THREE METHODS TO DETERMINE THE DENSITY OF MOIST AIR DURING MASS COMPARISONS

Transcription

1 THREE METHODS TO DETERMINE THE DENSITY OF MOIST AIR DURING MASS COMPARISONS A. PICARD, H. FANG + Bureau International des Poids et Mesures Pavillon de Breteuil F SÈVRES FRANCE Résumé Le présent travail a pour but de comparer trois méthodes utilisées pour la détermination de la masse volumique de l air : l application de la formule CIPM 81/91, la mesure directe grâce à deux artefacts de volume très différent et une méthode réfractométrique en exploitant la bonne corrélation entre l indice de réfraction de l air et de sa masse volumique. Les résultats obtenus montrent une bonne cohérence sur la sensibilité des trois méthodes. Les masses volumiques de l air déterminées sont en accord à kg m -3. La méthode utilisant les artefacts donne une incertitude relative sur la masse volumique de l air de Abstract The aim of our present work is to compare the performance of the following three methods used for air density determination: application of the CIPM-81/91 formula, direct determination using air buoyancy artefacts having a large volume difference, and refractometry, exploiting the high correlation between air density and air index of refraction. The response characteristics for the three methods are comparable and the agreement among the air density determinations is within kg m -3. The relative uncertainty obtained on the air density determination by using the air buoyancy artefacts is about Introduction The comparison of mass is usually carried out in air by means of gravimetric measurements. In air, a buoyant force which may be explained by Archimedes principle acts on the artefact; this force is proportional to the product of the air density and the volume of the artefact. During accurate mass comparisons it is necessary to take into account this force by applying an air buoyancy correction to the apparent mass measured. For the comparison between a stainless steel 1 kg mass against a 1 kg Pt-Ir mass standard this differential correction is about 95 mg, which is due to the large volume difference between the two masses. Generally, the volumes of the artefacts are well known by hydrostatic weighing to a relative uncertainty of a few parts in 10 6, which corresponds to an uncertainty of about 0.4 mm 3 in a kilogram of stainless steel. For the case where the air density is determined with a relative uncertainty of , the uncertainty for the comparison between one stainless steel 1 kg mass standards and one Pt-Ir mass standard is about 10 µg. The comparison of this uncertainty with the standard deviation of accurate commercial balances (0.1 µg) shows that the evaluation of the air density is a limitation on mass comparisons. An initial study has already been carried out in the context of Euromet project [1] based on the air buoyancy artefacts. The reproducibility of the difference between results obtained using the formula and those with air buoyancy artefacts was kg m -3. Encouraged by this result and to complete the study, a Fabry-Perot cavity, the heart of the refractometer [2], was placed inside the balance case in order to have an optical air density determination. In this manner we are able to determine simultaneously the air density inside the FB2 balance case. The objective is to follow the air density evolution in the short and long term with three independent methods. Basic principles CIPM 81/91 air density determination From the equation of state of a non-ideal gas and the experimental conditions the density of moist air can be evaluated using the CIPM 81/91 formula [3,4], pm M ρ = a x (1 a 1 ) ZRT M (1) a where the quantities and units are: p : Pressure [Pa] T : Thermodynamic temperature = t [K] t : Air temperature [ C] x v : Mole fraction of water vapour M a : Molar mass of dry air [kg mol -1 ] M v : Molar mass of water = [kg mol -1 ] Z : Compressibility factor R : Molar gas constant = * [J mol -1 K -1 ] + Hao Fang: Research fellow * The most recent CODATA recommendation [5] is only 5 ppm below the value adopted for the CIPM 81/91 formula.

