Measurement 169 (2021) 108511
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Measurement
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On the sensitivity of an air pycnometer
Thammarong Eadkhong, Sorasak Danworaphong ∗
Division of Physics, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand
ARTICLE
Keywords:
Pycnometer
Sensitivity
Energy conservation
Filling fraction
INFO
ABSTRACT
An air pycnometer is constructed based on the concept of energy conservation for the investigation of its
sensitivity. The apparatus assumes that the ideal gas law holds and energy transfer between the system and its
surroundings is negligible. It comprises two connected chambers—reference and sample. The addition of a ball
valve helps isolate them. The volume ratios between them, i.e., 0.52 and 1.00, are explored experimentally for
their sensitivity. The predictions for the ratio of 0.20 and 0.33 are also given. Acrylic disks of known volumes
are used for testing filling fractions under the volume-ratio conditions. The data are fitted to the model, derived
from energy conservation, for the atmospheric pressure and the initial pressure of the reference chamber. The
derivative of the model gives rise to sensitivity curves in terms of filling fractions for given volume ratios. Four
volume ratios, 0.20, 0.33, 0.52, and 1.00, are explored for their sensitivities and characteristics. The results
show that the sensitivity of the pycnometer relates to both the volume ratio between chambers and the filling
fraction. When graphed for different volume ratios, the sensitivity monotonically increases as the filling fraction
rises for all ratios. The system is more sensitive for a greater ratio only when the filling fraction is more than
75%. For lesser filling fraction, the sensitivity becomes subtle and is unlike that of the nearly-filled chamber.
Here, we reveal and discuss this phenomenon. Besides, the results also indicate that sensitivity customization
can be achieved. To clarify this statement, we apply the apparatus to determine the void volume, for four
filling fractions, of divided solids—full and broken grains of rice and sugar beads—and a fibrous sample—a
palm wood disk. The results agree with those of the water saturation approach. Our apparatus gives reliable
results, and its sensitivity dependence is elucidated. Locations of the sensitivity degeneracy are identified. The
correlation between the filling fraction and volume ratio of the system cavities is described.
1. Introduction
A pycnometer is an apparatus used for estimating the specific
gravity and volume of materials. Its uses possibly dated back to the
early nineteenth century. It was utilized for determining alcohol level in
the brewery business [1,2] and became a crucial volumetric tool [3,4].
However, they worked like a hydrometer, which required liquid, often
mercury, immersion in the measurement [5]. In 1936, the gas pycnometer was, possibly the first, applied to the determination of porosity
of soil samples using Boyle’s law [6]. The tool underwent constant
modifications to comply with particular measurement demands. For example, it was used for determining the air-filled porosity in unsaturated
organic matrices [7]. The porosity of agglomerated seeds and grains can
be measured by a pycnometer [8]. It was also used for measuring the
porosity in nanofiber sheets [9]. In parallel, the gas-based pycnometer
still kept its original goals in finding the density of matters, e.g., the
density of atmospheric aerosol particles [10], and soil particles [11]. Its
applications were extended to the estimation of API gravity, a density
index, of crude oils [12], and the density of plutonium alloy [13]. It
was also developed for measuring the density of volatile liquids [14].
The gaseous medium used can be air, helium, or other gases, depending
on samples or experimental requirements. Helium gives more accurate solid volume since it can fill in small crevices on the sample
and show the least adsorption than other gases [15,16]. A helium
pycnometer was applied to determine the soil–water density [17] and
particle density of fibrous materials and perlite [18]. Though the gas
pycnometer went through a long period of development and became
commercially available, it can be generally divided into three kinds—
constant-volume [19], variable-volume [20], and comparative [21,22].
A typical constant-volume pycnometer comprises a sample chamber
and a buffer cavity that their volumes are invariable. For a variablevolume pycnometer, one of the cavities allows a volume adjustment.
Both types register the absolute pressure equilibrium, using a pressure
transducer due to gas expansion into or out of the sample chamber.
The changes relate directly to the sample volume, according to Boyle’s
law. Being slightly subtle, the setup for the comparative pycnometer
incorporates a differential pressure sensor required for leveling the
∗ Corresponding author.
E-mail addresses: thammarong@mail.wu.ac.th (T. Eadkhong), dsorasak@mail.wu.ac.th (S. Danworaphong).
https://doi.org/10.1016/j.measurement.2020.108511
Received 13 May 2020; Received in revised form 28 August 2020; Accepted 22 September 2020
Available online 29 September 2020
0263-2241/© 2020 Elsevier Ltd. All rights reserved.
