TP003

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Measurement Canada

Volumetric Engineering

Technical Paper 003

Gravimetric Calibration of

Volumetric Standards

Issued: June 2, 1999

Revised Dec 1st, 2009

By: Christian Lachance, P. Eng.

Senior Engineer - Liquid Measurement

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Measurement Canada, Volumetric Engineering Page 2 of 25

Technical Paper 003

Gravimetric Calibration of Volumetric Standards Dec 1st, 2009

Table of Contents1.0 The Gravimetric Calibration of Volumetric Standards . . . . . . . . . . . . . . . 3

2.0 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Water Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Mass Determination of the Delivered Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Method A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.2 Method B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.3 The Use of Apparent Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.0 Gravimetric Calibration Uncertainty Overview . . . . . . . . . . . . . . . . . . . . 12

4.0 Uncertainty of the Base Volume Determination . . . . . . . . . . . . . . . . . . . . . 134.1 Volumetric Standard Temperature Uncertainty, u(Tshell) . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Cubical Coefficient of Expansion Standard Uncertainty, u(") . . . . . . . . . . . . . . . . . 154.3 Water Density u(D(Tw)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3.1 Water Temperature Standard Uncertainty, u(Tw) . . . . . . . . . . . . . . . . . . . 16

4.4 Mass Determination Standard Uncertainty, u(Mw) . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.1 Air Density Standard Uncertainty, u(Da ) . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4.1.1 Air Temperature Uncertainty, u(Ta) . . . . . . . . . . . . . . . . . . . . . . 20

4.4.1.2 Relative humidity uncertainty u(RHa) . . . . . . . . . . . . . . . . . . . . . 20

4.4.1.3 Atmospheric pressure uncertainty, u(Pa) . . . . . . . . . . . . . . . . . . . 21

4.4.2 Reference Mass Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4.3 Reference Mass Density Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4.4 Comparator Uncertainty and Method Repeatability . . . . . . . . . . . . . . . . . . 21

4.4.5 Water Density Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.0 Volumetric Standard Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1 Leveling Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Scale Plate Reading Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 Wetting Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.0 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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V V T T Tref T ref shell 1

V M T TTref w

Tw

ref shell 1

1.0 The Gravimetric Calibration of Volumetric Standards

Common large volumetric standards are calibrated to deliver a known volume. Gravimetric

calibration of volumetric standards consist of weighing water delivered by the volumetric

standard and then deriving the volume from the mass and density of the water. Since the volume

of the standard is dependent on temperature, the reported value is corrected to a reference

temperature.

2.0 Mathematical Model

2.1 General

Volume is determined from mass and density by the following equation:

(2.1)VM

T

T

where T = temperature

VT = volume @ T

M = absolute mass

DT = density @ T

The volume of the standard is a function of the temperature such that:

(2.2)

where VTref = volume of standard @ Tref

VT = volume of standard @ T

Tref = reference temperature

" = cubical coefficient of expansion of the standardTshell = temperature of the standard

Combining (2.1) and (2.2) we obtain:

(2.3)

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SMOW Tw aTw a Tw a

a Tw akg m

5

1

2

2

3 4

31

where Mw = absolute mass of waterTw = temprature of water

= density of water @ Tw Tw

2.2 Water Density

Water density values are obtained from published formulation by Tanaka 1 where the density is

provided as a function of temperature and isotopic content. The formulation provides the density

values and their uncertainties for Standard Mean Ocean Water (SMOW) at a pressure of 101 325

Pa.

(2.4)

where a1 = -3.983 035

a2 = 301.797

a3 = 522 528.9

a4 = 69.348 81

a5 = 999.974 950

DSMOW(Tw) = Standard Mean Ocean Water density

SMOW is air free water characterized by isotopic content. Most sources of water will have

isotopic content that differ from SMOW. When the highest accuracy is required, the isotopic

abundance of the water source must be determined and the density correction applied.

The expected difference between aerated and air free water density is calculated as per Bignell2

with the following equation:

Daerated = -0.004612 + 0.000106 Tw (2.5)

Where Tw is in degrees C and Daeratedis in kg/m3.

