# Soft Sensor Technology for the Determination of Mechanical
## 1 Introduction
### Development of the Soft Sensor for Friction Power Performance Determination  
The model shown in Figure 1b is also used to derive the soft sensor for determining the frictional power (\(P_{R}\)) of a mechanical seal. Friction power can be represented as the sum of all heat flows transferred to the pumped medium, as shown in Figure 1b:  
\[P_{R}:=\dot{Q}_{10}+\dot{Q}_{12}+\dot{Q}_{30} \tag{11}\]  
The frictional power can be calculated, in accordance with Equation (12), when Equation (5) is taken into account in Equation (11). Here, \(\underline{A}\) denotes the column matrix from Equation (13). It contains the form coefficients. The column matrix, \(\underline{\Theta}\), in Equation (14) summarises the temperatures required for the calculation. See Appendix A for the complete derivation. \[P_{R}=\underline{A}\cdot\underline{\Theta} \tag{12}\]  
\[\underline{A}:=[(a_{10}+a_{30}),\;a_{12},\;-(a_{10}+a_{12}+a_{30})] \tag{13}\]  
\[\underline{\Theta}:=[\theta_{0},\;\theta_{2},\;\vartheta_{\text{M}}]^{T} \tag{14}\]  
All temperatures in \(\underline{\Theta}\) can be measured by sensors with little effort, except the sliding surface temperature. However, Equation (9) can be used to calculate the sliding surface temperature. Equation (12) also requires the calculation of each form coefficient. The number of calculation steps can be significantly reduced by replacing the form factor matrix (\(\underline{A}\)) and reducing the temperature matrix (\(\underline{\Theta}\)). This can be achieved with a small number of transformations. In this way, the form factor matrix can be reduced to an equivalent form coefficient (\(a_{\text{neq}}\)):  
\[a_{\text{neq}}:=(a_{10}+a_{30})(a_{0}+1)+a_{12}. \tag{15}\]  
The advantage of defining the equivalent form coefficient is that it eliminates the need for matrix operations in Equation (12), thus simplifying the calculation of the frictional power, as shown below:  
\[P_{\text{R}}=a_{\text{neq}}(\hat{\theta}_{2}-\vartheta_{\text{M}}). \tag{16}\]  
See Appendix B for the full derivation of the equivalent form coefficient. Note that the heat transfer coefficient also applies to the equivalent form coefficient, so \(a_{\text{geq}}:\mathbb{R}^{2}\rightarrow\mathbb{R}\), as described in Section 2.1. The condition of the flow around the mechanical seal can be described by the speed of the primary ring (\(n\)). With little effort, the temperature of the pumped medium and the temperature of the mating ring can be measured in addition to the speed. For the visualisation of the equivalent form coefficient, a dimensional analysis was performed in accordance with the description of [PERSON]. [PERSON] et al. (2018). Equation (17) gives the equivalent dimensionless form coefficient (\(A_{\text{neq}}\)), where \(\eta\) is the dynamic viscosity of the medium, \(c_{\text{v}}\) is the specific heat capacity of the medium, and \(D_{\text{GR}}\) is the outer diameter of the mating ring. \[A_{\text{neq}}:=\frac{a_{\text{neq}}}{\eta c_{\text{v}}D_{\text{GR}}}. \tag{17}\]  
It can be deduced that \(A_{\text{neq}}\propto\overline{Nu}\) due to the dependence of the equivalent form coefficient on the heat transfer coefficient. This allows a regression function to be defined in accordance with the structure shown in Equation (4). Algorithm 1 below summarises the soft sensor principle described. The index \(t\) denotes the discrete time steps. Algorithm 1 incorporates a low-pass filter to attenuate sensor signal noise. The damping characteristicsare controlled by the damping factor, \(d_{\rm f}\). The soft sensor requires input variables, namely the measured signals for speed, medium temperature, and mating ring temperature at a discrete time step, \(t\) (\(n_{\rm exp}^{t},\ \theta_{\rm M,exp}^{t},\ \theta_{\rm 2,exp}^{t}\)). The outer diameter of the counter ring (\(D_{\rm GR}\)) is used as the characteristic length to calculate the Reynolds number (\(Re\)):  
\[Re:=\frac{\pi mD_{\rm GR}}{\
u}. \tag{18}\]  
This kinematic viscosity is known as \(\
u\).