The positive displacement method for calibration of gas flow meters

Applied Thermal Engineering 41 (2012) 111e115
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Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
The positive displacement method for calibration of gas flow meters.
The influence of gas compressibility
Carlos Pinho*
CEFT-DEMEC, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 25 March 2011
Accepted 6 December 2011
Available online 13 December 2011
An easy technique to calibrate small gas flow meters is the positive displacement method, whose
application requires a simple setup and laboratory procedure. With this methodology there is an
unknown gas flow coming from a given gas source that must be known by the action of a gas flow meter.
The gas flow to be measured is sent to a reservoir with rigid walls and full of water. As gas enters the
reservoir water flows out and the amount of water exiting the reservoir in a given time interval can be
connected with the average gas flow in that same time interval. In simple terms the volume flow rate of
water leaving the reservoir is equal to the gas volume flow rate entering it. The water being incompressible, the density variation is meaningless, however the same cannot be assumed for the gas.
Considerations on the simple techniques to be used to minimize the importance of gas compressibility
are presented in the paper.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
Positive displacement
Flow meters
Calibration
Compressibility effects
1. Introduction
A simple technique to calibrate small gas flow meters is the
positive displacement method, whose application requires the
setup shown in Fig. 1. With this methodology there is an unknown
gas flow coming from a given source G, gas flow that must be
known by the action of a gas flow meter MC. The flow meter can be
a rotameter, an orifice plate, a venturi or another flow measuring
device. The pressure on the gas flow coming from the source G has
to be adjusted to a previously defined value, the working pressure
of the flow meter. This pressure must be high enough to assure that
pressure drops downstream the flow meter are unimportant for the
measurement uncertainty, or in another words the gas absolute
pressure while flowing through the gas meter must be well above
the pressure drops to be expected. This pressure is regulated by
means of the pressure regulator R and measured by pressure gauge
M. After the flow meter there is the gas flow control valve and
beyond it there is the normal gas consumer installation or alternatively the calibration setup. As the gas flow passing the flow
meter increases with the opening of the controlling valve V1 the
absolute pressure of the gas reduces, unless the pressure reducer
acts to compensate its decrease. In the case of semi-automatic gas
pressure reducers they are manually adjusted to account for pressure variations with changes in the gas flow.
* Tel.: þ351 225081747.
E-mail address: ctp@fe.up.pt.
1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.applthermaleng.2011.12.012
Measured gas flow can be sent either through valve V4, in the
case of normal operating conditions, or through valve V2 under
calibration conditions. In the present situations it is the calibration
procedure that is under analysis and then the gas flow is sent to
a reservoir with rigid walls D, inside which there is water. As gas
enters the reservoir water is pushed out of it through pipe T,
provide valve V3, the water feeding valve, is closed. The amount of
water exiting the reservoir can be easily weighted in a given time
interval thus allowing the calculation of an average water mass and
volume flow rate.
2. The importance of initial gas volume
In simple terms the volume flow rate of water leaving the
reservoir is equal to the gas volume flow rate entering it. The water
being almost incompressible, the density variation is meaningless,
however the same cannot be assumed for the gas. The gas pressure
inside the reservoir increases as the water level lowers and the
liquid must be raised to a larger relative height before leaving the
reservoir through the siphon shaped pipe T. To take into account
gas compressibility effects its pressure is continuously measured
through the manometer MU. Then, knowing the atmospheric
pressure and the gas temperature, quite often assumed equal to
ambient temperature, the thermodynamic state of the gas inside
the reservoir is known and thus the gas density may be determined.
Considering ideal gas behavior but being careful enough to
consider simple compressibility effects through the compressibility
112
C. Pinho / Applied Thermal Engineering 41 (2012) 111e115
Fig. 1. Schematic layout of the experimental setup for calibration purposes.
factor [6], a simple equation to determine the gas mass flow rate
dependent on the knowledge of the mass of water collected in
a given time period can be deduced.