Water that has been distilled and not exposed to air is considered air-free water while water that

has been exposed to air will become air saturated. The gravimetric calibration process and

distilled water storage at MC laboratories are such that the water can be assumed to be fully

saturated. The density of the water at Tw is then:

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Tw Tw TwSMOW aerated

Where

)D(Tw) = is the deviation in the actual water density from SMOW with correction foraeration.

2.3 Mass Determination of the Delivered Water

Mass can be determined by the single substitution method using an electronic single pan balance.

Two methods will be presented. Both methods require a container which can be the volumetric

standard or a separate container into which the water is delivered.

2.3.1 Method A

Method A is a general purpose method where the weight of the water is obtained by subtracting

the empty container weight from the weight of the filled container.

This method is a 4 step process:

1. Weighing of the empty container, record:

IE

2. Weighing of reference weights close in value to that of empty

container, record:

IREMREDR

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3. Weighing of container and water, record:

DaIFD(Tw)

4. Weighing of reference weights close in value to that of container and the water, record:

IRFMRFDR

Where

Mw = absolute mass of water delivered or contained

ME = absolute mass of container empty

MF = absolute mass of container filled

MRE = absolute mass of reference weight for empty container

MRF = absolute mass of reference weight for filled container

D(Tw) = density of water @ Tw

DR = density of reference weights

Da = density of air during weighing

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W gMa

x

1

Kxx

a

1

IE = scale indication of empty container

IRE = scale indication of reference weights for container

IF = scale indication of a filled container

IRF = scale indication of reference weight for filled container

Ww = weight of water

WF = weight of filled container

WE = weight of empty container

The mass of water can then be calculated from the recorded data.

The weight of an object is related to its mass by

(2.7)

where

W = weight of objectg = acceleration due to gravity

M = absolute mass of object

Da = density of air surrounding objectDx = density of object

let

(2.8)

For an electronic single pan balance the indication, I, is proportional to the weight applied, suchthat:

I = $ M K (2.9)

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M KI M K

IE E

E RE RE

RE E

1

I M KE I E EE

I M KRE I RE RERE

M KI M K

I M KE EE RE RE

RE RE RE E

where $ is a calibration factor which includes g.

For the weighing method presented we have for the weighing of the empty container:

(2.10)

(2.11)

The ideal balance will have a value of$ that is constant for all weighings. In practice, however,

a number of factors will cause this parameter to vary. Of concern are the effects of the balancerepeatability, non- linearity, stability and off-center loading. These effects and their contribution

to the weighing uncertainty can be mitigated by proper weighing method. Other factors such as

ambient temperature, loading method and magnetism may be significant in certain applications

or with certain equipment. These should be investigated to ensure that they are accounted for in

the determination of the uncertainty if required.

In order to account for the effects of linearity, eccentricity and stability in the estimate of the

uncertainty, the difference between the sensitivity coefficients of the artefact and the subsequent

reference mass weighing for the empty container is written as:

(2.12) I I ERE E

Combining (2.10), (2.11) and (2.12) we obtain:

(2.13)

Since the )$ terms is a small value and MREKRE is close in value to IRE, (2.13) can beapproximated by:

(2.14)

the same is true for MF, where:

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M KI M K

IF FF RF RF

RF F ( )1

M KI M K

I

I M K

IW WF RF RF

RF F

E RE RE

RE E

( ) ( )1 1

MK

K

I M

I

I M

IWR

W

F RF

RF F

E RE

RE E

( ) ( )1 1

M K M K M Kw w F F E E

W W Ww F E

K K KRF RE R

(2.15)

The weight of the water delivered is obtained by:

(2.16)

Combining (2.7) , (2.8) and (2.16) we obtain:

(2.17)

Combining (2.17), (2.14) & (2.15) we obtain:

(2.18)

If reference weights of the same density are used for the filled substitution and the empty

subsitution, then:

(2.19)

By combining (2.18) and (2.19), we obtain:

(2.20)

The mass of the water is obtained by assuming a value of 0 for the )$ terms.

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I I M KF E I I W W F E

I I I IRF E F E

I I M KRF E I I R RRF E

MI I

I I

K

KM

W

F E

RF E

R

W

R

( )

( ) 1

2.3.2 Method B

Method B tares the weight of the empty container thus only requiring one substitution weighing.

