Pitot Tube Standart

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है”ह”ह

IS 14973 (2001): Measurement of Fluid Flow in ClosedConduits - Velocity Area Method Using Pitot Static Tubes[WRD 1: Hydrometry]

Is 14973:2001ISO 3966:1977

Wm%?m’m

~< diGmm’-EmMFT+--aTamPla-a\m U* ?W?l-m*

Indian Standard

MEASUREMENT OF FLUID FLOW IN CLOSEDCONDUITS — VELOCITY AREA METHOD USING

PITOT STATIC TUBES

ICS 17.120.10

0 BIS 2001

BUREAU OF IN DIANSTA Ni)ARDSMANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

June 2001 Price Group 12

Fluid Flow Measurement Sectional Committee, WRD 01

NATIONAL FOREWORD

This Indian Standard which is identical with ISO 3966:1977 ‘Measurement of fluid flow in closedconduits — Velocity area method using Pitot static tubes’ issued by the International Organization forStandardization (ISO) was adopted by the Bureau of Indian Standards on the recommendations of theFluid Fiow Measurement Sectional Committee (WRD 01) and approval of the Water ResourcesDivision Council.

In the adopted standard, certain conventions are, however, not identical to those used in IndianStandards. Attention is especially drawn to the following:

a) Wherever the words ‘International Standard’ appear referring to this standard, they should beread as ‘Indian Standard’.

b) Comma (,) has been used as a decimal marker while in Indian Standards, the current practiceis to use a point (.) as the decimal marker.

REFERENCES TO ERRORS AND CLARIFICATIONS IN TEXT

The Technical Committee while adopting the text of this International Standard identified certaintextual error to the following clause and felt necessary to correct this in the Indian context

Clause CorrectionReference

1, 3’d para The value of ~ 2’%o may be read as ~ 3%. This change in claim of accuracy has

been made to account for the displacement effect.

Is 14973:20011s0 3966:1977

Indian Standard

MEASUREMENT OF FLUID FLOW IN CLOSEDCONDUITS — VELOCITY AREA METHOD USING

PITOT STATIC TUBES

1 SCOPE AND FIELD OF APPLICATION

This International Standard specifies a method for thedetermination in a closed conduit of the volume rate offlow of a regular flow (see 5.1) :

of a fluid of substantially constant density orcorresponding to a Mach number not exceeding 0,25;

with substantially uniform stagnation temperatureacross the measuring cross-section;

running full in the conduit;

— under steady flow conditions.

In particular it deals with the technology and maintenanceof Pitot static tubes, with the calculation of local velocitiesfrom measured differential pressures and with thecomputation of the flow rate by velocity integration.

The method of measurement and the requirements definedin this International Standard aim at reaching, at the 95 %confidence level, an uncertainty on flow rate not greaterthan f 204. To attain this result it may be necessary,according to measurement conditions, to take into accountthe corrections given in clause 11. If any of therequirements of this International Standard are notfulfilled, ~is method may still be applied in special casesbut the uncertainty on flow rate will be larger.

2 SYMBOLS AND DEFINITIONS

2.1 Symbols

symbol

A

a,a’

D

d

d’

di

Quantity

Cross-sectionalarea of theconduit

Distance of the extrememeasuring csoint to the nearestwall

Pipe diameter

Heed diameter

Stem diameter

Total pressure tapping holediameter

)imensions

~z

L

L

L

L

L

corre-sponding

S1 unit

m2

m

m

m

m

m

H

h

kb

k9

kt

L

1

M

m

Ma

P

q“

R

Rr

Res

T

uu

v

x

Y

z

CY

T

AP

e

(1-.1

Quantity

~ectangular conduit height

+eight of a particular pointtbove the bottom

310ckage coefficient of a:yl ind rical stem

coefficient depending of theIose shape

coefficient of turbulence:orrect ion

Rectangular conduit width

> istance from a particular~oint to the side-wall

Ofolar mass of fluid

Roughness coefficient

Mach number

tisolute ststic pressure ofthe f Iuid

Volume flow rete

Molar constant of gas

Pipe radius

Measuring circle radius

Reynolds number

Frental projected area of thestem inside the conduit

Absolute temperature

Discharge velocity

Mean velocity along acircumference or a measure-ment line

Local velocity of the fluid

Pipe dimension

Distance of a measuring pointto the wall

Gas law deviation factor

Calibration factor of the pitoltube

Ratio of the specific beetcapacities

Differential pressure measure{by the Pitot tube

Expansibility factor

Compressibility correctionfactor

imensions

L

L

—

—

L

L

M

—

—

ML-I T-2

L3T-1

,L2T-2(3-1

L

L

—

Lz

QLT.l

~T-l

~T-l

L

L

—

—

—

~L-lT.2

—

Corre-spondingS1 unit

m

m

—

m

m

kg

—

—

Pa

m3/s

mOI-7 .K-’

m

m

m2

K

mls

rlsls

mls

m

m

—

.

Pa

1

Is 14973:2001ISO 3966:1977

2.24 relative velocity : The ratio of the flow veltity at

CwSntity

Universal coefficient for headloss

Dynamic viscosity of the fluid

Kinematic viscosity of the fluid

Head loss

Density of tha fluid

Pitot tube inclination

~L.lT.l

L2T-1

~L-l 7-2

~ ~4

—

corre-

spondingS1 unit

Pas

m2/~

Pa

kglm3

2.2 Definitions

The definitions in the following sub-clauses are given onlyfor terms used with a special meaning or for terms themeaning of which might be usefully recalled.

2.2.1 Pitot static tube : A tubular device consisting of acylindrical head attached perpendicularly to a stemallowing measurement of a differential pressure from whichthe flow rate of the fluid in which it is inserted can bedetermined. It is provided with static pressure tappingholes ‘(drilled all around the circumference of the head atone or more cross-sections) and with a total pressure hole

(facing the flow direction at the tip of the axiallysymmetrical nose of the head).

NOTE – Throughout this International Standard the expression“Pitot tuba” is uaad without amplification to designatea “Photstatic tube” sinceno confusionispossible.

2.2.2 static pressure tapping : A group of holes for themeasurement of fluid static pressure.

2.2.3 total pressure tapping : A hole for the measurementof fluid stagnation pressure (the pressure produced bybringing the fluid to rest without change in entropy).

2.2.4 differential preasura : The difference between thepressures at the total and static pressure taps.

2.2.5 stationary raka : A set of Pitot tubes, mounted onone or several fixed supports, which explore the wholediameter or measuring section simultaneously.

2.2.6 peripheral flow rate : The volume flow rate in thearea located between the pipe wall and the contour definedby the velocity measuring points which are the closest tothe wall.

2.2.7 discharge velocity : The ratio of the volume rate offlow {integral of the axial component of local velocitieswith respect to the cross-sectional area) to the area of themeasuring cross-section.

the considered point to a reference vel~ity measured dt”thesame time and being either the velocity at a particular point(for example, the centre of a circular conduit) or thedischarge velocity in the measuring section.

2.2.9 straight length : A conduit section the axis of whichis rect iIinear and the surface and cross-section of which areconstant.

NOTE – The shape of this section is usually circuler, but it may berectangular or annu)ar.

2.2.10 irragsalarity : Any element or configuration of aconduit which makes it different from a straight length.

NOTE – For the purpose of this International Standard, thoseirregularities which create the moat significant disturbances arabands, valves, gates and sudden widening of the section.

3 PRINCIPLE

3.1 Genaral principle

The principle of the method consists of :

a) measuring the dimensions of the measuring section,which must be normal to the conduit axis; thismeasurement is necessary for defining the area of thecross-section (see 3.2);

b) defining the position of the measuring points in thecross-section, the number of measuring points having tobe sufficient to permit adequate determination of thevelocity profile;

c) measuring the differential pressure existing betweenthe total and static pressures of the Pitot tube placed atthese measuring points (see 3.3) and determining thedensity of the fluid in the test conditions;

d) determining the local velocity of the flow, fromgiven formulae, on the basis of previous measurements(see clause 7);

e) determining the discharge velocity from these values;

f) calculating the xoltime rate of flow equal to theproduct of the cross-sectional area and the dischargevelocity.

Errors in tha techniques described in a) to f) contribute tothe error in the flow rate measurement; other sources oferror (such as the shape of the velocity distribution and thenumber”of measuring points) are discussed in clause 12.

2

Is 14973:2001

ISO 3966:1977

This International Standard presents three types ofmethods for determining the discharge velocity :

Graphical integration of the velocity area

(see clause 8)

This method consists in plotting the velocity profile on agraph and evaluating the area under the curve which isbounded by the measuring points closest to the wall. Tothe value thus obtained is added a calculated term whichallows for the flow in the peripheral zone (the areabetween the wall and the curve through the measuringpositions closest to the wall) on the assumption that thevelocity profile in this zone satisfies a power law.

For this method the measuring points may be located at

whichever positions are required in order to obtain a

satisfactory knowledge of the velocity profile.

Numerical intqvation of the velocity area

(see clause 9)

The difference between this method and the previousone lies in the fact that the graphical velocity profile isreplaced by an algebraic curve and the integration iscarried out analytically.

Arithmetical methods (see clause 10)

The arithmetical methods assume that the velocitydistribution follows a particular law and the meanvelocity in the conduit is then given by a linearcombination of the individual velocities measured at thelocations specified by the method.

For the arithmetical methods described in clause 10, theassumption is made that in the peripheral zone thevelocity distribution follows a logarithmic law as afunction of the distance from the wall.

3.2 Measurement of th(~ measuring cross-section

3.2.1 Circular cross-sections

The mean diameter of the conduit is taken as equal to thearithmetical mean of measurements carried out on at leastfour diameters (including the traverse diameters) atapproximately equal angles to each other in the measuringsection. Should the difference between the lengths of twoconsecutive diameters be greater than 0,5 %, the number of

. measured diameters shall be doubled.

3.2.2 Rectangular cross-sections

The conduit width and height shall both be measured atleast on each straight line (at least four) passing through themeasuring points. Should the difference between the widths

(or heights) corresponding to two-successive measuring linesbe greater than 1 %, the number of measured widths (orheights) shall be doubled.

3.3 Meeeuremrsnt of 100al velocities

3.3.1 Method of exploring traverse section

It is sometimes proposed that several Pitot tubes bemounted on a stationary rake in order to exploresimultaneously the whole measuring cross-section.“However, the experimental data at present available areinsufficient to allow the design of certain details (such asshape of head and of stem) which would ensure thatmeasurements by a rake would achieve the accuracyrequired by this International Standard.

Therefore, this International Standard deals only withvelocity area methods using a single Pitot tube placedsuccessively at each measuring point.

3.3.2 Reference measurement

Reference measurements shall be made in order to checkthe steadiness of flow and to correct individual velocitymeasurements for slight changes in flow rate duringtraversing; any reference measuring device inserted in theconduit shall be placed in such a way that there is nointeraction with the traversing Pitot tube. The referencemeasurement shall be made as far as possiblesirqultaneously with aech velocity measurement.

However, if only one measuring device is available, thesteadiness of the flow shall be checked by repeati~gmeasurements at the reference point after each Iota Ivelocity measurement.

It is essential that the shape of the velocity profile in themeasuring cross-section remains stable and is not affectedby possible variations of the flow rate whilst measurementsare being taken.