2 The molar mass of dry air, M a, is an average molar mass, the calculation of which is based upon the molar masses of various constituents and their respective mole fractions (neglecting, however, those which are present only in trace amounts): M a = kg mol -1. (2) The mole fraction of carbon dioxide must be measured to obtain a more accurate value of the molar mass of dry air: M a = [ (x co )]10-3 kg mol -1. (3) This equation treats the mole fraction of O 2 and CO 2 as correlated due to processes of combustion, respiration, photosynthesis, etc. The mole fraction of water vapour x v in moist air is not measured directly but is determined from the relative humidity h or from the temperature t d [ C] of the dew point. Because moist air does not behave as a perfect gas, a correction factor f called the enhancement factor must be introduced, which is a weak function of temperature and pressure: p ( t) x h f( pt, ) sv v = p (4) p ( t ) x f( pt, ) sv d v = d p (5) where p sv (t) and p sv (t d ) are the saturation vapour pressure at the air temperature t and at the dew point temperature t d. By measuring the air temperature, pressure, dew point temperature and mole fraction of carbon dioxide and from the relations (1) to (5), an evaluation of the density of moist air required for the air buoyancy correction can be made. Refractometry method Our colleagues at the BNM/INM-CNAM (France) have demonstrated that changes in air density can be determined with good precision using an optical method based on the high correlation between the air density and the air index of refraction. Specifically, the relation between ρ, the density of air, and n, the refractive index of air is: ρ = R'(n 1) 3 2 (6) where the ratio R' is called the specific refraction or the refractional invariant. Although the correlation is not exact (R' varies slightly as a function of the composition of air and the local atmospheric conditions), it is sufficiently close that a refractometer may be used to follow changes in air density within the limits of excursion of environmental parameters. The introduction of this new instrument allows one to monitor simply and independently the changes of the density of air. In order for R' to be treated as a constant (relative change less than 10-4 ), maximum individual variations are as follows (with ppm representing parts per million): pressure: ± 3500 Pa, temperature: ± 0.6 K, relative humidity: ± 1.8 % mole fraction of carbon dioxide: ± 800 ppm. The combined variations of these individual parameters give a limit to the air density variation of roughly ± kg m -3. With the new refractometer developed at the BIPM, n is determined by a simple ratio of laser frequencies: n= v (7) where v is the laser frequency locked to one transmission peak of the interferometer under vacuum and a is the frequency locked to the same peak of the interferometer placed in air. The values of v and a are measured, using a heterodyne technique, by comparison with a frequency reference. After preliminary calibration in vacuum, the air index of refraction is measured in real time by an optical beat-frequency measurement. Air buoyancy artefacts method The method is based on the weighing of two artefacts having the same mass and the same surface area but with very different volumes. Two weighings are necessary to determine the air density, one in air and one in vacuum as show the following equations. The weighing in air gives with: a m air = e 1 e 2 + ρ(v m1 V m2 ) (8) e 1 and e 2 : balance readings in air of m 1 and m 2 Vm 1 and Vm : volumes of m1 and m 2 2 ρ: air density and for the weighing in vacuum m vacuum = e 3 e 4 (9) where e 3 and e 4 are the balance readings in vacuum of m 1 and m 2, respectively. Taking into account the adsorption of water vapour, and if the two artefacts have same surface properties (alloy, roughness, cleanliness), then the relation between the difference of mass in air and in vacuum is given by: m air = m vacuum + σ S (10) where S is the surface difference between the two artefacts and σ the mass of adsorption per unit of area. Finally, the air density can be deduced from (8) to (10) as follow evacuum ( eair σ S) ρ = (11) V with e air = e 1 -e 2 e vacuum = e 3 -e 4 V = Vm 1 - V m 2 In this method there is a great interest in minimizing the surface area difference between the two artefacts in order to achieve a maximum reduction in the surface effects (gas adsorption or outgassing) on the artefacts.

