A Modified Model for the Temperature Effect-Induced Error in Hydrostatic Leveling Systems

Abstract

To investigate systematically how the temperature effect influences the measurement accuracy of a hydrostatic leveling system (HLS), ambient-temperature and temperature-gradient test platforms were built in the laboratory. The temperature of each measuring point of the HLS was collected by a high-precision temperature sensor, and the influences of the ambient temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}_{\mathrm {A}}$ </tex-math></inline-formula> , temperature gradient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}_{\mathrm {G}}$ </tex-math></inline-formula> , heating/cooling rate, communicating-pipe expansion coefficient, and measuring-point distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${D}_{\mathrm {test}}$ </tex-math></inline-formula> on the HLS measurement accuracy were analyzed quantitatively. The results show that the error of the two-HLS setup was caused by the expansion of the communicating pipe and the liquid between the datum point and the measuring point both under ambient-temperature and temperature-gradient conditions. The relative ambient-temperature influence coefficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {P}$ </tex-math></inline-formula> and temperature gradient influence coefficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {P}_{\mathrm {t-G}}$ </tex-math></inline-formula> , respectively, had a linear correlation with the difference between the expansion coefficients of the communicating liquid and pipe ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma _{\mathrm {di}} - \gamma _{\mathrm {pi}}$ </tex-math></inline-formula> ). Moreover, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {P}_{\mathrm {t-A}}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {P}_{\mathrm {t-G}}$ </tex-math></inline-formula> had a negative power relationship with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${D}_{\mathrm {test}}$ </tex-math></inline-formula> for given communicating pipe and liquid. A temperature-modified model was developed theoretically based on test results, and its reliability was verified via on-site tests.