Dynamic pressure corrections in a clearance-sealed piston prover for gas flow measurements

Abstract

The dynamic pressure effects and their corrections in a high-speed, clearance-sealed realization of a piston prover for gas flow measurements are discussed. The experimental results show the deterministic, rather than stochastic, nature of the dynamic pressure conditions and, consequently, the repeatable nature of their influence on the flow measurements. The experimental validation proves the advantage of the polytropic/adiabatic pressure correction model, which was proposed by the authors, as compared with the isothermal pressure correction model. The paper ends with an estimation of the measurement uncertainty related to the pressure corrections using either the adiabatic or isothermal model.

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Dynamic pressure corrections in a clearance-sealed piston prover for gas flow measurements
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IOP PUBLISHING METROLOGIA
Metrologia 50 (2013) 66–72 doi:10.1088/0026-1394/50/1/66
Dynamic pressure corrections in a
clearance-sealed piston prover for
gas flow measurements
Joˇ
ze Kutin, Gregor Bobovnik and Ivan Bajsi´
c
Laboratory of Measurements in Process Engineering, Faculty of Mechanical Engineering,
University of Ljubljana, Aˇ
skerˇ
ceva 6, SI-1000 Ljubljana, Slovenia
E-mail: joze.kutin@fs.uni-lj.si
Received 27 September 2012, in final form 11 December 2012
Published 28 January 2013
Online at stacks.iop.org/Met/50/66
Abstract
The dynamic pressure effects and their corrections in a high-speed, clearance-sealed
realization of a piston prover for gas flow measurements are discussed. The experimental
results show the deterministic, rather than stochastic, nature of the dynamic pressure
conditions and, consequently, the repeatable nature of their influence on the flow
measurements. The experimental validation proves the advantage of the polytropic/adiabatic
pressure correction model, which was proposed by the authors, as compared with the
isothermal pressure correction model. The paper ends with an estimation of the measurement
uncertainty related to the pressure corrections using either the adiabatic or isothermal model.
(Some figures may appear in colour only in the online journal)
1. Introduction
The piston-prover concept is widely used for primary standards
in the field of gas flow measurements [1–7]. The general
principle of operation is based on determining the time interval
that a piston needs to pass a known volume of gas at a defined
pressure and temperature.
The experimental work of this paper was performed on
a commercially available, clearance-sealed realization of the
piston prover [8–10]. The piston is made of a graphite
composite and the cylinder is made of borosilicate glass. The
piston and the cylinder are closely fitted, with a clearance of
the order of 10 µm. The passage of the piston is detected by
infrared light emitters and sensors. The base of the piston
prover holds the timing crystal, the barometric pressure sensor
and the computer. The gas gauge pressure is measured by a
fast-response pressure sensor, which is connected to the flow
system at the entrance to the cylinder employing the pressure
impulse line. The gas temperature is measured at the same
location using a thermistor temperature sensor. Due to its
relatively high heat capacity (a probe diameter of about 1 mm
to 2 mm), the measured temperatures represent nearly time-
averaged values during the high-speed operation of the piston
prover.
The piston prover under discussion employs the following
measurement model for the volume flow rate at the barometric
pressure Paand the gas temperature T:
qv(Pa,T) =Vm
t +qv,lεp,(1)
where Vmis the measuring volume of the gas collected by
the piston prover during the timing cycle t =t2−t1,qv,l
is the clearance leakage flow (more precisely, its Poiseuille
component, whereas its Couette component is considered as
the reduced effective diameter of the cylinder in Vm)and εp
is the pressure correction factor, which takes into account
the deviations of the actual pressure conditions from the
barometric pressure. The equation for εp, which is originally
employed in the piston prover and termed the isothermal
pressure correction in this paper, has the form
εp=1+ p2
Pa
+p2−p1
Pa
Vd
Vm
,(2)
where p1and p2are the gauge pressures at times t1and
t2, respectively, and Vdis the connecting volume of the gas
between the selected inlet transfer point and the piston at
time t1.
