8 Introduction to Mass Calibration

8/10/2019 8 Introduction to Mass Calibration

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Contents

1. Traceability of Mass Determination

2. Dissemination of the Mass Unit

3. OIML Classification Scheme

4. Handling and Cleaning of Weights

5. Conventional Value of the Result ofWeighing in Air

6. Mass Calibration

7. Evaluation of Uncertainty of Mass Calibration

8. Balance Calibration

Mass is a quantity that is familiar to everybodyprimarily for its importance in commence.

Moreover, it is not only one of the traditional

uantities of metrolo but also of science ingeneral, e.g. its involvement in the calculation ofkinetic energy and potential energy in physics.

Mass is a measure of the amount of material in an

object, being directly related to the number andtype of atoms present in the object.

Mass Laboratory

Responsible for calibrations pertaining to massand related quantities such as :

pressure

density

volume

torque hardness

as well as laser frequency for practical realisationof the definition of the metre, nanometrology androtational speed

1. Traceability of Mass Determination

The 11th General Conference on Weights andMeasures held in 1960 adopted the nameInternational System of Units (abbreviation : SI)for the recommended practical system of units ofmeasurement.

This system is based on several specific units,base units, to form other units, derived units, bycombining base units according to the algebraicrelations linking the corresponding quantities.

There are seven base units :

metre (m)

Unit of len th

The metre is the length of the path travelled bylight in vacuum during a time interval of 1/299 792

458 of a second.

kilogram (kg)

The kilogram is equal to the mass of theinternational prototype of the kilogram.


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Unit of time

second (s)

The second is the duration of 9192 631 770periods of the radiation corresponding to thetransition between the two hyperfine levels of theground state of the caesium 133 atom.

Unit of electric current

ampere (A)

e ampere s a cons an curren w c ,maintained in two straight parallel conductors ofinfinite length, of negligible circular cross-section,and placed 1 metre apart in vacuum, wouldproduce between these conductors a force equalt o 2 x 1 0-7 newton per metre of length.

Unit of thermodynamic temperature

kelvin (K)

The kelvin, is the fraction 1/273.16 of the

thermodynamic temperature of the triple point ofwater.

Unit of amount of substance

mole (mol)

e mo e s e amoun o su s ance o asystem which contains as many elementary

entities as there are atoms in 0.012 kilogram ofcarbon 12. When the mole is used, theelementary entities must be specified and maybe atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

Unit of luminous intensity

candela (cd)

The candela is the luminous intensity, in a givendirection, of a source that emits monochromaticradiation of frequency 540 x 1012 hertz and thathas a radiant intensity in that direction of 1/683watt per steradian.

Hence three other base units are affected by thedefinition of mass:

the ampere, whose definition refers to the

the mole, whose definition refers to 0.012 kg ofcarbon-12 and

the candela, whose definition refers to the watt.


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Derived SI base

quantity Name units

Examples of derived units in association with mass :

-

-

pressure pascal m-1kgs-2

density - m-3kg

torque - m2kgs-2

viscosity (dynamic) - m-1kgs-1

The unit of mass is the kilogram which has beendefined to be equal to the mass of theinternational prototype of the kilogram (IPK) at the

st

The kilogram is the only base unit which nameincludes a SI prefix, namely kilo.

Measures since 1889.

1 l of water

Standard Mean Ocean Water (SMOW)

at its maximum density at 4

w ou a r

under standard atmosphere (101 325 Pa)

1 l = 0.999 972 kg

The kilogram is the only remaining base unit of SIto be defined as a material artefact, rather than interms of a naturall -occurrin constant for instance , ,in the way that the metre is related to the speed of

light.

Redefining the kilogram

First alternative:

The kilogram, unit of mass, is the mass of exactly5.01845166 1025 free carbon 12 atoms at restand in their ground state

- this fixes the value of the mass of a carbon atom, and the Avogadroconstant NA if the current definition of the mole is retained.

This definition would be realized by any experiment that might be usedtoday to measure the mass of an atom, or the value of the Avogadroconstant, such as the XRCD Si crystal density experiment, or the wattbalance experiment combined with the relation between hand NA.

Redefining the kilogram

Second alternative:

The kilogram, unit of mass, is such that the value of

the Planck constant is 6.626 0693 1034 kg m2 s1

-this fixes the value of the Planck constant h.