3 Balance Experimental Details The FB2 balance, designed and constructed at the BIPM, is a double-pan servo-controlled balance based on the use of a flexure strip [6]. This balance is designed to compare up to eight 1 kg standards by substitution on one pan, the second pan holding a counterweight tare. The balance is fully automated and enclosed in a chamber suitable for vacuum operation. The full range is about 200 mg and the standard deviation of the balance is generally within 0.1 µg. To reduce air convection, any heat sources inside the enclosure are minimized. Optical detectors (diodes and photodiodes) used to control the exchanger positions are placed outside the balance case. Excellent conditions of air stability are achieved inside the balance case. Typically, for a period of 15 hours (time necessary for a series of comparisons), the temperature inside the weighing chamber drifts within 10 mk and the air pressure varies less than 5 Pa. The vertical thermal gradient inside the balance housing is less than 30 mk and the dew point temperature variation is within 3 mk (see below). A vacuum of 0.01 Pa can be achieved inside the balance case ( 1m 3 ) using an oil-free pump system composed of a magnetic-bearing turbo-molecular pump in series with a scroll primary pump. To change the relative humidity inside the balance case, a chemical membrane pump is used (in closed circuit) with vapour from a bottle of distilled water or silica gel (dehydrating agent) depending on the humidity desired for the study. In addition, the air can be changed by modifying the atmospheric pressure by several thousand pascals inside the enclosure, using the same chemical pump. Refractometer A heterodyne refractometer, similar to the BNM/INM- CNAM one, was developed at the BIPM. To recall the principle, the main element is a double plane-plane Fabry- Perot interferometer both cavities of which are illuminated independently with a DBR (Distributed Bragg Reflector) laser diode, tuneable in frequency. The shorter cavity allows unambiguous identification of the transmission peak of the longer one, to which the laser frequency is servolocked. The laser frequency is determined via a heterodyne comparison with a second laser locked to one hyperfine component of the rubidium transition. Contrary to classical refractometers, after preliminary calibration under vacuum, no part needs to be evacuated during operation and the air index of refraction is determined in real time by an optical beat frequency measurement. For the study reported here, the Fabry-Perot interferometer was placed inside the FB2 balance. It is linked to the rest of the refractometer (optical and electronic servo-control), which are located in a laser laboratory, by using a mono-mode optical fibre and two electrical cables. With such a device, changes in the index of refraction smaller than could be detected which, at standard atmospheric pressure and temperature, corresponds to a relative variation in air density of Devices used for ambient parameter measurements The balance is equipped with a dew-point gauge, pressure gauge, CO 2 gas analyser, air temperature thermometers and few thermocouples. Air temperature measurement The temperature is obtained by reading a commercial bridge with a resolution equivalent to 0.1 mk, which determines the ratio between a 25 Ω platinum resistance thermometer and a 10 Ω standard resistance. The result is corrected for the self-heating effect and the thermometer is placed as close as possible to the test mass at the mid-height of the mass. Calibrations of the thermometer in terms of ITS-90 were done before and after the study. Atmospheric pressure measurement The pressure is measured by using a transducer gauge with a resolution of 0.1 Pa. This instrument is calibrated in situ each month against the primary BIPM manobarometer. An air column correction is applied to the pressure reading by taking into account the elevations of the gauge, the artefacts and the interferometer. Similar corrections are, of course, applied to the calibration of the gauge. Mole fraction of carbon dioxide measurement The CO 2 concentration measurement is carried out by using an Infrared Gas Analyser with 1 µmol/mol resolution (1ppm). The instrument is calibrated in situ just before each measurement (after closing the enclosure) by adjusting the zero and the span at 900 ppm and checking the linearity at 600 ppm. For the span calibration, a mixture of %-pure carbon dioxide and %-pure nitrogen were used at 900 ppm and 600 ppm concentration. For zero calibration we used a bottle of %-pure nitrogen with less than 1 ppm of carbon dioxide. Humidity measurement Measurement of humidity inside the enclosure is made by using a commercial airflow dew point temperature instrument with a resolution of 0.1 K. The calibrations in situ against our dew point reference instrument were made before and after the study. During the weighing a capacitance hygrometer is used in order to know the air humidity variation as a function of time. This probe was calibrated in October 1998 against four saturated salt solutions and in situ against our dew point instrument in June Located close to the test mass this probe is used only to evaluate the drift within a series of weighings. Thermal Gradient measurement Several copper-constantine thermocouples are used to determine the horizontal and vertical gradient inside the balance case. A low thermal EMF switch box is used to select and reverse the thermocouple desired. The reference junctions of the thermocouples are placed inside a large block of copper set on the baseplate of the enclosure; the gradient is obtained directly by reading the EMF of each thermocouple and assuming its coefficient to be 40 nv/mk.

