## 2 Study on the Working Principle of an Online Vibrating Tube Liquid Densitometer  
### Working Principle of the Online Vibrating Tube Liquid Densitometer  
A vibrating tube liquid density meter, according to the number and shape of the vibrating tube, can be divided into a single straight-tube type, double straight-tube type, multi-straight-tube type, or U-type vibrating tube. Regardless of the kind of online vibrating tube liquid density meter, the working principle is the same [3,4]. The vibration tube material and size of its intrinsic vibration frequency and mode are pre-established. When the fluid enters into the vibration tube, the total mass of the liquid changes the vibration. The fluid will vibrate in the vibration tube; the intrinsic vibration frequency in the vibration tube changes. The vibration tube's volume is fixed. The vibration frequency can be measured by the change in the density of fluid in the tube [4,5]. Figure 1 shows the working principle of the on-line vibrating liquid densitometer. The structure of single straight tube type on-line vibrating tube liquid densitometer is shown in Figure 2. When the vibrating tube is used, a free vibration is performed, according to the composition and working principle of the vibrating tube. The vibrating tube can be regarded as a spring-damping-mass system [5]. When no fluid passes through the vibrating tube, the vibrating tube vibrates at the natural frequency, and the vibration period can be measured by the sensor. When the fluid flows through the vibrating tube, the mass and stiffness of the vibrating system will change, the intrinsic frequency of the vibrating tube system will change, and the frequency-density system equation of the vibrating tube can be obtained after appropriate compensation and correction [5,6]. According to the principle of elastic mechanics, the two ends of the fixed spring support the vibration tube  
Figure 1: Working principle of online vibrating tube liquid densitometer. system's transverse parallel vibration of the intrinsic angular frequency, which can be expressed as [7,8]. \[\omega_{0}=\sqrt{\frac{k}{M}} \tag{1}\]  
\(\omega_{0}\): Angular frequency of transverse vibration of the vibrating tube system (rad/s); \(k\): modulus elasticity (Pa); and \(M\): system quality (kg). From (1), it can be seen that the vibration tube system's spring modulus of elasticity, \(k\), is a constant, and its natural frequency changes only with the system mass, \(M\). The vibration tube material and volume are certain, so the quality of the fluid is determined only by the density of the fluid, that is, changes in the fluid density change the natural frequency of the vibration tube system. The vibration tube's vacuum-state-resonance angular frequency can be used (2) expressed as follows:  
\[\omega_{1}=\sqrt{\frac{k}{m}} \tag{2}\]  
\(\omega_{1}\): Resonant angular frequency in the vacuum state of the tube (rad/s); \(m\): vibration tube system quality (kg). At room temperature and pressure, the air mass inside the vibrating tube is much lower than the mass of the vibrating tube, so the air mass inside the tube can be ignored [9]. When calculating the mass of the empty tube in use, the resonant angular frequency under the empty tube state can also be expressed as (2). When fluid runs through the vibration tube system, the resonance angular frequency can be expressed as:  
\[\omega_{x}=\sqrt{\frac{k}{m+\Delta m}} \tag{3}\]  
\(\omega_{x}\): Resonant angular frequency of the vibrating tube after liquid flows through it (rad/s); \(\Delta m\): mass flowing through the vibrating tube (kg). The combination of Equations (2) and (3) results in:  
\[\rho_{x}=\frac{\omega_{1}^{2}}{\omega_{x}^{2}}\rho-\rho \tag{4}\]  
\(\rho_{x}\): The flow through the vibration tube of the liquid density (kg/m\({}^{3}\)); \(\rho\): the vibration tube material density (kg/m\({}^{3}\)). When the vibration tube material volume is certain, the vibration tube density and intrinsic frequency are constant. \[\left\{