When the curve of reference velocity variation v, has beenplotted against time, this curve is used to relate all traversemeasurements to the same reference flow rate 90(preferably that which corresponds to the mean of velocitymeasurements at the fixed point). For comparatively smallchanges of the reference velocity, the velocity vi,t measuredat any point i at time tcan be transposed by multiplicationby the ratio of velocity V,,O at the reference pointcorresponding to flow rate q. at velocity Vr,t at thisreference point at time t:

NOTE – Where the reference measurement is e quantity directl~proportional to the flow rate (for instance, the rotational spaed of ashaft driving a fan or a pump), this measurement can be subatitutaddirectly for Vr,o and Wr,t in the above equation. Where the referencereading is in the form of a pressure difference (for instance across afixed feature of the flow circuit, or the differential pressureof areference Pitot tuba), the square root of each reference reading canbe substituted for Vr,o and Vr,t in the abova equation.

3

Is 14973:2001lso 3966:1977

However, it must be noted that velocity profile fluctuations

may occur with cut creating flow rate fluctuations. In sucha case the use of reference point velocity may lead to errorsand it is preferable to check steadiness of flow by means ofany pressure difference device (standardized pressuredifference flow meter, piezometric control on a convergent,bend, spiral casing, peculiar pressure loss, etc.), even if it isnot calibrated, provided that its reliability and adequatesensitivity have been ascertained. In this case the above-mentioned proportional correction will relate to the dif-ferential pressure and not to the velocity.

3.3.3 Checking of velocity distribution

It is recommended that the regularity of the velocity dis-tribution be checked either by plotting or by other means,regardless of whether or not the plotting is necessary forcalculating the discharge velocity.

In the same way, when several measurements are made onthe same cross-section at different flow rates it isrecommended that the velocity profiles be plotted in a non-dimensional manner (i.e. by using the relative -velocities;see 2.8) to check their consistency with each other andhenc~ to ensure that there are no abnormal features at

particular flow rates (thus, the profiles shall not changeerratically as the flow rate varies over a wide range ofReynolds numbers).

It may also be useful to plot the velocity distribution curvesas indicated above in order to detect any error in themeasurement of a local velocity. The doubtful measure-ment shall be repeated whenever possible; when this cannotbe done, it shall be ignored and the velocity profile drawnon the basis of the previously obtained profiles providedthere are independent reasons for believing the doubtfulmeasurement is false.

3.4 Location and number of measuring points in thecross-section

3.4.1 General requirements

The rules to be followed for locating the measuring pointsdiffer according to the methods of determination of thedischarge velocity as specified in this InternationalStandard. These rules are given in clauses 8, 9 and 10respectively.

Whatever method is used, the distance between the axis ofthe head of the Pitot tube and the wall shall not be lessthan the head diameter d.

The location of the Pitot tube shall be calculated from theactual dimension of the conduit along each traverse line(rather than from the mean dimension) and shall bemeasured to :

* 0,005 X, where X is the dimension of the duct parallelto the measurement of Pitot tube position, or

* 0,05 y, where y is the distance of the Pitot tube fromthe nearest wall, whichever is the smaller.

Sub-clauses 3.4.2 and 3.4.3 prescribe ~ minimum numberof measuring points applying in particular to smalldimension conduits. As it is necessary to define the velocityprofile as accurately as possible, the number of measuringpoints can be advantageously increased provided that this isallowed by the operating conditions and steadiness of theflow.

When a single Pitot tube is traversed across the duct, thedistance between a reference point (from which -eachposition is measured) and the wall of the duct must first beobtained. This may introduce a relatively large systematicerror in all position measurements. In such instances it isrecommended that complete diameters be traversed (ratherthan opposite radii on each diameter) since the systematicerror will then tend to cancel out on the two halves of thetraverse.


The measuring points shall be located at every point ofintersection between a prescribed number of circlesconcentric with the pipe axis and at least two mutuallyperpendicular diameters.

The measurements shall be carried out in at least threepoints per radius, so that there is a minimum of twelvepoints in the cross-secticm. An additional measuring pointat the centre of the conduit is desirable to check the shapeof the velocity profile and is necessary for the calculationof the stem blockage correction, where applicable

(see 11.1.2).


The minimum number of measuring points shall be 25.Unless a special layout of measuring points is requiredfor the use of an arithmetical method, their position shallbe defined by the intersections of at least five straight linesrunning parallel to each wall of the conduit.

4 DESIGN OF PITOT TUBES

4.1 General description

The use of one of the types of Pitot tube described inannex A, all of which fulfil the requirements of 4.2, is ‘recommended; this avoids the necessity of making severalcorrections to the measurements. The use of any otherPitot tube which fulfils the requirements of 4.2 is permittedprovided that its calibration is known.

The Pitot static tubes dealt with in this internationalStandard consist of a cylindrical head attachedperpendicularly to a stem which usually passes through thewall of a conduit. The length of the head is generallybetween 15 and 25 times the head diameter.

Is 14973:2001

1s0 3966:1977

At one or two cross-sections along the head, static-pressureholes are drilled around the circumfiwence, so that, in theabsence of leakage, the registered pressure is transferredthrough the head and stem to a point outside the conduit.

A smaller tube, concentric with the head and stem,transfers the total pressure, registered by a hole facing theflow direction at the tip of an axially symmetrical noseintegral with the head, to a point outside the conduit,

An alignment arm, fitted to the end of the stem, facilitatesalignment of the head when this is obscured by the conduitwall.

4.2 Criteria to be fulfilled by the Pitot tube

The nose (including the total pressure hole) shall be

designed in such a way as to comply with the followingrequirements :

a) The response of the differential pressure toinclination of the head relative to the flow shall meetone of the following two conditions according to thecircumstances (in both cases it is necessary to know theresponse curve of the Pitot tube) :

1) if precise alignment of the Pitot tube with theconduit axis is not possible but there is no swirl, thedifferential pressure should be as independent aspossible of the yaw of the head in uniform flowl );

2) if precise alignment of the Pitot tube with theconduit axis is possible but swirl is present, thevariation of the differential pressure recorded by thetube in uniform flow with yaw angle Q shall beapproximately proportional to COSZ14.If the head isperfectly aligned axially and if swirl is less than f 3°,the differential pressure shall not deviate from thislaw by more than 1 %.

It should be noted that rqisalignment and swirl can occursimultaneously and efforts shall be made to minimizeeach of them.

b) The calibration factors for different specimens oftubes to a particular specification shall be identical, towithin t 0,25 %, and shall remain so for the working lifeof any such tube. If the user has any doubt upon thispoint, an individual calibration of each Pitot tube shouldbe made.

c) When used in a,liquid, any cavitation from the noseshall not cause a significant error in the static crressurereading of the tube.

d) The static-pressure holes shall be :

1) not larger than 1,6 mm in diameter;

2) at least six, and sufficient in number for thedamping in the static pressure circuit to be as nearly

as possible equal to that in the total-pressure circuit; ifnecessary, on Pitot tubes the diameter of which issmall, the orifices may be placed in two planes;

3) placed not less than six head-diameters from thetip of the nose;

4) placed not less than eight head-diameters fromthe axis of the stem.

e) If the stem is enlarged to a diameter d’, there shall bea length of stem not less than 7 d’, between the axis ofthe head and the commencement of the enlargement, forwhich the stem-diameter is equal to the head-diameter.

f) The junction between the head and stem shall be

either mitred or curved to a mean radius equal to3 ~ 0,5 times the head-diameter.

g) . An alignment arm shall be fitted to the end of thestem away from the head, to ensure precise alignmentand positioning within a conduit.

Three types of Pitot tubes which are currently used andwhich comply with these criteria are described as examplesin annex A,

5 REQUIREMENTS FOR USE OF PITOT TUBES

5.1 Selection of the measuring cross-section

5.1.1 The cross-section selected for measurements shall belocated in a straight @ipe length and shall be perpendicular

to the direction of flow. It shall be of simple shape, forexample either circular or rectangular. It shall be locatedin an area where the measured velocities fall within thenormal working range of the apparatus used (see 5.3.2).

5.1.2 Close to the measuring cross-section, flow shall besubstantially parallel to and symmetrical about the conduitaxis and contain neither excessive turbulence nor swirl; themeasuring cross-section shall thus be chosen far enoughaway from any disturbances that could create asymmetry,swirl or turbulence (see 5.1.4).

1) The Pitot tubes described in annex A allow independence of the differential preaaure to within * 1,5 % up to 14” yew in uniform flow.

5

Is 14973:20011so 3966:1977

The length of straight pipe that may be required to achievethese conditions will vary with the flow velocity, upstreamdisturbances, level of turbulence and the degree of swirl, ifany.1 )

5.1.3 Although measurements with the Pitot tube inoblique or converging flow should as far as possible beavoided, these may however be carried out provided thatthe-maximum flow deviation with respect to the Pitot tubeaxis does not exceed 3°.

For guidance, it can also be considered that a swirl is smallenough not to increase the confidence Iimits given in thisInternational Standard on the measured flow rate if theresulting gradient of local velocities to the pipe axis is lessthan 3°.

5.1.4 Preliminary traverse tests shall be made to ascertainthe regularity of flow.

If these traverses show that flow is not satisfactory, this cansometimes be remedied using one of the devices describedin 5.2.

Once these devices are in place it shall be checked that theflow complies with the requirements of this InternationalStandard. If not, a more detailed traverse of the measuringcross-section is necessary, and reference shall then be madeto a separate document which will be published later.

5.2 Oevices for improving flow conditions

5.2.1 If swirl is present in the flow, it can often besuppressed by means of an anti-swirl device consistingeither of several adjacent pipes parallel to the flow directionor of a honeycomb with square or hexagonal cells. Which-ever type is used the whole device shall be rigorouslysymmetrical and the following requirements shall be met :

— the maximum transverse dimension a of a channelshall be less than 0,25 D;

– length shall be greater than 10a.

5.2.2 If the velocity distribution is unacceptably irregular,it can often be remedied by means of a profile developerconsisting of, for example, one or more screens, grids orperforated plates. It must be noted, however, that suchdevices are only effective at the price of a rather high headloss.

5.2.3 The devices described in 5.2.1 and 5.2.2 shall belocated at the greatest possible distance upstream of the

measuring cross-section and in any case at a distance of atleast five diameters of a circular cross-section (or 20 timesthe hydraulic radius of a conduit of any cross-sectionshape). Furthermore they shall not be located immediatelydownstream of a disturbance.

5.2.4 If the velocity distribution is unacceptably irregularor if the flow is not parallel enough, but if it has beenpossible to check that no swirl is present, it is sometimespossible to remedy the situation by means of a provisionalguiding installation. The latter will consist of a slightlyconverging entrance, connected in such a way as to ensurethat no separation occurs, to a straight pipe length, thelength of which shall be at least twice the larger dimensionof the conduit.

5.3 Limits of use

5.3.1 Nature of the fluid

The fluid shall be a continuous single-phase fluid or shallbehave as if it were such a fluid. Liquids shall be Newtonianand shall not exhibit anomalous viscosity or thixotropicbehaviour.

