean
Reynolds Numbers of the tests with the small orifice were
lower than for the others.
113, '
2.4.1/2 Reynolds Number
Fig. 37 shows the results of e series of tests made at various
flow rates for the three sizes of orifice plates when (Arms/As) was
1.5. The results for the 2" orifice at Reynolds Number 10.5 x 104
and for the 2.5" orifice at Reynolds Number 15.8 and 10.1 x 104
fall convincingly along the predicted curve. All Reynolds Numbers
quoted here are based on the meter bore. The results of tests at
lower Reynolds Numbers for the same orifices however, tend to be
scattered above this curve, i. e. at a given Strouhal Number the
negative residual errors are not so large. The same effect is
repeated in tests at lower pulsation amplitudes shown in Figs. 40
and 41.
This effect is perhaps most clearly shown on Figs. 45 and 46.
This shows results for the 2" orifice as plotted in Figs. 37,38
but with pulsation frequency as the abcissa. At a given frequency
Strouhal Number-increases as Reynolds Number decreases and thus,
according to the theory, the curves on Figs. 45,46 should fall one
below the other. In fact the lowest Reynolds Number (5.2 x 104)
curve is noticeably not in the same 'family'. In Fig. 44 the
same kind of effect to a lesser extent is apparent with the lowest
Reynolds Number curve (7.1 x 104) obtained with the 2.5" orifice.
It thus appears that there might be a critical Reynolds Number in
the region of 70 000 below which the pulsation errors tend not to
conform to a predictable behaviour.
In view of this observation it is perhaps less surprising that
the results obtained with the smallest orifice (m = 0.103) were
so scattered. The Reynolds Numbers for these tests ranged from 3.5 to
6.6 x 104. The reason why tests were not carried out at higher
Reynolds Numbers was because it was not possible to generate the
114
required amplitude of pulsation over a reasonably wide frequency
range at the higher flow rates. It is interesting to note, however, that
raising the Reynolds Number means raising the differential pressure
with the_attendant possibility of compressibility effects. At
Reynolds Number of 66 000 the equivalent steady flow expansibility
factor of the small orifice was 0.993. At a Reynolds Number of 124 000
the expansibility factor would be 0.978 and according to Figs. 19 and
21 residual errors due to compressibility effects would amount to
3% when the pulsation amplitude (ASS /as) is 1.5.
If there is minimum Reynolds Number for stable behaviour under
pulsating conditions it is difficult to put forward an explanation
without making detailed measurements of the flow patterns through
the orifice in this apparently critical regime.
An explanation may come to light on measuring the flows in
the jet issuing from the orifice. Sparks70 has already reported jet
instability under pulsating conditions. It is possible that the
effective vena contracta diameter does not have a stable time-mean
value under pulsating conditions at low Reynolds Numbers. The
effects of a cyclic variation in the diameter has been investigated
in the theoretical analysis in Chapter 1 and appear to be negligible
as far as residual errors are concerned. The analysis, however, did
assume a quasi-steady variation of contraction coefficient with
Reynolds Number. This assumption may be invalid particularly at
low Reynolds Numbers.
In the above discussion we have assumed that the low Reynolds
Number points are disposed above the others in Figs. 37 - 39
because there is an effect producing a positive residual error
which partially offsets the negative error due to the temporal inertia
term. An alternative explanation is that at low Reynolds Numbers the
1151
residual error is less because the temporal inertia is less.
There is a possibility that such a reduction may be due to a
decrease in the effective length of the restriction.
2.4.1/3 Effective Length of the Restriction
The solid line curves in Figs. 37 - 40 are the theoretical
residaul errors due to temporal inertia effects calculated for the
2.0" orifice and venturi nozzle assuming values for the restrictions'
effective lengths of twice the bore diameter. For the 2.0" orifice
additional curves were calculated assuming that the length was equal
to the bore diameter.
The results indicate that at Reynolds Numbers above about
70 000 the effective length ratio (Le/d) -s between 1 and 2 for an
orifice and about 2 for a venturi nozzle. At lower Reynolds Numbers
the length appears to decrease assuming that only temporal inertia
causes significant residual errors.
The majority of published data on orifice effective lengths
is due to Thurston and his co-workers ?4 78
. In an early work 76 he measured I
the acoustic impedance of a small orifice in the wall of a vessel
through-which passed a 'sinusoidal alternating flow without any steady
flow component. Thurston found that the effective length of the
56 orifice was as predicted by Lord Rayleigh
i. e. Le =L+ 3n
, provided the flow through the orifice had a
simple piston like motion. At large amplitude alternating flows a
vortex structure in the issuing jet was observed and this coincided
with a slight changed in the effective length. In a further
investigation Thurston and Wood77 discovered that when the orifice
was in a. circular tube a further correction to the effective length
of the orifice was necessary to allow for the effect of the tube
116 ,
walls. Moseley51 expressed Thurston's and Wood's results by
Ld ät(
3n ' l. 06ß ) ..... 2/1
where ß (orifice/pipe diameter ratio) has the limits 0<$<0.7.
In further investigations Thurston, Hargrove and Cook7S added a
steady flow component to the alternating flow and measured the
variation of the orifice effective length with flow velocity and the
pulsation parameters. They found that despite the variations the
effective length was still comparable to the actual bore length rather
than to the bore diameter. Unfortunately the maximum value of the
steady flow Reynolds Number (Red) did not exceed 3000 and the results
are thus hardly relevant to fully developed turbulent flow conditions.
Nevertheless they provide further evidence that effective lengths
are likely to decrease with decreasing Reynolds Number.
According to Mottram's results, Le/d 1 when Red > 70 000
According to Thurston's results LeId 0.1 when Red< 3000
Further work is required to determine the variation of (Le/d) at
intermediate Reynolds Numbers.
2.4.2 Frequency Effects
When the pulsation errors are plotted against Strouhal Number
the results show a definite trend in line with the quasi-steady and
inertia term theory. There are other minor effects, however, which
show-up when the results are plotted against pulsation frequency.
Figs. 41,42 and 43 show total and residual errors for the
three orifices at various pulsation amplitudes.
If the square root error and inertia effects were the only
ones present curves of total error should be flat at low frequency and
117-
decline graudally at higher frequencies, curves of residual error
should start at zero and show a gradual increase in negative error
at high frequencies. Instead, however, although the downward
trend at high frequencies. is apparent the curves show a number of
humps and troughs which consistantly occur at the same frequencies,
whatever the size of orifice or the amplitude of pulsation.
Figs. 44 - 47 show the residual erros alone plotted against
frequency with Reynolds Number as parameter. At a given frequency
Strouhal Number increases as Reynolds Number decreases and, therefore,
according to theory the lower Reynolds Number curves should correspond
to the largest residual errors. Generally this is the case but as
discussed previously the curves at Reynolds Numbers of 70 000 and
below tend not to follow this rule. On comparing the humps and
troughs in the various curves one notices that these seem to be
more pronounced at the lower Reynolds Numbers and indeed the lowest
Reynolds Number curves tend to have some shapes not present in
the others. These effects tend to bear out the conclusion already
reached that below a certain Reynolds Number pulsation effects
tend to be less predictable.
The fact that there are certain effects dependent purely on
frequency (not Strouhal Number) suggests that the acoustic
characteristics of the system are significant.
2.4.2/1 Acoustic Characteristics
The acoustic characteristics of a system can be determined by
varying the frequency of a constant amplitude disturbance and
measuring the amplitude of the pressure and velocity pulsations at
points within the system. At resonance frequencies a strong
standing gave pattern forms with nodes and antinodes of velocity
118
pulsation occuring alternately at a wave-length intervals along
the pipe. Since there is a source of forced pulsation in the flow
rig there is a standing wave formation at all frequencies. As the
frequency varies the positions of the nodes and antinodes move along
the pipe.
Figs. 48 and 49 show the non-dimensional amplitudes of pulsation
in the region of the orifice for the 1.0" and 2.0" diameter sizes
respectively. The amplitudes measured were of the velocity and
static pressure three pipe diameters upstream, the differential
pressure, and the static pressure three pipe diameters downstream
of the orifice. The pulsator stroke was constant at about i" for
all the measurements and the flowrate was constant for each series of
tests.