0026-1394/13/010066+07$33.00 © 2013 BIPM & IOP Publishing Ltd Printed in the UK & the USA 66

Dynamic pressure corrections in a clearance-sealed piston prover
The derivation from the law of conservation of mass
points out that the validity of the pressure correction model
in equation (2) depends on the following assumptions [11]:
(i) ideal gas: this is valid due to the relatively small pressure
changes from the barometric pressure and so the gas
compressibility effects can be considered as negligible;
(ii) spatially homogeneous pressure changes: this is at least
valid for the compact internal volume of the piston prover,
the linear dimensions of which are small compared with
the wavelengths of the pressure oscillations;
(iii) isothermal system: this is only valid if the dynamic
processes in the gas are slow with respect to the heat
exchange.
Because the pressure oscillations in the piston prover under
discussion are of the order of a few tens of hertz, the validity of
the isothermal assumption is questionable; fast processes with
respect to heat exchange can often be considered as polytropic
or quasi-adiabatic. In [11] the authors of this paper derived
and theoretically validated an equation for polytropic/adiabatic
pressure correction, which can be written as
εp=1+ ¯p12
Pa
+1
γp2−¯p12
Pa
+p2−p1
Pa
Vd
Vm,(3)
where ¯p12 is the time-averaged value of the gauge pressure
during the timing cycle and γis the polytropic index. In
the limit of adiabaticity, γis the ratio of the specific heats
or the adiabatic index. For dry air the adiabatic index is
approximately γ=1.4. If γ=1 is considered, the
polytropic/adiabatic measurement model (3) transforms into
the isothermal measurement model (2).
The aim of this paper is a metrological analysis
of the pressure correction factors in the piston prover.
The main contribution is an experimental validation of
the proposed polytropic/adiabatic measurement model for
pressure corrections and the corresponding decrease in its
contribution to the uncertainty of the gas flow measurements.
Section 2describes the measurement system, which is then
used for measuring the pressure variations at different flow
rates. These pressure variations are discussed in terms of
their deterministic or stochastic dynamic nature. Section 3
deals with the properties of the pressure correction factors.
The experimental validation test proves the advantage of the
polytropic/adiabatic model in comparison with the isothermal
model. Section 4presents an analysis of the measurement
uncertainty of the pressure correction factors.
2. Pressure characteristics
Measurements of the pressure characteristics of the piston
prover (Sierra Instruments, Cal=Trak SL-800-44, measuring
range 0.5 sl min−1to 50 sl min−1)were carried out for different
flow rates within its measuring range using clean, oil-free, dry
air. The measurement system is schematically presented in
figure 1. The stable mass flow rate through the piston prover
was set with the help of a pressure regulator and a selected
critical nozzle (TetraTec Instruments, array of five Venturi-
shaped critical nozzles) operating under sonic conditions.
All the flow rate values given in this paper represent flow
measurements by the piston prover calculated as the standard
volume flow rate at Ps=101.325 kPa and Ts=293.15 K:
qs=ρ(Pa,T)
ρ(Ps,T
s)qv(Pa,T), (4)
where the REFPROP database [12] was used to determine the
air densities ρ(Pa,T)and ρ(Ps,T
s). The flow measurements
of the piston prover were controlled using an RS-232 serial
communication with the ASCII protocol, which initiates the
measurement and acquires the data query stream, including
the uncorrected volume flow rate Vm/t, the barometric
pressure Pa, the temperature T, the gauge pressures p1and
p2, etc [13]. In order to study the pressure variations during
the piston prover’s operation and to have the possibility
to determine the time-averaged pressure ¯p12 we used an
additional ‘external’ gauge pressure sensor (Validyne, P855,
measuring range −1.4 kPa to 1.4 kPa, voltage output −5V to
5 V, low pass filter at 250Hz/–3 dB), which was connected
in parallel with the internal gauge pressure sensor. Its voltage
output was measured with a DAQ board (National Instruments,
USB-6251 BNC). The processing of the measurement signals
was realized with LabVIEW software (National Instruments,
Ver. 10.0).