It can be used when the substitution weight can be weighed with the empty container.

This method is a 3 step process:

1. Weigh the empty container, record Ie ( please note that this step is last when the standard test

measure is used as the container).

2. Weigh the filled container, record: IF , Da.

3. Weigh the empty container and reference weight close in value to the water delivered, record:IRF , MR , DR.

Because reference weights are added directly to the tare, the filled container indication less the

tare indication will be proportional to the reference weights such that:

(2.21)

and

(2.22)

In order to account for the effects of linearity, eccentricity and stability in the estimate of the

uncertainty, let:

(2.23)

combining (2.21), (2.22) and (2.23) and since the )$ term is a very small value and MRKR isclose in value to ( IRF - IE), we obtain the following:

(2.24)

The mass of the water is obtained by assuming a value of 0 for the )$ term.

Since the weighing is done at filled load, the error introduced by the scale linearity, stability

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and eccentricity MR/(1-)$) can be approximated with MF/(1-)$F). This is convenient if the

estimate of)$F is available while )$ is not.

2.3.3 The Use of Apparent Mass

Apparent mass is related to absolute mass by the following:

(2.25)M Mapparenta

apparent

a1 1

Where: M = absolute mass

Mapparnt = apparent mass

Dapparnt = apparent densityDa = density of airD = density of mass

The apparent mass of an object is thus a function of air density. The difference in the value of

the apparent mass due to a difference in air density can be obtained by:

(2.26)

M M

Mapparent

a

apparent

a apparent

1 1

The difference of the apparent mass value, or resulting error, due to differing air density in

calibration and in use is:

(2.27) Errorapparent

ai r du ri ng ca l ai dur in g use

1 1

_ _ _ _

This error is small in relation to the uncertainty of the mass. For example for a situation wherethe apparent density is 8000 kg/m3 but the actual mass density is 8600 kg/m3 and where the

difference between the air density during the calibration of the mass and it use differs by 0.1

kg/m3 the error will be less than 1 ppm. Therefore the following approximation can be made:

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(2.28)M Muse

apparent

use

apparrentcal

1 1

,

Where Mapparnt, Dcal = apparent mass determined in calibration air density.

3.0 Gravimetric Calibration Uncertainty Overview

Gravimetric calibration gives rise to uncertainty due to two key factors: base volume

determination and the volumetric standard reproducibility.

The mathematical model for the base volume determination is provided by equation (2.3), it

covers the following influence factors:

1) uncertainty in the mass determination;

2) uncertainty in the water density determination;

3) uncertainty of the cubical coefficient of expansion; and

4) uncertainty of the temperature of the volumetric standard.

The uncertainty of the base volume determination is obtained by root sum square of the influence

factors such that:

(3.1)u u u u uT Mshell W W Volume determination2 2 2 2 2 ( )

The amount of volume delivered by a volumetric standard can be expected to vary, this

reproducibility characteristic is due to:

1) leveling of the standard (u leveling);

2) reading of the standard (u reading); and

3) wetting and drip variances.

The uncertainty due to the reproducibility of the standard will be the combined effect of theabove. Since these effects are independent they can be combined by a root sum square:

(3.2)u u u ureproducibility leveling reading wetting2 2 2 2

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u VfV

uc Tref i

nTref

i

i

2

1

2

2( )

The combined standard uncertainty is the reproducibility of the standard combined with the

uncertainty of the base volume determination. These can be combined by root sum square as perthe following equation:

(3.3)

u u u u u

u u u

V T M

leveling reading wetting

Tref shell W W

2 2 2 2 2

2 2 2

( )

4.0 Uncertainty of the Base Volume Determination

The mathematical model for the base volume determination was given in (2.3). Although Tw and

Tshell are assumed to be the same value, we can also assume the error of the two values to beindependent since the error of Tshell is mostly due to the measurement process while the error of

Tw is mostly due to the thermometer reading uncertainty. We will thus consider all influence

factors independent. The combined standard uncertainty can then be calculated by the

following:

(4.1)

where:

uc = combined standard uncertainty

xi = influence quantities

u(xi) = standard uncertainties attributed to the influence quantities

The following table lists the influence quantities, their sensitivity coefficients and the square of

the uncertainty contribution of the influence factor. The uncertainty terms have been

approximated, where appropriate.

