A Hydrodynamic Model for Measuring Fluid Density and Viscosity by Using Quartz Tuning Forks

Abstract

A hydrodynamic model of using quartz tuning forks (QTFs) for density and viscosity sensing, by measuring the resonance frequency and quality factor, has been established based on the cantilever beam theory applied to the atomic force microscope (AFM). Two examples are presented to verify the usability of this model. Then, the Sobol index method is chosen for explaining quantitatively how the resonance frequency and quality factor of the QTFs are affected by the fluid density and viscosity, respectively. The results show that the relative mean square error in viscosity of the eight solutions evaluated by the hydrodynamic model is reduced by an order of magnitude comparing with Butterworth-Van Dyke equivalent circuit method. When the measured resonance frequency and quality factor of the QTFs vary from 25,800-26,100 Hz and 28-41, the sensitivities of the quality factor affected by the fluid density increase. This model provides an idea for improving the accuracy of fluid component recognition in real time, and lays a foundation for the application of miniaturized and cost-effective downhole fluid density and viscosity sensors.Sensors 2020, 20, 198 2 of 12 lithium niobate tuning fork were 10.48% and 2.89%, respectively. Although it is a physical fact for tuning fork sensors that the error on density is smaller than that on viscosity [22], we can still reduce the viscosity error to meet the measurement requirements.Tuning fork sensors, microcantilever beam, AlN resonator, torsional resonators and so on, can all be used as sensitive components for the D-V sensor [23][24][25]. On the one hand, quartz tuning forks (QTFs) studied in this paper have high Curie temperature, good stability and high accuracy, which is suitable for downhole high temperature and high pressure environment, and, on the other hand, a millimeter-sized QTF can be integrated in small-scale measurement platforms for downhole deployment. In addition, the QTFs are low-cost and commercially used as frequency standards in watches, working at 32.768 kHz. The QTF is widely used in gas sensing based on photoacoustic spectroscopy and photothermal spectroscopy [26][27][28], and scanning probe microscopy applications such as atomic force microscopy (AFM) [29][30][31] and near-field scanning optical microscopy (NSOM) [32][33][34]. In order to estimate the density and viscosity of a liquid at the same time, the hydrodynamic model based on the work of Sader, J.E. [35] for the atomic force microscope (AFM) is established, and two examples are verified by using this model.

Higher-Order Models for Resonant Viscosity and Mass-Density Sensors

“…The determination of the fluid parameters from the fluid loaded resonator is difficult, as and appear also in the arguments of two Bessel functions. The fluid models presented in this work, are based on an expansion the hydrodynamic function in Equation ( 2 ) into a power series around , yielding For the reduced order models [ 22 , 23 , 24 , 25 , 26 ], the approximation of the fluid force in Equation ( 7 ) is added to the equation of motion of the resonator (see Section 2.3 ), yielding models for the resonance parameters of the fluid loaded resonator shown in Equation ( 12 ). A consequence of the agreement between Equation ( 7 ) and Equation ( 2 ) being best for large is that the model approximation is more accurate for thicker cylinders vibrating at higher frequencies.…”

“… and are functions of the resonance parameters which correspond to the real and imaginary parts of the hydrodynamic function. It is apparent that the number of parameters of the models for [ 23 ], for [ 22 , 25 ], for [ 24 ] or for [ 26 ] differ. For instance, the model from Heinisch et al [ 23 ] where the vacuum resonance parameters and are fitted in the calibration procedure ( ), are considered known in the model of Zhang et al [ 26 ].…”

“…It is apparent that the number of parameters of the models for [ 23 ], for [ 22 , 25 ], for [ 24 ] or for [ 26 ] differ. For instance, the model from Heinisch et al [ 23 ] where the vacuum resonance parameters and are fitted in the calibration procedure ( ), are considered known in the model of Zhang et al [ 26 ]. Furthermore, and are allowed to differ, while according to Equation ( 7 ), the same dependency in real and imaginary part of the hydrodynamic function suggests that and should be equal as in the model of Zhang et al [ 26 ] (parameter A ).…”

“…For the reduced order models [22][23][24][25][26], the approximation of the fluid force in Equation 7is added to the equation of motion of the resonator (see Section 2.3), yielding models for the resonance parameters of the fluid loaded resonator shown in Equation (12). A consequence of the agreement between Equation (7) and Equation (2) being best for large β is that the model approximation is more accurate for thicker cylinders vibrating at higher frequencies.…”

“…The task of the sensor designer is to guarantee that the resonance parameters can be correctly measured and that a fluid model is available which can be transformed to yield unique and accurate fluid parameters from the resonance parameters. As the exact hydrodynamic function is only known for some special cases, the conventionally used fluid models derived, e.g., by [22][23][24][25][26] are all based on an approximation of the hydrodynamic function of a vibrating prismatic beam featuring a cylindrical cross-section by means of a truncated power series. A remarkable fact, shown by a many researchers in experimental work [22][23][24][25][26][27][28][29][30][31], is that the resulting reduced order models with adjustable parameters are suitable for representing various different types of fluid loaded vibrating structures, including cantilevers of different cross-sections, large steel and miniaturized quartz tuning forks, spiral springs, platelets, U-shaped wires, piezo buzzers in fluids, etc.…”