The determination of the volume of weights in the range of 1 g – 5 kg: a comparison of hydrostatic weighing and double weighing in air using the Monte Carlo simulation
ACTA IMEKO
ISSN: 2221-870X
March 2020, Volume 9, Number 1, 61 - 68
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 61
The determination of the volume of weights in the range of
1 g – 5 kg: a comparison of hydrostatic weighing and double
weighing in air using the Monte Carlo simulation
Yi Su1, Kilian Marti2, Christian Wüthrich2
1 SIMT Shanghai Institute of Measurement and Testing Technology, Shanghai, China
2 Federal Institute of Metrology METAS, Lindenweg 50, 3003 Bern, Switzerland
Section: RESEARCH PAPER
Keywords: volume determination; double weighing; hydrostatic weighing; Monte Carlo
Citation: Yi Su, Kilian Marti, Christian Wüthrich, The determination of the volume of weights in the range of 1 g – 5 kg: a comparison of hydrostatic weighing
and double weighing in air using the Monte Carlo simulation , Acta IMEKO, vol. 9, no. 1, article 10, March 2020, identifier: IMEKO-ACTA-09 (2020)-01-10
Section Editor: Yon-Kyu Park, KRISS, Republic of Korea
Received August 15, 2019; In final form February 25, 2020; Published March 2020
Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the National Key R&D Program of China for National Quality Infrastructure under Grant 2017YFF0205006.
Corresponding author: Kilian Marti, e-mail: kilian.marti@metas.ch
1. INTRODUCTION
There are six accepted methods for the determination of the
density of weights, which are described in the recommendation
of the Organisation Internationale de Métrologie Légale (OIML)
R111 and are denoted in Methods A to F [1]. These methods can
broadly be classified into three categories: the hydrostatic
method, the geometric measurement, and density estimation.
Here, the hydrostatic method, which traces the volume/density
to the reference volume weight or the water density, is
considered the most accurate measurement method. It is called
the reference method of volume/density determination and has
been implemented by many National Metrology Institutes
(NMIs) worldwide [2]. Even though hydrostatic weighing is the
most accurate method among those described in OIML R111 for
determining the volume of the weights, it does have some
disadvantages. The method is time consuming, it is expensive,
and it changes the surface of the weight. After immersion in
water, it can lead to instabilities in the weight. Furthermore, the
immersion of stainless steel weights in water can remove the
adsorbed contaminants or the surface oxide layer, thus changing
the surface properties [3]. However, immersion in water is
unlikely to cause significant corrosion of stainless steel weights
[3], [4]. Besides the hydrostatic method, there are some
alternative methods: optical interferometry [5], which is probably
the most accurate method available; weighing with a balance
immersed in fluorocarbon fluid [5]; using an acoustic volumeter
[7]–[9]; or double weighing in air [10]–[12]. Like hydrostatic
weighing, double weighing in air is based on Archimedes’
principle. However, the main difference between the two
methods is the medium to which the weight is exposed. Although
the double weighing in air has the advantage of being clean and
efficient, its consistency with the hydrostatic method requires
verification by means of experiments. Previous publications by
Clarkson [10], Malengo [11], and Ueki [12] have described the
principle and feasibility of determining the volume of weights by
double weighing in air. Clarkson described a method of volume
determination of weights of 1 kg nominal mass. Malengo
adopted a modified version of Clarkson's method by considering
mass and volume as two measurands that are determined
simultaneously in a multivariate context. He validates the
calculations experimentally by using stainless steel and platinum-
ABSTRACT
We have investigated two methods for the determination of the volume of weights in the range of 1 g – 5 kg: double weighing in air and
hydrostatic weighing. We present the mathematical equations of both methods, showing that the Monte Carlo simulation is a suitable
way of determining the measurement uncertainties and of overcoming the difficulties in dealing with correlated variables. We found
that the measurement uncertainties of the two methods are comparable and that double weighing in air is an efficient method of
determining the volume of weights below 1 kg.
mailto:kilian.marti@metas.ch
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 62
iridium weights of a 1 kg nominal mass. Ueki investigated the
simultaneous calibration of mass and volume of weights in the
range of 1 kg to 20 kg. We extend the range from 5 kg down to
1 g and compare the results from double weighing to those from
hydrostatic weighing. In addition, we explain how to use support
weights both in double weighing and hydrostatic weighing when
measuring test weights that are too small to be loaded on the
weighing pan. We compare the measurement results and
uncertainties of these two methods and show that the Monte
Carlo simulation is a suitable and simple way to overcome the
difficulties of laboriously calculating correlations between input
quantities.