As said before, the water volume flow rate V_ w can be easily
calculated, if the mass of water mw, with density rw, that leaves the
reservoir during a time interval Dt is measured
whereas the variation of liquid mass contained in the same control
_ w,
volume dmwcv is dependent upon the outlet mass flow rate m
mw
V_ w ¼
Dt rw
dVw ¼ V_ w dt
(1)
_ g and volume V_ g flow rates are
On the other end, the gas mass m
connected through the equation of state, for a given pressure pg and
temperature Tg.
V_ g
R Tg
_g
¼ Zm
Mg pg
(2)
In the above equation R is the universal gas constant, Mg is the
molecular mass of the gas and Z is the compressibility factor of the gas.
Equalizing Equations (1) and (2) and considering that Tg ¼ Ta and
that pg ¼ pa þ Dp, where Ta and pa are the ambient temperature and
pressure and Dp is the pressure differential measured through the
manometer MU,
_g ¼
m
mw 1 Mg ðpa þ DpÞ
rw Dt Z R
Ta
(4)
or,
_ g dt
d Vg rg ¼ m
(6)
As the liquid is incompressible, the variation of volume of water
inside the reservoir dVw is given by
(7)
If the reservoir has rigid walls,
Vw þ Vg ¼ constant0dVw þ dVg ¼ 0
(8)
and introducing Equation (8) into Equation (7),
dt ¼
dVg
V_ w
(9)
Calculating the derivative of Equation (5) and combining the
result with Equation (9),
dV
rg dVg þ Vg drg ¼ rg V_ g _ g
Vw
(10)
and reworking this last equation,
(3)
It is however important to analyze how rigorous is the
assumption of equality of volume flows between the gas entering
the reservoir and the liquid leaving it.
The gas filling and liquid leaving the reservoir is to be analyzed
in small time steps dt. It is assumed that the gas is not soluble in the
liquid and that the process follows a uniform regime flow. So the
gas mass variation inside the reservoir dmgcv, i.e., the control
volume under analysis, is dependent upon the inlet gas flow rate,
_ g dt
dmgcv ¼ m
_ w dt
dmwcv ¼ m
(5)
drg
rg
¼
_
Vg
dVg
1
_
Vg
Vw
(11)
which can be integrated between the following limits,
Vg ¼ Vg1
Vg ¼ Vg2
rg ¼ rg1
rg ¼ rg2
and consequently,
¼ rg2=r ,
pg2=
pg1
g1
(12)
if
the
#
"
ln pg2 =pg1
_
_
Vg ¼ Vw 1 þ ln Vg2 =Vg1
gas
evolution
is
isothermal,
(13)
It can be immediately deduced that if Vg1 /0, i.e., at the start of
the calibration procedure the reservoir is full of water, V_ g /V_ w .
C. Pinho / Applied Thermal Engineering 41 (2012) 111e115
Then to minimize the calibration or measurement uncertainty of
the gas volume flow entering the reservoir there are two possibilities, one is to start the calibration procedure with the reservoir full
of water so that Equation (13) can be safely used and the other is to
know what is the initial value of the gas, i.e., the value of Vg1 s 0 for
each calibration step. Of course the simplest experimental procedure will be to fill up the reservoir of water at the beginning of each
calibration step.
Fig. 2 shows what happens when there are no precautions to
minimize the initial gas volume inside the water reservoir during
the calibration procedure.
Table 1
Properties of the gases under analysis.
Gas
u
Reference for u
Tc (K)
pc (Pa)
Air
CO
CO2
CH4
C3H8
C4H10
N2
O2
0.078
0.051
0.225
0.011
0.1524
0.2
0.038
0.0222
[1]
[2]
[8]
[8]
[2]
[8]
[8]
[2]
132.6
132.8
304.13
190.56
369.85
425.13
126.19
154.58
3.77
3.49
7.38
4.60
4.25
3.80
3.40
5.04
3. The importance of gas compressibility factor
Another aspect to be accounted for is the need to consider the
compressibility factor of the gas flowing in the flow meter undergoing the calibration procedure. Going back to Equation (3) it is
necessary to calculate the error obtained when for the sake of
simplicity it is assumed, for the gas under consideration, a value of
Z ¼ 1 instead of the correct value. So, the correct gas mass flow
rate is,
_g ¼
m
mw 1 Mg ðpa þ DpÞ
rw Dt Z R
Ta
(14)
and the approximated gas flow rate is,
_ gap ¼
m
mw Mg ðpa þ DpÞ
Ta
rw Dt R
(15)
The relative calibration error obtained through the use of
Equation (15) instead of Equation (14) is determined by,
_g ¼
Dm
_ g m
_ gap
m
¼ ðZ 1Þ
_g
m
(16)
To evaluate the result of such approximation to the perfect gas
behavior, errors for some gases that usually are used in the laboratory experiments are now quantified. The gases under evaluation
are presented in Table 1.