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M

T T

Twref shell

w

1

2

( ) MTw

u Tw

w

2

2

( )( )

VTref

i

u V ux Trefi

ii

2

2

2

MT T

Twwref shell

( )

M

T T

Twref shell

w

1

2

( )

1 T T

T

ref shell

w

( )

Influence quantity Sensitivity coefficient Square of the uncertainty

contribution of the

influence factor (the

sensitivity coefficient has

been replaced by an

equivalent approximation

where appropriate)

Pi

Tshell M

Tw

w

( )

(V " u(Tshell) )

2

" V T T uref shell ( )2

SMOW wT M

Twu T

w

SMOW w

2

2

( )( )

MW

u MTww

( )

2

)D(Tw)

4.1 Volumetric Standard Temperature Uncertainty, u(Tshell)

The volumetric standard temperature, Tshell, is assumed to be the same as that of the water Tw.

The uncertainty of this value is influenced by the thermometer calibration uncertainty,

thermometer drift and, for the most part, the uncertainty due to the temperature measurementmethod. The latter is a function of the following factors:

- the assumption that the shell temperature is the same as that of the water, this is influenced

by the temperature difference between ambient and the water;

- possible temperature non-uniformity in the standard.

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The volumetric standard temperature uncertainty is:

(4.2)u T u calibration u drift u method shell2 2 2 2( )

Where:

u calibration can be obtained from calibration certificate;

u drift can be obtained from manufacturer specifications, based on calibration history or based

on ice point verification tolerance;

u method is estimated based on expected ambient conditions and water temperature differential.

4.2 Cubical Coefficient of Expansion Standard Uncertainty, u( )

The uncertainty of the assumed cubical coefficient of expansion is estimated at +/- 10 %:

(4.3) u

01

3

.

4.3 Water Density u( (Tw))

The uncertainty of the water density determination is a function of the water temperature

measurement uncertainty, the density model uncertainty and the level of purity of the water

sample.

The uncertainty of the water density due to the measured temperature is obtained as follows:

(4.4) u u TSMOW w SMOW wT

w

w

( )

( )

The uncertainty of the SMOW density formulation is reported at 0.9 ppm. However, certain

isotopic ratio must be measured to obtain this level of uncertainty. Where the isotopic ratios

have not been measured a value of 20 ppm is suggested4 as appropriate.

The correction for aerated water density is approximately 2 to 3 ppm at room temperature. Sincethe correction is small relative to the 20 ppm uncertainty of air free water mentioned above, the

uncertainty of this correction can be neglected.

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u Tw kg m 0020 33

. /

In some instance, where large volumetric measure are calibrated and where the facilities do not

provide a large supply of purified water such as the Measurement Canada gravimetric calibrationlaboratory, purified water can be re-circulated and re-used. In order to assess the quality level of

the purified water used for gravimetric calibrations, the water density can be monitored by

comparing the density of representative samples to that of ultrapure water such as that produced

at the NRC facility. Comparisons should be conducted using a high precision densitometer. For

very high accuracy applications the results can be used to estimate a correction for the water

density when the reference water sample is traceable.

Unless a density correction is used, the uncertainty due to deviations from SMOW ,with

correction for aeration, is assumed to be +/- 20 ppm so that:

(4.5)

If a correction is used, the uncertainty of the correction must be estimated from the reference

sample stated uncertainty and the densitometer measurement repeatability.

4.3.1 Water Temperature Standard Uncertainty, u(Tw)

The uncertainty of this value results from the thermometer calibration uncertainty and the

uncertainty due to the temperature measurement method. The latter is a function of the

following factors:

- water temperature uniformity in the standard;- self-heating effect of the probe in static water.