2. SETUP
We now describe the measurement sites that we used to
perform the double weighing and hydrostatic weighing. All
measurement sites are located in the mass laboratory of METAS.
2.1. Hydrostatic method
The volume apparatus VK1 at METAS consists of an
AT1005 comparator from Mettler-Toledo with a maximum
capacity of 1011 g and a readability of 1 µg. The suspension
system is immersed in a water basin (Figure 1). The water comes
from an ultrapure water system (ELGA Labwater, PURELAB).
The suspension system carries both the reference weight and the
test weight and loads them individually on the weighing pan in
water. The weighing pan is connected with the comparator in air
through a thin metal duct. The comparator measures the
resulting force – the difference between the gravitational force
and the buoyancy due to the water. The buoyancy is determined
by the water density, which is measured experimentally by means
of the reference weight. We use a silicon sphere as the reference
volume. We do three weighings and measure (i) the reference
volume, (ii) the test weight, and (iii) the suspension only. To
compensate for the balance's linearity and for the different
buoyancy forces and masses during the three weighings, we use
auxiliary weights that are placed on the four-position weighing
carousel of the comparator. The temperature, pressure, and
relative humidity of the air are recorded during the measurement.
Additionally, four thermistors from YSI Inc. measure the water
temperature on the front and rear sides of the reference and test
weights.
The volume apparatus VK10 consists of a PR10003
comparator from Mettler-Toledo with a maximum capacity of
10100 g and a repeatability of 2 mg (Figure 2). The operating
principle is similar to that of VK1, except that the reference
volume (the middle image of Figure 1) is not present. The
traceability of the density is obtained by calculating the formula
of Tanaka for the density of water [13].
2.2. Double weighing in air
For the double weighing measurements, we use an AT10005
comparator (Mettler-Toledo) with a repeatability of 0.02 mg for
the 2 kg and 5 kg weights; an M_one (Mettler-Toledo) with a
repeatability of 0.5 µg for weights from 200 g to 1 kg; and an
AT106 (Mettler-Toledo) with a repeatability of 1.5 µg for weights
from 1 g to 100 g. The comparators are in airtight enclosures, in
which the air density can be varied by changing the air pressure
(Figure 3). To change the air pressure, we use a membrane pump
(813.3, KNF Neuberger).
3. MEASUREMENT PROCEDURES
In our experiment, we used OIML stainless steel weights with
nominal mass settings of 1 g, 10 g, 20 g, 50 g, 100 g, 200 g, 500 g,
1 kg, 2 kg, and 5 kg (Figure 4). For each weight, we first carried
out a double weighing measurement at 950 mbar and 750 mbar.
Then, we performed a hydrostatic weighing. Thereafter, we
carried out a second double weighing measurement at 950 mbar
and 750 mbar to check the stability of the weight and to observe
possible adverse effects after immersing the weight in water. For
both methods, we calculated the air density according to the
CIPM formula [14], which we verified by using air buoyancy
artefacts before comparing the double weighing and hydrostatic
weighing methods (CIPM = Comité international des poids et
mesures, the International Committee for Weights and
Measures).
Figure 1. Principle of the volume apparatus VK1 at the Federal Institute of
Metrology METAS for the determination of volume of test weights up to 1 kg
by using hydrostatic weighing. Left: the weighing of the suspension only.
Middle: the weighing of the reference weight (silicon sphere). Right: the
weighing of the test weight.
Figure 2. VK10 apparatus at METAS for volume determination of weights up
to 10 kg. Only the test weight is immersed in water.
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 63
3.1. Hydrostatic method
In the hydrostatic method, we used the silicon sphere in
gravimetric weighing to determine the water density. The water
density was then used to determine the volume of the test weight.
