## 4 Online Vibrating Tube Densitometer Measurement Performance Experiment
### Temperature Vibration Frequency Test
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Liquid Name** & \begin{tabular}{c} **Measuring Temperature (\({}^{\circ}\)C)** \\ \end{tabular} & \begin{tabular}{c} **Vibration Period** \\ **(us)** \\ \end{tabular} & \begin{tabular}{c} **Standard Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Fit Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} &
\begin{tabular}{c} **Fit Error** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} \\ \hline Tridecane & 15.55 & 3564.05124 & 759.442 & 759.510 & 0.07 \\  & 20.00 & 3563.23146 & 756.226 & 756.225 & 0.00 \\ Tridecane & 29.31 & 3561.62569 & 749.509 & 749.616 & 0.11 \\  & 40.19 & 3561.10569 & 745.123 & 745.239 & 0.12 \\  & 45.11 & 3559.74569 & 740.369 & 740.472 & 0.10 \\ \hline \hline \multirow{4}{*}{Anhydrous ethanol 1} & 14.33 & 3593.69892 & 833.236 & 833.312 & 0.08 \\  & 20.00 & 3593.46230 & 831.088 & 831.176 & 0.09 \\ \cline{1-1}  & 30.00 & 3593.40000 & 828.369 & 828.291 & \(-\)0.08 \\ \cline{1-1}  & 40.22 & 3593.69000 & 826.326 & 826.225 & \(-\)0.10 \\ \cline{1-1}  & 45.25 & 3592.55426 & 822.136 & 822.016 & \(-\)0.12 \\ \hline \hline \multirow{4}{*}{Anhydrous ethanol 2} & 14.54 & 3617.61236 & 892.909 & 892.950 & 0.04 \\  & 20.00 & 3616.55763 & 888.93 & 888.862 & \(-\)0.07 \\ \cline{1-1}  & 29.03 & 3614.44563 & 881.237 & 881.175 & \(-\)0.06 \\ \cline{1-1}  & 36.44 & 3614.21369 & 878.639 & 878.626 & \(-\)0.01 \\ \cline{1-1}  & 44.22 & 3613.68000 & 875.236 & 875.219 & \(-\)0.02 \\ \hline \hline \multirow{4}{*}{Water} & 14.21 & 3659.75236 & 999.191 & 999.188 & 0.00 \\ \cline{1-1}  & 15.11 & 3659.73000 & 999.009 & 998.906 & \(-\)0.10 \\ \cline{1-1}  & 20.00 & 3659.95351 & 998.221 & 998.254 & 0.03 \\ \cline{1-1}  & 29.15 & 3659.93123 & 995.898 & 995.917 & 0.02 \\ \cline{1-1}  & 43.54 & 3659.29000 & 990.651 & 990.698 & 0.05 \\ \hline \hline \multirow{4}{*}{Tungstate solution} & 14.65 & 3818.55000 & 1410.236 & 1410.162 & \(-\)0.07 \\ \cline{1-1}  & 20.00 & 3816.98624 & 1405.012 & 1405.004 & \(-\)0.01 \\ \cline{1-1}  & 29.33 & 3816.2600 & 1401.236 & 1401.294 & 0.06 \\ \cline{1-1}  & 40.02 & 3815.92360 & 1398.369 & 1398.496 & \(-\)0.13 \\ \cline{1-1}  & 45.44 & 3815.13123 & 1395.237 & 1395.104 & 0.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Curved-tube densitometer temperature experiments on density, frequency, and fit error at (15–45) \({}^{\circ}\)C.  
with the increase in temperature vibration frequency. The decrease in the trend of change and the theoretical derivation is basically consistent, indicating that the instrument has a good degree of differentiation and reasonableness, with the follow-up to carry out the experimental requirements. From the change in ATC frequency, it can be concluded that the environment maintained at 20 \({}^{\circ}\)C near the ATC frequency fluctuation is small; the resonance and the theoretical value of the difference are not significant. The straight-tube maximum ATC frequency fluctuation value was 0.009 us, the average value and the theoretical value was 0.11 us, the curved-tube maximum ATC frequency fluctuation value was 0.12 us, and the average value of the difference between the theoretical value was 0.27 us. The ATC frequency change is small and kept near the resonance frequency, indicating that the tubes are relatively clean and there is no risk of mixing in subsequent experiments. Figures 11 and 12 show the change rule of fitting error in the experimental process of curved tube type and straight tube type densitometer, respectively. The \(K_{\rm x}\) coefficients were derived from different liquids and vibration frequencies using the principle of least squares, and the experimental fit errors of the two densitometers were obtained by comparing the fitted densities derived from the fitting equations with the standard densities. From the fitting error distribution, it can be seen that most of the fitting error distributions fall within the nominal error.