# Soft Sensor Technology for the Determination of Mechanical
## 3 Evaluation of the Accuracy of the Soft Sensor Determination
### Quality of Soft Sensor Calculation for Stationary Operation of Mechanical Seals  
The soft sensor algorithm described in Algorithm 1 can be analysed under different operating conditions using experimentally determined approximation functions for the three seals. The following tests have been carried out on PO001. To evaluate the quality of the soft sensor, the root mean squared error (\(RMSE\)) is used according to Equation (24) [32]. \(P_{\text{R,exp}}\) is the measured frictional power and \(P_{\text{R}}\) is the frictional power determined by the soft sensor. \[RMSE:=\sqrt{\sum_{i=1}^{N}\frac{\left(P_{\text{R,}i}-P_{\text{R,exp},i}\right) ^{2}}{N}} \tag{24}\]  
Figure 7a shows the \(RMSE\) plotted in black against mechanical seal differential pressure (\(\Delta p\)). A preload of \(s=5\) mm was applied to the primary seal ring. During the measurements, the speed of the primary ring was \(n_{\text{exp}}=1480\) rpm. Besides the \(RMSE\), the measured friction power (\(P_{\text{R,exp}}\)) is shown in blue. The differential pressure was varied in the interval \(\Delta p\in\{0\text{ bar};5\text{ bar}\}\). When \(\Delta p\) was varied, \(P_{\text{R,exp}}\) changed only slightly. Given the numerous assumptions and the stationary model underlying the friction power calculation in Algorithm 1, an \(RMSE\leq 20\) W is considered a minor deviation. The \(RMSE\) for varying \(\Delta p\) was always \(RMSE<6\) W. Equalisation were been drawn through the measurement points for trend identification. The \(RMSE\) increases with the increase in \(\Delta p\). The \(RMSE\) was also analysed using varying \(s\) values. The results are shown in Figure 7b. Figure 6: Comparison of the calculated equivalent dimensionless form coefficients of the approach functions (black curves, labelled ‘Models’ [18; 27; 28; 29; 30; 31]) with the experimentally determined approach (blue curve, labelled ‘Approach’) plotted against the Reynolds number for the three mechanical seals, PO001, PO002 and PO003. \(RMSE\) (black) and \(P_{\text{R,exp}}\) (blue) are plotted against the preload (s). The speed of the primary ring was also set to \(n_{\text{exp}}=1480\) rpm. The differential pressure across the mechanical seal during the measurements was \(\Delta p=0\) bar. When \(s\) is varied, the \(RMSE\) also shows a weak dependence. It shows \(RMSE\) values that are similar to those obtained by varying \(\Delta p\). In addition to the variation in \(\Delta p\) and \(s\), the geometry around the mechanical seal was also varied. The heat transfer between the mechanical seal and the medium is influenced by the change in the geometry around the mechanical seal. For this purpose, fins were distributed around the mechanical seal, as shown in Figure 7c. The fins extend up to half the unloaded primary ring height. For the modified geometry around the mechanical seal shown in Figure 7c, an approximation function for the equivalent dimensionless form coefficient was again determined. The calculated curves of \(A_{\text{neq}}\) are shown as black curves in Figure 7d. These are called \"Models\". The blue curve labelled \"Approach\" represents the experimentally determined approximation function without fin geometry around the mechanical seal, while the red curve \"Approach Fin\" includes fin geometry. The experimental curves (Approach Fin) agree better with the calculated curves (Models). A possible reason is that the fin geometries improve the turbulence around the mechanical seal and thus the heat transfer. The average Nusselt number calculation models assume turbulent flow. It can be concluded that the geometry around the mechanical seal has a major influence on the heat transferred and therefore on \(A_{\text{neq}}\).