For the calculation of the compressibility factor, the following
equations were used [1],
113
Z ¼ 1þ
b1 ðTr Þ
Tr
pr
106
106
106
106
106
106
106
106
(17)
is a truncated version of the pressure series virial form of the p-v-T
equation of state, and b1(Tr) is given by
0:422
b1 ðTr Þ ¼ 0:083 1:6 þ
Tr
!
0:172
0:139 4:2 u
Tr
(18)
where Tr ¼ T=T and pr ¼ p=p are the reduced temperature and
c
c
pressure, while Tc and pc are the critical temperature and pressure.
Parameter u is the acentric factor [4], Table 1.
Figs. 3 to 6 present calibration error values when the
compressibility factor Z, calculated according to Equations (17) and
(18) and using data from Table 1, is replaced by 1 for the several
gases under analysis. In such circumstances such gases are being
considered as having ideal behavior at temperatures in the 0e45 C
range. The figures show that these errors are meaningless for air,
CO, CO2, N2, O2 and even for CH4, but are above 1% for propane
(C3H8) and butane (C4H10). In Fig. 3 there is a comparison for 8 gases
at 1 atm and for temperatures going from 0 to 45 C. The worst
situation is for propane and butane. Figs. 4 and 5 show what
happens for air and methane, in the 0.8e1.2 atm range, and it is
evident that both gases can be assumed as ideal. The same happens
for oxygen, nitrogen, carbon monoxide and carbon dioxide, whose
plots are similar to that for air.
For propane, and above all for butane, the assumption of ideal
gas behavior leads to large errors in the calibration procedure,
Figs. 5 and 6.
3
10
2.5
2.25
2
Error [%]
Mass flow rate - commercial propane [kg/h]
2.75
1.75
1.5
Air
O
1
2
N
2
CO
CO
1.25
2
CH
4
1
0.1
CH
3
0.75
8
C4H10
0.5
0.25
0
2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Tension at differential pressure transducer [mV]
Fig. 2. Calibration plot for an orifice plate flow meter. Gas being measured, commercial
propane [5]. Gray symbols Vg1s0. Black symbols, Vg1 /0.
0.01
0
10
20
30
Temperature [ºC]
40
50
60
Fig. 3. Calibration errors when real gases are supposed to have ideal gas behavior at
1 atm.
114
C. Pinho / Applied Thermal Engineering 41 (2012) 111e115
0.07
5
C H - 1 atm
0.06
0.01
2
40
50
4
3.5
2.5
30
10
C H - 1.2 atm
0.02
20
10
C H - 1.1 atm
4
3
10
4
4
0.03
0
10
C H - 0.8 atm
0.04
0
10
C H - 0.9 atm
4
Error [%]
Error [%]
0.05
4
4.5
Air - 1 atm
Air - 0.9 atm
Air - 0.8 atm
Air -1.1 atm
Air - 1.2 atm
1.5
60
0
10
20
30
10
40
50
60
Temperature [ºC]
Temperature [ºC]
Fig. 6. Calibration errors for butane at several pressures and temperatures.
Fig. 4. Calibration errors for air at several pressures and temperatures.
So, although the positive displacement method is a very
simple one to calibrate gas flow meters, some precautions need
to be taken when working with higher molecular weight gases,
and the true value of the gas compressibility factor must be
adopted.
Fig. 7 represents the result of the calibration of an orifice plate
flow meter for commercial butane and the corresponding correction carried out according to Equation (16). Although the composition of commercial products changes along time and the
compressibility factor Z should account such effect, in the present
explanation the calibration curves shown refer to the uncorrected
and the corresponding corrected values using only the compressibility factor for pure butane. The data in Fig. 7 were obtained for
a day with an ambient temperature of 22 C and pressure of
1.007 bar.