The most significant feature of the characteristics is the
velocity pulsation node at 21.5 Hz. Velocity antinodes which
correspond to resonance conditions as far as the orifice is concerned
occur at frequencies depending on the size of the orifice. For
the 1" orifice there are peaks at 17, and 37 Hz and for the 2"
orifice there were peaks at 10,30.5 and 46 Hz.
In Fig. 45 the length of stroke necessary to generate the
required constant pulsation amplitude (ems/As = 1.5) is plotted
0
in addition to the residual error curves for the 2.0" orifice. As
one might expect the minima of the stroke curves occur at the same
frequencies as the amplitude maxima in the constant stroke curves
on rig. 49. An interesting feature of rig. 45 is the similarity
in shape between the residual error and stroke length curves.
Large stroke lengths are required on either side of the nodal
frequency and there is also a noticeable upward trend in the
j
119-,
residual error curves. The same feature is repeated in many other
test series with 2.5 and 2.0" orifices and thus it would appear
that there is a connection between residual error and the acoustic
characteristics of the pipe system.
Clearly one or more of the pulsating flow parameters which
vary with the relationship of the frequency to the characteristic
acoustic frequencies of the system is affecting the meter behaviour.
The most likely parameters to have an effect are the profiles of
the steady and alternating components of velocity, and the amplitude
of the upstream and downstream static pressure pulsation.
2.4.2/2 Velocity Profile Variations
Reference to Ferron's23 results summarized in Table 2 (page 72)
indicates that when the pipe factor is increased from 0.91 to 0.96
orifice plates (of area ratio 0.55) with. flange taps and D and D/2
taps would be in error by about + 1.2%. Orifice plates with corner
taps are likely to be slightly more sensitive due to the contribution
of the impact pressure effect but reference to Fig. 11 shows that
extra errors due to 5% pipe factor change are unlikely to amount to
more than about 4%.
Figs. 50 and 51 show both residual errors and time-mean pipe
factors plotted against frequency for the 2.0" and 2.5" orifices.
The time-mean pipe factors were obtained by multiplying the ratio
of the centre line mean velocities ( 8/ Cp) by an assumed
steady flow pipe factor of 0.848. Values of pipe factor close or
equal to this figure were obtained on a number of occasions by
pitot-static tube traverse and were assumed to remain constant
over the whole range of operating conditions. It is noticeable
that pulsating flow pipe factors are consistantly higher than the
120 ,
steady flow value and that there is a slight tendency for the value
to rise near the nodal frequency of 21 Hz. The variations, however,
are hardly large enough to account for all the irregularities of
the residual error curves although they are probably a contributory
factor for the large area ratio meters. The pipe factor variations
which occurred with the 1.0" orifice are not plotted since small
area ratio meters are not sensitive to velocity profile changes.
Variations in the profile of the alternating velocity component
were not systematically measured. It is important that these should
be measured in the continuing programme of investigation to see
if their variations correlate with those of residual error.
2.4.2/3 Static Pressure Pulsations
In the analysis of the effects of upstream static pressure
fluctuations (section 1.5.2) it was demonstrated that if a standing
wave pattern existed in the pipework system then the static pressure
and velocity fluctuations would be out of phase by n/2. When this
is the case the effect of static pressure fluctuations on pulsation
errors would theoretically be negligible. In a system in which
there was no wave interaction with pipe ends or obstructions, and
therefore no standing wave pattern, the pressure and velocity fluctuations
would travel in phase with each other. In that case the static
pressure fluctuations would be of great significance and the resulting
errors should follow the pattern shown in Figs. 22 - 23.
An experiment was carried out to measure the phase relationship
between the pressure and velocity fluctuations. The hot wire probe
and pressure transducer giving the signals were positioned in the 0 same plane in the pipe about three pipe diameters upstream of the
flowmeter. The two signals were displayed on a two-beam storage
oscilloscope.
121
With a fixed pulsator stroke and flow rate the frequency was
varied over the range 0- 50 Hz while measurements of the phase
difference were made on the storage oscilloscope. The measurements
had an accuracy of only about ± 10 degrees at best but this was
thougit adequate. The results are shown on Fig; 52. As expected
the phase difference was n/2 near the nodal frequency of 21 Hz.
The interesting feature is that below the nodal frequency the
static pressure leads the velocity fluctuation by n/2 ( more like
n/4 for the smallest orifice) but above it the pressure lags
behind the velocity.
At higher. frequencies the phase relationship depends on the
area ratio of the orifice. With no orifice (m = 1.0) the phase lead
of the pressure changes abruptly from - n/2 to + it/2 at 42 Hz
(2 x nodal frequency). With the 2.0" orifice (m = 0.4) the change
over from -ve to +ve lead occurs gradually between 33 and 50 Hz.
With the 1.0" orifice (m = 0.1) the -ve phase lead, gradually
declines to zero from 23 Hz to 45 Hz.
According to the theory in Chapter 1 the effect of static
pressure fluctuations are negligible if they are in phase with the
veloicty. This theory assumes that the static pressure fluctuations
are only important in so far as the fluid density is a term in the
flow equation. There is still the possibility, however, that pressure
fluctuations in some way affect the flow patterns through the orifice.
For this reason it is interesting to see if there is any correlation
between the upstream and downstream pressure fluctuation amplitudes
with residual erros.
Figs. 53 - 55 show both residual errors and pressure fluctuation
amplitudes plotted against frequency. The steep rise of the pressure
amplitudes on either side of the nodal frequency is striking and in
122.
the case of the 2.0" and 2.5" orifices this seems to match the
upward trend of the residual error curves. The same cannot be said
of the results for the 1.0" orifice, however, and this makes it
difficult to draw any conclusion on the significance of the static
pressure fluctuation amplitudes. Further experiments involving
measurements in the vicinity of the jet are necessary to determine
if. or how the pressure fluctuations affect the flow patterns.
2.4.3 Venturi-Nozzles and Compressibility Effects
It has already been noted from Fig. 40 that venturi-nozzles
have similar residual error/Strouhal Number characteristics to
orifices. Fig. 40 also shows, however, that there are some effects
which cannot be accounted for by the simple quasi-steady and temporal
inertia theory. One of the noticeable features is the 4% bandwidth
of measured residual errors at low Strouhal Numbers. To a certain
extent this can be explained in terms of compressibility effects.
In Figs. 56 and 57 the unbroken curves are the. measured residual
errors plotted against frequency. The broken line curves are for
the same results but with the theoretical contribution of compressibility
effects subtracted from the measured residual error. The correction
was determined by means of Figs. 19 and 21 for the equivalent steady
flow expansibility factor for each test series. The highest. Reyrold
Number series for the 0.8" nozzle had an equivalent steady flow
expansibility factor of 0.975 and the theoretical compressibility
residual error amounted to +2%.
It should be noted that for a given differential pressure the
expansibility factor for an orifice is much nearer unity than it is
for a venturi or nozzle. For this reason compressibility effects
have been apparent in the tests on the venturi-nozzles but not in 'those
on orifices. Further evidence of compressibility effects is
shown on Fig. 62.
123,
2.5 CONCLUSIONS
The Quasi-Steady Theory Including
Temporal Inertia Effects
The results for the orifices and venturi-nozzles subjected to
sinusoidal pulsations indicate that provided the effective Strouhal
Number, (H fd1Ud), is sufficiently low most of the metering error is
due to the square root effect.
Residual errors due to non-quasi-steady behaviour but
excluding temporal inertia effects may amount to ± 1% when total
errors are about 12%, and up to ± 2% when total errors are 25%.
Suggested values for the upper limit of effective Strouhal
Number for which temporal inertia effects can be neglected are as
follows:
orifice meters with mß+0.39,0.61 (H fd/Ud)max " 0.036
corner tappings ) mov 0., 10 (H fd/U d)max ' 0.01
venturi-nozzles, all sizes (H fd/Ud)max " 0.01
The figure for orifices, with an m-ratio less than 0.4 is probably
over-pessimistic and may be revised when more results are available.
When the differential pressure pulsation amplitude is held
constant the total error decreases as Strouhal Number increases
above the limiting values. At the same time negative residual errors
increase if the square root error is eliminated. The trend indicates
that effective Strouhal Numbers of about 2 would be necessary before
total errors approached zero at values of (dam/As) up to 1.5.