Small weights from 1 g to 5 g could not be loaded directly on the
weighing pan because of their geometrical dimensions. Instead,
we used a 10 g stainless steel disc as a support weight (see section
4.1.1).
3.2. Double weighing method
In the double-weighing method, we compared every test
weight to at least one reference weight of the same nominal mass
and density. Weights smaller than 10 g could not be loaded
directly on the weighing pan because of their dimensions. For
this reason, we used a 100 g OIML weight as a support (see
section 4.2.1).
4. CALCULATIONS
We now explain the basic mathematical formulae we used in
the hydrostatic and double weighing methods. The uncertainty
calculations are presented in section 5.
4.1. Hydrostatic method
The basic force equations for the three weighings of the
suspension only (F0), the suspension plus the test weight (F1),
and the suspension plus the reference weight (F2) are
|
𝐹0 = 𝐺O − 𝐴O + 𝐺Z − 𝐴Z
𝐹1 = 𝐺T − 𝐴T + 𝐺Z − 𝐴Z + 𝐺B − 𝐴B
𝐹2 = 𝐺R − 𝐴R + 𝐺Z − 𝐴Z + 𝐺A − 𝐴A
|, (1)
where Gi and Ai denote the gravitational force and the buoyancy,
respectively. The indices represent the mass of the suspension
(Z) and its auxiliary weights (O); the mass of the test weight (T)
and its auxiliary weights (B); and the mass of the reference weight
(R) and its auxiliary weights (A). All the auxiliary weights are in
air.
As the weights are placed at different heights above the surface,
we correct the force equations for the gravitational acceleration
of each weight with 𝛼𝑖 =
𝑔E
𝑔𝑖
|
|
𝑚O
𝑔E
𝛼O
− 𝜌LO 𝑉O
𝑔E
𝛼O
+ 𝜆𝐾 = 𝑀0
𝑔E
𝛼H
Γ
𝑚T
𝑔E
𝛼T
− 𝜌WT𝑉T
𝑔E
𝛼T
+ 𝑚B
𝑔E
𝛼B
− 𝜌LB 𝑉B
𝑔E
𝛼B
+ 𝜆𝐾 = 𝑀1
𝑔E
𝛼H
Γ
𝑚R
𝑔E
𝛼R
− 𝜌WR𝑉R
𝑔E
𝛼𝑅
+ 𝑚A
𝑔E
𝛼A
− 𝜌LA 𝑉A
𝑔E
𝛼A
+ 𝜆𝐾 = 𝑀2
𝑔E
𝛼H
Γ
|
|, (2)
where gE represents the gravitational acceleration at the load cell
and λK equals 𝐺Z − 𝐴Z.
The uncorrected mass difference, i.e. the indication of the
balance, is denoted as Mi and is corrected by the standard
densities for air and material Γ = (1 - 1.2/8000) [14]. As the
suspension is present in every weighing, its influence is denoted
by 𝜆.
|
|
𝑀0 = [
𝑚O
𝛼O
−
𝜌LO𝑉O
𝛼O
+ 𝜆] 𝛼HΓ
−1
𝑀1 = [
𝑚T
𝛼T
−
𝜌WT𝑉T
𝛼T
+
𝑚B
𝛼B
−
𝜌LB𝑉B
𝛼B
+ 𝜆] 𝛼HΓ
−1
𝑀2 = [
𝑚R
𝛼R
−
𝜌WR𝑉R
𝛼R
+
𝑚A
𝛼A
−
𝜌LA𝑉A
𝛼A
+ 𝜆] 𝛼HΓ
−1
|
|
(3)
We can now solve the volume of the test weight, VT, at 20 °C by
using the thermal expansion coefficients Ci of the weights.