4 One thermocouple is spiralled around the 25 Ω platinum resistance thermometer and another one is placed in the interferometer. A correction is applied to the temperature to take into account the thermal gradient between the artefacts and the interferometer locations. Air buoyancy artefacts Two 1 kg stainless-steel artefacts were polished (Ra 0.01µm) and the masses were adjusted at the BIPM in early Designated Cc (hollow cylinder) and Cp (solid cylinder), these masses have the same nominal surface area (194 cm 2 with a small surface difference close to 1.7 cm²) but the volumes are quite different (V Cc = 207 cm 3 and V Cp = 124 cm 3 ). In this paper, the volumes were estimated by the airdrostatic method, which gives a high correlation between the air density and the volumes. For the Cc artefact the outer diameter is equal to its height to minimize the surface area. The advantage of having artefacts with the same surface area is that the surface effects on the mass difference between the two artefacts are minimized. The magnetic properties were measured, yielding a magnetic susceptibility χ = and a permanent magnetization lower than 0.1 µt. The magnetic properties at the welds (Cc) are indistinguishable from the solid material. The masses were cleaned late in April 2000 with mixture of alcohol and ether and rubbed using special tissue. Protocol Measurements were made alternately in air and in vacuum. The methods using air buoyancy artefacts and refractometry both require measurements from time to time in vacuum to control possible drift of the mass difference or of the frequency locked to the interferometer. During the weighing in vacuum (about one week for each set) the artefacts were measured against a 1 kg Pt-Ir mass standard {691}, the latter being calibrated in air against our 1 kg Pt-Ir mass reference {77} which always remains in air. In order to minimize the correction of the residual pressure in vacuum on the artefacts (2 µg/pa), active pumping was used to obtain a residual pressure less than 0.01 Pa. Unless otherwise specified, the masses were just dusted with a brush before putting them into the balance. Measurements in air were carried out under different atmospheric conditions. For each set of measurements (about 10 days) where the balance case remains closed, the mole fraction of carbon dioxide is considered constant. It was measured just after closing the balance case. Almost 20 series of comparisons were carried out during each set. During the weighing of the artefacts in air, the optical beat frequency and the environmental parameters were measured continuously to compare the air densities obtained simultaneously by the three methods. 1 ) The air buoyancy artefacts were measured against a 1 kg Pt-Ir mass standard {77}. Using the relation (11), the mass differences obtained in a series of measurements were used to calculate the air density at the artefacts location. 2 ) The input parameters necessary for air density determination using the CIPM 81/91 formula were measured as follows. The absolute temperature, pressure and humidity (using a capacitance humidity sensor) were measured every 2 minutes during the weighing. The absolute dew point temperature was determined from measurements before and after the series by using the dew point meter. Finally, the air density evaluation was calculated by using the recommended formula. 3 ) At the same time, the beat frequency associated with the refractometry method was measured continuously in the laser room by independent software. The air index of refraction, at the interferometer, can be deduced to determine the air density by using the relation (6). To compare the air densities determined by the three methods at the same location in the balance, it is necessary to take into account the air density gradient (temperature and pressure) between the location of the interferometer and that of the artefacts. This is obtained via the CIPM formula by using input parameters for the two locations. The mass difference in air between the two artefacts, the optical beat frequency as well as the environmental parameters were fitted using least squares in order to evaluate the measurement values at the weighing times and at the midpoint of the series. Results Four periods of weighing were made alternately in air and in vacuum. In total, 60 and 144 series of weighings were carried out in vacuum and in air, respectively. The air density varied from 1.15 kgm -3 to 1.21 kgm -3 in order to evaluate the linearity among the three air density determination methods near atmospheric pressure. For comparison of the three methods, the air density determined by using the CIPM formula is taken as the reference. For the first series of each set of weighings in air, the factor R' (relating the air index of refraction and air density) was constrained to have the same air density given by the CIPM formula. For this reason, the refractometry method is just relative measurement which may be used to follow the evolution of air density for each weighing set, provided the air density variation is within the limit mentioned above in the refractometry section. During a series of weighings Figure (1) shows an example of the results obtained during a series of comparisons. The balance was closed five days before these measurements began. The air density evolution inside the balance case during 15 hours of weighing was within kg m -3.

