ISO/TR 20461:2000
(Main)Determination of uncertainty for volume measurements made using the gravimetric method
Determination of uncertainty for volume measurements made using the gravimetric method
This Technical Report gives the detailed evaluation of uncertainty for volume measurements according to the Guide to the Expression of Uncertainty in Measurement (GUM) [1]. It uses the gravimetric method specified in ISO 8655-6 [2] as the reference method for calibrating piston-operated volumetric apparatus. It has been arranged in paragraphs to facilitate direct access to different aspects of this kind of evaluation as follows: _ modelling the measurement by describing the physical equations necessary to calculate the volume using the gravimetric method of measurement; _ determination of the standard uncertainty of measurement associated with the volume V20 by describing the calculation procedure according to the GUM; _ determination of the sensitivity coefficients with an example of the calculation of all sensitivity coefficients by using complete equations, approximations of equations and by giving numerical values for standard conditions; _ determination of the standard uncertainty associated with the volume delivered by a piston-operated volumetric apparatus giving the combination of the standard uncertainty associated with the volume V20 measured using the gravimetric measuring system and the experimental standard deviation associated with the volume delivered by the apparatus; _ determination of the standard uncertainties of measurement with a brief insight into the calculation of uncertainties of measuring devices according to GUM; _ determination of the expanded uncertainty of measurement associated with volume V20; _ example of the determination of the uncertainty for volume measurements.
Détermination de l'incertitude de mesure pour les mesurages volumétriques effectués au moyen de la méthode gravimétrique
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Standards Content (Sample)
TECHNICAL ISO/TR
REPORT 20461
First edition
2000-11-01
Determination of uncertainty for volume
measurements made using the gravimetric
method
Détermination de l'incertitude de mesure pour les mesurages
volumétriques effectués au moyen de la méthode gravimétrique
Reference number
ISO/TR 20461:2000(E)
©
ISO 2000
---------------------- Page: 1 ----------------------
ISO/TR 20461:2000(E)
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ii © ISO 2000 – All rights reserved
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ISO/TR 20461:2000(E)
Contents Page
Foreword.iv
1 Scope .1
2 Modelling the measurement .1
3 Standard uncertainty of measurement associated with the volume V .4
20
4 Sensitivity coefficients.4
5 Standard uncertainty associated with the volume delivered by a piston-operated volumetric
apparatus.6
6 Standard uncertainties of measurement.7
7 Expanded uncertainty of measurement associated with volume V .7
20
8 Example for determining the uncertainty of the measurement .7
Bibliography.10
© ISO 2000 – All rights reserved iii
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ISO/TR 20461:2000(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
The main task of technical committees is to prepare International Standards. Draft International Standards adopted
by the technical committees are circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that which is
normally published as an International Standard (“state of the art”, for example), it may decide by a simple majority
vote of its participating members to publish a Technical Report. A Technical Report is entirely informative in nature
and does not have to be reviewed until the data it provides are considered to be no longer valid or useful.
Attention is drawn to the possibility that some of the elements of ISO/TR 20461 may be the subject of patent rights.
ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 20461 was prepared by Technical Committee ISO/TC 48, Laboratory glassware and related apparatus,
Subcommittee SC 1, Volumetric instruments.
iv © ISO 2000 – All rights reserved
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TECHNICAL REPORT ISO/TR 20461:2000(E)
Determination of uncertainty for volume measurements made
using the gravimetric method
1 Scope
This Technical Report gives the detailed evaluation of uncertainty for volume measurements according to the
Guide to the Expression of Uncertainty in Measurement (GUM) [1]. It uses the gravimetric method specified in
ISO 8655-6 [2] as the reference method for calibrating piston-operated volumetric apparatus. It has been arranged
in paragraphs to facilitate direct access to different aspects of this kind of evaluation as follows:
� modelling the measurement by describing the physical equations necessary to calculate the volume using the
gravimetric method of measurement;
� determination of the standard uncertainty of measurement associated with the volume V by describing the
20
calculation procedure according to the GUM;
� determination of the sensitivity coefficients with an example of the calculation of all sensitivity coefficients by
using complete equations, approximations of equations and by giving numerical values for standard
conditions;
� determination of the standard uncertainty associated with the volume delivered by a piston-operated
volumetric apparatus giving the combination of the standard uncertainty associated with the volume V
20
measured using the gravimetric measuring system and the experimental standard deviation associated with
the volume delivered by the apparatus;
� determination of the standard uncertainties of measurement with a brief insight into the calculation of
uncertainties of measuring devices according to GUM;
� determination of the expanded uncertainty of measurement associated with volume V ;
20
� example of the determination of the uncertainty for volume measurements.
2 Modelling the measurement
The equation for the volume V of the delivered water at 20 °Cisgivenby
20
V = m� Z� Y (1)
20
with
m = m – m – m (2)
2 1 E
where
m is the balance reading of delivered water;
m is the balance reading of the weighing vessel before delivery of the measured volume of water;
1
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ISO/TR 20461:2000(E)
m is the balance reading of the weighing vessel after delivery of the measured volume of water;
2
m is the balance reading of the mass loss due to evaporation of liquid during the measurement;
E
Z is the combined factor for buoyancy correction and conversion from mass to volume;
Y is the thermal expansion correction factor of the delivering device.