Measurement 169 (2021) 108511
T. Eadkhong and S. Danworaphong
Fig. 1. The experimental setup. Initially, two chambers are kept at different pressures.
The reference pressure (𝑃𝑟 ) is 15.0 kPa while the sample cavity is open to atmosphere
(𝑃𝑎 ). Their volumes (𝑉𝑟 and 𝑉𝑠 ) can be varied to achieve particular sensitivity and
detection limit. Their initial volumes are identically 926.6 ± 0.22 cm3 .
Fig. 2. Variation of final pressure (𝑃𝑓 ) as filling fraction (𝜑) increases. Solid and clear
circles indicate the experimental results performed to validate Eq. (1) and the system
for 𝜏 = 0.52 and 1.00 [27]. The other line is given for 𝜏 = 0.20 and 0.33.
pressure between the reference and sample chambers. These chambers
are connected to two separated movable pistons allowing individual
volume adjustment for the pressure-balancing purpose, as mentioned
above. Among these, the constant-volume pycnometer is the only system having no moving parts. Its primary source of error comes directly
from the pressure sensor given that the cavity volumes are known
precisely, and the temperature is unchanged during measurement.
Therefore, it is the most straightforward pycnometry system, but the
specimen must nearly fill the sample chamber to achieve an accurate
result [7]. The law of propagation of uncertainty was extensively used
to numerically determine the measurement uncertainty [23–25].
Even though its uses are common and the sensitivity has been
proved to be filling-fraction dependence, the approach is still intangible, and the role of the volume ratio is still unsettled [26]. In this work,
we aim to explore the parameters affecting the sensitivity of an air
pycnometer, possibly a constant-volume one. The governing equation
of the pycnometer is derived based on the energy conservation model or
Boyle’s law. It is validated through experiments and nonlinear regression analysis, assuming that the gas behaves ideally and isothermally.
Experiments are conducted for two-volume ratios, the ratio between
the sample and reference cavity. The atmospheric pressure and a predefined pressure are the fitting parameters of the model. The governing
equation allows us to generate sensitivity curves for any volume ratios
in terms of the filling fraction by differentiating it with respect to the
grain volume. These curves depict the sensitivity of a pycnometer in
terms of the filling fraction. Moreover, it allows one to consider which
sensitivity to use as a function of the filling fraction and the volume
ratio. The sensitivity is limited not only by the filling fraction and the
volume ratio but also by the accuracy of the pressure sensor. Finally,
we carry out experiments showing the degeneracy of the sensitivities
for various filling fractions. We also apply the system to determine the
volume and porosity of divided solids and a fibrous sample. The results
are in good agreement with those obtained from the water immersion
method and reported in the literature. This work is essential for setting
up the experiment to comply with the desired sensitivity. Sensitivity
customization is, therefore, possible.
3. Validation and experiments
Fourteen acrylic cylinders of known volumes are used as samples
for system calibration. Each acrylic disk has the volume of 42.70 ±
0.04 cm3 . Initially, the pressure in the reference cavity, 𝑃𝑟 , is kept
at 15.0 kPa while the sample reservoir is open to atmosphere, 𝑃𝑎 .
The known-volume acrylic sample, 𝑉𝑔 is collectively placed in the
sample cavity and then tightly pressed with the acrylic lid for each
individual measurement. The ball value is then slowly open to allow
air flow due to pressure gradient between the two chambers. The final
or equilibrium pressure, 𝑃𝑓 , is recorded in terms of 𝜑. The data are
then fitted to Eq. (1) for the atmospheric pressure or 𝑃𝑎 and the initial
pressure of the reference cavity, 𝑃𝑟 , as the two fitting parameters where
other variables in the equation are fixed. Eq. (1) can be expressed as
(
)
(
)
𝑃𝑟 + 𝑃𝑎 𝑉𝑠 − 𝑉𝑔 𝑉𝑟−1
𝑃𝑓 = 𝜏
.
(1)
𝜏 + 1 − 𝑉𝑔 𝑉𝑠−1
Here, 𝜏 represents the volume ratio of 𝑉𝑟 and 𝑉𝑠 . The former is the
volume of the reference, and the later is that of the sample chamber.