Since the metre and the second are already defined, and since thevalue of his a universal constant, fixing the numerical value of hdefines the kilogram.

This definition would be realized by any experiment that may be usedat present to determine the value of h, such as the watt balance, orthe silicon crystal density experiment to measure NA combined withthe relation between hand NA.


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Property of the result of a measurement or thevalue of a standard whereby it can be related to

Metrological Traceability :

stated references, usually national or internationalstandards, through an unbroken chain ofcomparisons all having stated uncertainties andperformed by competent laboratories.

(Refer to www.nist.gov/traceability for furtherdetails)

IPK is being kept at Bureau International desPoids et Mesures in France.

All standards of mass must ultimately betraceable to the IPK.

IPK is a cylinder of 39 mm in height and 39 mm indiameter, made of an alloy consisting of 90 %platinum and 10 % iridium (Pt-Ir) with a density ofabout 21,500 kg/m3.

2. Dissemination of the Mass Unit

Since 1889, about fifty copies of the IPK havingthe same form and material have beendistributed to various countries or economies toserve as mass standards.

SCL is the holder of the copy no. 75.

BIPM referencestandards :

K1

BIPM workingprototypes :

No. 25 .

No. 8 (41)

No. 32

No. 43

No. 47

o.

No. 31

No. 77

National prototypes :

No. 2 Rumania

No. 3 Spain

No. 5 Italy

No. 6 Japan

No. 12 RussianFederation

No. 16 Hungary

No. 18 United

No. 38 Switzerland

No. 39 South Korea

No. 40 Sweden

No. 44 Aust ral ia

No. 4 6 Indonesia

No. 4 8 Denmark

No. 4 9 Austria

No 50 Canada

No.60 Peoples Repub licof China

No. 62 Italy

No.64 Peoples Repub licof China

No. 6 5 SlovakiaNo. 66 BrazilNo. 68 Peoples Republic

Kingdom

No. 20 USA

No. 21 Mexico

No. 23 Finland

No. 24 Spain

No. 34 Acadmie dessciences de

Paris

No. 35 France

No. 36 Norway

No. 37 Belgium

.

No. 5 1 Poland

No. 5 2 Germany

No.53 TheNetherlands

No. 54 Turkey

No. 5 5 Germany

No.56 SouthAf rica

No. 57 India

No. 58 Egypt

No. 69 Por tuga lNo. 70 GermanyNo. 71 IsraelNo. 72 South KoreaNo. 74 CanadaNo. 75 Hong KongNo. 76 ItalyNo. 78 Chinese TaiwanNo. 79 USANo. 80 Tha il andNo. 81 United KingdomNo. 82 United Kingdom


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To reduce the change of mass value of theprototype copy from wear due to usage, the mass

unit is disseminated to other weights of nominalvalue of 1 kg.

Reasons for choosing platinum-iridium as materialof IPK or its copies :

(i) Good chemical passivity

(ii) High density very small geometrical surface area

From practical and financial standpoints, weightsare usually manufactured of metals, e.g.

stainless steel, with a density significantly lessthan 21,500 kg/m3.

To compare prototype copy and stainless steelweights, corrections are required, and the twomain corrections are :

(a) Air buoyancy correction

100 mg

(b) Correction on gravitational configuration effect

The acceleration due to gravity decreases withthe inverse of the square of elevation, themagnitudes of the gravitational force on weightsof equal mass but of different size and shape willbe different.

The difference in geometry between the prototypecopy and stainless steel weight results in adifference in the relative locations of the centre ofmass. ong w e ex s ence o ver ca gra enof the acceleration due to gravity, there is a changein the measured mass that is proportional to thelocations of the two centres of gravity. For 20 mmheight difference in centres of gravity, the differenceis about 6g.


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Mass spectrum

Sun 2 x 1030 kg Car 1000 kg

Bag of rice 5 kg

$10 coin 0.011 kg

Hair 10-6 kg

Atom 10-26 kg

Mass unit above and below 1 kg is required to berealised.

Realisation of the mass scale below and aboveone kilogram is achieved by subdivision andmultiplication schemes respectively.

In each scheme, group intercomparisons areperformed within a group of weights pertaining toa decade.

These intercom arisons ma involve eithersingle weight or groups of weights. Thereby the

masses of the unknown weights can beobtained via a least squares analysis of thesystem of equations.

