THE NEW MIKES STANDARD FOR LIQUID DENSITY
S. Sillanpää, M. Heinonen
Centre for Metrology and Accreditation (MIKES), Flow Laboratory
P.O. Box 239, 00181 Helsinki, Finland
Résumé
Au cours des années 2003-2004, un nouveau système
d'étalonnage pour les mesures de densité de liquide entre
600 kg/m3 et 2000 kg/m3 a été construit et expérimenté au
MIKES. Le système basé sur la pesée hydrostatique est
raccordé directement aux étalons nationaux de masse et de
longueur. Basée sur les calculs d'incertitude et les mesures
expérimentales, l'incertitude relative élargie (k = 2) du
nouveau système d'étalonnage est de 1·10-5 dans le
domaine de température compris entre 10 oC et 40 oC.
The method is based on Archimedes’ principle. When a
body of a known volume is immersed in liquid, the loss of
its apparent mass is equal to the mass of the liquid displaced by the body. If the volume and mass of the body are
known, the density of liquid at the measurement temperature can be calculated.
In this paper, a description of a new MIKES liquid density
standard (LDCS) and a full uncertainty analysis for it are
presented. The apparatus and the measurement routines are
tested and validated with three different liquids: ethanol,
water and liquid consisted of perfluorocompounds (commercial name Fluorinert FC-77). The liquids covered the
density range from 800 kg/m3 to 1800 kg/m3.
Abstract
During the years 2003-2004, a new calibration system for
liquid densities between 600 kg/m3 and 2000 kg/m3 was
constructed and tested at MIKES. The system is based on
hydrostatic weighing providing traceability directly to the
national mass and length standards. Based on uncertainty
calculations and test measurements, the expanded relative
uncertainty (k = 2) of the new calibration system is 1·10-5
at the temperature range from 10 oC to 40 oC.
Introduction
In the international system of units (SI) [1], the unit of
density (kg/m3) is derived from two SI based units: the unit
of mass and the unit of length. Thus, the primary realisation of a density standard should be traceable both to the
mass standard defined by the international prototype of the
kilogram and to the length standard referred to the wavelength of lasers stabilised at recommended radiations. In
practise, a primary standard for density is an artefact,
which mass and volume can be related to the mass and
length standards with small uncertainty.
The standard mean ocean water (SMOW) is commonly
used as a reference standard of liquid density. To realise
the traceability link between SMOW and the mass and
length, a solid density standard with an appropriate measurement method is needed. A hydrostatic weighing method
is commonly used as a primary method for liquid density
in many national standard laboratories [2]. When using the
hydrostatic weighing method in routine work, relative
expanded uncertainties are usually around 1·10-5. However, uncertainties smaller than 1·10-6 are reported [3]. The
method is also commonly used for determination of a
density of solid substances.
Theoretical background
To find out buoyancy forces acting on the sinker, the following equilibrium equation set can be written
⎧S1 + ρ l g (Vs + V h ) − g (ms + m h ) + gδm1 = 0
, (1)
⎨
⎩S 2 + ρ l gV h − gm h + gδm 2 = 0
where S1, S2, ρl, Vs, Vh, g, ms, mh, δm1 and δm2 are the force
sensed by the balance during the weighing of the sinker
and empty suspension, density of the liquid sample, volume of the sinker, volume of the hanger, acceleration due
to gravity, mass of the sinker and hanger and the effect of
the meniscus with and without the sinker, respectively.
Due to the surface tension of the liquid sample, the liquid
surface rises up around the suspension wire of the sinker.
The effect of meniscus may slightly variate during the
weighing process. The effect is estimated by measuring the
mass difference ∆mm=δm1 - δm2.
The volume of the sinker depends on the temperature and
pressure at the immersion depth. The temperature and
pressure dependence of the volume can be expressed as
Vs (T , p ) = [Vs0 + ∆Vs (T )][1 − κ s ( p − p 0 )] .
