Physical apparatus parameters and model for vibrating tube densimeters at pressures to 140 MPa and temperatures to 473 K

Vibrating tube densimeters are well-established tools for measuring fluid densities precisely at elevated temperatures and pressures. However, the conventional method of calibrating them utilises a model in which the apparatus parameters are represented as polynomials of temperature and pressure that contain a variable number of terms. Here a robust, physically-based model is presented and demonstrated for six different instruments at temperatures from (273 to 473) K, pressures from (0 to 140) MPa, and densities from (0 to 1050) kg m(-3). The model's physical basis ensures that only seven apparatus parameters are required to relate the measured resonant period to fluid mass density with an average r.m.s. deviation of ±0.23 kg m(-3) across all six densimeters. Estimates for each of the apparatus parameters were made based on the geometry and material properties of the vibrating tubes, and these estimates were consistent with the parameter values determined by calibration with reference fluids. Three of the apparatus parameters describe the temperature dependence of the resonant period: for the six vibrating tubes tested, the relative standard deviations of these parameters were all within the range of values estimated from the thermoelastic properties of the Hastelloy tubes. Two distinct parameters are required to describe the pressure dependence of the vibrating tube's volume and effective spring constant, both of which are estimable from equations describing the elastic deformation of thick-walled tubes. The extensive calibrations conducted demonstrate that, for these densimeters, the variations with pressure of the tube's spring constant and its volume have a ratio that is neither 0 nor 1, as has been assumed previously. The model's physical basis allows vibrating tube densimeters to be calibrated accurately using fewer reference fluid measurements than required by the conventional method. Furthermore, use of the physically-based model reduces the uncertainty of measurements made at densities, temperatures, or pressures beyond the range of the calibration.