The effective lengths of the orifices (m = 0.393 and 0.612)
were between 1 and 2 bore diameters and about 2 throat diameters
.ý
for the venturi-nozzles (m = 0.0625,0.393). There was some
124 ''
evidence that the effective lengths decrease at throat Reynolds
Numbers below 70 000.
Frequency Dependent Effects
The behaviour of the meters is dependent to a certain extent
on frequency as well as Strouhal Number. The tendency is more
pronounced as meter area ratio decreases and as pulsation amplitude
increases. The behaviour suggests a sensitivity to changes in one or
more flow parameters dependent on tie acoustic characteristics of
the system. The results obtained showed no conclusive evidence as
to which parameters were significant in this respect.
The Behaviour of Venturi-Nozzles
Compared with Orifices
The tests showed no significant difference in the behaviour
of orifice and venturi-nozzle meters in pulsating flow conditions.
The wide bandwidth of residual errors (+2 - =2%) at low Strouhal
Numbers can be accounted for by compressibility effects. These
effects were only evident with the venturi-nozzles because they
have lower expansibility factors than orifices for the same
differential pressure.
6 Compressibility Effects
Compressibility effects were not systematically investigated
bit the analysis in Chapter 1 indicates that the above conclusions
on the experimental work only apply when these effects are negligible.
Compressibility effects are theoretically negligible when:
a) The equivalent-steady-flow expansibility factor, cs, > 0.99
b) The upstream static pressure pulsation amplitude, ( PUMS
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164
Chapter 3.
APPLICATION OF THE RESULTS TO THE PROBLEMS OF
METERING PULSATING FLOW
3.1 INTRODUCTION
Despite the large volume of work on pulsation effects that has
been completed in the last sixty five years only a very small part
of it has been of direct use co the engineer in the field actually
faced with the problem of metering a pulsating flow. Most of the work
has merely underlined the difficulties and uncertainties of metering
in these conditions and the only approach to the problem that has
been accepted and followed is that of damping the pulsation at source.
The topic of damping will naturally be fully discussed below but
in addition the alternative approach of coping with the pulsation by
modifying the meter installation or allowing for the errors in the
indicated reading will be thoroughly explored.
Another topic which is discussed is the problem of defining when
pulsations become too severe for the flow to be regarded as effectively
steady.
When the pulsating flow threshold has been crossed it becomes
essential to measure-the differential pressure with a secondary device which is not susceptible to pulsation errors.
3.2 THE CORRECT DESIGN OF THE SECONDARY
DEVICE FOR USE IN PULSATING FLOWS
Some of the more remarkable pulsation effects can be due to the
malfunction of the secondary device. Metering errors of the order of
100% from this source have been reported by Williams80 - 82.
Dr. T. J. Williams has made a study of the causes of manometric
errors and as a result of his findings has drawn up a list of
165,
recommendations for the proper design of the secondary device when
this is to be used in pulsating flows.
The main causes of manometric errors are as follows: -
(i) Wave action in the connecting leads leading to distortion of
the pressure wave. The wave action effects are due to friction
variable propagation velocity and wave reflection.
(ii) Wave action in gas filled volumes in the connecting leads
or in the pressure sensing element.
(iii) Resonance of the fluid in the connecting leads, of the
" manometric fluid, or of the transducer diaphragm etc.
(iv) Non-linear flow through restrictions throttles, piezometer
rings etc. Such restrictions are sometimes mistakenly used
to damp manometer oscillations.
Dr. William's recommendations for the correct design of the
secondary device are listed below.
(1) The connecting leads should be of uniform bore and as short as
possible.
(2) No gas filled volumes to be included in the system.
(3) No restrictions or, throttles which would lead to non-linear damping
to be allowed.
(4) The natural frequency of the sensing element should be much lower
than the pulsation frequency in the case of a slow response instrument
or much higher in the case of a fast response transducer. (The slow
response instrument would be used to indicate the instantaneous
differential pressure. )
(5) If the above recommendations cannot be followed a slow response
instrument can be effectively isolated from the pulsation by the use
of viscous flow damping plugs inserted in the pressure tapping bores.
166
The importance of eliminating secondary device errors cannot be
overstressed. It is clearly pointless to attempt to correct for
pulsation errors in the primary device unless there is confidence in
the accuracy of the secondary system.
In the experimental work the differential pressure transducer
and its connecting leads conformed to Dr. William's design recommendations
in every way. The device was completely symmetrical, the dead space
on either side of the diaphragm was negligible, the i" bore connecting
leads were only about 6" long, the diaphragm natural frequency was
about 4 kHz, and the Helmholtz resonance frequency of the dead space
was also about 4 kHz.
The U-tube water manometer used as a rough check however, was
not an ideal device. The manometer itself had limbs 5ft long and
its connecting leads were nearly Oft long. The worst feature was that
the bore of the tube and leads was only I" to match the pressure tapping
diameter. The tapping diameter as stipulated in BS 1042 should not
be greater than 0.03D when d/D is greater than 0.67. Therefore, even
at I" the bore diameter was too big according to the standard.
A comparison of the transducer and manometer readings taken on
tests with orifice meters is shown on rigs. 58,59(page 194). Approximately
85% of the readings agree within 5%. Fig. 59 shows the ratio of the two readings plotted against frequency for differential pressures of 7 cms WG and above. It is noticeable that the manometer tends to read high over most of the frequency range. Jeffery35 showsa similar comparison between transducer and manometer readings in which the
agreement is much better. The main difference between the two manometers is that Jeffery's had bore of ," and the tube and the leads were shorter.
It seems likely that frictional wave distortion in the long
leads and that possibly capilliarity in the narrow bore manometer limbs
167
were the cause of the errors. It was observed that capilliarity effects
could cause a zero reading error on the U-tube of up to jcm of water.
This could be reduced by adding a few drops of detergent.
3.3 THE PULSATION THRESHOLD
The national codes of practice3' 8
and other flow measurement
hand books are not very helpful on the subject of pulsating flow
measurement. A summary of their recommendations would be that
metering should not be attempted if the flow is pulsating. The
recommendation in BS 1042 is that if necessary damping chambers should
be installed in the line between pulsation source and meter.
The information which would be of real benefit to the engineer
and which, so far, is not in the codes, would be a definition of
the threshold between pulsating and effectively steady flow.
Situations often arise when the source of flow is known to be
pulsating (e. g. the output of a reciprocating compressor) but it is
not convenient because of bulk, space or time to install damping
chambers. In other cases the volume enclosed by the pipe work might
appear to be sufficient to satisfy the Hodgson criterion of adequate
damping but because of possible acoustic effects there would be doubt
concerning its effectiveness. The obvious requirement is that the
pulsation amplitude should be measured at the flow meter and that the
likelihood of measurement errors be quantitatively assessed.
In years past the objection to such a procedure was that there
was no simple means of measuring pulsation quite apart from the problem
of interpreting the measurement. A mechanical pulsometer was described
by Beitler, Lindahi and McNichols6'2 nearly 30 years ago but its
use does not appear to have been widespread.
This mechanical pulsometer indicated the peak pressure in the
differential pressure cycle. Lindahl42 obtained a relationship between
168
the maximum permissible peak pressure and the mean differential
pressure for a metering error of 1% in a series of tests on orifice
meters. The agreement between his results and theoretical relationships
for various waveforms was good. Nevertheless Lindahl himself expressed
doubt that the same results would necessarily be obtained with a
pulsometer of different design.
A different approach was suggested by Head30. He proposed that
the velocity pulsation amplitude should be used as a measure of
intensity on the grounds that there was a unique relationship between
such an intensity and meter error.
The main objection to Head's proposal is that the only kind of
instrument that could be used to measure velocity pulsations is a
hot wire (hot film for liquids) anemometer. These instruments are
very delicate and can only be used for clean fluids. Another drawback
is that they only indicate a local velocity. Traverses across the
pipe section would be necessary to determine accurately the bulk
velocity pulsation amplitude.
A possible alternative is to define the threshold intensity in
terms of the differential pressure pulsation amplitude. A fast
response pressure transducer which complied with William's82 design
recommendations would be required. In these days such instruments
are readily available. In high pressure applications it may be
necessary to use two separate transducers and electronically produce
a differential pressure signal.