3. OIML Classification Scheme

International Recommendation OIML R111-1,e g s o asses 1, 2, 1, 2, 1, 2, 3,

prepared by International Organization of Legal

Metrology (OIML), presents the principalcharacteristics and metrological requirements forweights that are used :

(i) For the verification of weighinginstruments

(ii) For the tolerance verification ofweights of a lower class of accuracy

(iii) With weighing instruments

Accuracy class :

meets certain metrological requirements intendedto keep the mass values within specified limits.


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The approach in OIML R111-1 to the accuracyclassification of weights as mass standards is to determineand set limits for

(a) the error in mass a weight may have, which is measuredat verification, together with

(b) a number of specified factors and influence quantities(shape, surface roughness, etc), which together determinethe variation in error in mass on use of the weight in servicefollowing verification in a way so as to ensure that the errorof a weight in service does not exceed required limits.

The OIML recommendation R111 includes not only theselimits but also relatively detailed instructions of how todetermine the various influence quantities.

The recommendation applies to weights (of nominalmass from 1 mg to 5 000 kg) in classes ofdescending order of accuracy : E1, E2, F1, F2, M1, M2,and M3.

Class E1 weights are intended to ensuretraceability between national massstandards (with values derived from theInternational Prototype of the kilogram)

and weights of class E2 and lower.

Class E2 weights are intended to be used for theinitial verification of weights of class F1.E2 weights can be used as E1 weights if

surface roughness and magnetic

susceptibility of class E1 weights.

Class F1 weights are intended to be used forthe initial tolerance verification ofweights of class F2.

Class F2 weights are intended to be used for theinitial tolerance verification of weightsof class M1 and possibly M2.

Class M1 weights are intended to be used forthe initial tolerance verification ofweights of class M2.

Class M2 weights are intended to be used forthe initial tolerance verification ofweights of class M3.


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Weight :

A material measure of mass, regulated in regardto its physical and metrological characteristics :

(i) dimensions,

(ii) shape,

(iii) material,

(iv) surface quality,

(v) nominal value,

(vi) density,

(vii) magnetic properties,

(viii) maximum permissible error,

(ix) construction,

(x) presentation

Maximum permissible error

Nominal

Value

Class ( m in mg )

E E F F M M M

50 kg 25 80 250 800 2500 8000 25000

20 kg 10 30 100 300 1000 3000 1000010 kg 5 16 50 160 500 1600 5000

5 kg 2.5 8.0 25 80 250 800 2500

2 kg 1.0 3.0 10 30 100 300 1000

Nominal

Value

Class ( m in mg )

E1 E2 F1 F2 M1 M2 M3


1 kg 0.5 1.6 5 16 50 160 500

500 g 0.25 0.8 2.5 8 25 80 250200 g 0.10 0.30 1.0 3.0 10 30 100

100 g 0.05 0.16 0.5 1.6 5 16 50

50 g 0.030 0.10 0.30 1.0 3.0 10 30

Nominal

Value

Class ( m in mg )

E E F F M M M


20 g 0.025 0.080 0.25 0.8 2.5 8 25

10 g 0.020 0.060 0.20 0.6 2 6 20

5 g 0.016 0.050 0.16 0.5 1.6 5 16

2 g 0.012 0.040 0.12 0.4 1.2 4 12

1 g 0.010 0.030 0.10 0.3 1.0 3 10

Nominal

Value

Class ( m in mg )

E1 E2 F1 F2 M1 M2 M3


500 mg 0.008 0.025 0.08 0.25 0.8 2.5

200 mg 0.006 0.020 0.06 0.20 0.6 2.0

100 mg 0.005 0.016 0.05 0.16 0.5 1.6

50 mg 0.004 0.012 0.04 0.12 0.4

20 mg 0.003 0.010 0.03 0.10 0.3


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NominalValue

Class ( m in mg )E1 E2 F1 F2 M1 M2 M3


. . . . .

5 mg 0.003 0.006 0.020 0.06 0.20

2 mg 0.003 0.006 0.020 0.06 0.20

1 mg 0.003 0.006 0.020 0.06 0.20

The values in this table are taken form OIML InternationalRecommendation OIML R111.

These maximum permissible errors relate to

conventional mass.

For each wei ht the ex anded measurement ,uncertainty U for k = 2 of the conventionalmass, shall be less than or equal to one-thirdof the maximum permissible error.