For each of the seven flow rates (at about 1.5 sl min−1,
2.7 sl min−1, 4.8 sl min−1, 8.6 sl min−1,15slmin
−1,27slmin
−1
and 46 sl min−1)20 consecutive repetitions of the measure-
ments with a simultaneous acquisition of the readings of the
piston prover and the external pressure sensor were carried out.
Figure 2shows an example of the measured pressure response
during one measurement cycle of the piston prover for a flow
rate of 8.6 sl min−1. It schematically demonstrates the extrac-
tion of the timing cycle from this external pressure sensor’s
signal. Although the length of the timing cycle t is known
from the piston prover’s measurands, its position in time is un-
determined (the output signals of the optical sensors, which are
used in the piston prover’s electronics to trigger the time values
t1and t2, are not easily accessible to the user). The real-time
estimation of the t2value was achieved by a software trigger,
the level of which was set at approximately 80% of the average
pressure change at the end of the measuring cycle, and by an
additional subtraction of a time lag of about 0.015 s from the
corresponding trigger time. Using the described procedure we
achieved a suitable synchronization of the measured pressures
p1and p2from the external and internal pressure signals, with
differences of less than 10 Pa (it has to be emphasized that even
a less accurate estimate of the timing cycle on the external pres-
sure signal would not affect the estimate of its time-averaged
value ¯p12, which will only be used in the polytropic/adiabatic
model).
Figure 3shows the extracted timing cycles of the
pressure responses for two different flow rates of 8.6sl min−1
and 27 sl min−1. Each graph consists of four selected
measurements from 20 repetitions (5th, 10th, 15th and 20th
series). It is evident that the pressure variations in the
piston prover are not stochastic but deterministic, because the
measured pressure signals practically overlap (see [11] for a
Metrologia,50 (2013) 66–72 67

J Kutin et al
Figure 1. Scheme of the measurement system.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-0.3
0.0
0.3
0.6
0.9
1.2
t2
t1
time delay
trigger level
timing cycle
∆t
Time / s
Pressure / kPa
Figure 2. Example of the pressure response during the piston
prover’s measurement cycle (flow rate of 8.6 sl min−1).
frequency analysis of the observed pressure oscillations and a
discussion of the reasons for their occurrence).
The predominantly deterministic nature of the dynamic
pressure conditions is, consequently, also shown in the values
of the characteristic pressures: figure 4shows the pressures
p1and p2(internal pressure sensor) and the time-averaged
pressures ¯p12 (external pressure sensor) for all 20 repetitions
of the measurements at a certain flow rate. The entire range
of pressure oscillations in the timing cycle is illustrated by
the values of the maximum and minimum pressures. The
largest difference between p2and p1, which approaches the
span between the maximum and minimum pressures, occurs
at a flow rate of 8.6 sl min−1. As already reported in [11], this
flow range represents the most intensive resonance effects of
the piston oscillator, which are excited by the self-sustained
oscillations of the inlet flow in the cylinder below the moving
piston. At higher flow rates, p1remains smaller than ¯p12 and
p2remains almost equal to ¯p12. Such repeatable pressure
characteristics indicate a certain degree of synchronization
between the pressure response and the piston movement along
the cylinder (other positions of the optical sensors would
change the values of p1and p2).
3. Isothermal versus adiabatic pressure corrections
We assume that the presented dynamic conditions are fast
enough to consider a quasi-adiabatic process. The isothermal
and adiabatic pressure correction factors εpare calculated
using equations (2) and (3), respectively. We take into account
the measured pressure values from figure 4, the measured
barometric pressures Pa(≈98.5kPa), the dimensionally
calibrated measuring volume Vm=118.2 ml, the estimated
connecting volume Vd=200 ml (in this value there is about
30 ml of the volume between the critical nozzles and the
entrance to the piston prover, which will be considered as the
source of the uncertainty; see section 4) and the polytropic
index of dry air close to γ=1.4 for the adiabatic model. The
relative values of the pressure corrections (εp−1)determined
for both models are presented in figure 5. They range from
about 0.15% at the smallest flow rate to about 1% at the largest
flow rate. The dashed line in the graph shows the share of
the influence of the time-averaged pressure ¯p12 /Pa. Figure 5
68 Metrologia,50 (2013) 66–72

Dynamic pressure corrections in a clearance-sealed piston prover
(a) Flow rate of 8.6 sl/min.
