# Soft Sensor Technology for the Determination of Mechanical
## 3 Evaluation of the Accuracy of the Soft Sensor Determination
### Soft Sensor Behaviour with Variation in the Mechanical Seal Stationary Operating Conditions  
A notable feature of the heat flow model derived in Section 2.1 is the assumption of stationary temperature fields. However, the operating conditions of mechanical seals in centrifugal pumps can change. This is especially true when the pumps are speed-controlled. For example, the centrifugal pumps used in swimming pool technology are operated at two fixed points of operation. At night, the pumps are throttled back to a lower speed compared with the daytime operating point to save energy. The amplitude spectrum of the measured frictional power (\(\left|F\left(P_{\text{R,exp}}\right)\right|\)) and speed (\(\left|F\left(n_{\text{exp}}\right)\right|\)) of PO003 is shown in Figure 8a. The spectra were generated for a step response of PO003. For the step response from \(n_{\text{exp}}=1500\) rpm to \(n_{\text{exp}}=2100\) rpm, the primary ring speed (\(n_{\text{exp}}\)) was increased abruptly. The spectra were not filtered. A Hanning window was used to window the data. As Figure 8a shows, other frequencies are much more prominent in the spectrum of \(P_{\text{R,exp}}\) than in \(n_{\text{exp}}\). Figure 8b also shows this behaviour. The amplitude spectrum of the measured frictional power (\(\left|F\left(P_{\text{R,exp}}\right)\right|\)) and speed (\(\left|F\left(\Delta\beta_{\text{exp}}\right)\right|\)) of PO003 is shown. The variables are therefore not dependent on each other in a linear way. The dominant frequencies are the rotational frequency and its harmonics when analysing the spectra for constant speeds. Figure 9a shows the step response of PO003 as a function of time. The red line shows the speed of the primary ring. The measured friction power (\(P_{\text{R,exp}}\)) is shown as a black line, and the friction power (\(P_{\text{R}}\)) calculated using Algorithm 1 is shown as a blue line. The differential pressure across mechanical seal was \(\Delta p=0\) bar during measurement. \(P_{\text{R,exp}}\) follows with an overshoot and a slight delay in changing speed. Algorithm 1 also reacts to the change in speed. Algorithm 1 will stabilise at its new stationary operating point after seven seconds. This is due to the slow change in \(\Delta\theta\). Algorithm 1 cannot model the frictional power between the two stationary operating points. Compared with that at the stationary operating points, the \(RMSE\) increases significantly. The \(RMSE\) for the step response was found to be \(RMSE=11.28\) W.  
Figure 8: (**a**) The amplitude spectrum of the measured friction power (black line) and the amplitude spectrum of the measured velocity (red line) for the step response of PO003; (**b**) the amplitude spectrum of the measured frictional power (black line) and the amplitude spectrum of the temperature difference between the mating ring and the medium temperature (red line) for the step response of PO003. In Figure 9b, it is shown that \(n_{\text{exp}}\) varied in steps in an interval of \(n_{\text{exp}}\in\{500\text{ rpm};3000\text{ rpm}\}\) for PO003. The differential pressure across the seal was also \(\Delta p=0\) bar. The coloured representation of \(P_{\text{R}},P_{\text{R,exp}}\) and \(n_{\text{exp}}\) is the same as that in Figure 9a. There was a fifteen-second interval between two speeds. If the speed changes, \(P_{\text{R}}\) will be different from \(P_{\text{R,exp}}\). If there is a change in speed, there will also be a change in \(\alpha\) and the amount of heat dissipation into the medium will change. This behaviour is also shown in Figure 9c. There, \(n_{\text{exp}}\) for PO002 varied in a stepwise manner over an interval of \(n_{\text{exp}}\in\{0\text{ rpm};3000\text{ rpm}\}\). The differential pressure across the seal was \(\Delta p=0.4\) bar. The colouring of the curves of \(P_{\text{R}},P_{\text{R,exp}}\) and \(n_{\text{exp}}\) also corresponds to that in Figure 9a.