A correction factor to compensate the indicated flow rate for
pulsation effects is defined by equation 1/35:
FT = D{1 + (1 - (Arm/Ap)2/(1 + H2J2) )I}a-l
169
When ( /AP)2 «1
FT =1- JAMS/ßp)2/(1 + H2J2)
Now if we require the maximum permissible value (& s%Kp)
for a
1% error, (FT = 0.99), we see that the worst case occurs when temporal
inertia effects are negligible.
The definition of a maximum permissible (rms/Ap) become, in theory:
/)2: 8E = 8ET
We can see how closely experimental results follow the theoretical
relationship of equation 1/35 from Figs. 60 - 62.
They show results of tests on all the orifices and on the small
area ratio venturi nozzle. The total pulsation error correction
factor (As/Ap is plotted against the pulsation amplitude (G s%6p)
for a large number of tests over the frequency range 0- 50 Hz.
The results for the small area ratio orifice and venturi nozzle
(Figs. 61 and 62) show more scatter than for the larger area ratio
orifices (Fig. 60) but nevertheless, when the pulsation error is 1%
the scatter band is'only ± 1/3% about the theoretical square root
error curve.
The results shown on these figures were for tests at low Strouhal
Numbers at which inertia effects were negligible. Figs. 35 and 36
however, show that as Strouhal Number increases total pulsation error
decreases and therefore it is safe to assume that the above results
represent the maximum likely pulsation errors.
On the basis of these results- it would be reasonable to define the
threshold between steady and pulsating flow by a limiting value of
, pulsation amplitude associated with a given acceptable metering error
due to the pulsation.
170
It is suggested that the following values can be justified by
the results:
For ET <t MSIA 0.1
For ET <A Arms/Ap e. 0.2
Until further tests are completed the following limitations should
be observed:
(1) The expansibility factor should be greater than 0.975
(2) The waveform should be basically sinusoidal until tests with
non-sinusoidal waveforms have been completed.
From a theoretical point of view waveform effects on the square
root error are accounted for by measuring the r. m. s. value of the
pressure fluctuations and thug it is not expected that modifications
to the threshold amplitudes will be necessary.
Clearly it would be desirable to have confirmatory results
obtained on different flow rigs and with different working fluids -
particularly with water,
Quite apart from additional testing, however, other factors must
be considered. A definition of a limiting acceptable pulsation
amplitude will only be useful if it is practicable to apply.
Fortunately electronic transducers and r. m. s. meters are much more
readily available now and their requirement is not the obstacle it
would have been 20 years ago.
In an industrial situation where the question might arise as to
whether the presence of pulsation precluded the use of a normal flow
meter installation the following steps would have to be taken:
(1) A fast response differential pressure transducer, or separate
transducers with a signal differencing unit would have to be temporarily
installed. The design of this secondary device would have to conform
171,
to Dr. William's recommendations.
(2) The normal permanent secondary device would have to be
isolated from the pulsations by viscous damping plugs inserted in the
tapping bores, as it is likely that the design of an industrial unit
would be subject to pulsation effects.
(3) A true r. m. s. voltmeter would be required temporarily to
measure the value of AMs. Alternatively, the differential pressure/time
trace would have to be recorded and the r. m. s. value of the alternating
component computed from ordinate measurements. The value of Ap could
be measured using the permanent secondary device. Fortunately the
value of A %ap does not have to be accurate.
(4) If the pulsation amplitude is found to be acceptable the
fast response transducer(s) and the true r. m. s. voltmeter can be
disconnected and removed.
It is believed that the above procedure is practical and should
present no great difficulties to'large concerns with instrumentation
departments. Smaller firms could either borrow the necessary
equipment or consult a specialist.
3.4 SQUARE ROOT ERROR ELIMINATION
If, in a given metering installation, the pulsation intensity
was measured, as described in section 3.3 and found to exceed the
maximum acceptable value the possibility of eliminating the square
root error ought to be considered.
It has already been shown (see rigs. 37 - 40) that below a
certain value of the Effective Strouhal Number pulsation errors can
mostly be accounted for by the square root effect. The square root
error can be eliminated either by using an electronic square rooting
device or by applying a correction factor to the indicated flow rate.
172
3.4.1 Using an Electronic Square Rooting Circuit
Throughout the experimental programme a fast response pressure
transducer in conjunction with a square rooting circuit (see Fig. 30)
was used for tests on the orifices and venturi nozzles. The results
of those tests in which the temporal inertia effects were negligible
are shown in Figs. 63 - 65 Residual error is plotted against total
error. Fig. 63 shows results for orifices of area ratio, m=0.335,
0.393 and 0.612 when Hfd/ud < 0.045
Fig. 64 shows results for the smallest orifice, m=0.103 when
Hfd/uud < 0.025
Fig. 65 shows results for the venturi nozzle m=0.0625 when
Hfd/ud < 0.013
In all cases the equivalent steady flow expansibility factor,
es, was greater than 0.99.
The reduction in error is remarkably good for the orifices.
The residual error is well within ±1% except for two results with the
smallest orifice, for total pulsation errors up to 15%. Even when
the total error is 25% the residual error is not more than ±2%.
The residual errors for the venturi nozzle are a bit larger
being up to ±2% for total errors up to 15%.
It would appear to be a feasible proposition to use a square
rooting device with orifice meters provided the Effective Strouhal
Number is' sufficiently low. The recommended maximum permissible values
of (Hfd%ud) are shown plotted against total error in Fig. 66. The
values are smaller than those permitted in plotting rigs. 60 - 65
to give an additional margin of safety.
Until further work is done (Hfd/üd) should be less than 0.01
for venturi nozzles.
173
The real difficulty which will occur in practice is the determination
of the Effective Strouhal Number. An accurate estimation of the
Harmonic Distortion Factor is essential and this requires a harmonic
analysis of the velocity waveform unless this can be seen to be
basically sinusoidal. A rough guide to the extent of the harmonic
distortion can be seen from the square-rooted differential pressure
trace.
The procedure in an industrial situation would be as follows:
(a) fit a fast response pressure transducer conforming to the design
rules (see 3.2) and incorporating a square rooting circuit.
(b) check the waveform of the square-rooted differential pressure and
if this is sinusoidal calculate (Hfd%ud)
(c) if the above waveform is not sinusoidal and accurate results
are required a linearized hot wire (or hot film) anemometer must
be used to indicate the velocity waveform for harmonic analysis.
H must be evaluated and (Hfd%ud) calculated. If (Hfd%ud) is
sufficiently low according to Fig. 66 and if cs > 0.99, residual
errors should be comparable to those shown on rigs. 63,64 and 65.
3.4.2 Applying a Correction Factor
The possibility that the indicated flow rate of a pressure
difference meter could be corrected for pulsation errors was investigated
by Earles and Zarekl and by Jeffery35. It was hoped that a correction
factor based on the differential pressure peak-to-trough amplitude and
the Strouhal Number could be determined empirically. The main
difficulty with this approach was that it was not able to account for
waveform effects. The analysis developed in Chapter 1 shows that it
should be possible to determine a theoretical expression based on the
174
differential pressure r. m. s. amplitude, the Strouhal Number and a
Harmonic Distortion Factor. According to this theory waveform effects
should be accounted for.
It is possible to form three expressions for pulsation error
correction factor. The most basic expression derives from equation 1/22
FT = (As/Ap)i _ (1 t( UrMS
)2)- U
..... 3/1
This equation should automatically account for temporal inertia
effects as well as the square root effect. The practical difficulty
with this equation is that the velocity fluctuations can only be
measured by a separate instrument such as a hot wire anemometer. An
alternative expression derived from equation 1/30 which involves the
differential pressure fluctuations is
FT = El + 4(Arms/As)2/(i + H2J2)J -i 9 .... 3/2
The obvious objection to this expression is that it involves
A. the equivalent steady flow differential pressure. If this
quantity was known there would be no need for the correction factor.