(
b
)
Flow rate of 27 sl/min.
0.0 0.2 0.4 0.6 0.8
0.0
0.1
0.2
0.3
0.4
Time / s
Pressure / kPa
0.00 0.05 0.10 0.15 0.20 0.25
0.0
0.2
0.4
0.6
0.8
Time / s
Pressure / kPa
Figure 3. Pressure responses during the timing cycle at two
different flow rates (four repetitions).
1 10 100
0.0
0.2
0.4
0.6
0.8
1.0
min. pressures
max. pressures
p1
p2
p12
Flow rate / sl/min
Characteristic pressures / kPa
Figure 4. Characteristic pressures in the timing cycle.
also presents the difference between the pressure correction
factors for the isothermal and adiabatic models, which can be
written as
δT−A=γ−1
γp2−¯p12
Pa
+p2−p1
Pa
Vd
Vm.(5)
110100
0.0
0.2
0.4
0.6
0.8
1.0
isothermal ε
ε
p
adiabatic p
p12 / Pa
difference δT
–
A
Flow rate / sl/min
100 × (1– εp
)
Figure 5. Relative values of the pressure corrections.
The largest value of the difference δT−Ais about 0.1% at a flow
rate of 8.6 sl min−1.
An important question that remained open after deriving
and theoretically validating the polytropic/adiabatic model in
[11] was the experimental confirmation of its validity. For this
reason, the following validation experiment was prepared:
(i) the sonic nozzles were used to ensure a constant mass flow
rate to the piston prover;
(ii) the outlet flow conditions were modified to influence the
dynamic conditions in the piston prover and so the values
of the dynamic pressure corrections;
(iii) if the pressure correction model of the piston prover is
appropriate, it would take into account the variation in the
pressure conditions and therefore the output of the mass
flow calculation would not change after the modification.
The following results were obtained by the validation
experiment at a flow rate of 8.6 slmin−1. A modification of the
dynamic conditions was made by a restriction at the outlet of
the piston prover giving an additional pressure drop of about
0.5 kPa. The flow restriction was realized by connecting a
tube with an inner diameter of 4 mm. The measured values
of the characteristic pressures before the modification (left)
and after the modification (right) are presented in figure 6(a).
In addition to the higher value of the average pressure, the
flow restriction significantly reduced the influential dynamic
components p2−¯p12 and p2−p1, and thereby the magnitudes
of the dynamic pressure corrections. The corresponding
relative changes in the piston prover’s mass flow readings
are shown in figure 6(b). While the modification of the
dynamic pressure conditions caused a systematic change in the
measured flow rate of about 0.06% when using the isothermal
model, no notable systematic changes were observed in the
case of the adiabatic model. This confirms the advantage
of the adiabatic pressure correction model in comparison
with the isothermal pressure correction model. The same
conclusion regarding the adiabatic model was also found on
the basis of other measurements at a flow rate of 8.6 sl min−1
using different lengths of the outlet connecting tube (the
gauge pressures in the piston prover were kept below the
recommended maximum value of 1.25 kPa [14]).
Metrologia,50 (2013) 66–72 69

J Kutin et al
(a) Characteristic pressures.
(
b
)
Relative error of the flow rate.
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
p1
p2
p12
after modification
before modification
Number of measurements
Characteristic pressures / kPa
0 5 10 15 20 25 30
-0.04
0.00
0.04
0.08
0.12
after modification
before modification isothermal ε
ε
p
adiabatic p
Number of measurements
100 × Relative error
Figure 6. Validation of the adiabatic model by modifying the
dynamic pressure conditions at a constant inlet mass flow rate.