5.3.2 Range of velocities

Pitot tubes shall not be used with flow velocities less thanthe velocity corresponding to the lower limit of theReynolds number (see 7.1] or greater than the velocitycorresponding to a Mach number of 0,25.

5.3.3 Nature of the flow

The formulae given (see 7.1 and 7.2) are accurate only forsteady flow without transverse velocity gradient orturbulence, In practice both are always present in closedconduits. Clause 11 and annexes B and C give indicationsof the magnitude of the corresponding errors.

5.3.4 Dimensional limitations

The ratio d/D of the Pitot tuba diameter d to the conduitdiameter D shall not exceed 0,02 with a view to keepingnegligible the error on the rate of flow resulting from thevelocity gradient and from the stem blockage effect (seeclause 11). In difficult flow conditions, a ratio of up to0,04 may be admissible provided that the necessarycorrections for blockage effect and velocity gradient aremade; this limit value may indeed be necessary to avoidvibration of the tube in very high velocity flows. On theother hand the requirements mentioned in clause 4 shall besatisfied.

1) For guidance it is normally assumed that to comply with these conditions there should be a length of upstream conduit between thebaginning of the working section and any significant upstream irregularity (see 2.2.10) of at least 20 diameters of a circular cross-section (or 80times the hydraulic radius of a conduit of any cross-section shape). Similarly there should be at least 5 diameters of a circular cross-section (or20 times the hydraulic radius of a conduit of any cross-section shape), between the measuring cross-section and any significant downstreamirregularity.

6

IS 14973:2001ISO 3966:1977

!5.3.5 Influence of turbulence

Turbulence has a twofold influence in the case of anexploration by means of a Pitot tube, i.e. :

a) on the total pressure reading;

b) on the static. pressure reading.

Turbulence of flow leads to an overestimation in thedetermination of velocity, which is a function of the degreeof turbulence.

Detailed study of the turbulence correction is given inannex C.

5.4 Performance of measurements

5.4.1 Measurement of differential pressure

The device chosen for the measurement .of differentialpressure shall be capable of measurement of a steadydifferential pressure equal to the maximum value recordedduring the traverse with an uncertainty not exceeding 1 %

(at 95% confidence level).

5.4.2 Differential pressure fluctuations

In order to obtain, from the measurements, time-averaged

values which are representative in spite of random

fluctuations of the flow rate, it is necessary :

a) that the differential pressure fluctuations be dampedby applying to the measuring apparatus the minimumdamping allowing easy reading without concealinglonger-term fluctuations. The damping of the apparatusshall be symmetrical and linear; this can be achieved bymeans of a capillary tube located in the manometriclimb in accordance with the requirements of annex D;

b) that readings at each measuring point shall berepeated a certain number of times, preferably atunequal time intervals. A sufficient number of readingsis reached when suppressing any one of them (exceptthose which present an abnormally high error and areexcluded automatically) does not modify the mean bymore than + 1 %.

However, if damping condition a) has been satisfied

sufficiently well so that the instantaneous readings of

differential pressure do not fluctuate by more than 12 %of the mean differential pressure over a sufficiently longperiod of time (for example ten maximum and tenminimum values to be observed), then a-visual averaging ofthe measurement is permissible.

NOTE – The final tolerance applicable to the rate of flow onaccount of random fluctuations of the readings w-ill be a function ofthe total number of readings made during an exploration.Consequently if the total number of measuring points is high, thenumber of readings at each point mav be comparatively small.

5.4.3 Determination of fluid density

The fluid density shall be determined in such a way as toensure that the uncertainty in the value obtained does notexceed * 0,5 ‘A (at 95 % confidence level).

When the fluid density is obtained from the absolute static

pressure and static temperature, these quantities may

generally be taken from single readings made at a pointlocated at 0,75 times the pipe radius from the wall. Never

theless, for measurements in a compressible fluid where theratio of the maximum differential pressure to the absolutestatic pressure in the plane of the traverse is greater than0,01, the procedure described in 7.2 and in E.3 of annex Eshall be followed.

5.5 Inspection and maintenance of the Pitot tube

The Pitot tube does not require any Specialmaintenance,.but it shall be ensured, before and after the measurements,that the tube used complies with the criteria specified inclause 4.

The following points in particular shall be checked :

— the pressure sensing holes and their connecting tubesare not blocked;

– there is no leakage between the chambers inside thePitot tube which receive the total pressure and the staticpressure;

– the tube has not been strained, or its nose damaged;

– the tube is clean;

— the head of the Pitot tube is truly perpendicular tothe supporting stem.

Furthermore, Since the determination of the velocity isrelated to the differential pressure, it shall also be checkedthat :

6

– the connections to the pressure gauge are as short aspossible and that they are absolutely leak-tight (porous

or cracked rubber tubes, etc., are not permissible);

– they are in general in accordance with ISO 2186,Fluid flow in closed conduits – Connection for pressuresignal transmissions between primary and secondaryelements;

— where damping of the differential pressure gauge isnecessary, it is symmetrical and linear (see annex D).

POSITIONING OF PITOT TUBE

The axis of the Pitot tube head shall be set parallel to thepipe axis; an alignment arm shall be provided to assist indoing this.

The Pitot tube shall be rigidly fixed during themeasurements.

7

Is 14973:2001ISO 3966:1977

The Pitot tube shall be positioned in the pipe in accordancewith the requirements of 3.4.1 and clause 8 or 10.

The device which holds the Pitot tube in the pipe shall besuch that no leak can occur into or out of the pipe.

7 VELOCITY COMPUTATION

7.1 Verification of conditions for a measurement

Pr&vided that the Reynolds number based on the diameterof the total pressure hole of the Pitot tube is in excess of200, and that the local Mach number (for measurementsin a compressible fluid) does not exceed 0,25, the localvelocity may be calculated. However, annex E givesindications on the method of carrying out velocitymeasurements in the case of a compressible fluid at a higherMach number.

The first condition is equivalent to a requirement thatAp is never less than

()2x104 P 2.— —P &di

where

Ap is the differential pressure measured by the Pitottube;

p is the density of the fluid;

L is the dynamic viscosity of tbe fluid;

di is the diameter of the total pressure hole of the Pitottube;

a is the calibration factor of the Pitot tube : to betaken as 1 for this calculation.

.

The second condition requires that, for measurement in a

compressible fluid, the ratio of the differential pressure tothe absolute value of the pressure recorded by the staticpressure tapping of the Pitot tube shall never exceed alimiting value, which varies with ‘y (the ratio of the specificheat capacities of the gas) according to table 1.

+ABLE I

~

7.2 Formulae for velocity computation

The local velocity of a fluid in a steady flow withouttransverse velocity -gradient or turbulence at Reynolds

numbers,pressure

based on the internal diameter of the totaltapping, greater than 200 is given by the

in which (1 - e) is a compressibility correction factor. In aliquid, e = O so that no compressibility correction isrbquired, but in a compressible fluid at low Mach numbersthe factor ( 1 -e) may be determined by the relationship

[ ()]1 Ap 7–1 ‘Ar”2 “2(1–d= l-——+——

2-y p 6YZ P

where

~ is the ratio of specific heat capacities;

p is the local statio pressure;

p is the local density of the fluid;

AP is the differential pressure indicated by the Pitottube;

a is ‘the calibration factor of the Pitot tube (under theabove-mentioned conditions and for Pitot tubesdescribed in this International Standard, it is Practicablyequal to 1,00).

The density of the compressible fluid is determined fromthe following equation :

PM

‘==

where

R = 8,3143 J.mol-l K-l, the molar mass being

expressed in kilograms per mole and having a value0,02895 for air;

Z is the gas law deviation factor; it is insignificantlydifferent from unity for air at absolute pressures lessthan ten times atmospheric and temperatures between273 and 373 K (it should be distinguished from (1 - e),the compressibility correction factor);

T is the local static temperature given by the formulal J

+’=[1+=’:1TO being the total temperature measured on the axis of

the duct using an ideal total temperature probe. Theeffect of using any non-ideal temperature probe isdiscussed in annex D.

For selected values of ~ and Ap/p, values of (1 – e),together with T/TO, are shown in table 2.

1) This formula is an approximation which is adequately precise for tha purposas of this International Standard.

8

Is 14973: 2WI1s0 3966:1977

TABLE 2

‘@—P

0,01 0,988 0,998 0,998 0,898 0,986 0,988 0,887

0,02 0,898 0,996 0,997 0,886 0,995 0,996 0,994

0,03 0,997 0,993 0,995 0,884 0,993 0,994 0,992

0,04 o,9e6 0,991 0,984 0,982 0,991 0,993 0,889

0,05 — — — 0,989 0,991 0,866

8 DETERMINATION OF THE DISCHARGE VELOCITYBY GRAPHICAL INTEGRATION OF THE VELOCITYAREA

The general principle of this method is specified in 3.1.

The measuring points shall be located along straight lines,and in order to determine m accurately, two measuringpoints shall be placed on each straight line as close aspossible to the wall (see annex F).

The number and position of the other points shall beselected in such a manner that the velocity profile can bedetermined satisfactorily. They will usually be distributedin the cross-section in such a way as to divide it into areas,each of which has the same flow rate in order 10 attach

approximately the same importance to ail measuring points.

Reference should be made to 3.4 when determining thenumber and location of measuring points, and to clause 11when it is considered necessary 10 make some correction tolocal velocity measurements or to the position of measuringpoints.

8.1 Circular cross-section

If v is the flow velocity at a point of polar co-ordinates rand 0, and if R is the mean radius of the measuring section,the discharge velocity is

2W R

f(

1u=~

f ()

2v(r,(l ) r dr d6J= ud ~

rrR2, o , ~ R90

,4

(1–E)

0,886

0,887

0,985

0,983

0,881

I 1,5

TtTI TO (1–e) T/TO

0,997 0,888 0;886

0,993 0,997 0,883

0.980 0,995 0,869

0,987 0,994 0,885

0.984 0,992 0,882

1,6 1,1

(1 –c)

0,886

0,887

0,885

0,894

0.882

T/TO (1-.E)

0,986 0,988

0,992 I 0,997

0,968 I 0,996

0,984 I 0,984

where

u is the spatial mean velocity along the circumferenceof radius r;

rn is the radius of the circle defined by the measuringpoints closest to the wall.

The method used consists in :

a) taking UC (arithmetical mean of the velocities at themeasuring points located on one circle of radius r=) asthe value of u;

b) plotting UC against (rC//?)2 between r = O and.r= rn (see figure l)l);

c) graphically determining the value of the includedarea below this curve (see figure 1);

d) adding to this value a calculated termz ) correspond-ing to the peripheral zone and equal to :

m

()

r“ 2—u” l–~m+l

where

Un k the value of the arithmetical mean of thevelocities at the measuring points located on the circle ofradius rn (i.e. the closest to the wall);

1) To facilitate plotting in the vicinity of the measuring point closest to the wall, the tangent line to the cuwe for r = rn will be drawn with eslope equal to :

()

;Uc - ‘nzr=rn=

2m~()1–~

denoting (r/R)2 asX.R

The stope of the curve is derived from Karman’s conventional law, for the varia+imn nf the fluid velocities in the peripheral zone :

()

R-r ‘s~mu.”” —

R–rn

2) This simplified expression omits the other term

-m

()

rn2

u“ 1 -—(rrr+lt(2rrr+l) R

in the result of the integration (within the peripheral zone) derived from Karman’s conventional law : this latter term only represents about

1 – {m/R).4m+2

times the flow in tha peripheral zone.