𝑉T,water ∶= 𝑉T = [
(𝑀0−𝑀1)
𝛼H
Γ +
𝑚T
𝛼T
+
𝑚B
𝛼B
−
𝑚O
𝛼O
−
𝜌LB𝑉B20𝐶B
𝛼B
+
𝜌LO𝑉O20𝐶O
𝛼O
]
𝛼T
𝜌WT𝐶T
(4)
The coefficient Ci is given by the linear thermal expansion
coefficient β and the measured temperature in air or in water
𝐶𝑖 ∶= 1 + 3 𝛽 (𝑇meas − 20 °C). (5)
We must know the density of water, 𝜌WT, at the test weight's
position inside the water basin. To calculate 𝜌WT, we determine
the density of water, 𝜌WR, at the position of the silicon sphere in
two ways: (i) we use the well-known silicon sphere as the volume
reference in gravimetric weighing
𝜌WR = [
(𝑀0−𝑀2)
𝛼H
Γ +
𝑚R
𝛼R
+
𝑚A
𝛼A
−
𝑚O
𝛼O
−
𝜌LA𝑉A20𝐶A
𝛼A
+
𝜌LO𝑉O20𝐶O
𝛼O
]
𝛼R
𝑉R20eff𝐶R
(6)
Figure 3. AT106 mass comparator with an airtight enclosure in the mass
laboratory at METAS.
Figure 4. OIML stainless steel weights used for the comparison.
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 64
and (ii) we calculate the density �̃�WR by using Tanaka's formula
[13]. We also calculate the water density at the test weight's
position, �̃�𝑊𝑇, by using Tanaka's formula. Now, we can establish
the following relation
𝜌WT =
�̃�WT
�̃�WR
𝜌WR . (7)
The difference between the calculated (�̃�WR) and measured (𝜌WR)
water densities is about 2.5 ppm, which is in line with the
uncertainty contributions related to the temperature and the
isotopic distribution. The volume, VR20eff, of the silicon sphere
was determined by optical interferometry and ellipsometry at the
National Metrology Institute of Japan in 2006.
4.1.1. Hydrostatic method with support
Weights smaller than 10 g cannot be measured directly on our
volume comparator because their geometrical dimension is too
small. Instead, a support weight must be used. By doing this, we
measure the total volume of the support weight and the small
test weight. The mass mT in Equation (4) becomes
𝑚T = 𝑚small + 𝑚support . (8)
The volume of the small test weight is simply the difference
between the total volume and the volume of the support weight.
𝑉small = 𝑉total − 𝑉support . (9)
The latter was a 10 g stainless steel disc and was determined
separately through hydrostatic weighing.
4.2. Double weighing in air
The weighing equation for mass determination in air on a
comparator is given by
𝑚T = 𝑚R + 𝜌𝑎 (𝑉T − 𝑉R) + Δ𝑚𝑤 (1 −
𝜌0
𝜌𝑐
) (10)
where mR and VR are the true mass and volume of the reference
weight, and VT is the volume of the test weight. Δmw is the
uncorrected (conventional) weighing difference indicated by the
balance, which needs to be corrected for the reference air density
ρ0 = 1.2 kg/m³ and the reference material density
ρc = 8000 kg/m³ to calculate the true mass mT of the test weight
[15], [16]. After a double weighing measurement at two different
air densities, ρa1 and ρa2, we can write the following system of
equations
|
𝑚T = 𝑚R + 𝜌a1(𝑉T − 𝑉R) + Δ𝑚w1 (1 −
𝜌0
𝜌c
)
𝑚T = 𝑚R + 𝜌a2(𝑉T − 𝑉R) + Δ𝑚w2 (1 −
𝜌0
𝜌c
)
| (11)
from which we can derive the volume VT of the test weight
𝑉T,air ∶= 𝑉T = 𝑉R +
Δmw2−Δmw1
𝜌a1−𝜌a2
(1 −
𝜌0
𝜌c
). (12)
4.2.1. Small weights with support
If the test weight is too small to be loaded on the weighing
pan, a support weight is necessary. In our experiment, we used a
100 g OIML weight as support and put the small test weight (1 g
or 10 g) on top of the OIML weight. This combination can be
compared to another 100 g reference weight, providing that the
weighing difference is still within the weighing range of the
balance. Firstly, we performed a double weighing with the
support weight only and calculated its volume according to
Equation (12). Then, we added the small test weight, repeated
the double weighing, and calculated the volume of the weight
combination according to Equation (12), which is the sum of the
two volumes. The volume of the small test weight is given by the
total volume of the weight combination (small weight plus
support weight) minus the volume of the support weight
𝑉small = 𝑉total − 𝑉support . (13)
The support weight can either be determined separately by
double weighing or by any other method, such as hydrostatic or
interferometric measurements.