5 Air density / (kg.m -3 ) Fig 1: Comparison among the three air density evolution methods as a function of time during a series of weighings. The data determined are fitted by using the least squares method. As can be seen, the evolution of the air density was determined with a satisfactory coherence among the three methods, within kg m -3. This indicates that the responses of each method to small changes are essentially equivalent. During a set of comparisons The following figures show the evolution among the three methods under two different conditions for air stability during a period of about two weeks. a). Air density difference / ( kg.m -3 ) b) :00 18:00 20:00 22:00 0:00 2:00 4:00 6:00 8:00 10:00 5E-5 4E-5 3E-5 2E-5 1E-5 0E+0-1E-5-2E-5-3E-5-4E-5-5E kg.m -3 Formula air density determination. Data of artefacts air density determination. Refractometry air density determination. Time / (H:Min) 17/07/ /07/ /07/ /07/ /08/ /08/ E-4 Artefacts method - CIPM method. Refractometer method - CIPM method. 5x10-5 kg.m -3 Balance case closed on 17 July at 1 PM. Time / (day/month/year) Artefacts method - CIPM method. Refractometer method - CIPM method. p = 150 Pa t = 0.30 C Rh % = 1.1% ρ = 7x10-4 kg.m -3 Fig 2: Evolution of the air density (during a set of weighings after closing the balance case) obtained with the artefacts and the refractometer methods against the CIPM formula method. a): Measurements achieved with a small air density evolution. b): Measurements achieved with a large air density evolution. In Figure (2a) the air density evolution was about kg.m -3 and the agreement among the three determinations was within kg m -3. No significant drift was observed for the different methods. This demonstrates a good repeatability of measurements when changes in the air density are within the above-specified limit. In the case of figure (2b), the individual input parameters to the CIPM formula varied less than in the previous set, but the resulting value of ρ varies much more ( kg m -3 ). That explains the divergence of air density determination using the refractometer and confirms that the limit of the applicability of this method is of the order of Note, however, that the results given by the artefact method remain stationary during this set of measurements. Overall results The following table summarizes the results obtained during eight months. The values given in the table represent the mean of results for each period of comparisons. For the refractometer, after each new set of air conditions (new set of measurements) the coefficient R' was calibrated against air density calculated by using the CIPM formula. For the first period of weighing, artefact comparisons were not successful because the temperature stability inside the balance case was not sufficient (the air conditioning in the laboratory was out of order during this period, and the room temperature rose to about 25 C). Therefore, only the CIPM and the refractometry methods were compared for the first period of comparisons. n Humidity (%) Air density (kg.m -3 ) Air density differences Artefacts- CIPM (kg.m-3) Air density differences Refractometry-CIPM (kg.m -3 ) Air density difference / ( kg.m -3 ) 1.0E-4 5.0E-5 0.0E+0-5.0E-5-1.0E-4-1.5E-4 1x10-4 kg.m -3 Balance case closed on 5 January at 9 AM. 07/01/ /01/ /01/ /01/ /01/2001 Time / (day/month/year) p = 36 Pa t = 0.2 C Rh % = 0.9 % ρ = 1.2x10-3 kg.m -3 For a large variation in air conditions (0.04 kg m -3 ) measured during the eight months, the air buoyancy artefacts method and the CIPM method agreed to within kg m -3 with a standard deviation of kg m -3. The results for the refractometry method and CIPM determination give satisfactory agreement, within a few parts of 10-6 kg m -3, provided the variation of air density is less than kg m -3 inside a set of measurements. For the last period, where the air density variation was as shown in figure (2b), satisfactory agreement was also obtained for the artefacts method ( kg m -3 ), but a drift was

6 observed in the refractometer measurements, which explained the large standard deviation. Uncertainties The uncertainty on the air density determination when using the CIPM formula comes mainly from the formula itself and from the type B uncertainty of dew point measurement. The relative combined standard uncertainty obtained is ,which is difficult to reduce. Such an uncertainty can be reached if the calibrations of the accurate instruments are performed in situ and frequently. The relative combined standard uncertainty for the air buoyancy artefacts is The main contribution comes from the uncertainty in the volume difference between the two artefacts. This particular result illustrates the benefit gained by employing the artefacts method as compared with that using the CIPM formula. For the refractometry method, the uncertainty of the air density determination is associated with the factor relating the air density and the air index of refraction. The absolute combined uncertainty of the air index of refraction determined by our refractometer is a few parts times In our case, the refractometer was just used for relative control with a relative uncertainty of the