9

IS 14973:2001ISO 3966:1977

m is a coefficient depending on the wall roughness andon the flow conditions, the value of which can bedetermined in accordance with the indications givenin annex F and is generally between4 (rough wall) and10 (smooth wall).

FIGURE 1 – Computation of the discharge velocity in a cirmdarconduit - Graphical integration in the mea oxplomd by

the Pitot tubes

8.2 Rectangular cross-sactions

The computation of the discharge velocity requires a

double integration across both dimensions of the conduit.

Measurement shall be started either on the vertical lines?)

or on the horizontal Iinesl j. The matter is developed herestarting with horizontal line measurements.

The formula for the discharge velocity is :

u=~’p(:)43where

L is the conduit width in the measuring cross-section(arithmetical mean of the widths measured on at leasteach horizontal measuring line);

H is the conduit height in the measuring cross-section

(arithmetical mean of the heights measured on at leasteach vertical measuring line);

1 is the distance from a particular point to the side-wallchosen as origin;

h is the height of a particular point above the bottom

The method used consists in :

a) plotting the variation curve of the vetcscity on eachhorizontal line between the extreme measuring points,as a function of the relative distance l/L (see figure 2)2};

b) determining graphically the value of the includedarea below this curve between the extreme measuringpoints (see figure 2);

c) adding to this value two terms corresponding to theperipheral zones, both being equal to

(fi)(:)(va)

(the sum so obtained is the mean velocity ui on thehorizontal measuring line concerned);

d) plotting the variation curve of ui between theextreme horizontal measuring lines as a function of therelativa height h/H of the corresponding horizontal line(see figure 2)3);

1) Throughout this sub-clause, a “vertical line” will mean a line parallel to the conduit height and a “horizontal” line will mean a line parallelto the conduit width.

2) To facilitate plotting in the vicinity of the extreme measuring points, the tangant line to tha cuwe at each of tham will be drawn with adope the absolute value of which is equal to

VaL—ma

where

v= is the velocity at the considered extreme measuring point (at a distance a from the neerest wall);

m is a coefficient depending on the wall roughness end on the flow conditions, the value of which can be determined in accordance withthe indications given in annex F and is generally between 4 (rough wall) and 10 (smooth wall).

The slope of this curve is derived from Karman’s conventional law, for the variation of the fluid velocities in the peripheral zone,

()~I/m

vx=va —a

3) To facilitate plotting in the vicinity of the peripheral zones, the same procedure shall be followed #s in determining the mean velocity alongeach horizontal line (see 8.2 a)).

.

10

Is 14973:2001ISO 3966:1977

e) graphically determining the value of the included velocity. Both terms are equal to :

area below this curve between the extreme horizontalmeasuring lines;

(+)($ (”a’)

f) adding to this last value two terms, corresponding to where Ua, is the mean velocity on the horizontal measurin9

the peripheral zones, in order to obtain the discharge line closest to the wall (at a distance a’ from the wall).

0

0

/

o

-5

1

2

+’ //

H 1

‘m I

,+

-QJ

--L

9

Ui

F IG U R E 2 – Computation of the discharge velocity in a ractarsgular conduit -Graphical integration in tha area axplorad by Pitot tubas

11

Is 14973:2001ISO 3966:1977

9 DETERMINATION OF THE DISCHARGE VELOCITYBY NUMERICAL INTEGRATION OF THE VELOCITYAREA

The general principle of this method is specified in 3.1.

The formulae proposed below are derived from inter-polations between successive pairs of measuring pointsalong third-degree curves in (r//?)2 for a circular cross-section conduit, or in l/L or h\H for a rectangularcross-section conduit. The separate individual arcs combineto form a continuous -curve with a continuous derivative.

In the peripheral zone the same law as indicated in thepreceding clause is applied.

For the number and position of maasuring points, referenceshall be made to the indications of clausa 8 and 3.4.Reference shall be made to clause 11 when it is considerednecessary to apply certain corrections to local velocitymeasurements or to the position of measuring points.

9.1 Circular cross-sections

If VO is fhe velocity at the conduit centre, and u,, U2, .... Unare ttie mean velocities [calculated as indicated in 8.1 a)]along the circumferences with increasing relative radii r;, r;,.... rj (with r,* = rilR, where R is the radius of the cross-section)r the dischaFge velocity in the cross-section is givenby the formula :

[U= v. –Ar”2 1

1r~3+~r”2+i2~,2 2 ,2 1

2

+U1[

2 1Lr*2+_r21

*2— _r*2613 ,* 3

[1r*3

– U2~

12r:

j=”-z

+ m 1 ●2 2*22*2u; ——

‘fir(i+2)+~r[i+l) Sr(i– l)+i=2

+1 ●21

12r(i -2)

[

1+U – r’2 + ~r*2

2 ●2 1

“’–’) 2 n 112(n-1) –~r(n–2)+ #-3)

+ u“[

x (1 –rjz) +(r~2 – ●2r(n-1))2

+~r~2–m+l 12m(l–rj2) 12

When n = 3, the term on the fwrth line of the equetion abovedisappears and the formula is simplified es foll~s :

[ 1U.vo –Jr”2++2r;2 +&-$-,~22

[2 1 192 +_r*2__r.2

+“16132 ,2 3

[

1 r“3 2 1 1+ Liz 1

1__r.2+—r*z+—r:212r~ 31 1222 —

[

[,32 – q2)2+ U3 & (1 –r$) + ●zr*2_i .2+ ~ .2

12rn(l-r~2) 12 3 3r2 ~rl 19.2 Rectangularcross.sections

In the following formula, U represents :

– either the mean velocity along a measuring line, inwhich case v,, V2, .... Vn are the velocities measured atpoints located at distances J,, 12,..., /n from the referencewall; L is the distance between the wo walls on the

considered line :

11 12-11v;=~v; =—, . . .

L

‘n-l(n-l). . .. V.=

L-ln

L.zv;n+l)=~

— or the discharge velocity in the measuring cross-section, in which case v,, V2, . . . . Vn represent the mean

velocities u,, U2, .... Un alohg the measuring lines locatedat distances h,, /s2, .... hn from the reference wall; Histhe height of the measuring cross-section.,

h, h2 -h,V:=<, v;= —,...

H

hn–h(n_l) H-hn/v;=

H,v(; +1, =~

NOTE – When n = 4, the fourth line is evaluated only for i =“2. NOTE – When n = 5, the third line is evaluated only for i = 3.

12

10 DETERMINATION OF THE DISCHARGE -VELOCITYBY ARITHMETICAL METHODS

The general principle of these methods is specified in 3.1.

For each method the measuring cross-section is divided intoa small number of section elements. The measuringlocations are predetermined for each section elementfrom :

a) an assumption of the mathematical form of the veloc-ity distribution law in the section element concerned;

b) a choice of the weighting coefficients.

The various curves corresponding to each section elementdo not need to constitute a continuous curve with acontinuous derivative.

In the peripheral zone, a logarithmic law is assumed forvelocity distribution with respect to the distance from thewall. In the arithmetical methods described hereafter, theweighting coefficients have been chosen to be equal in thecase of circular cross-sections and the section elements haveareas proportional to the number of measuring points inthe element concerned; reference shall be made to clause 11when it is considered necessary to make certain correctionsto local velocity measurements or to the positions ofmeasuring points.

10.1 “Log-linear” method

By hypothesis the mathematical form of the velocity

distribution law for each element is :

u= Alogy+By+c

where

y is the distance to the wall;

A, f? and C are any three constants (except for theexternal ring element where B is zero).

Is 14973:2001

1s03966:1977


The location of the measuring points corresponds to thevalues of the relative radius r/R; or of the relativa distance

to the wall ylDi shown in table 3.

TABLE 3

Number I Iof meaauring

pointsrlRi ylDi

per radius

0,3586 t 0,0100 0,3207 f 0,00503

5

0,7302 * 0,01000,9358 * 0,0032

0,2776 ? 0,01000,5658 t 0,01000,6950 t 0,01000,8470 * 0,0076

0.9622 t 0.0018

0,1349 i 0,00500,0321 t 0,0016

0,3612 * 0,0050

.0,2171 t 0,00500,1525 t 0,0050

0,076 5.? 0,0038

0.0189 * 0.0009

The mean velocity on each radius is taken as equal to thearithmetical mean of the velocities determined at themeasuring points located on the radius concerned, and thedischarge velocity is equal to the arithmetical mean of themean veloci~ies on each radius. The discharge velocity istherefore given by the arithmetical mean of local velocities.


Different layouts may be developed to apply the log-linearmethod in a rectangular cross-section, using a variety ofnumbers of measuring points. This International Standardis limited to the method using 26 points, for which thelocation is given in figure 3.

In addition to the location of the measuring points given byVL and h/H, the table in figure 3 gives the weightingcoefficients for each measured velocity.

13

Is 14973:2001ISO 3966:1977

I II Ill Iv

h/H

0,034 2 3 3 2

0,092 2 — 2

0,250 5 3 3 5

0,3675 – 6 6

0,600 6 — — 6

0,6325 – 6 6 —

0,750 5 3 3 5

0,908 2 — — 2

0,966 2 3 3 2

FIGURE 3 – Location of nsaasuring points ins ractangulw cross-seotion conduitin tfsa casa of the “log-linear” mathod using 26 points

The discharge velocity is equal to the weighted mean of the 10.2 “Log-Tchebycheff methodmeasured local velocities :

Zki vi By hypothesis the mathematical form of the velocityu=—

Zkidistribution law as a function of the distance from the wallis logarithmic in the outermost elements of the section and

For the method using 26 points Zki = 96. polynomial in the-other elements.

14

102.1 Circular cross-sections

Is 14973:2001

ISO 3966:1977

TABLE 5

The position of the measuring points corresponds to the

values of the relative radius r/Rj or of the relative distanceto the wall y\Di shown in table 4.

TABLE 4

Number

of measuringrlRj

pointsVIDj

per radius

3

0,3754 +.0,01000,7252 ! 0,0100

0,9358 ~ 0,0032

0,3123 k 0,0050

0,1374 * 0,0050

0.0321 t 0.0016

4

!5

0,3314 f 0,01000,6124 ! 0,01000,8000 t 0,0100

0,9524, 0,0024

0,2866 ! 0,01000,5700 ~ 0,01000,6892 , 0,0100

0,8472 ! 0,0076

0.9622 + 0,001 a

0,3343 i 0,0050

0,1938 ~ 0,00500,1000 ~0,0050

0,02384 0,001 2

0,3567 ~ O,OO5 0

0,2150 ‘. 0,0050

0,1554? 0,00500,0764 I 0,00380.0189 f 0,0009

As the weighting coefficients have been chosen to be equal,the discharge velocity is equal to the arithmetic mean of themeasured local velocities.


A number (e) of traverse straight lines, at least equal to five,

are selected parallel to the smaller side of the rectangle; on

each of them a number (f) of measuring points, at least

equal to five, are located. (See figure 4.)

NOTE – For the example chosen, f = 5 and e = 6.