The calibration experiments are done under two different 𝜏’s, 0.52 and
1.00, represented in solid and clear circles in Fig. 2, respectively [27].
The resulting atmospheric pressures with one standard deviation of the
two sets yield 101 090 ± 620 and 101 160 ± 228 Pa while the fitted
outcomes, solid and dashed lines, for the initial pressures are 15 207 ±
396 and 15 203 ± 272 Pa. These values are within 0.25% deviation of
the standard atmospheric pressure, 101 325 Pa at 22 ◦ C [28], and
1.50% deviation of the user-defined initial pressure of the reference
volume. By using Eq. (1), the equilibrium pressure as a function of grain
volume can be graphed for any values of 𝜏. As an example, also for later
d𝑃
treatment, a trend line for 𝜏 = 0.20 is displayed in Fig. 2. The slope, d𝑉𝑓 ,
𝑔
of each curve is determined and displayed in Fig. 3 . This expression is
equivalent to the sensitivity of the system. It should be noted that the
volume of the reference cavity must always be lesser than that of the
sample cavity. The violation deteriorates the sensitivity, particularly
for the filling fraction of 0.80 or higher since a larger volume requires
more gas to cause pressure change. Moreover, the initial user-defined
pressure can also be reduced to slightly improve the sensitivity.
2. Experimental setup
The system is composed of two connected polyvinyl chloride cavities of identical volume, 926.6 ± 0.22 cm3 . The chambers are isolated
from one another by a ball valve. One of the cavities, a reference
chamber, is connected to a 0.1-kPa resolution pressure gauge, a rotary
pump, and a pressure relief valve. The other cavity is called a sample
chamber that one of its ends is open to surrounding. A 1.00-cm thick
acrylic lid is gouged for 5.0-mm deep circular groove of 88.0-mm
diameter for snugly securing an o-ring. This lid is used for covering
the sample cavity during measurement. The schematic of the system is
displayed in Fig. 1.
4. Results and discussion
Since the system and all calculations thus far are assumed isothermal, it is necessary to explore the temperature variation within the
system. By using a portable Bluetooth-enabled digital thermometer, we
find that during the measurement, the fluctuation inside both chambers
is read in real-time to be less than 0.3 ◦ C or ∼ 0.24 J given that
2
Measurement 169 (2021) 108511
T. Eadkhong and S. Danworaphong
Fig. 4. Samples for measurements. (a) a palm wood disk, (b) sugar beads, (c) sugar
beads in a dedicated housing, (d) full-grain jasmine rice, and (e) broken seed jasmine
rice.
Fig. 3. The derivative of equilibrium or final pressure with respect to the filling
fraction (𝜑) in the sample chamber. Four different 𝜏’s are displayed to illustrate how
the sensitivity or the derivative varies in terms of 𝜑. Here, 𝜏 = 0.52 and 1.00 are those
belonging to experiments, whereas 𝜏 = 0.33 is stated to be most effective ratio [26].
𝜏 = 0.20 displays better sensitivity for the filling fraction greater than 0.74. The
inset depicts the intersections of the sensitivity curves. They represent the common
minimums of sample volumes of which can be detected. The oval, star, and hexagon
are the intersections of 𝜏 = 1.00 and 0.52, 𝜏 = 1.00 and 0.20, and 𝜏 = 0.20 and 0.52,
respectively. 𝛥𝑉𝑔 of each mark is calculated from 𝛥𝑃 of 0.1 kPa, the deviation of the
pressure sensor. Points A and B indicate the filling fraction region where 𝜏 = 0.33 shows
greater sensitivity than those of other volume ratios. However, the sensitivity outside
the AB region is mundane.
soil porosity [26]. This 0.33-curve is superior to others for 0.59 < 𝜑 <
0.74 (gray arrow line), denoted by A and B. For 𝜑 > 0.74, the 0.20-curve
shows a steep rise of sensitivity. As a result, one can adjust the filling
fraction and the volume ratio to find the desired sensitivity.