The conventional mass, mc (determined with anexpanded uncertainty, U), shall not differ by morethan the difference of the maximum permissibleerror, m, minus expanded uncertainty from the

,

m - (m - U) mc m + (m - U)

where m = nominal value of the weight

Construction

e.g. each class E1 or E2 weight shall consistof a single piece of material.

Material

The weights shall be corrosion resistant.

For classes E1 and E2 weights, the hardness ofmaterial and resistance to wear shall be similar orbetter than that of austenitic stainless steel.

For classes F1 and F2 weights, the hardness andbrittleness of the material shall at least equal tothat of drawn brass.

For class M1 cylindrical weights of below 5 kgshould be made of material similar or better thanthat of brass. Rectangular weights from 5 to50 kg shall be made of material having aresistance to corrosion at least equal to that ofgrey cast iron.


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Surface condition

The surface of the weights shall be smooth andthe edges shall be rounded.

Class E1 E2 F1 F2

Rz(m) 0.5 1 2 5

Ra(m) 0.1 0.2 0.4 1

Maximum values of surface roughness :

Magnetism

Magnetic susceptibility,

A measure of the ability of a medium to modify a.

For weights of 20 g and above:

class E1 weights, 0.02 class E2 weights, 0.07

class F1 weights, 0.2

Density

The density of the material used for

% from the specified air density (1.2 kg/m3)does not produce an error exceeding one-quarter of the maximum permissible error.


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4. Handling and Cleaning of Weights

The need for the care of weights is.

A scratch, dirt or trace of oil or grease cansubject the weight to be out-of-tolerance.

Classes E1 to F2 weights and class M1 weightsof nominal value 500 g and below should bekept in their boxes for storage.

Classes E1, E2, F1 and F2 weights should notbe handled with bare hands.

Smallest weights should be handled usingtweezers (with non-metallic end).

Large weights can be handled with clean washedgloves.

When taking out of their containers, they shouldbe placed on a clean surface, e.g. surface

covered with acid-free tissue.

Chamois leather is better than cotton because itis lint free. Nevertheless, clean cotton glovescan be used with weights of lower classes.

Classes M1, M2 and M3 weights should not beslid across abrasive surfaces but placed on dryclean surface.

Large weights should be handled with correctposture and lifting equipment to avoid injuries.

Prior to use of the weights, the weights arerequired to be acclimatised to the ambientconditions.

Cleaning of weight is normally confined to lightdusting with a suitable brush.

If dusting is not successful, solvent cleaning (e.g.s eam c ean ng, w p ng e we g w n - reecloth soaked in alcohol) is required.

The solvent cleaning will most probably alter themass value of the weight, and recalibration isrequired.

Before using solvent, review material safetydata sheet to ensure safety use of the solvent.

When using solvent, prevent ingress of solventinto the adjusting cavity, if any, of the weight.

Solvent may also degrade the coating and causethe weight to tarnish.


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5. Conventional Value of the Result of

Weighing in Air

Normally a weighing is a force measurement.

If good weighing practice has been followed,other factors such as magnetic force,electrostatic force and force due to convectioneffect will become insignificant when comparedwith gravitational force and buoyancy force.

The net force exerts on the balance pan will be :

Fg Fb = mg m( a/ m)g

= m (1 - a/ m)g

where m = mass of weighed object

g = gravitational acceleration

a = air density

m = density of weighed object

The quantity (1 - a/ m) is the air buoyancycorrection factor and m( / ) is the air buoyancycorrection which can be written as aVm.

The weighing result is proportional to mass,density of the weighed object, air density andgravitational acceleration.

rav a ona acce era on can eas y e an eby a one-off adjustment of the balance which is

usually installed at a fixed location.

Other three quantities will be related with theweighed object or the environmental conditionsduring weighing.

For example, when 1000 g of object is weighed at an airdensity of 1.18 kg/m3 (air density at sea level). For object ofdensity 7800 kg/m3, the mass supported by the weighing panwill be : 1 kg (1 1.18/7800)= 999.85 g

For object of density 1000 kg/m3, the mass supported bye pan w e :

1 kg (1 1.18/1000)= 998.82 g

Similarly, in air density of 0.91 kg/m3 (air density at highmountain)

mass supported by the pan for object of density 7800kg/m3 = 999.88 g

mass supported by the pan for object of density 1000

kg/m3 = 999.09 g

It has been shown that dependent on the airdensity and density of object different weighing

,actual mass with an error of about 0.1 %, unlessbuoyancy corrections have been applied.