The volumetric standard temperature and water uncertainty is obtained by the root sum square of

the above mentioned factors:

(4.6)u T u calibration u drift u method W2 2 2 2( )

Where:

Ucalibration can be obtained from the thermometer calibration certificate;

Udrift can be obtained from thermometer manufacturer specifications, calibration history or it

can be estimated from ice point check tolerance;

Umethod can be estimated from temperature gradients in the standard and self heating

characteristics of the thermometer.

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I M

I

K

K

F RF

RF

R

wF

1

12

I M

I

K

K

E RE

RE

R

wF

1

12

M uE E2 2

cMw

i

i

Mu

w

i

i

2

2

M

I

K

K

RF

RF

R

w

u IF2

M IK

K IRF F

R

W RF

1 2

u IRF2

M

I

K

K

RE

RE

R

w

u IE2

M I KK I

RE E

R

w RE

1

2

u IRE

2

4.4 Mass Determination Standard Uncertainty, u(Mw)

The uncertainty of the mass determination is again determined as per equation (2.20) and (2.24)

for methods A and B respectively.

The following table lists the influence quantities and their associated sensitivity coefficients for

method A:


contribution of the

influence factor (the

sensitivity coefficient has

been replaced by an

equivalent approximation

where appropriate)

i

IF

IRF

)$F M uF F2 2

IE

IRE

)$E

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I

I

K

K

F

RF

R

w

I

IM

I

IME

RE

REF

RF

RFw r w

r w a

2

M uWw R

a

2

2

21 1

I

IM

I

IM

K

E

RE

RE

F

RF

RF

a

R W

2

u MR W aR

2

2

2

I

I M

I

I M

E

RERE

F

RFRF

R a

W a

2

M uWa

ww

2

2

2

u MRF2

I

I

K

K

E

RE

R

w

u MRE2

Da

Dr

Dw

MRF

MRE

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I uRF F2 2

Mw

i

Mu

w

i

i

2

2

M

I I

K

K

R

RF E

R

W( ) u IF2

M I I

I I

K

K

R F E

RF E

R

w

( )

( )

2 u I

RF

2

M I I

I I

K

K

R RF F

RF E

R

w

( )( )

2 ( )I IM u IRF F

RFE

2

2

2

I I

I IM k

F E

RF E

R R

w r w

r w a

2 M uR

w R

a

2

2

21 1

u MR RF aR

2

2

2

M K

I I

I IR R

F E

RF E

a

W a

2

M uR aW

W

2

2

2

F E

RF E

R

w

a

R

M

K

2

The following table lists the influence quantities and their associated sensitivity coefficients for

method B:


contribution of the

influence factor

i

IF

IRF

)$F see text section 2.3.2

IE

Da

Dr

Dw

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I I

I I

K

K

F E

RF E

R

w

u MR

2

u RHaTolerance

3

MR

4.4.1 Air Density Standard Uncertainty, u(Da )

Air density is calculated using the equation provided by Davis3. The uncertainty of the air

density is a function of air temperature, humidity and barometric pressure.

The uncertainty of the air density can be estimated by the following equation:

(4.7)u u T u P u RH a a a a2 2 2 5 2 2 4 2 2

0 00415 118 10 122 10( ) ( . ) ( ) ( . ) ( ) ( . ) ( )

The above influence factors are addressed individually below.

4.4.1.1 Air Temperature Uncertainty, u(Ta)

The uncertainty of the air temperature is a function of the following factors:

- u thermometer calibration is obtained form calibration certificate;

- u thermometer drift can be obtained from thermometer manufacturer specifications,

calibration history or it can be estimated from ice point check tolerance;

- u temperature measurement method is estimated from the expected temperature fluctuation

during the calibration.

(4.8)u T u calibration u drift u method a2 2 2 2( )

4.4.1.2 Relative humidity uncertainty u(RHa)

The magnitude of relative humidity uncertainty can be large without significantly affecting

results. It can be estimated from manufacturers specifications as a type B and or from calibration

process limits.

(4.9)

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u M ToleranceR

3

4.4.1.3 Atmospheric pressure uncertainty, u(Pa)

The uncertainty of the air pressure is a function of the following factors:

- u calibration;

- u drift can be obtained from manufacturer specifications or calibration history;

- u pressure measurement method is estimated from the expected barometric pressure

fluctuation during the calibration.