A more practical expression derived from equation 1/35 is:
FT ={ [l + (ý. _( arms
)2 1 )J }1 ..... 3/3 ep (i+x2J2)
This equation involves (Arms/op) which can be measured using a
fast response transducer and a true r. m. s. voltmeter. Both equations
3/2 and 3/3 have the further disadvantage, however, that they involve
the quantity HJ. To calculate J (equation 1/18) the Strouhal Number
and the effective length is required. At the present time there is
insufficient data to calculate effective length accurately and H, the
harmonic distortion factor, can only be determined after analysis of the
175
velocity waveform. Clearly it is impractical to attempt to allow for
temporal inertia effects in most situations likely to arise in
industry.
. From the above discussion it is apparent that the only practical
proposition is that square root errors alone could be corrected by
the following expression:
Ar FSR _ {jrl t (1 -(
ms )2) T} 1.
Q ,1..... 3/4 P
The agreement between this expression and the experimental
results shown in Figs. 60 - 62 indicates that its use could be
successful within certain limits.
All the results plotted on Fig. 61 were obtained using the smallest
orifice (m = 0.104) for Strouhal Numbers less than 0.025 whereas the
results plotted on Fig. 60 were obtained for three larger orifices
(m = 0.335,0.393,0.612) for Strouhal Numbers less than 0.065.
Fig. 62 shows results for the small venturi nozzle (m = 0.0625) for
Strouhal Numbers less than 0.013. The velocity waveform in all cases
was essentially sinusoidal. As has been remarked previously the
smallest orifice was subject to flow noise and also tended to behave
less predictably than the larger sizes.
To give a greater margin of safety it is recommended that
equation 3/4 should be used only when the effective Strouhal Number
is somewhat lower than those allowed in plotting Figs. 60 - 62.
Fig. 66 shows maximum permissible values of (Hfd%ud) for negligible
temporal inertia effects for orifices with corner tappings. For
venturi nozzles (Hfd%ud) < 0.01 should be a safe criterion.
With reference to Figs. 60 - 62 and 66 equation 3/4 could probably
be used to obtain results with an accuracy better than tl% under the
176
following conditions: -
m<0.3s` (ems /AP) < 0.55, (H fd) < 0.01
ud
0.3 <m<0.62 (firms/Ap) < 0.70, (H fd) < 0.04
ud
* Further tests on orifices with area ratios between 0.1 and
0.3 may allow this figure to be reduced.
Before the use of equation 3/4 could be recommended with confidence
further tests with non-sinusoidal waveforms should be undertaken.
Tests to determine the effects of compressibility should be carried out
and until this is done it should be assumed that the above results are
only valid when es > 0.99. Results on Fig. 62 show that FT falls with ss.
Although it has been demonstrated that the application of
equation 3/4 as a pulsation correction factor is feasible the writer
is of the opinion that this is not the best solution to the problem.
In his view the use of an electronic square rooting device is preferable
since this would eliminate tha tedious process of making separate
readings with a true r. m. s. meter. By comparing rigs. 60 - 62 with
Figs. 63 - 65 it can be seen that the square rooting device is likely
to give accurate results for a larger range of pulsation amplitude
than could be obtained by using the correction formula.
The major difficulty which applies equally to both techniques
is the determination of the Harmonic Distortion Factor and hence the
effective Strouhal Number since it is essential to ensure that
temporal inertia effects are negligible.
3.5 PULSATION DAMPING
3.5.1 A Review of Existing Techniques
Although the difficulties of metering pulsating flow were being
reported as early as 1905 by Venable79 it was not until the 1920s
that the nature of the problem was being properly appreciated. The
most important contributor to the state of understanding at this time
7J 1/7
was J. L. Hodgson. Hodgson 32,33 had come to the conclusion
that it was not possible to measure pulsating gaseous flows directly
with an orifice meter and only possible to measure pulsating liquid
flows when the meter was fitted with a fast response secondary
device. He advocated the use of damping chambers and throttles placed
between the source of pulsation and the flow meter to reduce the
pulsation amplitude to such a level that only an acceptable error
was incurred. He calculated values of a dimensionless group, VfLý 4 p
since called the'Hodgson number, from which an adequate damping
chamber volume (V) and system pressure loss (L) would be determined
for a given pulsation waveform such that a certain value of pulsation
error would not be exceeded. His published work (recently reviewed
by Zanker86) shows that he'was well aware that his criterion for
adequate damping only took account of square root errors. He was
aware that he was neglecting effects due to the temporal inertia of
the fluid being accelerated between the two pressure tapping positions.
He was also aware that quite apart from the square root effect
manometers could give spurious readings due to the effects of pressure
pulses travelling up the connecting lines. He described how to
eliminate the effect of these pressure pulses by using viscous flow
elements in the pressure lines.
Hodgson's method of calculating the metering error that would
result when pulsations had been attenuated but not entirely eliminated
involved laborious step-by-step calculations. He split up the
pulsation cycle into a number of small intervals and calculated the
changes in the flow through the damping chamber. He assumed a quasi-
steady relationship for flow through the orifice.
178
Hodgson presented his results as a series of curves of metering
error versus Hodgson Number each curve applying for a specific
waveshape and amplitude. The waveforms he chose related to various
cut-off ratios for the induction of steam in low speed reciprocating
engines.
Later both Lutz 45 and Herning and Schmidt31 analysed the
smoothing effect of a receiver on pulsating flows in the intake to
an internal combustion engine (Fig. 67 below).
pulsation orifice source
Md PV Mp
pat pd M =1oV
FIG. 67 DAMPING CHAMBER SYSTEM
They considered the continuity of flow through the receiver
Vdp t Md = Mp
..... 3/5
and by assuming quasi-steady incompressible flow through the flowmeter
Po - pd = Kd2 Md2 ..... 3/6"
and isentropic compression and expansion in the receiver
p/pr = const ..... 3/7
developed relationships between Hodgson Number and the metering error.
179
Suffix p refers to the pulsation source and suffix d refers
to the damped conditions at the orifice.
Lutz derived the following relationship between the damped
and undamped pulsating components of the flowrate.
and =e at (C ta fo t mp eat dt) 3/8
where m=M-M..... 3/9
af where =2H ..... 3/10
and C= integration constant.
Herning and Schmidt, by representing the flowrate at the
pulsation source in the form of a Fourier series
M=M Cl + (an sin n wt + bn cos n wt)] ..... 3/11 P n=l
derived the following expression for metering error
fd an2 + bn2 (- 1) =n..... 3/12
As 1+ (4Yn NH)2
Unfortunately the above expression is of limited usefulness since
it can only be evaluated if an analysis of the pulsation source
waveform is available.
It is interesting to note that the Hodgson Number comprises the
group V. f ), a form of Strouhal Number, and the pressure ratio Q
(L/p). The Strouhal Number part shows that for a given flow rate the
required damping chamber volume decreases as the pulsation frequency
rises. It might appear from this that at a sufficiently high frequency
the volume enclosed in the pipe work itself would form an adequate
capacitance. This is not necessarily the case, however, since this
180'
supposition neglects possible acoustic effects. These will be
significant when the axial lengths in the system are no longer small
compared with the pulsation quarter-wave length (approximately c/4f).
The effects of acoustic resonances occurring in systems including
volumes intended for pulsation damping were considered recently in
a paper by Fortier25. Like Lutz and Herring and Schmidt, Fortier
considered the continuity of flow through the receiver (equation 3/5).
Instead of making the quasi-steady flow assumption for the meter,
however, Fortier included the temporal inertia term in the momentum
equation for incompressible flow.
A similar approach has been adopted in the analysis set out
below but whereas Fortier presented separate formulae for specific
waveforms the writer has developed a universal expression applicable
to every waveshape.
3.5.2 An Analysis of the Hodgson Number
and Pulsation Error Relationship
In this section a theoretical analysis of the relationship
between Hodgson Number and metering errors due to flow pulsations is
presented. The effects of acoustic' resonance in the damping chamber
and of temporal inertia of fluid being accelerated through the flowmeter
are accounted for in the manner suggested by-Fortier. The effects
of waveform variation,. will also be considered but whereas Fortier
developed separate equations for sinewaves and for rectangular waves
a universal relationship will be derived which can be applied to
any waveform.