(2)
In equation (2) Vs0, κs and p0 are the volume of the sinker
at some reference temperature, isothermal compressibility
of the sinker and reference pressure, respectively. The
volume change of the sinker as a function of temperature
can be written as follows
[
∆Vs (T ) = Vs0 A 1 (T − T0 ) + A 2 (T − T0 )2 + A 3 (T − T0 )3 +
]
+ A 4 (T − T0 )4 ,
(3)
where coefficients A1, A2, A3 and A4 are calculated from
Okaji’s equation [4], T is the temperature of the sinker in
Kelvins and T0 = 293.15 K. For silicon, which is used at
MIKES for material of density standard
A1 = 7.674·10-6 1/K, A2 = 1.341·10-8 1/K2,
A3 = -2.862·10 11 1/K3 and A4 = 4.965·10-14 1/K4.
sphere
ple vessel and density standard sphere during weighing.
The cooling thermostat is situated in a different room to
minimise the heat load to the laboratory.
To find out the buoyancy forces acting on the sinker as
precisely as possible, the comparative weighing method is
used. In the method, reference weights and the sinker are
weighed alternately. The apparent mass of the sinker immersed in the liquid sample is compared to the mass of
reference weights. By choosing appropriate set of reference weights, the apparent mass difference between the
sinker and weights is minimized.
By introducing corrections due to air buoyancy and the
gradient of gravitational force to the balance indication, an
equation for ρl is derived from equations (1) to (3)
⎡
⎛
ρ a ⎞⎤
1
⎪⎧
⎟⎥ ⋅
ρ l (T , p ) =
⎨m s − ⎢m w − V w ρ a + ∆B⎜⎜1 −
Vs (T , p ) ⎪⎩
ρ c ⎟⎠⎥⎦
⎢⎣
⎝
⋅ (1 − ς∆h ) + ∆m m } ,
(4)
In equation (4) ∆B is the comparative weighing result
∆B=Ir - Is. Ir and Is are the balance indication when the
reference weights are weighed and when the sinker is
weighed, respectively. The air density ρa is calculated
according to equations [5], [6] and ρc is the density of
reference weights. ζ is the gradient of gravitational force.
To calculate the density of the liquid sample at target temperature and pressure, following equation is used
ρ l0 (T0 , p 0 ) = ρ l (T , p )[1 − γ l (T0 − Tl )][1 + κ l ( p 0 − p s )] , (5)
where γl, κl, Tl and ps are the cubic thermal expansion coefficient, isothermal compressibility, temperature and pressure of the liquid sample, respectively.
Figure 1. Schematic diagram of the new LDCS apparatus
at MIKES. 1 balance, 2 firm table, 3 stone plate, 4 water to
and from a cooling thermostat, 5 lifting mechanism, 6
wind shield of a suspension wire, 7 sample vessel, 8 water
bath, 9 rails, 10 rollers
Description of the apparatus
For weighing, a commercial balance with a maximum load
of 220 g and a resolution of 0.01 mg is used. Air pressure,
temperature and humidity are measured with a PTU device, calibrated at MIKES.
A silicon sphere is used as a density standard at MIKES.
Its diameter is 55 mm and according to the calibration
certificates, the true mass of the sphere is ms = (201.94117
± 0.00020) g and volume at 20 oC Vs0 = (86.70389 ±
0.00050) cm3. Stated uncertainties are expanded uncertainties with 95 % confidence level, which leads to the coverage factor k = 2.
The sample vessel was designed to hold the liquid sample
in such a way that its exposure to air is minimal. It is made
of borosilicate glass. The silicon sphere can be lifted on
and off the suspended hanger with an automatic lifting
device. A diagram of the vessel is shown in figure 2. The
stainless steel lid slopes on the underside so that air bubbles are not trapped when the vessel is filled with liquid.
One of the most significant uncertainty component in the
former version of MIKES liquid density measurement
system was the temperature gradients in the sample vessel
[7]. In the new LDCS apparatus the sample vessel is immersed in a large insulated water bath to minimise the
gradients. Water in the bath is circulated and its temperature is controlled with an external cooling thermostat.
