# 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 mating ring length to width ratio (\((L/W)_{\text{GR}}\)) for seals PO001 and PO002 is approximately 1.6 times that of the primary ring (\((L/W)_{\text{GL}}\)). For PO003, \((L/W)_{\text{GR}}\approx 1.7(L/W)_{\text{GL}}\). Given that \(E_{10}\approx E_{30}\), comparable transmission efficiencies can be achieved in the mating ring with lower heat transfer coefficients than in the primary ring. As shown in Figure 1b, this is due to the larger area of \(A_{1}\) compared with that of \(A_{3}\)  
Figure 4: Numerically calculated transmission efficiencies (\(E_{10}\), \(E_{12}\), \(E_{30}\) as black curves and \(E_{20}\) as red curve) of the three mechanical seals analysed with different heat transfer coefficients and a constant friction power of \(P_{\text{R}}=100\) W, plotted against the parameter \(\dfrac{L}{\text{W}}\sqrt{\frac{\alpha\,\text{W}}{\lambda}}\) in accordance with [PERSON] and [PERSON] [17]. The differences in transmission efficiency between PO001 and PO002 can be considered small. The mating ring material is different for these two seals. In addition to resolving the temperature field, the model for calculating the averaged Nusselt number (\(\overline{Nu}\)) must be known to calculate the equivalent dimensionless form coefficient. [PERSON] et al. compared several models for the determination of \(\overline{Nu}\) with their own model [27]. These are divided into analogue model approaches [28; 29] and mechanical seal approaches [30; 31]. By varying the \(\overline{Nu}\) approach, the equivalent form coefficient was calculated for the three mechanical seals. Equation (15) was used to calculate the equivalent form coefficient. All equations used to calculate \(\overline{Nu}\) were taken from [PERSON] et al. [27]. These include the approaches of [PERSON] [28], [PERSON] and [PERSON] [29], [PERSON] and [PERSON] (2008) [31] and [PERSON] et al. [30]. The approach presented by [PERSON] and [PERSON] was also used [18]. For all three mechanical seals, the calculated equivalent dimensionless form coefficients (\(A_{\text{neq}}\)) are plotted against the Reynolds number in Figure 5. The individual graphs were labelled according to the first authors' surnames of [18; 27; 28; 29; 30; 31]. The equivalent dimensionless form coefficient is significantly influenced by the approach used for characterising the heat transfer. Since the equivalent dimensionless form coefficient depends on the heat transfer coefficient and thus on the average Nusselt number, the function from Equation (4) can be used. Factors \(h\) and \(b\) from Equation (4) were set to \(h=1/3\) and \(b=0.25\). The fitted coefficients of the approach functions for the equivalent dimensionless form coefficients as a function of the approach functions of the averaged Nusselt numbers are shown in Table 2. In addition to the coefficients of the calculated approximate functions, the coefficients of equivalent functions determined from measurements of the three mechanical seals (Approach) are also given. Figure 5: Calculated equivalent dimensionless form coefficients (\(A_{\text{neq}}\)) of the three mechanical seals analysed as a function of the approach functions for the calculation of the averaged Nusselt number (\(\overline{Nu}\)) for the determination of the heat transfer coefficient (\(\alpha\)) of different approach functions, denoted by the surname of the first authors [PERSON] [27], [PERSON] [28], [PERSON] [29], [PERSON] [30], [PERSON] [31], and [PERSON] [18]. The coefficient \(m\) in the models of [PERSON] (2001) and [PERSON] (2002) is different from that in the other models. For the experimental determination of the equivalent dimensionless form coefficients, the differential pressures across the mechanical seal (\(\Delta p\)) and the primary ring preload (\(s\)) were kept constant. In the interval \(n_{\text{exp}}\in\{0\) rpm; \(3000\) rpm\(\}\), the speed of the primary ring (\(n_{\text{exp}}\)) was varied.