The positions of (efl measuring points (abscissa Xi and

ordinate Yj in relation to the centre of the section) aredefined from table 5.

—.—

eorf Values of XjlL Or Yj/H

5 0 * 0,212 * 0,426

6 ? 0,063 t 0,265 t 0,439

As the weighting coefficients have been chosen to be equal,the discharge velocity is equal to the arithmetic mean of themeasured local velocities at the various measuring points.

11 CORRECTIONSMENTS

OF LOCAL VELOCITY MEASURE-

The measurement of local velocity is affected by errorswhich are due in particular to the blockage effect, to thevelocity gradient, to turbulence and to the head loss whichwould require corrections whose amount is unfortunatelynot always exactly known. These corrections are in factonly applied when very accurate measurement is requiredor if errors are very large.

The following sub-clauses give indications of the valueswhich can be expected from these corrections. Thetheoretical bases of the estimation are given in annexes Band C.

11.1 Correction for stare blockage

When a Pitot static tube is used in an infinite stream, thecalibration factor takes account of the stem influence onthe pressure readings at static pressure tappings.

!,‘~b- ‘] ‘- A

—xy+

x x

x x

q

‘o x x

1 m Hx

x x x x x x—

—-x—— x x x x— v

FIG UR E 4 – Position of the measuring pointa in a rectangular conduitin the CSJWof the “lorTchebychaff’ mathod

15

Is 14973:2001ISO 3966:1977

When the tube is used in a conduit, the velocity increasewhen the flow passesbetween the stem and the walls causes

a decrease in the recorded static pressure without affectingthe total pressure. The differential pressure Ap shalltherefore be reduced to take account of the stem blockageeffect.

11.1.1 Case where the correction can be neglected

When \he ratio d/D is less than or equal to 0,02 as specifiedin 5.3.4, and when the Pitot tube is consecutivelyintroduced into two diametrically opposite insertion holesin such a way that traverse only occurs across one radius,the correction for blockage effect can be neglected. In theother cases the correction to be made shall be estimatedbefore it is determined whether or not it can be neglectedin view of the required accuracy.

11.1.2 Estimation of the correction of local velocitymeasurement t

‘The correction for each individual measurement &s isgiven by :

6 (Ap) = –o,7kb (w) APmax

where

& is the recorded value of the differential pressure;

AP~,, is the corresponding value on the conduit axis;

S is the frontal projected area of that portion of thestem inside the conduit;

A is the cross-sectional area of the conduit;

kb is the blockage coefficient of a cylindrical stem

(see figure 5).

The value of 0,7 is an average (0,65 to 0,75) of the ratioof the mean value to the maximum value of the squaredvelocity in the conduit cross-section.

11.1.3 Estimation of the overall correction of the flow-rate value (application to arithme tical methods)

The relative error caused in ignoring the correction of localvelocity for the point situated at a distance y from the wallcan be calculated as follows for a velocity profile of the

v

()

Y1 /m

shape— = —vmax R

~ (v/vmax) (m+2)(m+l)(v/vm*X) = – 2 rn2 (R/Y) 2/m kb(S/A)

If the result is applied to each point, the position of whichis predetermined by arithmetical methods (clause 10), the

overall corrections for flow rate are practically identical forall arithmetical methods. Their values are given in figure 6.

11.2 Correction for transverse velocity gradtent

The stagnation pressure recorded by a Pitot tube in a fluidflow with a tranverse velocity gradient is always slightly

overestimated. Interaction between the nose of the Pitottube and the fluid flow causes a small displacement of theapproaching streamlines, so that the tube brings to rest astreamline originating in a higher velocity region ahead ofthe plane of measurement. Annex B gives a few justifica-tions of how the corrections indicated below are evaluated.

This influence can be taken into account in two ways,either by integrating the velocity area on the basis of thecorrected position of the measuring points (see 11.2.1) or,for arithmetical methods only, by keeping the pre-determined position and making an overall correction forflow rate (see 11.2.2).

11.2.1 Correction for measuring point position

11.2.1.1 CASE OF GRAPHICAL OR NUMERICAL

INTEGRATION

Meastirements of differential pressure ~ recorded at realdistances y from the wall are considered for calculation asbeing carried out at distances y + Ay, where Ay is thefictitious displacement of the measuring point which canbe calculated using the following formula :

Ay

(Y)[ [-]—=k~–0,195k~ ~ l–

d

9

If the value of kg has not been determined once for all,for any particular nose shape, kg can be taken as 0,10 f 0,02for all Pitot tubes meeting the requirements of 4.2. Table 6can be used for displacement evaluation of Pitot tubeswith kg = 0,10 and for a certain range of distance from thewall.

TABLE 6

t ,

$ 0,50 0,67 0,75 1,0 1,5 2 3 4 -

$ 0,069 0,075 0,077 0,082 0,068 0,091 0,094 0,095 0,100

The correction for displacement must not be forgottenwhen m k determined in accordance with the requirementsof annex F.

11.2.1.2 CASE OF ARITHMETICAL ME THOOS

When measuring positions are calculated in accordance withthe tables in 10.1.1, 10.1.2, 10.2.1 and 10.2.2 thecorrespondirlg y distances shall be reduced by Ay ascalculated above to obtain the real position yl to be usedfor measurements.

16

IS 14973:2001ISO 3966:1977

1,0.

0,9 \

-x = distanca from plane of static holes to axis of Pitot-stemA = cross-sectional area of conduit

0,8 ~ \

0,7\

k~ 0,6

0,5

0,4\

0,3F

0,20 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Xl+i

FIGURE 5 – Variation of blocka~ conatent with diatanca ahead of cylindri~l Mem in= ~Io~ ~~uit

Is 14973:2001ISO 3966:1977

–1,0

r

-0,8

– 0,6

%

– 0,4

– 0,2

0 E

;t 7Staticholes

+ 00

One insertionhole/d iemeter

Two insertionholsddiemeter “=8

o 0,01 0,02 0,03 0,04

dfD

FIGURE 6- Overeli oorreotion to be edded to volumeflow rete to ●liow for Pltot-stem blockege

1122 Overall correction of flow rata

Figure 7 gives an indication of tha corrections to be made

lo the flow-rate value in circular conduit when a Pitot tubewith kg = 0,10 is used for the measurements. Formeasurements with slightly different k values it is permitted

kto correct the values of the figure with a ratio of Ho.

11.3 Correction for turbulence

It may be assumed that for approximately 10% turbulencethe correction of the local velocity value is about – 0,5 to– 2 % according to the nose shape of the Pitot tube usedand the Reynolds number of the fluid flow (see annex C).

11.4 Correction for head loss

Since the static pressure tapping

18

is located at a distance

downstream of -the local pressure tapping, the pressuremeasurement is affected by an error which is equal to thefriction head loss in the conduit over this distance. Themeasured differential pressure is therefore slightly over-estimated.

This correction is generally negligible, but if it is considerednecessary to take it into account, the differential pressureat each measuring point shall be reduced by the quantity :

~=$?xp:

where

X is the universal coefficient for head loss;

nd is the distance of the total pressure tapping to theplane of static pressure tappings.

Is 14973:2001ISO 3966:1977

%

– 0,8

- 0,6

– 0,4

- 0,2

%

0,8

-0,6

- 0,4

– 0,2

TI Pcsintslradtusog-iinear,og-Tchebycheff

0,01 0I

2 0.03 I

5 pointe/radiuslog-linear, I I I 1

04

0 0,01 0,02’ 0,03 0,04

FIGURE 7 – Ovarall correction to ba addad to voluma flow rata to ●now for Pitotdiaplaoamont

19

Is 14973:20011S0 3966:1977

12 ERRORS

This clause defines a number of fundamental statistical

terms which are used in this International Standard and

describes the method employed to assess the accuracy of

the volume flow-rate measurement from a list of error

sources involved in local velocity measurement and flow-

rate calculation.

An example of calculation of the overall uncertainty is

given in annex G with the sole purpose of illustrating the

described method of calculation, but it does not give

typical values of the various errors. Each particular case

should therefore be studied carefully,

12.1 Definition of the error

The error in the estimate of a quantity is the differencebetween that estimate and the true value of the quantity.

No measurement of a physical quantity is free from un-certainties arising either from systematic errors or from therandom dispersion of measurement results. Systematicerrors cannot be reduced by repeating measurements sincethey arise from the characteristics of the measuringapparatus, the installation, and the flow characteristics.However, a reduction in the random error may be achievedby repetition of measurements, since the random error ofthe mean of n independent measurements is ~ timessmaller than the random error of an individual measure-ment.

12.2 Errors in the estimation of the local velocity

122.1 Random errors

12.2.1.1 ERROR IN THE MEASUREMENT OF DIF-

FERENTIAL PRESSURE

The measurement of differential pressure is necessarilyaffected by a random error 8 Ap which is due at the sametime to the pressure gauge, to the connecting pipes betweenthe Pitot tube and the pressure gauge and to the operator.This error does not include, however, some distrubances,such as fluctuations, which are considered ‘separately inthe following sub-clauses.

12.2.1.2 ERROR DUE TO SLOW VELOCITY FLUCTU-

ATIONS

A random error bf is made if the measuring period is notlong enough for a correct integration of slow fluctuatitms

8of the flow velocity to be made. This error decreases whenthe number and duration of the measurements at a givenpoint are increased,

12.2.1.3 ERROR IN DENSITY

An error i5P is made in the measurement of density becauseof inaccuracies in the temperature and pressuremeasurements and of the d:gree of cleanliness of the fluid.This error varies in importance according to be fluid natureand conditions.

12.2.1A ERROR IN THE CALCULATION OF THE

COMPRESSIBILITY CORRECTION

A random error 8G is made in the calculation of thecorrection factor for compressibility ( 1- e) according tothe indications of 7.2 and annex E.

12.2.2 Systamaticerrors

It is assumed in the following that the correctionsmentioned in clause 11 for blockage, the velocity gradient,turbulence and head loss have not been applied. If theyhave been, a systematic error will nevertheless be madebecause of the noticeable uncertainty of these correctionsbut this systematic error can be efiher positive or negativeand its absolute value is obviously far less than in the firstcase.

1222.1 ERROR IN THE PITOT TUBE CALIBRATION

Any error in the calibration factor of a pitot tubesystematically affects the measured velocity and introducesan error e=.

12.2.22 ERROR DUE TO TURBULENCE

The error increases with the increasing degree of turbulenceof the measured fluid flow and is always positive, i.e. themeasured velocity value is ‘always greater than the actualflow velocity. Indications on the estimation of this errorare given in 11.3 and annex C.

The resulting error et in the measured velocity will be thesame for all measurements at one and the same point andat the same velocity although errors vary with flow rate onthe one hand and the measuring position on’ the other hand.

12.2.2.3 ERROR DUE TO THE TRANSVERSE VEL

OCITY GRADIENT

The error eg depends on the diameter of the Pitot tube. Itis always positive. Indications on the estimation of thiserror are given in 11.2 and annex B.

12.2.2.4 ERROR DUE TO CONDUIT BLOCKAGE

This error eb increases with increasing blockage of theconduit by Pitot tubes and their supports. It is alwayspositive. Indications on the estimation of this error aregiven in 11.1.