The sample vessel hangs on with three stainless steel tubes
through the lid. Two of them are for holding a thermometer and a lifting rod. The third tube is used for pumping
liquid samples to and from the sample vessel. A glass tube
at the middle is for suspension wire. The wire with diameter of 0.127 mm is made of platinum-iridium alloy. The
glass tube is used also for monitoring the liquid surface
level.
Figure 1 shows a schematic diagram of the LDCS. A balance is situated above the water bath and rests on a 40 mm
thick stone plate. The bath can be rolled on rails from
underneath the balance to allow the sample vessel to be
lifted easily in and out of the bath. The bath is manufactured of stainless steel and its volume is approximately 55
litres. The insulation of the bath is 50 mm thick and it is
equipped with two glass windows for monitoring the sam-
The hanger holding the sphere is situated so that the suspension system remains vertical when the sphere is loaded
and unloaded. This prevents the suspension wire from
interfering with the narrow-necked section of the glass
tube.
A Pt-100 thermometer with a diameter of 1.6 mm is used
for measuring the temperature of the liquid sample at the
level of the centre of the sphere. It is connected to a digital
thermometer and calibrated at the MIKES thermometry
laboratory.
− ρl
∂ρ l
=
.
∂Vs (T , p ) Vs (T , p )
For the temperature of the sinker
− ρ l ∂Vs (T , p )
∂ρ l
=
.
∂T Vs (T , p )
∂T
For the atmospheric pressure
ρ ∂Vs (T , p )
∂ρ l
=− l
.
∂p
∂p
Vs
For the mass of reference weights
∂ρ l
− 1 + ζ∆h
=
.
∂m w
Vs (T )
For the density of air
∂ρ l ⎛
∆B ⎞ 1 − ς∆h
⎟
= ⎜⎜V w +
.
∂ρ a ⎝
ρ c ⎟⎠ Vs (T , p )
For the comparative weighing result
∂ρ l ⎛ ρ a ⎞ 1 − ς∆h
⎟
− ⎜1 −
.
∂∆B ⎜⎝
ρ c ⎟⎠ Vs (T , p )
Figure 2. Diagram of the sample vessel. 1 protective tube
of the lifting device, 2 protective tube of the thermometer,
3 glass tube for the suspension wire and for monitoring the
liquid surface level, 4 stainless steel lid, 5 hanger of the
density standard, 6 glass vessel, 7 lifting rod, 8 silicon
sphere (density standard), 9 platinum resistance thermometer
Uncertainty analysis
The density of a liquid sample at measurement temperature
and pressure is a function of 12 components
ρ l (T , p ) = f ( y1 K y12 ) ,
(6)
where y1=ms, y2=Vs(T, p), y3=T, y4=p, y5=mw, y6=ρa,
y7=∆B, y8=Vw, y9=∆mm, y10=κs, y11=ζ and y12=∆h. Assuming non-correlated variables, the combined standard uncertainty can be calculated according to the GUM [8]
11
2
⎛ ∂ρ l ⎞ 2
⎜⎜
⎟ u ( yi ) .
(7)
∂y i ⎟⎠
i =1 ⎝
The sensitivity coefficients ∂ρ l / ∂y i can be derived from
equation (4). The sensitivity coefficient for the mass of the
sinker is
∂ρ l
1
=
.
(8)
∂ms V s (T , p )
For the volume of the sinker at T = 293.15 K
u c2 ( ρ l ) =
∑
(9)
(10)
(11)
(12)
(13)
(14)
For the volume of reference weights
∂ρ l
1 − ς∆h
= ρa
.
(15)
∂V w
Vs (T , p )
For the effect of meniscus
∂ρ l
1
=
.
(16)
∂∆m m Vs (T , p )
For the compressibility of the sinker
∂Vs
∂ρ l
ρl
=−
.
(17)
∂κ s
Vs (T , p ) ∂κ s
For the gradient of gravitational force
⎛ ρ ⎞⎤ ∆h
∂ρ l ⎡
= ⎢m w − V w ρ a + ∆B⎜⎜1 − a ⎟⎟⎥
.