The rigorous calibration of a commercial gas mixture is very
complex because of typical composition variations along the year
and also when the gas comes for a gas cylinder. In this last situation
initially lighter hydrocarbons will be released whereas as time goes
by the percentage of higher molecular weight hydrocarbons will
raise, creating further difficulties for the assessment of the correct
compressibility factor.
A simple approach to determine the mixture compressibility
factor is to use an average molar composition of the commercial gas
mixture and determine the compression factor based on the
Amagat’s Law [7],
Z ¼
X
Xi Z i
(19)
where Xi is the molar fraction of given component of the mixture
having a compressibility factor of Zi. This might be a crude
approximation for a situation when the gas mixture composition
is well known, as shown for example by [3]; but nonetheless
small compared with the uncertainties on the composition of
a commercial gas as its reservoir is being emptied or even
according to fluctuations in the commercial gas composition
along the seasons of the year. These composition changes are
necessary to account for the influence of ambient temperature on
the saturation pressure of a liquefied petroleum gas inside its
reservoir.
3.5
Mass flow rate commercial butane [kg/h]
0.3
CH - 1 atm
4
CH - 0.9 atm
4
0.25
CH - 0.8 atm
Error [%]
4
CH - 1.1 atm
4
CH - 1.2 atm
4
0.2
0.15
0.1
3
2.5
2
1
2500
0
10
20
30
40
50
Temperature [ºC]
Fig. 5. Calibration errors for methane at several pressures and temperatures.
60
No correction
With correction
1.5
3000
3500
4000
4500
5000
5500
6000
Tension at differential pressure transducer [mV]
Fig. 7. Calibration plot for an orifice plate flow meter. Gas being measured, commercial
butane.
C. Pinho / Applied Thermal Engineering 41 (2012) 111e115
Because of such uncertainties the correction applied in Fig. 7 is
only based on the compressibility factor of butane and consequently Equation (19) was not used and it is here proposed as
a pragmatic recommendation based on [7].
4. Conclusions
The positive displacement method is a very simple methodology
to calibrate gas flow meters. However some simple precautions
need to be taken to minimize calibration errors.
First of all, to minimize the gas compressibility effect the initial
gas volume in the water and gas collecting tank must very small.
Secondly, the real value of the gas compressibility factor must be
taken into account, primarily when working with higher molecular
weight gases, as the displacement from ideal gas conditions can
lead to serious calibration errors.
Using these very simple experimental precautions can lead to
successful calibration procedures.
115
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[5] R.M.B.R. Pilão, 1996, Estudo do Comportamento Térmico de Esquentadores
Domésticos a Gás e Study on the Thermal Behavior of Gas Burning Domestic
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[7] P.S. Tow, Evidence of validity of Amagat’s law in determining compressibility
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Editorial
This special issue of Applied Thermal Engineering contains 12
selected papers presented at the 13th Brazilian Congress of Thermal
Sciences and Engineering – ENCIT-2010 held in Uberlandia, Minas
Gerais, located in the southeast of Brazil, December 05–10, 2010.
This conference was organized by the Faculty of Mechanical Engineering of the Federal University of Uberlandia, under the auspices
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We would like to thank the Elsevier editors, particularly the
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help from all members of the organizing committee of ENCIT-2010.
The next ENCIT will be held in Rio de Janeiro, Brazil, November
18–22, 2012.
Enio Pedone Bandarra Filho*
Energy and Thermal System Laboratory,
Faculty of Mechanical Engineering,
Federal University of Uberlandia,
Av. Joao Naves de Avila, 2121 Uberlandia, MG, Brazil
Dongsheng Wen1
School of Engineering and Materials Science,
Queen Mary University of London,
Mile End Road, London E1 4NS, UK
E-mail address: d.wen@qmul.ac.uk
* Corresponding author. Tel.: þ55 (34) 32394022.
E-mail address: bandarra@mecanica.ufu.br (E.P. Bandarra Filho)
Available online 24 March 2012
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