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Since commodities of the same mass buthaving different densities entail differentweight forces, the avoidance of makingbuoyancy corrections to commercial

impetus of the conventional mass approach.

This concept is presented in OIML D28Conventional value of the result of weighing inair.

The conventional value of the result of weighing abody in air is equal to the mass of a hypothetical

3 o ,in air of density 1.2 kg/m3, balances the weighedbody.

All balances are adjusted such that at thereference air density, o=1.2 kg/m

3, theyindicate the conventional mass, instead of thetrue mass of the object.

ACCORDINGLY MOST CERTIFICATES OFMASS CALIBRATIONS AND BALANCECALIBRATIONS INDICATE RESULTS INCONVENTIONAL MASS.

For scientific or technical work related withweighings of very high accuracy and for thecalibration of measurement standards, for instance,determination of capacity of volumetric measure,even when using conventional value of mass, it isadvisable to carry out a correction so as to takeinto account the difference between the density ofthe weighed object and its assumed density, andthe ambient air density.

Air density estimation

Air density can be determined precisely fromthe BIPM 1981/91 equation.

When the mole fraction of carbon dioxide is 4x 10-4, the equation is as follows :

3.48349 * 10-3 kg.K.J-1 * (1-0.3780 * XV ) * (P/(Z*T))

Where P = pressure in PaT = temperature in KelvinsXV = mole fraction of water vapour

= hf P t P t /P= f(P,td)Psv(td)/P

h = relative humidity in %/100td = temperature of the dew point


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f(P,t) = enhancement factor, which acts as acorrection factor for moist air not behaving as

a perfect gas and depends on thetemperature and the pressure

= + P + t2 when pressure between 60 kPaan a an empera ure rang ng

from 0C to 30C; P expressed in Pa and texpressed in degrees celsius

PSV(t) = saturation vapour pressure, expressed in Pa,as a function of thermodynamic

temperature T, expressed in kelvins= 1 Pa * exp (AT2 + BT + C +DT-1)

Z = compressibility factor

With pressure, temperature and relative humidityin mbar, C, % respectively, when the followingenvironmental conditions are met :

900 mbar p 1100 mbar

10 C t 30 C

h 80 %

A simpler formula can be used to calculate theair density in kg/m3

a= (0.34848 * p-0.009024 * h * exp(0.0612 * t)) / (273.15+t)

The relative error of the above formula does not

exceed 5 x 10-4 (1). Apart from the uncertaintyof the formula itself, the uncertainties of themeasured values of p, h and t must also beconsidered.

6. Mass Calibration

Calibration is performed by comparing the testweight with a reference weight/weights using asuitable balance.

The reference weight should generally be of ahigher class of accuracy than the test weight.

If the air density deviates from 1.2 kg/m3 bymore than 10 percent, mass value should beused in calculations and the conventional massshould then be calculated from the mass.

Two different weighing methods :

(i) Subdivision/multiplication method

This method is mainly used to calibratesets of class E1 weights where the highestaccuracy is required.


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(ii) Direct comparison method

The test weight is calibrated by comparison.

In each comparison, the nominal mass of thetest weight and the reference weight shouldbe equal.

The calibration of weights should be performed at steady ambient conditionsunder ambient atmospheric pressure. Typical recommended values aregiven below:

Ambient conditions during calibration

(Typical values recommended for obtaining successful results)

Weight

Class

Temperature change during calibration

E1 0.3 C per hour with a maximum of 0.5 C per 12 hours

E2 0.7 C per hour with a maximum of 1 C per 12 hours

F1 1.5 C per hour with a maximum of 2 C per 12 hours

F2 2 C per hour with a maximum of 3.5 C per 12 hours

M1 3 C per hour with a maximum of 5 C per 12 hours

Weight

Class

The relative humidity (rh) of air should be in the range

E1 40 to 60 % with a maximum of 5 % per 4 hours

E2 40 to 60 % with a maximum of 10 % per 4 hours

F 40 to 60 % with a maximum of 15 % per 4 hours

7. Evaluation of Uncertainty of Mass

Calibration

(i) Standard uncertainty of the weighingprocess, uw (type A)

(ii) Uncertainty of the reference weight, u(mcr)

If a combination of reference weights is usedfor a mass comparison and their covariancesare not known, a correlation coefficient of 1can be assumed. This will lead to a linear

summation of uncertainties: u(mcr) = iu(mcri)

where u(mcri) is the standard uncertainty ofreference weight i.