5. UNCERTAINTY CALCULATIONS
We used the Monte Carlo simulations to estimate the
measurement uncertainties in the hydrostatic weighing and
double weighing methods. We assume that the input quantities
xi in Equations (4) and (12) are normally distributed
𝑋~𝒩(𝜇, 𝜎 2). (14)
For the mean value, μ, and the standard deviation, σ, we use
the average value, �̅�, and the standard deviation, s, from the
measurement or certificate. For the simulation, we used N =
10000 iterations.
5.1. Hydrostatic weighing
Equation (4) can be written as
𝑉T,water (𝑗) ∶= 𝑉T(𝑗) = [
(𝑀0(𝑗)−𝑀1(𝑗))
𝛼H(𝑗)
Γ +
𝑚T(𝑗)
𝛼T(𝑗)
+
𝑚B(𝑗)
𝛼B(𝑗)
−
𝑚O(𝑗)
𝛼O(𝑗)
−
𝜌LB(𝑗)𝑉B20(𝑗)𝐶B(𝑗)
𝛼B(𝑗)
+
𝜌LO(𝑗)𝑉O20(𝑗)𝐶O (𝑗)
𝛼O(𝑗)
]
𝛼T(𝑗)
𝜌WT(𝑗)𝐶T(𝑗)
. (15)
where j represents the iteration. If a support weight is used,
Equations (8)and (9) become
Table 2. Typical values of the uncertainty components in the double weighing
of a 1 kg OIML weight (k = 1).
Uncertainty component Value Unit
𝑢(𝑉R) 0.20 mm³
𝑢(Δ𝑚w1 ) 12.90 ng
𝑢(Δ𝑚w2 ) 28.25 ng
𝑢(ρa1) 0.0001 kg/m³
𝑢(ρa2) 0.0001 kg/m³
Table 1. Typical values of uncertainty components in the hydrostatic
weighing of a 1 kg OIML weight (k = 1).
Uncertainty component Value Unit
𝑢(𝑀0) 0.66239 mg
𝑢(𝑀1) 0.82361 mg
𝑢(𝑚T) 0.00658 mg
𝑢(𝑚B) 0 mg
𝑢(𝑚O) 0.16279 mg
𝑢(𝛼H) 0.032965 × 10
-6 (m/s²) / (m/s²)
𝑢(𝛼T) 0.017215 × 10
-6 (m/s²) / (m/s²)
𝑢(𝛼B) 0 (m/s²) / (m/s²)
𝑢(𝛼O) 0.032622 × 10
-6 (m/s²) / (m/s²)
𝑢(𝜌LB) 0.002 kg/m³
𝑢(𝜌LO ) 0.002 kg/m³
𝑢(𝜌WT ) 0.00108 kg/m³
𝑢(𝑉B20) 0 mm³
𝑢(𝑉O20) 1.1344 mm³
𝑢(𝐶T) 0.0223 × 10
-6 mm³/ mm³
𝑢(𝐶B) 0.1187 × 10
-6 mm³/ mm³
𝑢(𝐶O) 0.1187 × 10
-6 mm³/ mm³
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 65
𝑚T(𝑗) = 𝑚small(𝑗) + 𝑚support (𝑗)
𝑉small (𝑗) = 𝑉total(𝑗) − 𝑉support (𝑗).
(16)
The volume of the test weight we are looking for and the
estimated standard uncertainty are given by the mean and
standard deviation of the simulated values VT(j) and Vsmall(j). The
typical values of the uncertainty components in hydrostatic
weighing of a 1 kg OIML weight are listed in Table 1.
5.2. Double weighing
To estimate the measurement uncertainty in the double
weighing method, we proceed in the same way as for the
hydrostatic weighing. Equations (12) and (13) can be written as
𝑉T,air(𝑗) ∶= 𝑉T(𝑗) = 𝑉R(𝑗) +
Δmw2(𝑗)−Δmw1(𝑗)
𝜌a1 (𝑗)−𝜌a2 (𝑗)
(1 −
𝜌0
𝜌c
) (17)
𝑉small (𝑗) = 𝑉total(𝑗) − 𝑉support (𝑗). (18)
The typical values of the uncertainty components in the double
weighing of a 1 kg OIML weight are listed in Table 2.
