Applied Thermal Engineering 41 (2012) 111e115 Contents lists available at SciVerse ScienceDirect 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 References [1] G.A. Adebiyi, Formulations for the thermodynamic properties of pure substances, Journal of Energy Resources Technology 127 (2005) 83e87 March. [2] O. Chouaieb, J. Ghazouani, A. Bellagi, Simple correlations for saturated liquid and vapour densities of pure ﬂuids, Thermochimica Acta 424 (2004) 43e51. [3] E. Heidaryan, J. Moghadasi, M. Rahimi, New correlations to predict natural gas viscosity and compressibility factor, Journal of Petroleum Science and Engineering 73 (2010) 67e72. [4] J.R. Howell, R.O. Buckius, Fundamentals of Engineering Thermodynamics, McGraw-Hill Book Company, New York, U.S.A, 1987. [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 Water Heaters, M. Sc. Dissertation in Mechanical Engineering (In Portuguese), Faculty of Engineering, University of Oporto, Portugal. [6] R.E. Sonntag, C. Borgnake, G.J. Van Wylen, Fundamentals of Thermodynamics, John Wiley and Sons, New York, U.S.A, 1998. [7] P.S. Tow, Evidence of validity of Amagat’s law in determining compressibility factors for gaseous mixtures under low and moderate pressures, The Journal of Physical Chemistry 68 (7) (1964) 2021e2023. [8] H.W. Xiang, U.K. Deiters, A new generalized corresponding-states equation of state for the extension of the Lee-Kesler equation to ﬂuids consisting of polar and larger nonpolar molecules, Chemical Engineering Science 63 (2008) 1490e1496. Applied Thermal Engineering 41 (2012) 1 Contents lists available at SciVerse ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng 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 of the Brazilian Society of Mechanical Sciences and Engineering (ABCM). More than 500 researchers participated and 399 papers were presented in this edition. These papers are representative of the main topics presented in 72 sessions and 24 keynote lectures, and were accepted following the rigorous peer review process of the Applied Thermal Engineering. As in the previous editions of the conference, the papers focus on the fundamentals and applications associated with thermal and ﬂuid sciences. The papers represent a broad spectrum of research in the ﬁeld, covering topics such as modeling of refrigeration systems and absorption cycles, two-phase cooling for data centers and fuel cells, and performance test of tri-generation power plant, as well as nanoﬂuids for heat transfer intensiﬁcation applications. As such, this special issue reﬂects the breadth and depth of thermal science and engineering research currently conducting in Brazil. We would like to thank the Elsevier editors, particularly the Editor-in-Chief of the Applied Thermal Engineering, Prof. David Reay, for recognizing the opportunity of integrating Brazilian research into the international stage and graciously agreeing to publish this special issue. We sincerely thank all contributors for their overwhelming response to the call for papers for this special issue, and all reviewers for devoting their precious time to review the papers. Finally, we deeply appreciate the strong and consistent 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 1 1359-4311/$ – see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2012.03.031 Tel.: 020 78823232.

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