(iii) Uncertainty due to instability of thereference weight, us(mcr)

(iv) Uncertainty due to air buoyancy, ub

For conventional mass, the buoyancy correctiondoes not depend on the value of the absolutedensity of air, but on how much its value deviatesfrom the conventional value of 1.2 kg/m3 during thewe g ng process. nasmuc as e way aconventional mass is defined, when performing acomparison in air of density exactly equal to 1.2kg/m3, no buoyancy correction is required, even ifthe volumes of the weights being compared differgreatly.


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When the air density is not at 1.2 kg/m3, the appliedbuoyancy correction factor, C, will be :

)11

1.2)(( a rt

Buoyancy correction : mC

where m = mass value of reference weight

Uncertainty of air buoyancy correction :

)(u)]([m)]u(

)([mu

4

t

t

22

oacr

2

a

tr

trcr

2

b

)]2()[(

)(u)]([m oa1oa4

r

r

2

oa

2

cr

Where a1 is the air density during previouscalibration of the reference weight by use of ahigher order weight. Refer to Covariances inthe determination of conventional mass asublished in Metrolo ia, 2000,37,249-251 for

derivation of the above equation.

Generally the air buoyancy is not significant forweights of class F1 and below, the uncertainty fornot applying the buoyancy correction must beconsidered in the uncertainty budget.

Some publications give examples that no aircorrection is made, instead an uncertainty limit

test for class M1 mass calibration) is used to

estimate the uncertainty, ub.

(v) Uncertainty due to the balance used, uba

The recommended approach is to calibrate thebalance at regular time intervals and use the

evaluation.

(a)Uncertainty due to resolution of the balance

(b)Uncertainty due to linearity of the balance

(c)Uncertainty due to eccentric loading

(vi) Uncertainty due to deviation from thermalequilibrium between weights, balanceand environment, ut.

Deviation from thermal equilibrium will createvarious forces which can bias the balancereading.


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The revision of OIML R111-1 givesrecommendations on thermal stabilisation time.

conventional mass of the test weight is given by :

222222 )()(tbabcrscrw

uuumumuuu

(8) Balance Calibration

Calibration procedure should be similar to theway in which the balance will be used.

similar to the conditions under which thebalance will be used.

Generally the uncertainty of the weight/weightsused for calibration will not exceed one-third toone-tenth of the readability of the balance.

Use of Balance

Balance Location :

Free from airmovement.

Free from effects of heat source.

Free from machinery.

Free from electric and electromagnetic interferences.

ree rom stur ance orner o a u ng.

Free from sunlight localised temperature fluctuation.

Controlled temperature environment 0.1 mgper100g/change of1 C.

Sturdy weighing table.

Sufficient warm-up time.

Note the balance display before and after weighing.

Place theweighing objectin thecentre of balance.

Handle weighing object with tweezers.

Keepthe balance and its surrounding environment clean.

Calibration of a balance should involve sufficientmeasuremen s o prove e per ormance o a a anceto be adequate for its specific operations.

Generally the calibration procedure may include thefollowing tests :

(i) Repeatability

Repeatability test can be performed by repeatinga lication of a wei ht on the balance. In eneralten applications will be carried out for balance of lowcapacity,say,under 50 kg.

(ii) Linearity of scale

The test can be performed by checking the balance atabout ten incremental points up to the maximumcapacity of the balance.

(i ii) Eccentric loading

The test can be performed by placing a weight withnominal value about 30 % of the balance capacity atthe pan centre and then at extremes of the pan, andthen comparing the other results with the result at thepan centre.


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References :

(i) OIML International RecommendationD 28, Conventional value of weighing ina r, on

(ii) OIML International Recommendation

111, Weights of Classes E1 to M3, Edition 2004

(iii) Handbook of Mass Measurement, F.E.

Jones and R.M. Schoonover (iv) Comprehensive Mass Metrology, Edited

. .

(v) Guide to Mass Determination with Highaccuracy, R. Schwartz

(vi) Guide to the Measurement of Mass andWeight, The Institute of Measurementand Control

(vii) Covariances in the Determination ofConventional Mass, M. Glser,Metrologia, 2000, 37

(viii) Equation for the Determination of theDensity of Moist air, R. Davis,Metrologia,1992, 29

Documents

8 Introduction to Mass Calibration