Finite element method used for calculation of the distortion coefficient and associated uncertainty of a PTB 1 GPa pressure balance—EUROMET project 463

The national metrology institutes and their partners participating in EUROMET Project 463 developed finite element methods (FEM) for calculation of the pressure distortion coefficients, including their uncertainties, of pressure balances operated at pressures up to 1 GPa and applied them to a PTB 1 GPa piston–cylinder assembly. The methods use axisymmetric models developed and analysed on the basis of the experimental data including the elastic properties of the piston–cylinder materials, pressure-dependent density and viscosity of the pressure-transmitting fluid, dimensions of the piston and cylinder and the piston–cylinder clearance as well as the conditions at the piston–cylinder boundaries. Results such as pressure distributions and radial distortions along the piston–cylinder engagement length, pressure distortion coefficients and their uncertainties as well as piston fall rates dependent on pressure are presented for the free deformation (FD) and the controlled-clearance operating modes of the assembly. The theoretical results are verified by comparing them with the distortion coefficients determined by an experimental method and with the jacket pressure distortion coefficients. The participants' results demonstrate good agreement of the distortion coefficients up to 1 GPa but rather large differences in the uncertainties of the distortion coefficients as well as in the pressure distributions, gap profiles and piston fall rate at maximum pressure. The FEM distortion coefficients obtained for the real piston–cylinder gap profile are in good agreement with the coefficients determined by the experimental method; the FEM values obtained for the ideal gap agree well with the distortion coefficients furnished by the simplified theory. For the real gap model, the uncertainty of the gap geometry is the main uncertainty source. The total standard uncertainties of the controlled-clearance distortion coefficient obtained by different methods lie between (0.078 and 0.17) × 10⁻⁶ MPa⁻¹ at 400 MPa and between (0.04 and 0.098) × 10⁻⁶ MPa⁻¹ at 1 GPa.