6. RESULTS
6.1. Verification of the air density
We verified our air density calculation according to the CIPM
formula [14] by using air buoyancy artefacts. The pair of artefacts
consisted of a tube and a hollow body with volumes of
(124.80558 ± 0.00023) cm³ and (412.63374 ± 0.02563) cm³,
respectively. We used the artefacts to determine the air density
experimentally in the range of 650 mbar to 1060 mbar and
compared the results to the air density obtained using the CIPM
formula (Figure 5). The results are in good agreement. For the
Figure 5. Air density determined experimentally by using buoyancy artefacts
and calculated according to CIPM formula. Error bars represent expanded
uncertainties with a coverage factor k = 2.
Figure 6. Stabilization time of a 1 kg OIML weight after hydrostatic weighing
and the change in mass measured gravimetrically in the air before (black
triangle) and after (red circle) hydrostatic weighing. The error bars represent
expanded uncertainties with a coverage factor k = 2. The dashed line is an
exponential fit.
Figure 7. The simulated values of the volume are normally distributed in all
four scenarios. (a) 1 kg in double weighing without support, (b) 1 kg in
hydrostatic weighing without support, (c) 1 g in double weighing with
support, and (d) 1 g in hydrostatic weighing with support.
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.0004
0.0006
D
e
g
re
e
o
f
e
q
u
iv
a
le
n
c
e
(
k
g
/m
³)
11001000900800700600
Air pressure (mbar)
CIPM BA
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 66
subsequent calculations in the comparison of the double
weighing and hydrostatic weighing, we use the CIPM formula.
6.2. Stabilization time
After we complete the hydrostatic weighing, the test weight
was taken out of the water. At that moment, the surface of the
weight is chemically unstable. Gravimetric measurements in the
air immediately after the hydrostatic weighing show the change
in mass and thus the instability of the weight very clearly (Figure
6). The weight must achieve thermal equilibrium and stable
surface conditions. Therefore, a stabilization time of several
hours after hydrostatic weighing is necessary (also observed by
Malengo [11]).
6.3. A comparison of double weighing and hydrostatic weighing
We measured our test weights in air and in water by using
double weighing and hydrostatic weighing, respectively. Then,
we used Monte Carlo simulations to determine the volume and
uncertainty of each test weight by varying the input quantities
according to Equation (14). The output value, i.e. the volume of
the test weight we are looking for, is normally distributed as well.
This applies to all four scenarios: double-weighing with/without
a support weight and hydrostatic weighing with/without a
support weight (Figure 7).
To compare the results between the two methods, we
calculated the degree of equivalence and its standard uncertainty
according to Cox [17].
𝑑𝑖 = 𝑉𝑖 − 𝑉ref (19)
𝑢2(𝑑𝑖 ) = 𝑢
2(𝑉𝑖 ) − 𝑢
2(𝑉ref) (20)
where i represents the measurement in air or in water. The
comparison reference value, Vref, and its standard uncertainty,
u(Vref), are given by
𝑉ref =
𝑉air
𝑢2(𝑉air)
⁄ +
𝑉water
𝑢2(𝑉water)
⁄
1
𝑢2(𝑉air)
⁄ +
1
𝑢2(𝑉water)
⁄
(21)
1
𝑢2(𝑉ref)
=
1
𝑢2(𝑉air)
+
1
𝑢2(𝑉water)
, (22)
where we use the average value of the volumes and their
uncertainties from the double weighing before and after the
hydrostatic measurement:
𝑉air =
𝑉air,before+𝑉air,after
2
(23)
𝑢(𝑉air) =
𝑢(𝑉air,before)+𝑢(𝑉air,after)
2
. (24)
The analysis shows that the volume of the test weights
determined by double weighing and hydrostatic weighing are in
good agreement for all the weights in the range of 1 g to 5 kg
(Figure 8). The standard uncertainties are comparable between
the two methods. The relative uncertainties are the largest for the
1 g weight and decrease with an increasing nominal mass up to 1
kg. Above 1 kg, the relative uncertainties increase again (Figure
9).