Similar validation experiments were also performed at
higher flow rates. Although the corresponding dynamic
conditions have similar frequencies and so the validity of the
adiabatic model is expected, the properties of the piston prover
under discussion did not enable us to attain representative
validation results. The main problems were related to the
smaller dynamic pressure corrections (smaller differences
between the adiabatic and the isothermal models), reduced
repeatability of the mass flow readings (mainly related to the
time discretization effects that, for example, reach 0.1% at
the largest flow rates) and the inability to generate sufficient
modifications of the dynamic pressure conditions by an
additional pressure drop at the outlet.
4. Measurement uncertainty of the pressure
corrections
In this paper, the uncertainty analysis of the flow rate measured
by the piston prover is limited to the uncertainty of the pressure
correction factor εp. The main contributions can be divided
into the measurement uncertainty of the characteristic pressure
values p1,p2and ¯p12, the uncertainty related to the connecting
volume Vdand the uncertainty related to the polytropic index
γ. The contributions from the parameters Paand Vm—both
the relative standard uncertainties u(Pa)/Paand u(Vm)/Vmare
about 0.025%—can be neglected due to their minor influence
on the uncertainty of the pressure correction (they have to
be accounted for in the entire measurement uncertainty of
the mass flow rate where they appear in the estimate of
the gas density ρ(Pa,T) and the uncorrected volume flow
rate Vm/t, respectively). The measurement uncertainty
evaluation presented here refers to each single realization of the
20 repetitive measurements at each flow rate and is performed
in accordance with the GUM [15].
Considering the adiabatic equation (3) and uxi(εp)=
(∂εp/∂xi)u(xi), the contributions of the standard measurement
uncertainty of the characteristic pressures can be written as
up1(εp)=−1
γ
Vd
Vm
u(p1)
Pa
,
up2(εp)=1
γ1+ Vd
Vmu(p2)
Pa
,
u¯p12 (εp)=γ−1
γ
u(¯p12)
Pa
.
(6)
The standard uncertainties of the pressure values p1and p2
measured by the internal pressure sensor of the piston prover
are estimated to be u(p1)=u(p2)=5 Pa, and the standard
uncertainty of the time-averaged pressure ¯p12 measured by
the external pressure sensor is estimated to be u( ¯p12)=
2 Pa. These pressure uncertainties include the contribution
of the measurement errors and their uncertainties from the last
calibration of the pressure sensors, the contribution of their
time drift and, for the instantaneous pressure values p1and
p2, also the contribution of the dynamic errors of the pressure
measurements. The internal pressure sensor of the piston
prover is connected to a pressure tap at the entrance to the
cylinder with a pressure impulse line of length about 220 mm
and an inner diameter of about 2 mm. The natural frequency
of such a fluid oscillator is about 350 Hz [16]. Taking into
account that the largest amplitudes of the pressure oscillations
are about 200 Pa and that the main frequency components fall
in the frequency range up to 40 Hz [11], the maximum dynamic
measurement error can be estimated as 2.5 Pa.
The combined uncertainty of the pressure correction factor
caused by the uncertainties of all the characteristic pressure
values is calculated as
up(εp)=u2
p1(εp)+u2
p2(εp)+u2
¯p12 (εp), (7)
where the individual contributions are assumed to be
uncorrelated. (This assumption may lead to some
overestimation of the combined uncertainty. If a certain
correlation of the pressure values p1and p2, which are
measured with the same pressure sensor, is assumed, the part
of the pressure correction model including (p2−p1)would
have a smaller contribution to the combined uncertainty.)
The contribution of the standard uncertainty of the
connecting volume to the standard uncertainty of the pressure
70 Metrologia,50 (2013) 66–72

Dynamic pressure corrections in a clearance-sealed piston prover
correction factor can be written as
uVd(εp)=1
γ
p2−p1
Pa
u(Vd)
Vm
.(8)
With the uncertainty of the connecting volume we want to
encompass the uncertainty of the estimation of the connecting
volume and the uncertainty of the model’s assumption of
homogeneous pressure changes within the connecting volume.
The estimate of Vd=200 ml is considered to be the mid-point
of a rectangular distribution with a half-width of 30 ml and
therefore the standard uncertainty is u(Vd)=30/√3 ml. The
volume of 30 ml corresponds approximately to the connecting
volume outside the piston prover.