(18)
∂ς ⎣⎢
⎝ ρ c ⎠⎦⎥ Vs (T , p )
For the height difference between reference weights and
the sinker
⎛ ρ ⎞⎤
∂ρ l ⎡
ς
= ⎢m w − V w ρ a + ∆B⎜⎜1 − a ⎟⎟⎥
. (19)
∂∆h ⎢⎣
⎝ ρ c ⎠⎥⎦ Vs (T , p )
The density of the liquid at the target temperature and
pressure is a function of five parameters.
ρ l0 = ρ l (T0 , p 0 ) = f ( ρ l , γ l , Tl , κ l , p s ) ,
(20)
Sensitivity coefficients can be derived from equation (5).
For the density of the liquid sample at the measurement
temperature and pressure
∂ρ l0
= [1 − γ l (T0 − Tl )][1 + κ l ( p 0 − p s )] .
(21)
∂ρ l
For the cubic thermal expansion coefficient
∂ρ l0
= [ρ l (− T0 + Tl )][1 + κ l ( p 0 − p s )] .
(22)
∂γ l
For the temperature of the liquid sample
∂ρ l0
= ρ l γ l [1 + κ l ( p 0 − p s )] .
(23)
∂Tl
For the compressibility
∂ρ l0
= ρ l [1 − γ l (T0 − Tl )]( p 0 − p s ) .
(24)
∂κ l
For the pressure of liquid at the level of the sinker
∂ρ l
(25)
= − ρ l κ l [1 − γ l (T0 − Tl )]
∂p s
The volume of the sinker is correlated with temperature
and pressure. However, the contribution of the correlation
to the combined standard uncertainty is negligible.
Table 1. Example of an uncertainty budget for water at
temperature 40 oC
yi
Unit
ci
u(yi)
ciu(yi)
ms
kg
11500
0.0023
0.2⋅10-6
Vs
m3
11400000
0.0057
5⋅10-10
Ts
K
0.153
0.0001
0.4⋅10-3
kg
11500
0.0002
mw
2⋅10-8
kg/m3
0.168
0.0017
10⋅10-3
ρa
0.0015
kg
11500
∆B
1⋅10-7
-10
m3
13600
0.0000
Vw
7⋅10
-9
kg
11500
0.0002
2⋅10
∆mm
9
-13
1/Pa
0.0003
3.4⋅10
κs
1⋅10
-8
0.0000
1/m
401
1⋅10
ζ
m
0.003
0.005
0.0000
∆h
K
0.0004
0.02
0.0000
Tl
ps
Pa
100
0.0000
4.6⋅10-7
1/Pa
297240
0.0000
2⋅10-11
κl
1/K
0.05
0.0000
4⋅10-6
γl
kg/m3
0.0066
uc
U (k=2)
kg/m3
0.0132
Test measurements
Before starting measurements, the sample vessel was bundled up and the liquid sample was pumped in the vessel.
The desired water bath temperature was set and the water
circulation pump of the cooling thermostat was switched
on. Typical cooling or heating times for the water bath
were three hours from 20 oC to 10 oC and two hours from
20 oC to 40 oC. Usually, the cooling thermostat was
switched on at the morning and measurements were performed in the afternoon.
To determine possible leaks to the measurement vessel, the
empty vessel with density standard was immersed into the
bath. If no water leaks were detected, sealants of the vessel
were close-fitted and it was safe to pump the liquid sample
into the vessel.
The ambient air temperature, humidity and pressure were
measured near the weighing pan of the balance before and
after each measurement set. One measurement set consists
of 12 comparative weighings between density standard
immersed in the liquid sample and reference weights.
To evaluate the operation of the apparatus, tests were carried out with three different liquid samples each at three
different temperatures. Tests on a low density range were
performed with ethanol (about 800 kg/m3), in a mid range
with water (about 1000 kg/m3) and in a high density range
with FluorInert FC-77 (about 1800 kg/m3).