Let the mass flow rate through the flow meter in Fig. 67 be
represented by:
Md =M Cl + fit)) ..... 3/13
isi
As was demonstrated in Chapter 1 (equation 1/17) instantaneous
differential pressures measured under pulsating flow conditions
can be expressed as follows:
Ad = As [(1 + ß(t))2 + 2J (0'(t)/w)1 ..... 3/14
where Ad is the meter differential for the damped pulsation.
_, '2 D=M (1 -C 2m2)/2pC 2(11(1-)2
..... 3/15 sCC4
and where J= 27rC Le fd
(1 - CC m)dü..... 3/16
d
Co If Jar sin (r wt + ar) ..... 3/17
then d= +'(t) =I ar rw cos(rwt +a)..... 3/18
The time average value of Ad is given by
Ad = AS (i + (t) )= Os (1 + jii art) .... 3/19
Considering continuity of mass in the damping chamber we have
Vä+ tzd = MP ..... 3/20
where Md is the flow rate on the meter side of the damping chamber.
Mp is the flow rate on the side of the pulsation source.
If Mp =M (1 +I br sin (rwt + er)) ..... 3/21
Equation 3/20 can be written
V V. ="A (In b sin (rwt + er) - In a sin (rwt + o) ..... 3/22 1" r1rr
Now if the processes in the damping chamber can be assumed to
be isentropic
dCd where C= �-ý ..... 3/23 P
_ __.. ... ý_... _"__"-" -, _ý - týRlek+-..., -. ++rý++.. +-. w+... r ^+ ... +-. aus, ý*ý. w... w0. ý.......... +.... «... ........ .. ý.
182
We shall assume that d can be determined from equation 3/14
if Ap = &at Pdl
This is the assumption inherent in the analyses of Hodgson 32,33
45 31 38 25Lutz, Herning and Schmidt, Kastner and Fortier. Strictly
speaking their results can only be applied when either the upstream
or downstream side of the meter is open to the atmosphere or to a
very large reservoir. Otherwise their results can be safely applied
only when the pulsation frequency is so low that acoustic effects are
completely absent. In BS 10428 Appendix E, it is stated that Hodgson's
curves should only be used when the frequency is less than 100 cycles
per minute. Inspection of Figs. 53 - 55 shows that at higher
frequencies the amplitude of the static pressures on either side of
the orifice varies considerably despite the fact that the differential
pressure amplitude is constant.
However, assuming that the pressure is constant on one side of
the orifice
d=. 2t tý'(t) (1 + fi(t)) +W ý11(t)] ..... 3/24
co where ¢"(t) = -11 ar r2w2sin (rwt + ar) ..... 3/25
If we assume that ¢(t) «1 which would be the case when the
pulsations were sufficiently damped such that only small metering
errors would be incurred then equations 3/18,3/22,3/23,3/24 and 3/25
can be combined to give:
CIO ýl br sin (rwt tar) = Zý ar sin (rwt + cr r)
tV E2
As [E1 ar rw cos (rwt + Or) -w1 ar r2w2sin (rwt t vr)] in (30
M
..... 3/26
183 r
Both the left and right hand sides of equation 3/25 are Fourier
series and the mean square values of each must be equal, hence
br2{ ar2C(1 -S Jr2w)2 t rw)2J} ..... 3/27
mm
Using equations 3/15 and 3/16 we can obtain
v 24 Jr2w =* "r2w2
..... 3/28 A M
C2 Wo2
where 1=L
wog C2(CCA) ..... 3/29
and wo is the natural frequency of a Helmholtz resonator comprising
a chamber of volume V and a neck of conductivity (CCA L)
e Now 2e a v
c2 rw = 4rtf r=Y NH r
mQ
where NH =Vpf Q
Re-writing equation 3/27 we have
Iar 2
` b2 a2 (1 -
r2W 2)2+( 47iNH )2_?
rW2Y O a2
Zart.
r2w2 2 Zat (1 -2 We will call the quantity
r wo Iar2
... to 3/30
..... 3/31
an Anti-Resonance factor. Its value is always positive but tends to
zero at resonance conditions when the waveform is purely sinusoidal.
184
0
When calculating the minimum Hodgson Number for adequate damping
the worst condition that must be allowed for is that of resonance of
the 1st Harmonic when the Anti-Resonance factor tends to zero.
r2a 2 The quantity (r) is the Harmonic Distortion Factor
La2 r
which has been discussed previously. Its minimum value is unity
which occurs when the waveform is purely sinusoidal. It must be
noted that in this case it refers to the pulsation waveform at the
meter and not at the pulsation source. The effect of the damping
chamber on the waveform is to act like a low pass filter. The higher
harmonics will be the most severely attenuated. The value of the
Harmonic Distortion factor will therefore tend to unity.
Pulsation Error at the meter is ET and from equations 1/21 and 1/23
we have
ET=( 0
)-1=(l+jar2)-1=jar2whena «1 s
Assuming that Anti-Resonance Factor is zero and the Harmonic
Distortion Factor is unity then:
ýb2 r
..... 3/32 ET Y NH ý2
For a given error, ET, not to be exceeded m2rms
M ..... 3/33 NH >4 *2-
ýr -E- .T
For incompressible flow
mans grms Urms
"QU
185
U where rms is the dimensionless r. m. s. value of the velocity
U pulsation amplitude at the pulsation source.
( 's )1..... 3/34 Therefore NH >yU 4Uc
The disadvantage of equation 3/34 is that it requires the value
of. a velocity pulsation amplitude for its evaluation. The difficulties
and inconvenience of using a device such as a hot wire anemometer
have already been discussed in section 3.3 and it was suggested that
the orifice meter itself could be used to measure pulsation amplitude.
If fitted with an appropriately designed fast-response secondary-
device it could be used to measure the quantity (Arms%Ap)
The required velocity amplitude is related to (Arms/Ap by the
following equation obtained by combining equations 1/23 and 1/35.
(Uri/U)2 = {[[{1 + [l - (A / )2/(1 + H2J2)]1}J-1 - 11 ..... 3/35 rms
If the temporal inertia term H2J2is neglected (Urs%U) will tend
to be overestimated which will increase the safety margin of the
damping criterion.
It must be realised that if an orifice plate is placed in the
undamped pulsating flow it will tend to have a slight damping effect
on the pulsation and also the acoustic characteristics will not be
the same when the damping chamber is eventually interposed between
source and orifice meter. Nevertheless since we are already assuming
that acoustic effects are negligible it is still a reasonable
proposition to estimate (Urms/U) from a measurement of (Asp
across an orifice plate.
1
186
I
3.5.3 Comparison of the New Damping
Relationship with Existing Data
Fortier25 compares values of Hodgson Number given by Hodgson
himself with those calculated by Fortier from his own derived
expressions.
For sinusoidal waveforms the expression given by Fortier is:
NH > 18 1
00000 3/36
when y is assumed equal to 1.4.
This is identical with equation 3/34 since for a sine wave
1 Us0 and when y= 1.4 y=9
U T2
Tables 6 and 7 show Hodgson's values of NH and those calculated
from expressions 3/34 and 3/36. As Fortier points out Hodgson's own
values are smaller since he did not allow for the effects of acoustic
resonance in the damping chamber.
Tables 8,9 and 10 show values of NH for rectangular waveforms
of the form shown below
zo
6 Zu
Qmax _.
I __I___ý______ýw___{. S______}___ý__ý_
t Table 8 shows the values calculated from equation 3/34 assuming
Y=1.4
187,
It can be shown that Ur's
=Z as follows: U
LZ o
u
Q2max (Zo - Zu) = Q2 Zo ..... 3/37
2
also 02 Z0 [Q2max - Q2] 2(Z0 - Zu) + Q2 Zu
Hence q2- Ums 1 _2-_--z 04,3/38
UZ-1) Qw Z
u
Table 9 shows values of NH calculated by Fortier from an expression
he derives as 10irtZu/Zo
NH "3 ..... 3/39 9
Table 10 shows values given by Hodgson.
It can be seen that the universal expression (equation 3/34)
gives results comparable to those derived by Hodgson and Fortier
except in the extreme case when Zu/Zo = 1.