The contribution of the standard uncertainty of the
polytropic index can be written as
uγ(εp)=−1
γp2−¯p12
Pa
+p2−p1
Pa
Vd
Vmu(γ )
γ.(9)
The incomplete knowledge about the thermal conditions in
the device, which affect the validity of the adiabatic model’s
assumption with γ=1.4 for air, is encompassed by a
rectangular distribution with a half-width of 0.1 that leads
to a standard uncertainty of u(γ ) =0.1/√3. For the
isothermal pressure correction the uncertainty contribution of
γis considered as the systematic error, which is estimated as
the difference between the isothermal and the adiabatic models
δT−A(see equation (5)).
Using the estimated values of the individual contributions,
the combined standard uncertainty of the adiabatic pressure
correction factor can be calculated as
u(εp)=u2
p(εp)+u2
Vd(εp)+u2
γ(εp). (10)
The systematic error δT−Aof the isothermal pressure correction
is included in the measurement uncertainty using the
SUMUmax method, where the absolute value of the systematic
error is added to the value of the expanded measurement
uncertainty [15,17]. The equivalent combined standard
measurement uncertainty of the isothermal pressure correction
factor can therefore be estimated as
u(εp)=u2
p(εp)+u2
Vd(εp)+|δT−A|/2,(11)
where the coverage factor k=2 is assumed.
Figure 7shows the values of the individual contributions
and the combined standard uncertainty of the pressure
correction factors for the isothermal and adiabatic models.
In the case of the isothermal model (2), the combined
standard uncertainty exceeds 0.07%. The largest contribution
is represented by the systematic error, because the adiabatic
nature of the dynamic pressure conditions is not considered.
The use of the adiabatic model (3) reduces the largest combined
standard uncertainty by more than three times, to about
0.023%.
(a) Isothermal model.
(b) Adiabatic model.
1 10 100
0.00
0.02
0.04
0.06
0.08
up(ε
ε
ε
δ
p)
uVd
(p)
| T
–
A| / 2
u(p)
–
Eq.(11)
Flow rate / sl/min
100 × Standard uncertainty
1 10 100
0.00
0.02
0.04
0.06
0.08
up(p)
uVd
(p)
u(p)
u(p)
–
Eq. (10)
Flow rate / sl/min
100 × Standard uncertainty
ε
ε
ε
ε
Figure 7. Individual contributions and the combined standard
uncertainty of the pressure correction factor.
5. Conclusions
The purpose of this paper was to present the dynamic pressure
conditions in a high-speed, clearance-sealed realization of
the piston prover and to analyse their influence on gas flow
measurements. Here, we summarize some of the most
important findings.
(i) The experimental results show that the dynamic pressure
conditions are principally of a deterministic nature.
Consequently, the uncorrected pressure dynamic effects
are not reflected mainly as a reduced repeatability of
the flow meter, but as systematic errors. The largest
time-averaged gauge pressures during the piston prover’s
operation reach values of 0.75 kPa and the largest
amplitudes of the pressure oscillations are about 0.2 kPa.
(ii) The experimental validation proves the advantage of
the polytropic/adiabatic model of the pressure correction
factor by a comparison with the originally employed,
isothermal model. The measurement uncertainty analysis
of the pressure correction factor shows that its maximum
contribution, expressed as the standard uncertainty, was
reduced by nearly three times, from about 0.07% to
0.023%, when using adiabatic instead of isothermal
corrections.
Metrologia,50 (2013) 66–72 71

J Kutin et al
In addition to the further optimization of the correction
algorithms for the pressure effects in high-speed realizations
of piston provers, one possible direction for their improvement
would be a decrease in the pressure changes. Here it
would be useful to work on reducing the amplitudes of the
pressure oscillations (at least partially related to the resonance
effects of the piston oscillator excited by the inlet flow
instabilities) as well as on reducing the magnitudes of the time-
averaged pressure changes (related to the effects of the piston
weight, the clearance-fluid damping and the outflow pressure
losses).
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