Calculation of elastic distortion and associated uncertainty in piston cylinders operating up to 1 Gpa

Project Description

As a follow-up to EUROMET Project number 256, attention will be given to the calculation of elastic distortions for controlled-clearance piston-cylinder units in liquid media up to 1 GPa. The project will be mainly focused on the estimate of all the possible uncertainty components for the elastic distortion coefficient calculation.

Particular attentions will be given to the following contributions:

elastic constants of materials
fluid properties
real gap profile
geometry of the unit, including scaling
boundary conditions on the overall piston-cylinder assembly

The project is divided in two phases. In the phase 1 a study will be made of the available systems at BNM-LNE and PTB with a complete analysis of available results (including evaluation of pressure distortion coefficient) and data for the full definition of the model to be used in calculations. Results obtained in the previous EUROMET project 256 will be useful to identify parameters and conditions of particular importance.

In phase 2 the calculations will start in all participating laboratories using the same model as defined in the phase 1. Results will be compared between them and possibly with experimental results.

Results of this project will be beneficial for users to improve knowledge of pressure distortions at pressures up to 1 GPa, with expected information useful to consolidate the uncertainty of the pressure scale also for lower pressures.

Final Report 2006-03-02

The participants performed finite element analysis (FEA) of PTB and LNE 1 GPa piston-cylinder assemblies operated in free-deformation (FD) and controlled-clearance (CC) modes using information on the shape of the gap between undistorted piston and cylinder, their dimensional and materials’ elastic properties. Two models with the real and ideal piston-cylinder gaps were applied. The radial elastic distortions in the piston and cylinder, the pressure distributions along the gap, the FD, CC and jacket pressure distortion coefficients (l), as well as the piston fall rates were determined. Uncertainty budgets for the FD and CC coefficients at different pressures were established. The results of the study were presented at the 4^th CCM Pressure Conference in April 2005. The main results and conclusions are:

For all quantities the results are in better agreement for the ideal than for the real gap model with very good agreement between FEA and Lamé results for ideal gap model.
Agreement between participants’ l (in 10^-6/MPa) is for ideal gap: 0.04 (PTB unit), 0.07 (LNE unit), real gap: 0.05 (PTB unit), 0.2 (LNE unit). These differences as well as the uncertainties can be considered as a measure for possible errors when determining the distortion coefficients by FEA.
Differences between l obtained for real and ideal gap models are 0.1·10^-6/MPa (PTB unit) and 0.25·10^‑6/MPa (LNE unit).
Standard uncertainty is u(l) = (0.04-0.12)·10^-6/MPa in dependence on pressure and operating mode.
Gap dimensions were identified as a main uncertainty source.
Claiming 2u(l) = 0.1·10^-6/MPa when using Lamé equations or any other method assuming p.-c.-unit to be geometrically perfect (common CMC praxis) appears too optimistic.
No model describes well experimental piston fall rates.
The reasons for the differences between the theoretical and experimental results obtained for the LNE unit require further clarification. A pressure comparison of the PTB and LNE 1 GPa assemblies studied by FEA within the scope of this project appears useful for a verification of the theoretical results.

FEA provide a useful tool for distortion coefficient prediction, but further work is needed before FEA can completely be relied on for uncertainty determination especially when trying to model the behaviour of real piston-cylinders. In particular, application of FEA becomes more difficult with increasing distortions and decreasing piston-cylinder clearance.

[1] Sabuga W., Bergoglio M., Buonanno G., Legras J.C., Yagmur L. Calculation of the distortion coefficient and associated uncertainty of a PTB 1 GPa pressure balance using Finite Element Analysis – EUROMET Project 463. In: Proceedings of International Symposium on Pressure and Vacuum, IMEKO TC16, Beijing, September 22-24, 2003, Acta Metrologica Sinica Press, 92-104.

[2] Sabuga W., Molinar G., Buonanno G., Esward T., Rabault T., Yagmur L. Calculation of the distortion coefficient and associated uncertainty of PTB and LNE 1 GPa pressure balances using finite element analysis – EUROMET project 463. Metrologia, 2005, 42, pp. 202-206.

Progress Report 2005-02-25

PTB, IMGC-CNR in cooperation with Cassino University (Unicas) and BNM-LNE have performed FE analyses of BNM-LNE 1 GPa piston-cylinder assembly #7 operated in free-deformation (FD) and controlled-clearance (CC) modes using information on the shape of the gap between undistorted piston and cylinder, their dimensional and materials’ elastic properties. Two models with the real and ideal piston-cylinder gaps were applied. By these analyses, the radial elastic distortions in the piston and cylinder, the pressure distributions along the gap, the FD, CC and jacket pressure distortion coefficients, as well as the piston fall rates were determined. Uncertainty budgets for the FD coefficients at 200 MPa and for the CC coefficients at 200 MPa and 800 MPa were established.
For all quantities the results of PTB and IMGC-CNR/Unicas are in a close agreement. It is better for the ideal than for the real gap model. The results of BNM-LNE at pressures higher than 200 MPa significantly deviate from the results of the other institutes. However, their distortion coefficients for the real gap model are in a better agreement with the experimental data than those of PTB and IMGC-CNR/Unicas.
In the FD mode, the piston fall rates calculated with the real gap model agree well with the experiment. For the CC mode, neither real nor ideal gap model led to values compatible with the experimental fall rates.
With the ideal gap model, the pressure distortion coefficients determined for the FD and CC operation modes are in a good agreement with each other, with the simplified theory and with the experimental jacket pressure distortion coefficients.
Although all participants had received the same information on the assembly’s material and dimensional properties, their treatments of this information to create a full dimensional data set for the gap, the uncertainty models, the input quantities, the uncertainty contributions and, finally, the combined uncertainties of the distortion coefficients frequently were different. The final differences of the distortion coefficients and the associated uncertainties can be considered as a measure for possible errors when determining the distortion coefficients by a FE analysis.
The reasons for the differences between the theoretical and experimental results require further clarification. A pressure comparison of the assemblies studied by FEM within the scope of this project, the PTB and BNM-LNE 1 GPa, appears useful for a verification of the theoretical results.
The results of the study are included to a contribution which will be presented at the 4^th CCM Pressure Conference in April 2005.

Further Information

Project expected to obtain results in about 3 years.

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