Further comparison with other investigators' published data is
provided on Fig. 68. The solid line curves are based on results of
Lutz45 as presented in Fig. 8 of Oppenheim and Chiltons' literature
survey54. Their Fig. 8 is reproduced on Fig. 66 for the purpose of
identifying the Lutz curves. The full line curves on Fig. 68
represent the universal waveform equation 3/34 calculated for each of
the Lutz curves. In each case equation 3/34 gives a larger Hodgson
Number than the Lutz counterpart. The closest agreement is for curve
2 which appears to be for a sinusoidal pulsation. 11
188'
Clearly expression 3/34 will tend to overestimate the required
damping when the source pulsation waveform contains significant high
harmonics since the Harmonic Distortion Factor which was in equation
3/31-was assumed to be unity, its minimum value.
Strictly speaking equation 3/34 should read
N. 1 __ýr ___ '
Urms 1 H'H.....
3/40 4 772 iJ ºtT
Where H is the harmonic distortion factor for the waveform on
the meter side of the damping chamber. This expression would give
more accurate results but has the disadvantage that an analysis of the
damped pulsation waveform is required for its evaluation. This could
only be obtained after the damping chamber had been designed and
installed.
All the techniques used over the years for determining the
appropriate Hodgson Number from which the necessary damping volume
and pressure loss can be calculated have required some knowledge of
the source-pulsation amplitude and waveform. The advantage of using
equation 304 is that only the source-pulsation amplitude is required. U
The value of this amplitude, ( rms ) can be determined from a
measured or predicted velocity-time history.
Alternatively, by using equation 3/35 an estimate of (UMs/Ü)
can be obtained by measuring (ASS %p) with an orifice in the
undamped flow. Comparison with the results of Hodgson, Fortier and
Lutz show'that equation 3/34 would provide a generous safety margin
with waveforms with high harmonic content.
Although it is believed that the use of equation 3/34 would make
the technique of pulsation damping easier to apply nevertheless it
is hoped that other methods of solving pulsation problems will become
.ý
189. '
more popular in the future.
The disadvantages of pulsation damping is that it involves bulky
and expensive vessels which may have to withstand high pressures.
The alternative to using large vessels is to achieve a high resistance
by throttling the gas. Obviously this entails higher running costs
for the plant.
A further difficulty is that all the analyses, including the
author's, assume that the pressure on the side of the orifice
furthest from the source of pulsation is constant. Until a technique
of taking into account pressure pulsations on both sides of the orifice
the criteria for adequate damping strictly only apply when the
orifice inducts or discharges the fluid directly to or from the
atmosphere or a very large reservoir.
In other situations some loss of accuracy must be expected.
Finally it must be stressed that the dimensions of pipework
between pulsation source, the chamber and the meter have to be small
compared with the pulsation quarter-wavelength.
r
3.6 A PROCEDURE FOR REDUCING
PULSATION ERRORS IN ORIFICE METERS
Let us suppose that in an industrial situation it is necessary
to measure the pulsating flow of a gas.
By applying the results of this investigation it is possible to
suggest a step-by-step procedure for determining a suitable method
of metering this kind of flow using a square edge orifice plate with
corner tappings.
Step 1 The first step is to determine whether or not the pulsations
are sufficiently severe to warrant special treatment. This can be done
with an orifice fitted with a fast response differential pressure
190
transducer. The orifice area ratio should prefereably be between
0.33 and 0.61. The transducer and pressure connections must conform
to the design rules of Dr. T. J. Williams. The amplitude, n /eprms
must be measured and compared with the threshold figures.
For ET < 1% Arms/Ap 0.1
For ET < 1% e /a& < 0.2 rms
Step 2 If the amplitude is very much less than the threshold value _
a normal industrial manometer can be fitted.
Step 3 If the amplitude is only just less a manometer which conforms
to the Williams' design rules must be fitted before the flow can be
treated as 'steady'.
Step 4 If the pulsation amplitude is too large two possible courses
of action should be considered:
- (A) A square-root device in conjunction with a fast response
differential pressure transducer could be fitted.
(B) A pulsation damping system could be installed.
Step 5A number of points should be considered before alternative
A can be safely selected.
(a) The harmonic distortion factor should be measured or estimated
so that the effective Strouhal Number can be determined. The limiting
values of (H fd/Ud) for negligible temporal inertia effects are shown
on Fig. 66. If the area ratio is in the range 0.3 to 0.7 then (H fd/Ud)
must-be less than 0.036 for large pulsation amplitudes.
(b) A check should be made that compressibility effects are absent.
The criteria derived theoretically are:
e>0.99 s
Purms/Pu < 0.035
d« c/4f
11
--_- .ý
191
(c) If possible the throat Reynolds Number should be greater than
70 000.
(d) If the velocity waveform is sinusoidal the accuracy should be
better than 1% up to total errors of 15% and better than 2% up to
total errors of 25%.
If the waveform is not sinusoidal the accuracy may not necessarily
be as good. Test results with non-sinusoidal waveforms are still
required.
Waveform can be checked by using a linearized hot wire anemometer. .
The square-rooted differential pressure waveform will be a good guide
if temporal inertia effects are negligible.
Step 6 If the above points are satisfied alternative A may be
selected.
Step 7 The following points should be considered before selecting
alternative B.
(a) The theoretical damping criterion,
(Urms /U) NH 18 ---i---
was derived assuming the static pressure on the side of the orifice
furthest from the pulsation source was constant. This will be so
only if the orifice is open to the atmosphere or a very large reservoir.
(b) The analysis of the damping criterion is only valid when the
lengths of pipe between the point where the velocity pulsation amplitude
is measured and the final meter installations are short compared with
the pulsation quarter wavelength. If there is doubt it would be
sensible to make the maximum permissible error ET a very small figure
and to check the pulsation amplitude after fitting the damping chamber.
Step 8 If alternative B is feasible the procedure for designing the
damping system is as follows: -
192'
(a) The amplitude of the undamped velocity pulsation (Urms/U)
should be estimated preferably by traversing the pipe section with a
hot wire anemometer, or alternatively, by using the value of (Arms/gyp)
already measured in Step 1 to calculate (U rms
/U) from equation 3/35.
It should be noted that if temporal inertia effects are neglected in
the calculation the safety margin is increased.
(b) The Hodgson Number should be calculated for a maximum
permissible pulsation error ET.
(c) The chamber volume for a given overall pressure loss can be
calculated from the Hodgson Number.
(d) After the damping chamber has been installed it would be
desirable to check the value of (e s/K
) with the orifice meter and
to repeat Steps 1,2 and 3 of the above procedure.
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i 198
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204
FINAL CONCLUSIONS
The following is a summary of the conclusions reached on the
results of the experimental work and their application to the problems
of metering pulsating flows.
It should be noted that although the extrapolation of the results
to non-sinusoidal pulsations can be justified by the theory tests
have not yet been carried out to supply experimental verification.
Experimental Resusits
1) When the effective Strouhal Number is sufficiently low and when
compressibility effects and manometer errors are absent the error
in the flow rate indicated by orifice and venturi nozzle meters is
mainly due to the square root effect.
The limiting values of effective Strouhal Number are as follows: -
Orifice meters with corner tappings
area ratios near 0.39,0.61 (H fd%Ud)max ` 0.036
area ratios near 0.10 (H fd/Ud)max = 0.010
Venturi nozzles all sizes (H fd%Ud)max = 0.010
2) When (H fd/Ud) is greater than the limiting values temporal
inertia effects are significant and they
a) reduce the total error at a given amplitude of differential
" pressure pulsation,
b) produce negative residual errors which increase with (H fd/Ud)
at a given amplitude of differential pressure pulsation.
1 3) The effective lengths of orifices and venturi nozzles are of the
order twice the throat diameter but probably decrease at throat
Reynolds Numbers less than 70 000.
205'
4) There is evidence that there are small variations in residual
error dependent on the acoustic characteristics of the pipework.
5) There is no significant difference in the behaviour of orifice
and venturi nozzles in pulsating flow conditions.
6) The above conclusions are only valid when compressibility effects
are negligible. Analysis shows that these will become significant
under the following conditions:
a) when the equivalent steady flow expansibility factor, es,
is less than 0.99.
b) when the amplitude of the upstream static pressure pulsation,
(Perms/Pu)' is greater than 0.035.
c) when the meter throat diameter, d, is no longer small compared
with the pulsation quarter wavelength c/4f.