12.2.2.5 ERROR DUE TO THE INCLINATION OF THE

PITOT TUBE WITH RESPECT TO THE FLOW

DIRECTION

This error ey increases with the inclination angle anddepends on the Pitot tube used. It is always positive wheninclination remains within the limits given in 5.1.3.

Indications on the estimation of this error are given inannex A.

20

I

Is 14973:2001

ISO 3966:1977

12.2.2.6 ERROR OUE TO THE HEAD LOSS BETWEEN

TOTAL AND STATIC PRESSURE TAP PINGS

This error e~ increases with increasing spacing of pressuretappings and with the conduit roughness. It is alwayspositive. Indications on the evaluation of this error aregiven in 11.4.

12.3 Errors in the estimation of flow rate

12.3.1 Random errors

12.3.1.1 ERROR OUE TO LOCAL VELOCITY

MEASUREMENTS

The errors in the local velocity measurements will not be

truly random, as they will in part depend on the positionof the measurement across the duct. However, the erroron each measurement will be different, and the majorcontributions to each error will be random jn nature, so

that the overall error 6 “t contributed to the estimationof flow rate may be regarded as random.

12.3.1.2 ERROR OUE TO GRAPH IN GRAPHICAL

INTEGRATION TECHNIQUE

When the graphical integration technique is used, an error

6, will be introduced in drawing the velocity profile and

evaluating the area under the central portion of the graph;

this is random in nature, and the magnitude will dependboth on the operator and on the shape of the velocitydistribution.

12.3.1.3 ERROR DUE TO EVALUATION OF POWER

LAW IN DE X,17S

If the power law index m is calculated by the graphicalmethod given in annex F then the error am from thissource will be random in nature.

12.3.1.4 ERROR DUE TO POSITIONING PIT OT TUBES

If the errors associated with the positioning of the Pitot

tubes are independent of each other (i.e. no large common

systematic error is present, see 3.4.1 ) then the overall effect

will be to introduce a random error 6/ in the flow-rate

estimation. However, provided the conditions of 3.4.1 are

met, this error is negligible.

12.3.2 Systems tic errors

12.3.2.1 ERROR DUE TO MEASUREMENT OF DUCT

OrMENSIONS

Although the area AI of the plane of flow-rate measurement

is evaluated from the mean of several measurements of the

duct dimensions (see 3.2), a systematic error eA still

remains in the calculated flow rate.

12.3.2.2 “ERROR DUE TO NWMERICAL OR ARITH-

METICAL INTEGRATION TECHNIQUES

The techniques given in clauses 9 and 10 either approximate

the velocity distribution or assume a velocity distribution.For a given velocity distribution there is therefore asystematic error ei introduced in the calculated flow rate.

12.3.2.3 ERROR DUE TO NUMBER OF MEASURING

POINTS

If the velocity distribution curve is not perfectly smooth,

the number of measuring points may not be .sufficiemt todefine it adequately, and systematic error ep will result.

12.4 Definition of the standard deviation )

12.4,1 If a variable X is measured several times, eachmeasurement being independent of the others, then thestandard deviation ax of the distribution of n measure-ments, Xi, is :

/;=n \l12

(“~“” (~-x )2i

;=1

‘Jx=\ n_, !where

~ is the arithmetical mean of then measurements ofthe variable X;

Xi is the value obtained by the ith measurement ofthe variable X;

n is the total number of measurements of X,

For brevity, ox is normally referred to as the standarddeviation of X.

12.4.2 If repeated measurements of a variable X are notavailable or are so few that direct computation of thestandard deviation on a statistical basis is likely to be un-reliable and if the maximum range .of the measur~mentsmay be estimated, the -standard deviation may be taken as1/4 of Ibis maximum range (i.e. as 1/2 of the estimateduncertainty above or below the adopted value of X).

12.4.3 If the various independent variables, the knowledgeof which allows computation of the flow rate, are Xl, X2,.... Xk, then the flow rate qv may be expressed as a cerlainfunction of these variables :

9. = f {X1,X2, .. ..Xk)

If the standard deviations of the variables Xl, X2, ..,, Xkare 01, 02, .... uk, then the standard deviation OQVof theflow rate is defined as :

aqv aqv aqvwhere — , —, .... — are partial derivatives.

ax, ax2 axk

1) The standard deviation as defined here is what is more accurately referred to as the “standard deviation estimation” by statisticians.

21

Is 14973:2001

ISO 3966:1977

12.5 Definition of the tolerance squares of the relative standard deviations arising from thesources listed in 12.2. Thus the result of the local velocitymeasurement is :

12.5:1 For the purpose of this International Standard thetolerance in a measurement of a variable is defined as twicethe standard deviation of the variable. The tolerance shallbe calculated and quoted under this appellation whenever ameasurement is claimed to be in conformity with thisInternational Standard.

12.5.2 When partial errors, the combination of whichgives the tolerance, are independent of one another, are

small and numerous, and have a Gaussian distribution, thereis a probability of 0,95 that the true error is less than thetolerance.

‘“(”23at a 95 % confidence level

where

ULP is the standard deviation arising from the error ondifferential pressure;

12.5.3 Having estimated the standard deviation Oqv of theflow-rate measurement 9V, the tolerance ~qv is given by :

8qv =*2aqv

The relative tolerance b~v is defined bv

UP is the standard deviation arising from the error ondensity;

of is the standard deviation arising from slow velocityfluctuations;

Oe is the standard deviation arising from compress-ibility;The result of a flow measurement shall

one of the following forms :

always be given in

0= is the standard deviation arising from the Pitot tubecalibration;

a) flow rate = q. f i5qv (at the 95 % confidence level);

b) flow rate = qv (1f b~v)(at the 95% confidencelevel); Ut is the standard deviation arising from high-frequency

velocitv fluctuations and turbulence;

c) flow rate = q, within f 100 6LV % (at the 95%confidence level ). UQ is the standard deviation arising from the velocity

gradient;

12.6 Calculation of standard deviation UP is the standard deviation arising from the inclinationof the Pitot tube to the flow direction;

12.6.1 Standanf deviation on local velocity measurement tUb is the standard deviation arising from the un-certainty in the correction for blockage;The standard deviation u. associated with a measurement

of local velocity v is obtained by combining the standarddeviations of errors arising from the sources described in12.2. Although “systematic” errors have been distinguishedfrom “random” errors, the probability distribution of thepossible values of each systematic component is essentiallyGaussian. The combination of the random and systematic

errors may therefore be treated as though all were trulyrandom, and the standard deviation for the systematiccomponents can be obtained by calculating a value fortheir standard deviations in the manner described in 12.4.2.Thus the standard deviation of a particular systematiccomponent is half of the plus or minus maximum

uncertainty on that component.

u: is the standard deviation arising from head loss.

12.6.2 Standard deviation on flow-rate measurement

Once again the possible values of the systematic errorswhich are listed in 12.3.2 have a probability distributionwhich is essentially Gaussian, so that all errors may betreated as random for the purpose of estimating thestandard deviation on the flow rate; the standard deviationson the systematic components are obtained in the sameway as in 12.6.1.

The relative standard deviation on the local velocitymeasurement is then the square root of the sum of the

The relative standard deviation on the flow-rate measure-ment is then the square root of the sum of the squares of

22

Is 14973:2001

the relative standard deviations arising from the sourcesin 12.3. Thus the result of the flow rate measurement is :

QV(1+)2+(:)2+(;)2 +(:)’+...+fi)’+($j’]’n)

‘4’2’3at the 95 % confidence level

where

0“ is the standard deviation on local velocity measure-ments, calculated as described in 12.6.1;

ISO 3966:1977

ui is the standard deviation arising from the use of theintegration technique;

am is the standard deviation associated with theestimation of the value of m;

01 is the standardpositioning;

uA is the standard

deviation arising

deviation on themeasuring cross-section areal j;

from Pitot tube

evaluation of the

UP is the standard deviation arising from the number ofvelocity measuring points.

1) It should be noted that the relative standard deviation in the evaluation of the measuring cross-section area is twica the relative standarddeviation on the length measurements from which the area is calculated.

23

IS 14973:2001ISO 3966:1977

ANNEX A

PITOT TUBES

A.1 DIFFERENT TYPES

t--

A

k 16d 8d

24

FIGURE 8 – AMCAtYPe

IS 14973:2001ISO 3966:1977

Head

II

I Ir & I I

B .’ K

A TTotal-Pressure ‘ :Y$E=%RII

holashole Modified

ellipsoidalP

fl.~-z~~ ~1nose

I j.n.tion .~$~. \ l’\ ~\l

I

Mitredjunction

Stem

illbStaticpreeaure

I

Preaeurepreseure

Ii

tapping

1-lFIGURE 9- NPL type with modified ellipsoidal mea

2d

I

A

~-0 ~-l ‘-’’’ -” ”-4 Ai I

/’ A

. ;,<~

!

.-1

/ ““/

FIGURE 10 – Profile definition of the ellipsoidal heed

IS 14973:2001ISO 3066:1977

Nose profile :

Two quarter-ellipses with major semi-axes 2c/, minor semi-axes 0,5 (d – d}), separated by distance di.

Diameter (d) :

Must not exceed 15 mm.

Total-pressure hole :

Diameter di within range 0,1 Od <di <0,35 dl ). This diameter must not-change within 1,5 di from tip.

Static-pressure holes :\

Diameter d, must not exceed 1 mm; depth of hole not less than 0,5 d,; number of holes not less than six; plane of holes atdistance 8 d from tip of nose.

Stem :

Diameter constant and equal to d; junction curved with mean radius 3 d * 0,5 d, or mitred; axis of stem to bend from planeof static-pressure holes, where n >8.

Calibration factor (defined in 7.2) :

Values within f 0,002 given by table below.

n 8 10 12 14 16

Curved junction 1,0015 1,0015 1,001 1,001 1,0005 0,998

Mitred iunction 1.003 1.002 1.0015 1,001 1.0005 0.998

A.2 SENSITIVITY OF PITOT TUBE TO INCLINATION

When the axis of the head of a Pitot tube under use is not aligned with the mean flow direction but forms an angle q with it,

the differential pressure recorded, Q ~, is different from the true differential pressure &o.

However, some shapes of tip (and this is the case for the three types described here above) may reduce the value (4U -*O)

49 -&owithin a wide range of values of angle p. Figure 12 gives the values of for these three types of tubes.

40

NOTE – The values-given bv the cuwes below depend slightly on the value of the Reynolds number of the flow related to the outer diameter dof the Pitot tube (Red). This variation, which arises from the response of the static pressure tap, is practically identical for the three Pitot tubesunder consideration.

1) The larger diameter holes are intended to be usedwith tubes of small diameter, to extend the lower velocity rangewithout introducingviscouseffects in the hole.

26

Is 14973:2001ISO 3966:1977

x

/ I

Sadon A-A saction B-B

8 holes@ 0,1 d 0,8 dt

w’

i

l-%

NOTE – Static pressure taps may be limited to those indicated on section A-A, in which casa section A-A shall be placed at 6 d from the tubetip.

FIGURE 11 ~ CETIAT tyfM

1) The radius is only useful when the Pitot tuba is used in liquids in order to avoidLevitation (see4.2).

27

N w

II

AL

@P

oin

%l

Jm

++o

..

0.

