The calibration of a pressure balance involves determining the values of A0 and the pressure distortion coefficient λ that define the effective area AP of the piston-cylinder assembly as a function of pressure. There exist various approaches to determine and estimate the uncertainties of these parameters, with results that may differ significantly. The purpose of this project is to verify numerically the performance of different methods by applying simulated data sets.
The values of the effective area parameters calculated by each participant including the evaluation of their uncertainties will be compared. The results will then be used to evaluate the appropriateness of the different methods and to make recommendations.
All data which are usually recorded during a cross-float and their associated uncertainties for two piston cylinder assemblies: a gas calibration in 7 MPa and an oil calibration in 500 MPa.
1- The effective area AP of the calibrated piston cylinder assemblies as a function of pressure.
2- The effective area parameters of the calibrated piston cylinder assemblies i.e. A0 and the pressure distortion coefficient λ including their uncertainties.
3- The variance V(ÂP) of an estimate of AP.
Final Report 2012-07-09
The project has been completed and the report can be downloaded here>>.
Cross-floating is the most common method used for calibrating a pressure balance. When the smallest uncertainty is required, it is recommended to determine the effective areas, of the piston-cylinder assembly under test as a function of pressure. Then, the values of the effective area at null pressure and, if significant, the pressure distortion coefficient are calculated. There exist various approaches to determine these two parameters and estimate their uncertainties, which leads to results that may differ significantly. The purpose of project was to verify numerically the performance of different methods used by NMIs by applying simulated data sets.
For the types of data provided for this project, the most used method, based on the ordinary least squares, gives the less robust results. This method can be significantly improved by not taking into account the lowest pressure points that have the highest uncertainties. The WLS method takes into account more formally the heteroscedasticity of the effective areas and gives results suitable for this type of analysis. The GLS method, which takes into account the uncertainties and the correlation of the effective areas, is more complex but gives more accurate estimates of the parameters. However, attention must be paid to the associated variance-covariance matrix. An interesting and robust methodology is used by PTB. This method assumes the existence of an additional unknown force F, which leads to a model whose parameters are determined by the ordinary least squares method. This approach gives in some cases the best unbiased results. This is a good alternative to the previous ones, however, it requires determining whether the additional force is intrinsic to the calibrated pressure balance or results only from the conditions of the performed calibration.
The estimates of the effective areas determined by all participants agree with the theoretical values within their standard uncertainties. On the other hand, there are several orders of magnitude of difference in the determination of the uncertainties of the parameters. More generally, the results, although consistent with the uncertainties, show relatively large differences in terms of accuracy and uncertainty while all participants had the same input data.
The results of the project were presented at the 5th CCM Pressure Conference in May 2011.
A detailed report is available.
Progress Report 2010-04-07
The calibration of a pressure balance involves determining the values of effective area at null pressure A0 and the pressure distortion coefficient "lambda" that define the effective area Ap of the piston-cylinder assembly as a function of pressure. There exist various approaches to determine these parameters and to estimate their uncertainties, with results that may differ significantly. The purpose of this project is to verify numerically the performance of different methods used by the NMI's by applying simulated data sets.
Therefore, theoretical values of the parameters have been fixed arbitrarily for four cases: two for a gas calibration in 10 MPa and two for an oil calibration in 500 MPa. Considering those values, calibration data sets have been simulated with random and systematic errors. The protocol gives the data which are usually recorded during a cross-float and their associated uncertainties: description of the mass pieces (mass, density) and of the piston-cylinder assemblies (PCS), environmental conditions (temperature of the room and of the calibrated and reference PCA, humidity, parameters of the reference balance etc.) and the properties of the fluids.
A first draft of the protocol has been given to all the participants and has been commented. The second draft will be sent at the middle of April. The comparison will begin at the beginning of May 2010.