Application to Metering Problems
7) A threshold between effectively steady and pulsating flow can be
defined in terms of the differential pressure pulsation amplitude
(erms%op )
For ET < 5p 0.1
For ET <, 1% Q /-Ap 5 0.2 rms
8) A square rooting circuit used in conjunction with a properly designed
fast-response differential pressure transducer can be used to eliminate
most of the metering error at low values of (H fd/Ud). The limiting
values are stated in (1) above.
9) The theoretical analysis used in this thesis enables the technique
of pulsation damping to be applied more easily in two ways:
a) a criterion of adequate damping has been developed in the
form of a single equation applicable to pulsations of any
waveform, viz.
Y (Ums/U)
NH > 18 I
206
b) the undamped velocity pulsation amplitude (Urins/U) can be
shown to be related to the differential pressure amplitude
(Arms%ep) Measured across an orifice plate, viz.
(Urms/U)2 ={ [[{1 + [1 - (Arms/Ap)2/(1 + H2J2)]i)]-1 -1)
The neglect of H2J2 leads to an increase in the safety margin
of the damping criterion.
As with all previous derivations of damping criteria the assumptions
have been made that
(i) the pressure is constant on the side of the orifice furthest
from the pulsation source.
(ii) the lengths of the damping installation are short compared
with the pulsation quarter-wavelength.
f
207
RECOMMENDATIONS FOR FURTHER WORK
Much progress towards an understanding of the pulsation problem
and towards at least its partial solution has been made in the work
so far completed. Even more work is required, however, before the
results can be applied in industry with any confidence. The following
is a list of some of the more urgent investigations that should be
undertaken.
l. - The behaviour of orifices and other meters in non-sinusoidal
pulsating flow. This work can be undertaken with the existing
pulsator with minor modifications.
2. The behaviour of orifices and other meters at high frequencies -
(in the range 50 - 500 Hz). A simple siren-type which has already
been made could be developed for this work.
3. The flow patterns in the region of the orifice under pulsation
conditions with particular reference to the unexplained effects
associated with the acoustic characteristics and with low throat
Reynolds Number flows. Mr. P. M. Downing has already embarked
on this project on the existing rig.
4. Compressibility effects in meters subjected to pulsating flows.
S. The behaviour of orifice and other meters in fluids other than air.
An obvious choice of fluid would be water.
6. The behaviour of orifice metes with flange tappings and D and
D/2 tappings. This investigation could be conducted at the same
time as Nos. 1 and 2 by operating one meter with three different
types of pressure tapping each with its own secondary device.
Ili
6
208 ,
REFERENCES
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209 ,"
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210
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211
24: FORTIER, A.
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Document 24 1SO/TC 30/WG 4. International Organisation for Standardisation, 1963.
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Aero. J. of R. Aero. Soc. V. 75 April 1971.
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Proc. I. Mech. E. V. 173. N. 36.1959.
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Instruments p. 763 July 1951.
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Trans. A. S. M. E. p. 1471 October 1956.
31. HERMING, F. und SCHMID, C.
Durchflussmessung bei pulsierender Strömung.
VDI - Zeitschrift Bd. 82. Nr. 38.17 September 1938
212
32. HODGSON, J. L.
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Selected Engineering Papers, No. 31, I. C. E. 1925.
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Trans. A. S. M. E. 1929.
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Phil. Mag. S. 7 Vol. 7, No. 45, p. 873 - 886,1929.
35. JEFFERY,. B. J.
"Pulsating Flow Through Orifices. "
Ph. D. Thesis, University of London, 1965.
36. JOHANSEN, F. C.
"Flow through Pipe Orifices at Low Reynolds Numbers. "
A. R. C. R. & M. No. 1252 HMSO June 1929.
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Trans. A. S. M. E. 1923.
38. KASTNER, L. J.
"An Investigation of the Airbox Method of Measuring the Air Consumption of Internal Combustion Engines. "
Proc. I. Mech. E. 157, pp. 387 - 404,1947.
39. KASTNER, L. J. and WILLIAMS, T. J.
"Pulsating Flow Measurement by Viscous Meters, with Particular Reference to the Air Supply of Internal Combustion Engines. ""
Proc. I. Mech. E., 169, p. 419,1955
213
40. KASTNER, L. J., TINDALL, M. J. and WILLIAMS, T. J.
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I. Mech. E. Symposium on Accuracy of Instrumentation in I. C. Engines. 1966.
41. KIRCHHOFF, G.
"Zur Theorie Freier Flussigkeitsstrahlen. "
Crelles Journal, 70, p. 289,1869.
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"Pulsation and its Effect on Flowmeters. "
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Proc. I. Mech. E. Vol. 184 Pt. 1 No. 8 1970
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"The Influence of Swirling Flow on the Discharge Coefficients
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Translated by BHRA. from Brennstoff-Wgrme-Kraft. 13 No. 3. pp. 121 - 125 1961.
45. LUTZ, 0.
"Über Gasmengenmessung bei Kolbenmaschinen mittels Düsen und Blenden. "
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214'
47. MINTON, P. and SELVALINGAN, S.
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Symposium on The Measurement of Pulsating Flow. Institute of Measurement and Control, April 1970.
48. MISES, R. von
"Berechnung von Ausfluss und Ueberfallzahlen. ".
Zeitschrift, V. D. I. Baid 61, Nr. 21, pp. 447 - 452,26 May 1917.
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Southwest Research Institute, May 1958.
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215
54. OPPENHEIr4, A. K. and CHILTON, E. G.
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Pergamon Press, 4th Ed. 1966.
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R
216
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219
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A. S. M. E. Flow Measurement Symposium, Pittsburgh, 1966.
"
220,
TABLES OF EXPERIMENTAL RESULTS
Explanatory Notes.
The tables include all the basic quantities measured in the tests
from which the results plotted in the graphs have been calculated.
Some of the more important derived results such as total and residual
errors and Strouhal Number etc. are also tabulated.
Some quantities measured but not tabulated here are as follows:
Pressure and temperature upstream of the critical nozzle.
Temperature upstream of the test meter during steady and pulsating
conditions.
1st and 2nd harmonic amplitudes of velocity, differential pressure,
and upstream and downstream static pressures.
Specific humidity of the air.
The upstream temperatures would only have been important if there
was a significant variation between steady and pulsating conditions.
The greatest difference only amounted to 2K under certain conditions
and normally the variation was within JK.
It was not possible to carry out a full harmonic analysis of the
waveform and instead the relative amplitudes of the first and second
harmonics were measured. It might have been possible to make a crude
estimate of the harmonic distortion factor on this basis if the a. c.
component of the velocity signal had been satisfactory. Unfortunately
this was not the case for the majority of the tests as a defective
filter was found to have reduced the frequency response. This was only
corrected for tests 1013 onwards.
The mrsasured values of UCrms have been omitted for tests up to
1013 for the reason given above.
Tests 809 - 858 were carried out by Mr. P. M. Downing alone after
installing the new roller bearings in the stroke changing mechanism
221.
making it possible to attain speeds up to 50 Hz for the first time.
Specimen Calculations
1. Reynolds Number
Red = Üdpd/u
where p=0.075 lb/ft3 (nominal value at 20°C, 14.7 lbf/in2)
and u=0.121 x 10-4 lb/ft s(nominal value at 20°C, 14.7 lbf/in2)
2Q and Üd = CD Is
p(1 - m2)
1C ft/s
D -(i - m2)
when As is measured in cm. wg. 'units.
2. Strouhal Number
NS. = fd%Ud ,
where (ld is calculated as above.
0
3. Total Error
ET = [( &p/Qs)' - 11 x 100%
Ap and AS measured in volts.
4. Residual Error
ER x 100%
AP' and aS measured in volts.
222 w
' S. Pipe Factor
F_ UP
_ P uc p
where both velocities are measured in volts,
and where U=U=Fu pss cs
Nominal value of Fs assumed for the calculations was 0.85
6. Static Pressure Pulsation Amplitude
(purms/pu) and (pdrms/pd)
where Fu is absolute pressure.
The static gauge pressure pug was added to the atmospheric
pressure.
The downstream gauge pressure pdg was measured but was found to
be effectively zero at all times. The pressure pd was therefore
assumed to be atmospheric pressure.
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