N

Is 14973:2001

ISO 3966:1977

ANNEX B

CORRECTION TO THE MEASURING POSITION OF PITOT TUBESUSED IN A TRANSVERSE VELOCITY GRADlENT

B.1 DETERMINATION OF THE DISPLACEMENT OF A MEASURING POINT

Provided that the measuring position is not too close to a wall, the magnitude of the displacement of streamlines due to thevelocity gradient on the Pitot tube (see 11.2) is constant for a given tube, regardless of the magnitude of the velocity gradient;it is moreover proportional to the head diameter. Given that the magnitude of this displacement isnot directly measurable,theoretical studies of the phenonienon of displacement have led to the conclusion that this displacement is proportional tothe effective drag coefficient of the nose.

As the measuring position approaches a wall, the magnitude of the displacement is progressively reduced, but not eliminated,as the result of an entirely different process caused by the proximity of the tube 10 the wall. This proximity displacement iscalculable and is also proportional to the diameter of the tube and the effective drag coefficient of the nose.

The velocity gradient displacement therefore takes the form

(AY1 )/~= kg

Where d is the head diameter and kg is the constant depending on the nose shape.

The wall proximity displacement is :+=-.(;)~-~-lWhere a k a constant depending on the nose shape and y is the distance of the axis of the head from the wall.

For the case of the plane-ended Pitot tube, experiments show that a/k~ = 0,195.

Because both effects depend on the effective drag coefficient of the nose, and because a is much smaller thank, the combinedvelocity gradient and wall proximity displacement of other tubes may also be written, with sufficient accuracy for the presentpurpose, in the form :

(Y)[y“-$=k9–0,195k~ ~ 1-

B.2 DETERMINATION OF THE OVERALL CORRECTION FOR FLOW RATE IN A CIRCULAR CONDUIT

In the case of arithmetical methods, instead of. applying a correction to the position of the measuring points, it can be enoughto calculate an overall correction for the volume rate of flow taking account of the various displacements due to the velocitygradient and the wall proximity. From the total displacement of a particular measuring point calculated as mentioned above,assuming a velocity profile of the form :

(ti)=(~)”m

the effect of the velocity gradient is expressed by a relative variation of the measured velocity in each measuring point :/

6 (ldvm.x) 1 (Ay/d) (d/D).—

(v/vmax) m (yID)

The variation of the discharge velocity is the mean of individual variations in each prescribed measuring point :

() ldl i Ayld V(5 L=. —— ——

mDi E‘max , vID v~a,

29

Is 14973:2001

ISO 3966:1977

The discharge velocity is given by :

U“v—=+i—Vmax

1 ‘max

The relative variation of the measured flow rate is therefore :

This expression was used to derive figure 7 (see 11.2.2).

30

Is 14973:2001ISO 3966:1977

ANNEX C

STUDY CONCERNING TURBULENCE CORRECTION

C.1 INFLUENCE OF TURBULENCE ON THE TOTAL PRESSURE TAP

C.1.l Pressure probe insensitive to orientation

If a pressure probe which is insensitive to orientation is placed in a turbulent flow, this probe will at all times receive the totalpressure p,, given by :

PTL= PI+ +Pv; (see figure 13)

L-—-

Jvi Vz

Vv

-b-+:~yV,=v +Vx+vv+vz

where

p, is the instantaneous static pressure;

v, is the instantaneous velocity;

77 7v., VYand VZ are the components of the vetocity fluctuation;

T is the mean velocity.

This is not, however, the value which, in general, it is desired to obtain.

If the damping achieved by the ducts and the pressure gauge is correct, apt value is received, such that :

.t

~=; I p,, dr‘o

with

where .t

J~=1P dt

-o

The correct determination of;, which is a necessary step in the rate of flow calculation, demands :

a)

b)

C.1.2

knowledge of the mean pressure Fat each measuring point;

knowledge of the values~, ~ and ~,

Pressuce probe sensitive to orientation

. (a)

In the case where the total-pressure tapping nose is sensitive to orientation, flow turbulence introduces an error into themeasurement read at the total-pressure tap even when the Pitot tube is parallel to the conduit axis.

31

IS 14973:2001ISO 3966:1977

C.2 INFLUENCE OF TURBULENCE ON THE STATIC-PRESSURE TAP

The influence of turbulence in this case is a very much more complex probtem.

Up to now, it has always been assumed that there is a relationship in the form of :

p~=~+k,p(~+~)

where

p~. is the mean measured pressure;

~ is the mean actual pressure;

k. is a factor of the order of 1/4.

It has however been proved that in most cases we have a relationship in the form of :

. . . (b)

. . . (c)

where kt is a factor of the order of 0,6 for a conventional cylindrical probe of diameter d, relatively small in comparison witha correlation length L characterizing the turbulent flow under study. This is nearly always the case for industrial types offlow.

Combining equations (a) and (c) gives the following new equation :

C.3 EVALUATION OF ERRORS FOR AN EXPLORATION WITH A PITOT TUBE

Some evaluations of the errors have been made by using the values Gtqnc, v 2 corresponding to steady flow in a

long conduit.

For this estimation, it has been erroneously assumed that one had the relationp~= ~+ k, P !~ + @ with k, = 0,25, whichminimizes the error on the determination of the mean local velocity.

Some authors state for

ReD = ~ = 40000

~ = 0,54% (this positive error seems under-estimated for the reasons mentioned above).9“

For the same Reynolds number, these authors computed the error resulting from the use of a Pitot tube situated at a distance

(Y. )Re = 40000 = 0,235 ~ from the wall (three-quarter radius flow meter, the distance at which the measured velocity

should be equal to the mean flow velocity), i.e.

: = 0,50%

As this result closely approximates the previous result relating to a more complete computation, these authors assume thatthe error thus calculated is characteristic for the entire range of Reynolds numbers under consideration; they provide thefollowing table :

Re 4X104 7 XI04 105 2 X105 5X105 106 3x106 107 3X107

()

Av Aqv— .— 0,0050 0,0043 0,0040 0,0035 0,0029 0,0026 0,0022 0,0018 0,0015v [Ym)Re ‘V

These Iimits of error actually seems to be rather low in view of the accepted type of correction regarding static pressureprobes, and values given in 11.3 seem more realistic.

32

Is 14973:2001

1s0 396s :1977

ANNEX D

DAMPING OF PRESSURE GAUGES

It is often necessary, in order to facilitate reading the pressure gauge, that the differential pressure random fluctuations shall

be damped without, however, concealing longer term fluctuations or falsifying the time average of the fluctuating pressure.

The damping of the apparatus shall therefore be symmetrical and linear. When a sudden change in pressure is applied, thepressure indicator shall register 99 % of that change in not more than 60 s. Damping shall not be used to conceal regularpulsations of the pressure which render the measurement beyond the scope of this International Standard.

D.1 DAMPiNG PROCEDURE

Damping of the apparatus shall be effected using a resistance which is linear (i.e. proportional to the velocity) andsymmetrical. Thus every precaution shall be taken to avoid bending or pinching the rubber connecting pipes, and asymmetricnozzles, needle valves or gate valves, etc. must not be inserted between the Pitot tube and the pressure gauge.

A capillary tube of adequate length (for example 1 mm in diameter and 100 mm in length if water is used as a manometricfluid) shall be incorporated preferably in we of the manometer limbs (to ensure complete tightness of the connection) or inone of the leads close to the manometer and care shall be taken to avoid any sudden reduction or ekpansion of theconnecting pipe which would involve an apprecaible head loss in comparison with the loss due to the capillary tube.

D.2 BALANCING OF THE DAMPING

When ths time constant of the pressure gauge is of the same order of magnitude as those of the two connecting circuits (fromthe static-pressure tapping to the pressure gauge on the one hand and from the total-pressure tapping to the pressure gauge onthe other hand), it is necessary to balance the clampings of the two circuits and it may therefore be of interest to use thedevice described by Ower and Pankhurst in “The measurement of air flow” (Pergamon press, 1966).

When the time constant of the pressure gauge is sensibly higher than those of the two circuits, the damping balancing is notcompulsory.

D;3 CHECKING OF THE DAMPING

To try to make sure that the pressure gauge resistance is linear (i.e. its operation then corresponds to a flow in theconnections which is Iaminar), it can be checked that the observed fluctuations correspond (for sinusoidal pulses) to a

mSAhdmaximum fictitious Reynolds number well under 2000. This fictitious Reynolds number is equal to — 7J

s

where

S is the surface area of. the meniscus;

s is the minimum section of the capillary tube used to damp the pressure gauge;

d is the diameter of the capillary tube;

Ah is the peak-to-peak magnitude of a fluctuation of the meniscus level;

t is the period of this fluctuation;

v is the kinematic viscosity of the fluid.

In order to check the damping more rigorously, a controllable source of fluctuating pressure (which can cause a sinusoidalpressure difference of sufficient amplitude and of zero mean value) shall be used. The mean position of oscillation of themeniscus then corresponds to the rest position (in the absence of driving pressure) if the resistance is actually symmetricaland linear.

33

Is 14973:2001ISO 3966:1977

ANNEX E

MEASUREMENTS WITH A PITOT TUBE IN A COMPRESSIBLE FLUID

●

E.1 GENERAL

As pointed out in 7.1, the formulae for velocity calculation given in 7.2 are valid for a compressible fluid only if the velocityis rather low and more precisely if the Mach number is less than 0,25.

When the speed of a fluid is great enough to cause compressibility to sensibly affect the total pressure indicated.by the Pitottube, the isentropic flow equations should be used. Pitot static tubes as described in this International Standard cannotgenerally be used at Mach numbers greater than about 0,8, and for best accuracies should be calibrated under the.conditionsin which they will be used. At Mach numbers between 0,9 and 1,0 most tubes of this type show an anomalous behaviour, someasurements cannot be made. In addition, these types of tubes are then more sensitive to misalignment with the flowdirection so that it becomes more important that the swirl of the stream be minimized and that the transverse static pressuredistribution be uniform to ensure axial flow. In supersonic flow these types of instruments should not be used to determinevelocity. In this case, the pressure should be measured with a probe which indicates true local pressure, and the determinationof the total pressure requires the use of the normal shock relationships to evaluate the measurements with the Pitot tube.

E.2 LIST OF SUBSCRIPTS USED IN THIS ANNEX

o stagnation conditions

c at stream centreline

w at the wall

E.3 DETERMINATION OF VELOCITY CALCULATION

In compressible flow, the relationship

()

‘2Ap ’12v.ff —

P. . . (1)

does not apply without a compressibility correction factor.

