## 2 Methodology
### Analytical Model of the Material Weighing System inside Process Equipment
The total mass \(m\) to be weighed matches the characteristics of the weighing sensor used. As long as the change of the material mass during the data collection interval is greater than the resolution of the load cell (the interval is determined by the requirements of the engineering application), the measurement can be effective. In addition, \(m_{D}\) is the additional mass caused by the additional load \(F_{D}\). In the weighing process, the signal of the support force \(F_{N}\) is converted into an output electrical signal by the load cell. For the versatility of signal acquisition, the most widely used resistance strain gauge type load cell in the industry is used. Figure 3 is a typical resistance strain gauge load cell. It is mainly composed of an elastomer, resistance strain gauge, and bridge circuit. When a load \(F_{N}\) is applied to the weighing sensor, the resistance strain gauge generates strain, which is then converted into an electrical signal by the electric bridge circuit. After signal conversion, the change of the measurement value is obtained and displayed [34]. Figure 2: Unit equipment weighing system with equivalent cantilever beam: (**a**) the simplified model of the system; (**b**) the force diagram of unit equipment. The strain of the resistance strain gauge is given as [35]:  
\[\varepsilon=\frac{6F_{N}L}{wH^{2}E_{1}} \tag{6}\]  
where \(L\), \(w\), and \(H\) are the structural parameters of the sensor. \(L\) is the length of the cantilever of the load cell, \(w\) is the width of the beam section of the load cell, \(H\) is the height of the beam section of the load cell, and \(E_{1}\) is the elastic modulus of the strain gauge material. If \(\delta_{5}\) is the strain of the resistance strain gauge along the direction of \(F_{N}\), then there is equality \(\delta_{5}=\delta\). Substituting Equation (5) into Equation (6) and combining with the geometric deformation conditions, it can be deduced that:  
\[\left\{ \begin{array}{l}K\varepsilon=\frac{\Delta L}{L}=\frac{\sqrt{L^{2}+ \delta_{5}^{2}}-L}{L}=\frac{\delta_{5}^{2}}{L(\sqrt{L^{2}+\delta_{5}^{2}}+L) }\approx\frac{\delta^{2}}{2L^{2}}\\ K\varepsilon=\frac{6 Kl}{wH^{2}E_{1}}\Big{(}(m-m_{D})_{\mathcal{B}}-\frac{3 EI} {l^{3}}\delta\Big{)}\end{array} \right. \tag{7}\]  
where \(K\) is the sensitivity of the resistance strain gauge and \(\Delta L\) is the absolute elongation of the strain gauge. Further derivation shows that:  
\[\left\{ \begin{array}{l}\delta=\sqrt{\text{A}(m-m_{D})+\text{B}^{2}}-\text{B }\,,\\ \text{A}=\frac{12 Kl^{3}\text{g}}{wH^{2}E_{1}}\text{, \ }\text{B}=\frac{18 Kl^{3}E_{1}}{wH^{2}E_{1}l^{3}} \end{array} \right. \tag{8}\]  
Given that \(E\), \(I\), \(l\), \(w\), \(H\), \(K\), \(E_{1}\), \(L\), and g are the known parameters, A and B are regarded as given values. Hence, there is a linear power relationship between the deflection of cantilever beam \(\delta\) and the mass of the unit equipment \(m\), which is affected by the parameters of the connected beam and the load cell. According to the bridge voltage characteristic of the load cell:  
\[V_{0}=K\varepsilon V \tag{9}\]  
where \(V_{0}\) is the output bridge voltage of the load cell and \(V\) is the supply voltage. By substituting Equations (7) and (8) into Equation (9), there is:  
\[V_{0}=V_{1}-\Delta V_{1}-\Delta V_{2} \tag{10}\]  
where:  
\[V_{1}=\frac{6 KVL\text{g}}{w\text{H}^{2}E_{1}}m\,,\,\,\,\Delta V_{1}=\frac{6 KVL\text{g}}{w\text{H}^{2}E_{1}}m_{D},\,\,\,\Delta V_{2}=\frac{6 KVL\text{g}}{w \text{H}^{2}E_{1}}\frac{3 EI}{gl^{3}}\delta \tag{11}\]  
In Equation (10), \(V_{1}\) is the output bridge voltage change caused by the material mass change, \(\Delta V_{2}\) is the output bridge voltage change caused by the internal material state and operating conditions of unit equipment, and \(\Delta V_{2}\) is the output bridge voltage change caused by the force of the connected piping. When there is no connected piping and the  
Figure 3: Structure of the resistance strain gauge type load cell.