6.4. The influence of the different material densities
So far, we compared test weights and reference weights of the
same nominal mass and density. Now, we compare the 1 kg
OIML test weight with a double weighing measurement to
reference weights of different material densities: an air buoyancy
artefact with (2424.69 ± 1.03) kg/m³ from Mettler-Toledo, an
OIML 1 kg stainless steel weight with (7914.37 ± 0.01) kg/m³
from Häfner and a platinum-iridium stack of discs with
(21535.19 ± 0.07) kg/m³ from the Bureau International des
Poids et Mesures. We get three values and their corresponding
measurement uncertainties for the volume of the test weight. The
three values are consistent. However, the measurement
uncertainties are different. To compare the results, we calculate
the degree of equivalence. The smallest measurement uncertainty
in the volume of the test weight is achieved when the material
density of the test and reference weights are the same (Table 3).
Figure 8. A direct comparison of the volumes of the test weights determined
by double weighing in air and hydrostatic weighing. Error bars represent
expanded standard uncertainties (k = 2).
Figure 9. The relative uncertainties (k = 2) of the volume of the test weights
determined in double weighing and hydrostatic weighing.
1
10
100
1000
10000
1g 10g 20g 50g 100g 200g 500g 1kg 2kg 5kg
U
(V
)/
V
(
p
p
m
)
Air Water
ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 67
7. DISCUSSION
We have demonstrated that the double weighing
measurement is a suitable alternative to the hydrostatic
measurement for the determination of the volume and density of
weights ranging from 5 kg down to 1 g. The achieved results of
the two methods are in good agreement, a result that was also
found by Malengo [11]. The measurement uncertainties given by
the two methods are comparable, and they are similar to those
found by Clarkson [10], who performed the double weighing
measurements between 950 mbar and 1050 mbar.
Double weighing offers several advantages: it is less time
consuming than hydrostatic measurement to prepare and
perform the measurement; it is easier to do the calculations; and
it keeps the weight stable, both thermally and chemically.
Furthermore, corrections for the volumetric expansion
coefficient are not necessary. Firstly, the air-tight enclosure of the
comparator is an isolated system that can be considered as a
canonical ensemble representing a system in thermal equilibrium
with the heat bath. Secondly, both the test weight and reference
weight (usually) have the same nominal density and are exposed
to the same environmental conditions. An expansion coefficient
of γ = 50 × 10-6 C-1 – as suggested for stainless steel weights in
OIML R-111 [1] – for the test and reference weight results in a
difference of less than 0.1 ppm between the corrected and
uncorrected volume.
The major contribution to the measurement uncertainty in the
volume of the test weight comes from the uncertainty of the air
density followed by the measured weighing difference and the
volume of the reference weight. For instance, reducing the air
density uncertainty by a factor of 100 from 10-4 to 10-6 kg/m³
leads to a decrease in the volume uncertainty of about 64 %. A
decrease of the uncertainties of the measured weighing
difference or the volume of the reference weight by a factor of
100 results in a decrease of the volume uncertainty of only 3.5 %
or 1 %, respectively (Figure 10).
Thus, the only way to significantly reduce the volume
uncertainty of the test weight is to reduce the uncertainty of the
air density. The CIPM formula provides an uncertainty of 22
ppm [14], whereas we used a more conservative value of 100
ppm. In future, we will try to reduce the air density uncertainty
by using additional air buoyancy artefacts. However, the air
density uncertainty is dominated by the standard deviation of the
weighing difference. So, the challenge is to improve the
repeatability of the comparator.
We have also demonstrated that the Monte Carlo simulation
is useful for estimating the measurement uncertainty of the
measurand by varying the input quantities. In this way, we can
overcome the difficulties in dealing with correlations among the
variables.
ACKNOWLEDGEMENT
This work was supported by the National Key R&D Program
of China for National Quality Infrastructure under Grant
2017YFF0205006.
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ACTA IMEKO | www.imeko.org March 2020 | Volume 9 | Number 1 | 68
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