Equation (1) combines the measurement results yielded by the balance (m), air and liquid densities yielded by
measurements of air and liquid temperatures, air pressure and relative humidity of air in conjunction with tables or
equations for the factor (Z), and parameters of the delivering device (Y).
Z is given by
�
a
1�
� ���
11
bba
Z�� � � (3)
�
� �� ��
a
wbwa
1�
�
w
where
� is the density of water;
w
� is the density of air;
a
� is the density of the standard weight used to calibrate the balance [according to OIML (Organisation
b
3
Internationale de Métrologie Légale), � = 8 000 kg/m for steel weights].
b
3
[3]
The density of water � (in kg/m ) is given by an equation which is a very useful approximation of the equation
w
[4],[5]
of Kell in the temperature range 5 °Cto 40 °C. The relative deviation between this equation and the original
equation of Kell (given in reference [5] in terms of the ITS-90 temperature scale and valid for temperatures between
–6
0 °C and 150 °C) is less than 10 in the temperature range 5 °Cto40 °C.
4
i
� � t (4)
a
� i w
w
i�0
where
t is the water temperature in degrees Celsius;
w
with the constants (ITS-90 temperature scale):
3
a is equal to 999,853 08 kg/m ;
0
–2 –1 3
a is equal to 6,326 93�10 °C kg/m ;
1
–3 –2 3
a is equal to 8,523 829�10 °C kg/m ;
2
–5 –3 3
a is equal to 6,943 248�10 °C kg/m ;
3
–7 –4 3
a is equal to 3,821 216�10 °C kg/m .
4
Any additional corrections for the pressure dependence and gas saturation of the water density are negligible as
they are very small.
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ISO/TR 20461:2000(E)
3
The density of air � (in kg/m ) is given by [5]:
a
kp��� kt k
��
1a 2a 3
� � (5)
a
tt�
aa0
where
t is equal to 273,15 °C;
a0
p is the pressure, expressed in hectopascals (hPa);
a
� is the relative humidity, expressed as a percentage;
t is the air temperature, expressed in degrees Celsius;
a
with the constants (ITS-90 temperature scale):
3
k is equal to 0,348 44 (kg/m ) °C/hPa;
1
3
k is equal to –0,002 52 kg/m ;
2
3
k is equal to 0,020 582 (kg/m ) °C.
3
The correction for the thermal expansion of the delivering device is given by
Yt��1(� �t ) (6)
cd d20
where
�1
� is the cubic expansion coefficient in °C ;
c
t is the device temperature in degrees Celsius;
d
t is equal to 20 °C.
d20
The temperatures t , t ,and t are assumedtobeuncorrelated, as theactual values of t and t do not only
w a d w d
depend on t , but also strongly depend on the handling by the user. Considerable effects of evaporation-cooling
a
and hand-warming when using handheld apparatus are to be taken into account. The resulting temperature
differences are often larger than the uncertainty in the temperature measurement.
Equations (1) to (6) show that one may write:
��
m
ba
�
��
�� �1(� tt� ) (7)
� dd20
��c
V
20
�� �
�
bw a
This model shows that the measured volume V is a function of m, t , t , p , �, � , t , and some constants.
20 w a a c d
��Fx( ) F(m,t ,t , p ,��, ,t ; constants) (8)
V
20 i w a a c d
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ISO/TR 20461:2000(E)
3 Standard uncertainty of measurement associated with the volume V
20
According to the GUM the standard uncertainty of measurement associated with the value V may be written as:
20
2
��� F
222 2
uV( )��c u ()x� �u ()x (9)
20 ii i
��
��
��� x
i
ii
2 22
2
�� �� � �
����FF �F �F
22 2 2 2
u()V��u(m)� �ut( )� �ut( )� �u(p)�. (10)
20 w a a
�� �� �� � �
����mt�� ���t ��p�
wa a
where
2
u (x ) are the standard uncertainties referred to the measurement of each quantity which contributes to the
i
final result (described by the model);
2
c are the sensitivity coefficients giving the weight of each individual standard uncertainty.
i
The sensitivity coefficients may be determined by calculating the partial derivatives as indicated in equation (9), by
numerical calculations, or by experiment.
As the uncertainties of the constants [equation (8)] and the uncertainties of equations (4) and (5) for � and � are
w a
very small compared to other uncertainties, they may be neglected in the evaluation of uncertainty.
4 Sensitivity coefficients
The evaluation of the uncertainty of measurement does not require such exact values and exact solutions of the
mathematical model for the measurement, as the determination of the volume V itself. Approximations are
20
tolerable, but they have to be used only for this uncertainty evaluation.
3
In the following the approximations � ���� � � , � ���� � � , � � 1000 kg/m,1 – � (t – t )� 1,
w a w b a b w c d d20
–3
and�� ���� � � are used without special notation. Keep in mind that the first approximations are of the order 10
b w b
–1
or less, whereas the last approximation is of the order 10 . This last approximation is justified as it is affecting only
the air buoyancy correction.
The sensitivity coefficients c in equation (9) are calculated as partial derivatives using equations (11) to (29).
i
The sensitivity coefficient c related to the balance reading m is calculated as follows:
w
V
� F
20
c�� (11)
w
�mm
� F
c�� � (12)
ww
� m
3
� F mnl
�3
c�� 10 � 1 (13)
w
� m kgg�
The sensitivity coefficient c related to the cubic expansion coefficient � of the piston-operated volumetric
�
...
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