order of 10-9 on the air index, corresponding to on the air density. This arises from the variation in the polarization of the optical fibre used, which did not incorporate technology to maintain constant polarization. Conclusions During this experiment (eight months), we have compared experimentally the three methods for air density determinations inside the FB2 balance case. To perform this study, accurate parameter measurement instruments, novel refractometer and high-quality mass artefacts weighed with the high sensitivity FB2 balance were used. By changing the atmospheric pressure or the relative humidity, the air density was varied from 1.15 kg m -3 to 1.21 kg m -3. Three air density methods were compared at the high level of a few parts in 10-5 kg m -3. A relative combined standard uncertainty of was obtained by calculation using the CIPM formula. It is difficult to reduce significantly this uncertainty with this method owing to limits imposed by the formula itself ( ). For the air buoyancy artefacts method, hollow and solid cylinder artefacts having the same surface area but with very different volumes were machined. The apparent mass difference in vacuum and in air between the two artefacts was measured using the FB2 balance with a reproducibility better than 0.5 µg. A relative combined standard uncertainty in the air density of the order of 10-5 was obtained, a result achieved mainly from accurate knowledge of the volume difference between the two artefacts. This uncertainty is smaller by a factor of 5 than that determined by the CIPM formula method. The difference between the two absolute methods observed in our case is kg m -3. We note that an airdrostatic determination for the volume difference of the artefacts results in a correlation between the CIPM formula determination and that using air buoyancy artefacts. This dependence will be eliminated by using a hydrostatic determination. The difference between the two absolute methods can then be expected to change from these preliminary results. The optical method, based on a compact heterodyne refractometer, was used to follow the air density simultaneously with the other two methods. It is a very sensitive, independent method that does not induce any air perturbations. The other advantage to using the refractometer, true also for measurements employing artefacts, is that this method takes into account the global changes of the ambient parameters. In our application, the evolution of the air density was detected with satisfactory agreement at a few parts in 10-6 compared with the two other methods provided the air condition remained within the limits acceptable for refractometry. The technique applied in our instrument shows satisfactory stability in both the short and long term. The fact that only a beat-frequency is measured is a major advantage of this instrument. For air index of refraction measurements, an improvement can be made by using a single-mode fibre with constant polarization and by better knowledge of the compressibility correction of Zerodur used for the cavity. To conclude, the accuracy of the CIPM formula is validated by the result obtained for the mass difference between the artefacts in air and in vacuum, which is about 1.5 µg with a combined uncertainty of 6 µg. The comparison among the three methods, two absolute and one relative, shows that in the short term the response characteristics and the repeatability of each method are equivalent within few parts in 10-6 kg m -3. For the long term the two absolute methods gave satisfactory agreement within kg m -3 with a reproducibility of kg m -3. For the balance, which has enough places for additional masses, the absolute air buoyancy artefacts could be used directly to determine the air density with low cost and high accuracy. The methods could also be used to check from time to time the validity of the application of the CIPM formula. The optical method could be used, in combination with one of the absolute methods, to follow with very high sensitivity the small evolution of the air density inside a balance case during a weighing. Acknowledgements We thank Dr. R. S. Davis for his advice and discussions on this field, Prof. P. Juncar and the BNM/INM-CNAM for the technical assistance on the optical part of this study, and the staff of the BIPM workshop for manufacturing the air buoyancy artefacts. References [1] Euromet Mass & Derived Quantities Project 144, Measurement of air density using specially designed masses. [2] H. Fang and P. Juncar, A new simple compact refractometer applied to measurements of air density fluctuations Review of Scientific Instruments, 70, pp , 1999.

7 [3] P. Giacomo, Equation for the Determination of the Density of Moist Air (1981), Metrologia, 18, pp.33-40, [4] R. S. Davis, Equation for the Determination of the Density of Moist Air (1981/91), Metrologia, 29, pp.67-70, [5] P. J. Mohr and B. N. Taylor, CODATA Recommended Values of the Fundamental Physical Constants: 1998, Journal of Physical and Chemical Reference Data, 28, pp , [6] T. J. Quinn, The beam balance as an instrument for very precise weighing, Measurement Science and Technology, 3, pp , 1991.