Combining the above by root sum square we obtain:

(4.10)u P u calibration u drift u method a2 2 2 2( )

4.4.2 Reference Mass Uncertainty

Reference mass uncertainty can be obtained from the calibration certificate. Otherwise, the

grading of the reference mass can be used to estimate the uncertainty.

Most often the gravimetric calibration will be performed with class F1 weights which have a

tolerance of +/- 0.0005 %. Assuming a square distribution this can be converted to uncertainty

by:

(4.11)

4.4.3 Reference Mass Density Uncertainty

The reference mass density uncertainty component may be included in the uncertainty statement

of the reference mass. In this case, this uncertainty term need not be calculated. When stainless

steel reference mass is used, a tolerance of +/-200 kg/m3 can be used as the density uncertainty.

4.4.4 Comparator Uncertainty and Method Repeatability

When an electronic balance is used as a comparator, the uncertainty of the balance indication is

due to its repeatability, eccentricity, stability and linearity which are dependent on the weighingmethod. Linearity errors can usually be reduced to a negligible amount by using reference

weights which are close in value to the object weighed otherwise it must be accounted for.

Stability errors are usually reduced by the appropriate use of a substitution weighing method.

The uncertainty due to the scale eccentricity, stability and linearity must be evaluated in

conjunction with the weighing method.

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The uncertainty due to the repeatability of the scale can be established as a type A uncertainty

after an analysis of the scale characteristics and the repeatability conditions under which it willbe used. This will provide the value of u(I). This should be used when a limited number of

calibration runs is carried out.

Alternatively, if many calibrations runs are carried out then it is possible to lump all repeatability

components under one factor which is estimated as a type A uncertainty from the standard

deviation of the calibration runs. Since the average of the determinations is reported then the

contribution of the repeatability is calculated as per:

urepeatability = Standard deviation of volume determination/ sqrt(n)

Where n = number of runs

This last approach will account for the scale repeatability error and therefore u(I) factors are not

counted.

4.4.5 Water Density Uncertainty

The density of the water during the weighing is very well known. As a result the uncertainty of

the mass determination due to the water density uncertainty is very small in relation to the

overall mass uncertainty. It can be neglected.

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x

2

y tan( )

V yd

2

2

5.0 Volumetric Standard Reproducibility

5.1 Leveling Error

When open neck volumetric measures are used, it is necessary to obtain a reading of the water

delivered in the measure. Two principal factors affect the accuracy of the reading, the operators

ability to read the scale and the proper leveling of the measure.

Improperly leveling of the volumetric measure will result in a reading error. This error is

dependent on the distance between the center of the measure to the sight tube and is dependent

on the neck diameter of the measure.

The following equations apply:

(5.1)

(5.2)

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Where:

2 = leveling angle errorx = distance between center of neck and scale plate

)V = volume errord = internal neck diameter of the volumetric standard

as a result the leveling uncertainty is:

(5.3)u levelingd

( )tan( )

3 2

2

5.2 Scale Plate Reading Error

When an open neck volumetric measure is used by filling to the nominal mark, the reading

accuracy is assumed to be +/- 0.5 mm of scale such that:

(5.4)u readingd cm

( ).

2

0 05

3

2

5.3 Wetting Variance

Volumetric standards that are calibrated to deliver a known volume will be subject to variances

in the amount of liquid volume which is delivered. This will be influenced by the liquid

properties, construction of the standard, drip time and method. Wetting variance is estimated

based on the standards calibration history or know characteristics for typical standard design and

construction.

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6.0 References

1.M. Tanaka, G. Girard, R.Davis, A. Peuto and N. Bignell,Recommended table for the density of

water between 0C and 40 C based on recent experimental reports, Metrologia, 2001,38,301-

309

2.N. Bignell, Metrologia 19, 1983, 57

3. R.S. Davis,Equation for the Determination of the Density of Moist Air, Metrologia, 29, 1992,

67-70

4. Elsa Batista, Richard Paton, The Selection of Water Property Formulae for Volume and Flow

Calibration, Metrologia 44, 2007, 453-463

TP003

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