## 4 Online Vibrating Tube Densitometer Measurement Performance Experiment
### Temperature Vibration Frequency Test
According to (6) and (7), the experimental straight-tube-type densitometer in the standard condition is (\(K_{19}\) = \(-\)3.908279 \(\times\) 10\({}^{-1}\)); different temperatures under the liquid density and vibration frequency relationship are:  
\[\rho_{x}=-3642.245+3.464284\times\ 10^{-4}\times\ \text{T2} \tag{12}\]  
\[\rho_{tx}=\rho_{x}[1+0.0001418323\times(t-20)]-0.3908279\times(t-20) \tag{13}\] \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 N-Tridecane & 20.00 & 1378.541607 & 756.226 & 756.226 & 0.00 \\ \hline Anhydroous ethanol & 20.00 & 1401.152815 & 823.136 & 823.136 & 0.00 \\ \hline Aqueous alcohol 1 & 19.99 & 1415.154277 & 865.123 & 865.123 & 0.00 \\ \hline Industrial white oil & 20.00 & 1403.815470 & 831.088 & 831.088 & 0.00 \\ \hline Aqueous alcohol 2 & 19.99 & 1423.031064 & 888.930 & 888.930 & 0.00 \\ \hline Pure water & 20.00 & 1458.638571 & 998.226 & 998.226 & 0.00 \\ \hline Tungstate solution & 20.01 & 1584.048720 & 1405.012 & 1405.013 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Straight-tube accuracy experiments on density, frequency, and fit error at (20 \(\pm\) 0.02) \({}^{\circ}\)C. \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 \multirow{4}{*}{Tridecane} & 15.50 & 1379.68479 & 759.442 & 759.526 & \(-\)0.08 \\  & 20.00 & 1378.53900 & 756.226 & 756.218 & 0.01 \\  & 29.41 & 1376.25237 & 749.509 & 749.630 & \(-\)0.12 \\  & 40.23 & 1374.62239 & 745.123 & 744.991 & 0.13 \\  & 45.21 & 1372.95320 & 740.369 & 740.170 & 0.20 \\ \hline \multirow{4}{*}{Anhydroous ethanol 1} & 14.25 & 1404.62236 & 833.236 & 833.425 & \(-\)0.19 \\  & 20.00 & 1403.81547 & 831.088 & 831.088 & 0.00 \\  & 30.12 & 1402.83650 & 828.369 & 828.295 & 0.07 \\  & 40.22 & 1402.19565 & 826.326 & 826.512 & \(-\)0.19 \\  & 45.24 & 1400.63691 & 822.136 & 821.925 & 0.21 \\ \hline \multirow{4}{*}{Anhydroous ethanol 2} & 14.95 & 1424.40283 & 892.909 & 893.021 & \(-\)0.11 \\  & 20.00 & 1423.03450 & 888.930 & 888.940 & \(-\)0.01 \\  & 29.10 & 1420.48395 & 881.237 & 881.340 & \(-\)0.10 \\  & 36.42 & 1419.56231 & 878.639 & 878.651 & \(-\)0.01 \\  & 44.22 & 1418.32352 & 875.236 & 875.011 & 0.22 \\ \hline \multirow{4}{*}{Water} & 14.36 & 1459.03692 & 999.191 & 999.382 & \(-\)0.19 \\  & 15.56 & 1458.90888 & 999.009 & 999.002 & 0.01 \\ \cline{1-1}  & 20.00 & 1458.63911 & 998.221 & 998.228 & \(-\)0.01 \\ \cline{1-1}  & 29.26 & 1457.84188 & 995.898 & 995.885 & 0.01 \\ \cline{1-1}  & 43.89 & 1456.12524 & 990.651 & 990.767 & \(-\)0.12 \\ \hline \multirow{4}{*}{Tungstate solution} & 14.65 & 1585.58569 & 1410.236 & 1410.112 & 0.12 \\ \cline{1-1}  & 20.00 & 1584.04872 & 1405.012 & 1405.012 & 0.00 \\ \cline{1-1}  & 29.25 & 1582.86922 & 1401.236 & 1401.194 & 0.04 \\ \cline{1-1}  & 40.12 & 1582.01231 & 1398.369 & 1398.496 & \(-\)0.13 \\ \cline{1-1}  & 45.37 & 1580.97900 & 1395.237 & 1395.104 & 0.13 \\ \hline \hline \end{tabular}
\end{table}