# Soft Sensor Technology for the Determination of Mechanical
## 3 Evaluation of the Accuracy of the Soft Sensor Determination
### Comparison of Equivalent Dimensionless form Coefficient between Numerical and Experimental Results
The experimentally determined equivalent dimensionless form coefficients were determined using Equation (23), where \(P_{\text{R,exp}}\) is the measured frictional power, \(\theta_{\text{2,exp}}\) is the corrected surface temperature at the measuring point, \(\theta_{\text{M,exp}}\) is the measured medium temperature in the pressure chamber and \(n_{\text{exp}}\) is the measured primary ring speed. \[A_{\text{aeq}}=\frac{P_{\text{R,exp}}}{\left(\theta_{\text{2,exp}}-\theta_{ \text{M,exp}}\right)}\cdot\frac{1}{\eta\left(\theta_{\text{M,exp}}\right)\cdot c _{\text{v}}\left(\theta_{\text{M,exp}}\right)\cdot D_{\text{GR}}} \tag{23}\]  
The material data of dynamic viscosity (\(\eta\)) and specific heat capacity (\(c_{\text{v}}\)) of the medium in the pressure chamber were determined as a function of the measured medium temperature (\(\theta_{\text{M,exp}}\)). The corrected surface temperature at the measurement point (\(\theta_{\text{2,exp}}\)) was obtained by rearranging Equation (20). An approximation function with the structure of Equation (4) was fitted to the measurement data points. Table 2 also shows the coefficients of the approximation functions of the three mechanical seals. Figure 6 shows the approximation function (labelled \"Approach\", blue curve) together with the calculated equivalent dimensionless form coefficients as a function of the approach models used to determine the averaged Nusselt number (black curves, labelled \"Models\") for the three mechanical seals PO001, PO002 and PO003. For \(Re\leq 8\times 10^{4}\), the approximation function (Approach) is different from the calculated \(A_{\text{aeq}}\) value for PO001. [PERSON] et al. state that for \(Re\leq 1\times 10^{5}\), a laminar model should be used for the determination of the averaged Nusselt number (2001). The models used to determine the averaged Nusselt number are all for turbulent flow only. This means that no laminar models were considered in the calculation of the equivalent dimensionless form factor. In Figures 4 and 5, the differences in \(E_{ij}\) and \(A_{\text{aeq}}\) between PO001 and PO002 were judged to be small. The mating ring material differs between these two POs. Table 1 shows the thermal conductivities used to calculate \(A_{\text{aeq}}\). The measurements also show a small deviation. As shown in Table 2, the exponent \(m\) of the approach in accordance with Equation (4) is well approximated by the models of [PERSON] et al. (2001) and of [PERSON] and [PERSON] (2008) (2002). This can also be seen in Figure 6. In the logarithmic display of Figures 5 and 6, the exponent \(m\) is used to determine the slope of the curves. PO003 behaves differently to PO001 and PO002. However, PO003 has a larger deviation of the exponent \(m\) compared with the calculations. To summarise, the differences between \(A_{\text{aeq}}\) calculated using transmission efficiencies and the measured \(A_{\text{aeq}}\) value are obvious. Deviations depend on the Reynolds number. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Approach Function**} & \multicolumn{2}{c}{**PO001**} & \multicolumn{2}{c}{**PO002**} & \multicolumn{2}{c}{**PO003**} \\ \cline{2-7}  & \(\mathbf{c}\) & \(\mathbf{m}\) & \(\mathbf{c}\) & \(\mathbf{m}\) & \(\mathbf{c}\) & \(\mathbf{m}\) \\ \hline [PERSON] (2017) & 0.155 & 0.498 & 0.154 & 0.495 & 0.187 & 0.445 \\ [PERSON] (2018) & 0.051 & 0.607 & 0.052 & 0.600 & 0.111 & 0.505 \\ [PERSON] (2019) & 0.049 & 0.601 & 0.050 & 0.595 & 0.084 & 0.518 \\ [PERSON] (2001) & 0.009 & 0.729 & 0.009 & 0.722 & 0.021 & 0.622 \\ [PERSON] (2002) & 0.008 & 0.701 & 0.007 & 0.706 & 0.009 & 0.652 \\ [PERSON] (2018) & 0.027 & 0.576 & 0.026 & 0.574 & 0.035 & 0.519 \\ \hline Approach & \(1.718E-3\) & 0.810 & \(2.512E-3\) & 0.773 & \(5.240E-7\) & 1.376 \\ \hline \end{tabular}
\end{table}