## 2 Methodology  
### Analytical Model of the Material Weighing System inside Process Equipment  
In Figure 1a, the material weighing system inside process equipment with connected piping mainly includes unit equipment with materials 1, weighing unit 2, connected piping 3, and pipe fix unit 4. Considering piping 3 is embedded to pipe fix unit 4, it can be equivalent to a cantilever beam with one end fixed. Here, pipe fix unit 4 is the fixed end, peripheral connected piping 3 is the beam [29; 30]. The material mass inside the unit equipment 1 can be regarded as a concentrated load applied on the other end of the cantilever beam, as shown in Figure 1b. Suppose that there are many cantilever beams connected to the unit equipment. Since all cantilever beams are in co-ordination with each other, the displacements at the free end are equal. With the deflection equation \(y=Px^{2}(3l-x)/6 EI\)[31], the concentrated load \(P_{i}\) on the end of the cantilever beam can be obtained:  
\[P_{i}=\frac{3E_{i}I_{i}}{l_{i}^{3}}\delta\quad(i\in 1,2,3,\ldots\ldots n) \tag{1}\]  
where \(n\) is the number of cantilever beams, \(\delta\) is the deflection of cantilever beams, \(E_{i}\), \(I_{i}\), and \(I_{i}\) represent the elastic modulus, the moment of inertia, and the length of the \(i\)th cantilever beam, respectively. The influencing factors of temperature on beam deflection include external factors and internal factors. The external factors include ambient temperature, solar radiation, and other climatic factors. When the structure and section of the beam are fixed, the internal factors are mainly material properties, including elastic modulus and thermal parameters. Since the common external temperature range for chemical vessels is \(-20\)-\(200\)\({}^{\circ}\)C, the deflection change caused by material change within this range is considered to be negligible in this study [32; 33]. It can be seen from Equation (1) that the concentrated load \(P_{i}\) is only related to the parameters of \(E_{i}\), \(I_{i}\), and \(I_{i}\), and has nothing to do with the position of the cantilever beams. Therefore, the multi-cantilever structure can be equivalent to a single cantilever structure, and the resultant force of the reaction force exerted by the cantilever beam on the unit equipment is:  
\[P=\sum_{i=1}^{n}\frac{3E_{i}I_{i}}{l_{i}^{3}}\delta=\frac{3 EI}{l^{3}}\delta \tag{2}\]  
Figure 1: The material weighing system inside unit equipment with connected piping: (**a**) physical model; (**b**) system diagram. where \(E\), \(I\), and \(l\) are the elastic modulus, the moment of inertia, and the length of the equivalent cantilever beam, respectively. Based on the above simplification, the equivalent physical model of the material weighing system inside unit equipment with connected piping is established, as shown in Figure 2a. For the unit equipment, an equilibrium equation of force in Figure 2b can be drawn as follows:  
\[F_{N}+F_{D}+P=G \tag{3}\]  
where \(F_{N}\) is the supporting force of the load cell, \(G\) is the weight of the materials and unit equipment, and \(F_{D}\) is the additional load generated by internal materials flow and pressure under the continuous operation condition. In the process of two-phase or multiphase flow of a chemical system, the additional load \(F_{D}\) represents the influence of the internal material state and the operating conditions on the mass measurement. According to Equation (2), when the concentrated load \(P\) is applied at the end of the cantilever beam, the maximum deflection \(\delta\) along the negative direction of the z-axis is generated. \[\delta=\frac{Pl^{2}}{6 EI}(3l-l)=\frac{Pl^{3}}{3 EI} \tag{4}\]  
Substituting Equation (4) into Equation (3), it can obtain:  
\[F_{N}=G-F_{D}-P=mg-F_{D}-\frac{3 EI}{l^{3}}\delta=(m-m_{D})g-\frac{3 EI}{l^{3}}\delta \tag{5}\]  
where \(m\) is the total mass of the materials and unit equipment; since the mass of the equipment itself is constant, the change of \(m\) can be regarded as the mass change of the materials inside the equipment.