## 4 Online Vibrating Tube Densitometer Measurement Performance Experiment  
### Temperature Vibration Frequency Test  
Straight-tube and curved-tube online vibrating tube liquid densitometers were selected for experimental validation. Solutions of n-tridecane, anhydrous ethanol, lye, pure water, and sodium tungstate were selected for the experimental validation. Standard temperature calibration experiments were performed at a constant temperature of (20 \(\pm\) 0.02) \({}^{\circ}\)C, and temperature tests were performed at (15-45) \({}^{\circ}\)C. The temperature of the liquid passing through the vibrating tube densitometer was measured by the inlet and outlet thermometers of the vibrating tube densitometer during the experimental process, the temperature was collected every 500 ms, the average value of the inlet and outlet temperatures was taken as the temperature of the liquid passing through the vibrating tube densitometer, and the standard density value was determined by [PERSON] 5000. In order to reduce the experimental errors caused by mixing the liquid in the experimental process, three thermostatic stirring tanks were used to hold water, mixed liquids and organic solvents, and each liquid after the completion of the experiments on the pipeline blown clean. The end of the blowing was compared to the frequency of 20 \({}^{\circ}\)C when the empty tube was used to verify whether it was blown clean. The straight-tube type and curved-tube type showed two typical densitometer nominal accuracies of \(\pm\)0.2 kg/m\({}^{3}\) and \(\pm\)0.1 kg/m\({}^{3}\), respectively. In practice, it is not possible to directly calculate the theoretical \(K_{\rm x}\) coefficient, which must be experimentally calibrated by a different density and frequency to calculate the above coefficients, and adjusted to the density of the difference between the density and the standard density to calculate the fit error. The system collects the vibration period of the vibrating tube, once every 500 ms, 10 times each time, and takes the average value of 10 times as a measurement value of the measured liquid. According to the measured frequency values and standard density values calculated using the least squares method, the straight-tube type values were \(K_{0}\) = \(-\)1.233610 \(\times\) 10\({}^{3}\), \(K_{1}\) = \(-\)4.782074 \(\times\) 10\({}^{-2}\); \(K_{2}\) = 1.081764 \(\times\) 10\({}^{-3}\); \(K_{18}\) = 8.58998 \(\times\) 10\({}^{-6}\); and \(K_{19}\) = 5.94276 \(\times\) 10\({}^{-3}\). According to (6) and (7), the experimental straight-tube-type density meter in the standard condition of the liquid density and vibration frequency of the relationship is:  
\[\rho_{x}=-1233.610-0.04782074\times T+0.001081764\times T^{2} \tag{10}\]  
\[\rho_{tx}=\rho_{x}\Big{[}1+8.58998\times 10^{-6}\times(t-20)\Big{]}+5.94276 \times 10^{-3}\times(t-20) \tag{11}\]  
Calculation of straight-tube on-line densitometer at (20 \(\pm\) 0.02) \({}^{\circ}\)C and (15-45) \({}^{\circ}\)C fitted with a density with error. Table 2 shows the fitting error and vibration frequency of straight tube densitometer at (20 \(\pm\) 0.02) \({}^{\circ}\)C.  
The experimental frequencies and fitting errors of straight tube densitometers at (15-45) \({}^{\circ}\)C are shown in Table 3. According to the measured frequency value and the standard density value calculated by the least squares method, the curved-tube-type online vibrating tube liquid densitometer coefficients were \(K_{0}\) = \(-\)3.642245 \(\times\) 10\({}^{3}\); \(K_{1}\) = 0;\(K_{2}\) = 3.464284 \(\times\) 10\({}^{-4}\); \(K_{18}\) = 1.418323 \(\times\) 10\({}^{-4}\); and \(K_{19}\) = \(-\)3.908279 \(\times\) 10\({}^{-1}\). According to (6) and (7), the experimental straight-tube-type densitometer in the standard condition presented different temperatures of liquid density, and the standard condition of vibration presented the different temperature and density of liquid.