Assuming no losses, the true velocity of a stream of a compressible fluid in a uniform flow remote from a boundary surface isgiven by the expression :

which may be written in the form

or alternatively

‘=k%)[($+-1]11’2

()v=~ AL’ “2(1-,)P

. . . (2)

. . . (3a)

()2APZRT ‘D~=~(l.e) — . . .PM

(3b)

in which ( 1 – e) is a compressibility correction factor which can be evaluated using several equivalent expressions, as forexample :

(l-)=~(~:)[(;+;=-l]l’” ...(4a)

34

Is 14973:2001

1s0 3966:1977

or approximated by the series expressions,

or alternatively,

in which

. . . [4b)

(4C)

. . . (5)

To determine the velocity from equation (3b) @ and p are measureddirectly and the remaining unknown is the local statictemperature, T. The static temperature generally cannot be measured directly by any instrument introduced into the flow.The interaction of the flow with the probe causes some of the kinetic energy of the stream in the immediate vicinity of theprobe to be converted into thermal energy; but special shielded stagnation thermometers are available which indicate TO withonly a small error. This can be corrected to true stagnation temperature by applying the predetermined calibration factor.Any other equilibrium temperature probe, calibrated to determine its recovery factor, could likewise be used to determinestagnation temperature; but since the recovery factor of any such instrument is a function of velocit~, this can be done only

by some iterative process. After the true stagnation temperature has been determined, the local static temperature T can be

computed by the relationship

TO=1+

7-1— Ma2

7 2. . . (6)

To eliminate the cliff icult computations indicated by these expressions, Ma, T/TO,and (1 – e) are tabulated on page 36 withAp/p as the argument and y as a parameter.

For the best accuracy in determining velocity, the stream temperature should be determined at each measuring point, but ifno heat transfer occurs, the temperature profile can be approximated with temperature measurements only at the conduitcentrelirw and the wall. The following relationships would then apply :

T–T=— = K(&c –@)

T

in which

Tw – T=K=—

TWAPC

. . . (7)

. . (8)

E.4 OPERATING PROCEDURE

To determine the velocity profile in a closed conduit with compressible flow, the following procedure shall be used :

Measurements

1 – Measure Ap at each measurement point,

2 – Measure p at each measurement point. These measurements should be examined for abnormalities. The pressure at agiven cross-section should be sensibly constant; any differences may indicate a possible swirling or other non-uniform flowcondition so that measurements at this station become suspect.

3 – Measure or determine stagnation temperature on the conduit centreline.

4 – Measure or determine the interior conduit wail temperature.

Computations

1 – Compute &/p at each measuring point.

2 – Determine Mach number at conduit centreline.

3 – Determine free stream temperature at conduit centreline.

4 – Measure or compute free stream temperature for each measuring point using equations (7) and (8).

5 – Determine compressibility correction factors from table in each measuring point.

6 – Compute velocity at each measurement point, using equation (3 b).

35

1,1

T/TO

1,2

Tf10

0,996

0,997

0,995

0,984

0,992

0,990

0,989

0,987

0,988

0,964

0,970

0,957

0,945

0,935

—

—

1,3 1,4

T/TO

0,997

0,994

0,9~2

0,989

0,986

0,984

0,981

0,978

0,976

0,973

0,949

0,928

0,906

0,891

0,874

—

1,5

TiTO

0,897

0,993

0,980

0,987

0,984

0,981

0,978

0,975

0,972

0,969

0,941

0,916

0,894

0,874

0,855

—

1,6

T/T.

0,996

0,893

0,989

0,985

0,982

0,978

0,975

0,972

0,966

0,965

0,934

0,908

0,882

0,859

0,836

0,820

1,7-

T/TO

0,996

0,992

0,988

0,984

0,980

0,976

0,973

0,969

0,965

0,982

0,928

0,896

0,871

0,846

0,824

0,804

l–e)

0,998

0,986

0,993

0,991

0.989

0,987

0,985

0,983

0,981

0,978

0,959

0,941

0,924

0,909

—

—

(1–e;

0,898

0,8W

0,984

0,992

0,990

0,988

0,986

0,984

0,982

0;960

0,962

0,945

0,930

0,916

—

—

l-e)

0,988

0,996

0,994

0,993

0,991

0,989

0,987

0,985

0,983

0,982

0,965

0,950

0,935

0,922

—

—

[1-6)

0,888

0,997

0,895

0,993

0,991

0,990

0,988

0,986

0,965

0,963

0,967

0,953

0,940

0,927

0,916

—

(1–e;

0,998

0,997

0,995

0,994

0,992

0,990

0,989

0,987

0,986

0,984

0,970

0,956

0,944

0,832

0,921

—

(1-E)

0,889

0,997

0,996

0,994

0,993

0,991

0,990

0,989

0,987

0,986

0,973

0,8610,950

0,939

0,930

0,921

Ma

0,135

0,190

0,232

0,267

0,298

0,326

0,351

0,375

0,397

0,417

0,578

0,695

0,788

0,867

—

—

Ma

0,124

0,175

0,214

0,246

0,275

0,300

0,324

0,346

0,366

0,365

0,535

0,645

0,734

0,809

—

—

Ma

0,119

0,168

0,206

0,237

0,265

0,290

0,312

0,333

0,353

0,372

0,517

0,624

0,710

0,784

0,848

—

Ma

0,115

0,163

0,188

0,229

0,256

0,280

0,302

0,322

0,341

0,359

0,501

0,605

0,689

0,761

0,824

—

Ma

0,112

0,158

0,193

0,222

0.248

0,271

0,293

0,312

0,331

0,348

0,486

0,587

0,670

0,740

0,802

0,857

Ma

O,la

o,153

0,187

0,216

0,241

0,263

0,284

0,303

0,s21

0,338

0,472

0,571

0,652

0,721

0,781

0,835

AP-F Ma

0,129

0,182

0,222

0,256

0,286

0,312

0,337

0,359

0,380

0,400

0,555

0,669

0,760

0,836

—

.

T/TO

0,998

0,885

0,993

0,991

0,989

0,867

0,985

0,982

0,980

0,978

0,959

0,841

0,925

0,911

—

—

l-e)

0,998

0,997

0,995

0,994

0,982

0,991

0,989

0,968

0,867

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,06

0,09

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,999

0,898

0,997

0,996

0,996

0,995

0,994

0,993

0,892

0,991

0,984

0,976

0,970

0,964

—

—

0,971

0,947

0,926

0,926

0,916

H+’+)=-]}12Ma = (1–,)=

!

Is 14973:20011s0 3966:1977

ANNEX F

DETERMINATION OF COEFFICIENT m FOR EXTRAPOLATION NEAR THE WALL

F.1 METHOD OF DETERMINATION OF m

To calculate the discharge velocity in the peripheral zone, coefficient m can be determined graphically from the measuredvelocities and traverse positions, corrected for the displacement effect according to 11.2.1.1.

Plot in logarithmic co-ordinates the curve of the measured

point velocities against the distance to the wall (figure

oPPosite). In the zone near the wall, this curve is a straightline, the slope of which is equal to l/rrr.

log u

tan ~ =.—

I

()log 1–;’

The two measuring points placed the nearest to the wall shall comply with the following requirements :

– the first point shall be placed as close as possible to the wall, and in any case at a distance not greater than 0,032

(1 being the smailest transverse dimension of the cross-section) taking account, however, of the minimum distance equal to

the diameter of the stem, to be respected between the Pitot tube axis and the wall;

— the second point shall be placed at a greater distance from the”wall than the previous one, this distance being, however,kept equal to or less than 0,081. In any case the measured velocity at this point should be less than 0,7 times themaximum velocity in the section.

F.2 INFLUENCE ON THE CALCULATION OF DISCHARGE VELOCITY

Bad assessment of coefficient m leads to reversing of the tangent at the last measuring point, thus resulting in partialcompensation of errors on flow rate on either side of this point. On the other hand, the error rapidly decreases with therelative importance of the peripheral zone (i.e. when the cross-section dimensions increase or the Pitot tube dimensionsdecrease).

In so far as the two measuring points close to the wall allow determination of m, the error due to the selection of factor m is

considerably reduced and considered as negligible.

37

Is 14973:2001ISO 3966:1977

ANNEX G

EXAMPLE OF CALCULATION OF THE UNCERTAINTY ON THE FLOW-RATEMEASUREMENT BY MEANS OF PITOT TUBES

Thevalues of the errors shall reestimated bytheuser ofthislnternational standard fore~h particular case.

The calculation below is an example based on the estimations of the different errors made during a flow-rate measurement

carried out under normal conditions. The values used are for the purposes of illustration only and shall not be regarded as

typical.

On the other hand, it is assumed in the calculation that the various corrections considered in clause 11 have not been made.

G.1 ERROR ON THE LOCAL VELOCITY MEASUREMENT

– Standard deviation of error arising from measurement ofdifferential pressure: formeasurements made with agood in-dustrial quality apparatus it can be assumed for example that :

u~p= 0,004

z

– Standard deviation arising from thedetermination of density :

~ = 0,002P

– Standard deviation of error arisinga= 0,01 ~t, it can be assumed that :

from slow fluctuations of the velocity : if the amplitude of the fluctuations is

; = 0,001

– Standard deviation oferror arising from compressibility correction :

; = 0,001

– Standard deviation oferror arising from calibration:

~ = 0,002v

– Standard deviation of error arising from high-frequency fluctuations and to turbulence : according to 11.3 itcanbe :

~ = 0,005v

– Standard deviation of error arising from gradient velocity : fora Pitot tube thediameter of which isequaltol/500fthe conduit diameter, itcanbe, according to 11.2:

~= 0,0015v

– Standard deviation of error arising from blockage effect : in the same conditions and according to 11.1 :

~= 0,0025v

– Standard deviation of error arising from Pitot tube inclination : for an orientation defect of 3° and according toclause A.4, it could be :

; = 0,0015

38

Is 14973:2001ISO 3966:1977

– Standard deviation of error arising from head loss : assuming always d/D = 0,02 and for k = 0,05, it can be

approximately, according to 11.4 :

Crg

— = 0,00240

The standard deviation on the local velocity measurement is therefore :

:=/(: )(4 ) ()x16 + lx, +1+1+4+25+2,25+6,25 +2,25+ ;x4 X10-3 = 0,007

G.2 ERROR ON THE FLOW-RATE MEASUREMENT

– Standard deviation of error on the measurement of local velocities from G.1 :

0“— = 0,007

v

– Standard deviation of error arising from the integration technique : for the minimum authorized number of measuringpoints, it could not be greater than :

~ = 0,001q,

– Standard deviation of error arising from m estimation :

f = 0,0005

– Standard deviation of error arising from the Pitot tube positioning : if positioning tolerances given in 3.4.1 arefollowed, it can be assumed :

:= 0,0005q“

– Standard deviation of error on the area measurement :

UA— = 0,002A

– Standard deviation of error arising from an insufficient number of measuring points : if the flow conditions given inthis International Standard are followed, it can be assumed :

~ = 0,001q“

The standard deviation on the flow-rate measurement is therefore :

% “—= <49+1 +0,25 -t0,25+4+l x10-3 =0,0074

q.

and the rate of flow shall be determined with a tolerance, at the 95’% confidence level,

The final value thus obtained confirms that provided the blockage effect correction has been applied, the tolerance od aflow-rate measurement, carried out in accordance with this International Standard, is generally less than * 2%.

39

E3ureau of Indian Standards

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Review of Indian Standards

Amendments are issued to standards as the need arises on the basis of comments. Standards arealso reviewed periodically; a standard along with amendments is reaffirmed when such review indicatesthat no changes are needed; if the review indicates that changes are needed, it is taken up for revision.Users of Indian Standards should ascertain that they are in possession of the latest amendments oredition by referring to the latest issue of ‘BIS Catalogue’ and ‘Standards: Monthly Additions’.

This Indian Standard has been developed from Doc : No. WRD 1 (243).

Amendments Issued Since Publication

Amend No. Date of Issue Text Affected

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