To this end, the system is applied to determine the apparent porosity
of various specimens by measuring their solid volume. Defining the
(
)
porosity or air volume fraction as 𝑉𝑏 − 𝑉𝑔 ∕𝑉𝑏 [30] where 𝑉𝑏 is the
solid volume of the sample, we can evaluate the porosity for all
subjects, i.e., a palm wood panel, sugar beads, full grain, and broken
jasmine rice, as shown in Fig. 4(a), (b), (d), and (e). All measurements
are governed by 𝜏 = 0.52. For the palm wood disk, 𝑉𝑏 = 26.30 ± 0.03
cm3 , the measurement is carried out by placing the panel in the sample
chamber and its solid volume is measured directly. Its filling fraction
is 0.59 corresponding to 𝛥𝑉𝑔 of 1.80 cm3 . For granular samples, a
rectangular container with the inner dimensions of 3.90 × 5.36 × 11.75
cm3 , is employed for sample accommodating purpose. An example of
a partly-filled container with sugar beads is shown in Fig. 4(c) with
its dimensions. The housing must be completely filled with granular
samples for their measurement so that 𝑉𝑏 corresponds to the container
volume, 245.60±0.08 cm3 . Other samples – sugar beads, long and broken
grains – are displayed in Fig. 4(b), (d), and (e). Their filling fractions
are 0.36, 0.56, and 0.56, respectively. Besides, they correspond to 𝛥𝑉𝑔
of 2.76, 1.92, and 1.86 cm3 of which are also the uncertainty in grain
volume measurements. The resulting grain volumes and porosities are
given in Table 1. The uncertainties of the porosities are calculated
directly from error propagation from those of 𝑉𝑏 and 𝑉𝑔 .
In addition, the table includes general information and porosity
measured by water immersion for comparison. Each water immersion
experiment is repeated for five times. Its uncertainty is one standard
deviation. It can be seen that the measurements carried out by our setup
and the water immersion method yield the values of apparent pores
that are in good agreement with each other. For the fibrous sample,
the palm disk, the apparent porosity is evaluated to be 0.593 ± 0.068
with the deviation of 0.17% from that of the water saturation method.
As expected from Fig. 3, the deviation is decreased as the filling fraction
(𝜑) increases. For sugar beads, the void space in the pack of sugar beads
was also previously reported to be 0.397 ± 0.008 by means of air volume
comparison technique [22]. Our setup and water saturation method
yield the values within 2.3% of that given by the air volume comparison
method. The estimated porosity derived from the two methods – our
pycnometer (air volume fraction) and water saturation – for the broken
and full grain rice are within 1.0% or less of one another. The pack of
full grains possesses larger void space compared to that of broken rice.
This is logical since the smaller grains can fill more space as opposed
to the larger ones. Similarly, the sugar beads have smaller grain size
(2.18 mm); it is therefore expected to have less empty space. This is
clearly shown in Table 1.
the specific heat capacity at constant volume of air is 0.72 J/kg K at
300 K [29]. The energy is approximately 1.73% of the lowest energy
of the system calculated from the multiplication of initial pressure
and volume of the reference chamber. Due to this slight change of
energy, the system may be considered to be thermally isolated from
its surroundings, assuring the validity of our calculation.
It can be seen that as 𝑉𝑔 approaches the volume of the sample
chamber, 𝜑 → 1, the sensitivity increases for all 𝜏’s. Fig. 3 illustrates
that the larger the grain volume or the filling fraction, the greater the
sensitivity. Besides, if the grain volume or filling fraction is small, the
sensitivities are slightly different and competitive for all 𝜏’s. It seems
explicit that the system only works well for large volume specimens.
However, it is feasible for small volume measurement to add a known
volume sample to bring the sensitivity to where it is needed. For
example, given the lowest possible pressure change that the system can
monitor is 0.1 kPa (𝛥𝑃𝑓 ), considering the oval marker (𝜏 = 0.52 and
1.00) in the inset of Fig. 3, the filling fraction of 0.27 corresponds to the
sensitivity of 0.031 kPa/cm3 . The lowest volume variation measurable
is 𝛥𝑉𝑔 = 0.10∕0.031 = 3.23 cm3 . For the star marker (𝜏 = 0.20 and 1.00),
𝛥𝑉𝑔 = 0.10∕0.044 = 2.27 cm3 , whereas 𝛥𝑉𝑔 for the hexagon marker
(𝜏 = 0.20 and 0.52) is 1.47 cm3 . All these markers display sensitivity
degeneracies where different volume ratios share a common sensitivity,
the crossings of the sensitivity curves in the inset of Fig. 3. The volume
ratio alone is insufficient to define sensitivity. For 𝜏 = 1.00 and 0.52,
one can see that the former possesses higher sensitivity for the filling
fraction lower than 0.27. Both 𝜏’s then share the sensitivity of 0.031
kPa/cm3 at the filling fraction of 0.27, corresponding to the lowest
possible volume of 3.23 cm3 for the system to notice the difference. For
𝜑 > 0.27, the latter becomes more sensitive. Moreover, the sensitivity
of 0.52-curve is higher than that of 0.20-curve from zero filling fraction
to 0.55. At this common sensitivity, the lowest possible volume the
system can detect is 2.27 cm3 . Beyond the filling fraction of 0.55,
the sensitivity of the 0.20-curve rises significantly and intersects with
that of 0.52-curve at 𝜑 = 0.68. Before the crossing, the sensitivity of
0.52-curve is higher than that of 0.20-curve. At the intersection, the
sensitivity is 0.068 kPa/cm3 , which corresponds to the lowest volume
of 1.47 cm3 . An addition curve for the volume ratio of 0.33 is provided
to elucidate the claim stating that this ratio is generally sufficient for
3
Measurement 169 (2021) 108511
T. Eadkhong and S. Danworaphong
Table 1
The resulting grain volumes and calculated porosities in comparison with those measured by water immersion method.
