Quantification of free convection effects on 1 kg mass standards

We determine the free-convection effects and the resulting mass differences in a high-precision mass comparator for cylindrical and spherical 1 kg mass standards at different air pressures. The temperature differences are chosen in the millikelvin range and lead to microgram updrafts. Our studies reveal a good agreement between the measurements and direct numerical simulations of the Boussinesq equations of free thermal convection. A higher sensitivity to the free convection effects is found for the spherical case compared to the cylindrical one. We also translate our results on the free convection effects into a form which is used in fluid mechanics: a dimensionless updraft coefficient as a function of the dimensionless Grashof number Gr that quantifies the thermal driving due to temperature differences. This relation displays a unique scaling behavior over nearly four decades in Gr and levels off into geometry-specific constants for the very small Grashof numbers. The obtained results provide a rational framework for estimating systematic errors in mass metrology due to the effects of free convection.