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

Abstract

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.

Numerical analysis of the accuracy of bivariate quantile distributions utilizing copulas compared to the GUM supplement 2 for oil pressure balance uncertainties

“…Our approach of disregarding the covariance term cov ( A 0 , l) for the generated pressure in the absence of specific known covariance information is consistent with experimental recommendations for national metrology institutes for pressure balances operated in free deformation mode as documented in for example EURAMET CG3 [12], and results from theoretical finite element based studies for pressure balances operated in both free deformation and controlled clearance modes such as the EUROMET Project 463 by Sabuga et al [13] and in controlled clearance modes only by Dogra et al [14] respectively. As a result of our choice to disregard covariance effects in the generated pressure calculation when implementing a GS1 Monte Carlo approach as discussed by Cox and Siebert [15] for the pressure generation calculations, it will then usually suffice to sample with appropriate numerical techniques for the inputs q i from univariate PDFs g i (j i ) associated with the corresponding input parameters q i where j i is a random variable of q i unless information of a joint PDF is specifically available.…”

Recent progress in high pressure metrology in Europe

“…As known from former FEA studies, the results of hydrodynamic modelling strongly depend on the initial real gap profile between the undistorted piston and the cylinder [6]. In particular, information about the cylinder bore profile near the exit is important because the gap in this region becomes the narrowest under high pressure and therefore has a strong effect on the pressure distribution, vf and λ.…”

“…Combining the structural FEA of the HP PCA with a hydrodynamic analysis of its piston-cylinder gap, λ and vf were calculated using the iterative method described in [6]. As a pressure-transmitting medium, two liquids were considered: di(2)-ethyl-hexyl-sebacate (DHS) at pH ≤ 0.5 GPa and polydiethylsiloxan PES-1 for pH ≤ 1.6 GPa, whose properties are presented in 3.2.3.…”

A comparative approach for the characterization of a pneumatic piston gauge up to 8 MPa using finite element calculations

“…Prior to computation, a model of the piston-cylinder assembly was designed. The piston-cylinder assembly is taken as a 2D model [2,3,13] assuming axial symmetry as shown in figure 2. Assuming point no 8 in figure 2 as the origin, the coordinates of the other key points were calculated.…”

“…All the components are interdependent and complex in nature, and very few available methods provide solution. It is for this reason a number of quite different techniques have been reported to be used, and among them application of the finite element method (FEM) has been quite significant and important [2,3]. The complexity of the geometry and the boundary conditions makes the FEM more reasonable in solving mechanical problems.…”