Samples
Information
Grain volume (𝑉𝑔 )
(cm3 )
Porosity
(
)
𝑉𝑏 − 𝑉𝑔 ∕𝑉𝑏
Water immersion
Palm disk
30 mm radius, 9.3 mm thickness, 31.50
g, 𝑉𝑏 = 26.30 ± 0.03 cm3
10.68 ± 1.80
0.593 ± 0.068
0.594 ± 0.026
Sugar beads
1.37 to 2.18 mm diameter range, 209.37
g, 𝑉𝑏 = 245.60 ± 0.08 cm3
151.17 ± 2.76
0.384 ± 0.011
0.393 ± 0.002
Long grains
7.2 to 7.5 mm seed length, 1.83 ± 0.05
average diameter, 213.74 g,
𝑉𝑏 = 245.60 ± 0.08 cm3
140.57 ± 1.92
0.428 ± 0.008
0.424 ± 0.013
Broken rice
1.3 to 3.5 mm seed length, 1.83 ± 0.05
average diameter, 221.63 g,
𝑉𝑏 = 245.60 ± 0.08 cm3
148.21 ± 1.86
0.397 ± 0.008
0.391 ± 0.005
5. Conclusion
References
In conclusion, we construct a measurement system whose sensitivity
can be tailored as needed for determining the grain volume of an object
of any shape as well as porosity if its bulk volume is known. The atmospheric pressure is used as a fitting parameter for the model constructed
from the energy conservation. The fit result yields ∼ 0.25% deviating
from the typical value at 22 ◦ C. Within the model, the ratio of reference
and sample volume (𝜏) is defined to help generating the sensitivity
curve. Each curve has its own sensitivity variation in terms of the filling
fraction. This relationship allows one to select the desired sensitivity for
specific measurements. An essential feature of the sensitivity curves is
that the volume ratio alone is insufficient to determine the sensitivity.
In the lower filling fraction region, the higher the volume ratio, the
greater the sensitivity. On the contrary, the lower the volume ratio,
the higher the sensitivity for the area of the large filling fraction.
Besides, the interception positions of the sensitivity curves indicate
their degenerate values. To test the system, we apply it to granular
samples—sugar beads, long and short grains of rice. Moreover, a fibrous
sample, a palm wood panel, is also measured. The results are compared
to those given by the water saturation technique and literature. Our
experiments have substantiated the proposed setup for its capability to
measure solid volume of arbitrary shape sample as well as to estimate
the porosity if its bulk volume is given. Even though the system is based
on a simple concept and built with low-cost parts, it performs with
comparable accuracy to other techniques without liquid immersion
or high pressure compression. Sensitivity can also be customized to
match demand of particular measurements by varying 𝜏, and filling
fraction. Intriguing features such as the degenerate sensitivity and the
correlation of the sensitivity curves in Fig. 3 allow one to set desired
sensitivity for one’s measurement.
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CRediT authorship contribution statement
Thammarong Eadkhong: Conceptualization, Methodology, Investigation, Formal analysis, Writing - original draft, Visualization. Sorasak
Danworaphong: Conceptualization, Methodology, Validation, Formal
analysis, Writing - review & editing, Visualization, Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgment
This work is financially supported by the Scholarship from Ministry
of Science and Technology, Thailand.
4
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