The ethanol sample was taken from a bigger vessel, where
the hydrometer calibration ethanol was stored. The measurement vessel, lid and lifting rod were cleaned with ethanol and dried with compressed air. After the cleaning ethanol was evaporated, the apparatus were erected, leak tested
and the ethanol sample was pumped into the measurement
vessel. The isothermal compressibility of ethanol
(κl = 11.19·10-10 Pa-1) and the cubic thermal expansion
coefficient (γl = 1.40·10-3 K-1) were taken from [9].
The water sample was originating regular tap water, taken
from the Helsinki area and distributed by Helsinki Water.
Before measurements, the water sample was distilled twice
and stored in a glass bottle. The sample was not de-aerated
before measurements and its isotopic composition was
unknown. The isothermal compressibility of water
(κl = 4.591·10-10 Pa-1) and the cubic thermal expansion
coefficient (γl = 0.206·10-3 K-1) were taken from [9].
The Fluorinert FC-77 sample was taken from a bigger
storage vessel. Unfortunately, the liquid had been stored in
the vessel for a quite long time and probability of solved
impurities from the storage tank was increased. So, it was
not possible to compare the determined density value with
the one reported by the manufacturer. The cubic thermal
expansion coefficient of Fluorinert FC-77
(γl = 1.38·10-3 K-1) was taken from [9].
The liquid samples were measured at three temperatures:
10 oC, 20 oC and 30 oC. For water, the highest measurement point was 40 oC. Those points covered the temperature range of the apparatus and results at each temperature
point are summarised in tables 2 to 4. The point 20 oC was
considered especially important, because it is near the
normal laboratory temperature. The temperature is also
used as a reference temperature in many industrial density
measurement solutions.
Table 2. Summary of test measurements at 10 oC
Liquid
ρ (10 oC) U (k =2)
Ethanol
Water
FC-77
kg/m3
819.962
999.689
1799.742
kg/m3
0.011
0.013
0.022
Table 3. Summary of test measurements at 20 oC
Liquid
ρ (20 oC)
3
Ethanol
Water
FC-77
kg/m
811.348
998.200
1774.467
U (k =2)
kg/m3
0.011
0.013
0.022
Table 4. Summary of test measurements at 30 oC
o
Liquid
ρ (30 C)
U (k =2)
Ethanol
kg/m3
802.608
kg/m3
0.011
Water
FC-77
992.235*)
1748.526
0.013
0.022
*)
–3·10-3 kg/m3 to 4·10-3 kg/m3 with the expanded (k = 2)
uncertainty of 13·10-3 kg/m3.
0,020
0,015
0,010
Measured at 40 oC.
ρ-ρref / kg/m
3
0,005
Comparison with reference data
The amount of impurities in the ethanol sample was unknown. The theoretical equation for ethanol density assumes pure air-saturated ethanol-water mixture. During
measurements, a small part of ethanol sample was abutting
by ambient air and diffusion of water in air through the
boundary surface to the ethanol sample was feasible.
0,025
0,020
0,015
0,010
ρ-ρref / kg/m
3
0,005
0,000
5
10
15
20
25
30
0
5
10
15
20
25
30
35
40
45
-0,005
For comparison, ethanol density values were calculated
using formulae presented in [10]. Measured and calculated
ethanol density values are compared in figure 3. The percentage by weight of ethanol in the sample was determined
by fitting the calculated ethanol density value at 20 oC
equal to the measured one by changing the mass percent of
ethanol. The defined mass percent was used in calculating
the density values at 10 oC and at 30 oC. Differences between measured and calculated ethanol density values
were –14·10-3 kg/m3 at 10 oC and 8·10-3 kg/m3 at 30 oC
with the expanded (k = 2) uncertainty of 11·10-3 kg/m3.
0
0,000
35
-0,005
-0,010
-0,015
-0,020
o
t / C
Figure 4. Difference between measured (ρl) and calculated
(ρref) water density values at different temperatures (◊).
The vertical lines represent corresponding expanded uncertainties (k = 2) of the measured values.
Because the isotopic composition of the water sample was
unknown, the general coefficient for tap water,
a5’ = 999,972 kg/m3 from [13] was used according to [14].
The amount of dissolved air was estimated and taken into
account in calculations. The conductivity of the water
sample was smaller than 1 µS/cm.
Within a EuropeAid/113156/D/SV/EE project, two vibrating tube densitometers (Anton Paar DMA 5000 and Anton
Paar DMA 5001) at AS Metrosert were compared with the
LDCS using FluorInert FC-77. Measurements were carried
out only at temperature 20 oC. Differences with FluorInert
FC-77 between the LDCS and METROSERT were –25⋅103
kg/m3 (DMA 5000) and 11⋅10-3 kg/m3 (DMA 5001). As it
can be seen, the differences were quite well inside the
calculated expanded uncertainty value, 22⋅10-3 kg/m3.
-0,010
Conclusion
-0,015
-0,020
-0,025
-0,030
o
t / C
Figure 3. Difference between measured (ρl) and calculated
(ρref) ethanol density values at different temperatures (◊).
The vertical lines represent corresponding expanded uncertainties (k = 2) of measured values.
Several equations for density of water based on a large
amount of measurement data have been published in [2],
[3], [11] and [12]. The recommended table for the density
of water has been presented in [13].
Figure 4 shows results of the comparison between water
density values and corresponding literature values calculated with formulae in [13]. As can be seen, the measured
and calculated values are very close to each other and well
within the estimated uncertainties. Differences between
measured and calculated density values lie from
As a conclusion, it has been shown that the new MIKES
apparatus for measuring the density of liquid samples
works properly and the calculated uncertainty values are
realistic. As can be seen from table 1, the largest contribution to the uncertainty budget is caused by the volume
determination of the sinker. In the future, if one needs to
improve the measurement uncertainty of liquid density
samples, the volume of the density standard sphere should
be determined with a better accuracy.
References
[1] International system of units (SI), Bureu international
des poids et mesures (BIPM), 1998.
[2] S. V. Gupta, Practical density measurement and hydrometry, 2002.
[3] R. Masui, K. Fujii and M. Takenaka, ”Determination
of the absolute density of water at 16 oC and 0,101325
MPa, Metrologia, Vol. 33, pp. 333-362, 1996
[4]
H. Watanabe, N. Yamada and M. Okaji, ”Linear
thermal expansion coefficient of silicon from 293 K to
1000 K”, International Journal of Thermophysics, Vol. 25,
pp. 221-236, 2004.
[5] P. Ciacomo, ”Equation for the determination of the
density of moist air (1981)”, Metrologia, Vol. 19, pp. 3340, 1982.
[6] R. S. Davis, ”Equation for the determination of the
density of moist air (1981/91), Metrologia, Vol. 29, pp. 6770, 1992.
[7] M. Heinonen and S. Sillanpää, ”The effect of density
gradients on hydrometers”, Measurement Science and
Technology, Vol. 14, pp. 625-628, 2003.
[8] Guide to the expression on uncertainty in measurement, International organization for standardization (ISO),
1993.
[9] D. R. Lide, CRC handbook of chemistry and physics,
1997.
[10] H. Bettin, F. Spieweck, ”A revised formula for the
calculation of alcoholometric tables”, PTB-Mitteilungen,
Vol. 100, pp.457-460, 1990.
[11] J. B. Patterson and E. C. Morris, ”Measurement of
absolute water density, 1 oC to 40 oC”, Metrologia, Vol.
31, pp. 277-288, 1994.
[12] K. Fujii, ”Present state of solid end liquid density
standards”, Metrologia, Vol. 41, pp. S1-S15, 2004.
[13] M. Tanaka, G. Girard, R. Davis, A. Peuto and N.
Bignell, ”Recommended table for the density of water
between 0 oC and 40 oC based on recent experimental
reports”, Metrologia, Vol. 38, pp. 301-309, 2001.
[14] P. Chappuis, Trav. Mem. Bur. Poids. Mes., Vol. 13,
p. D1, 1907.