# 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 temperature field of the mechanical seal must be resolved to calculate the transmission efficiencies. The temperature fields were calculated using the two-dimensional steady-state heat conduction equation, Equation (21). The equation is derived from the three-dimensional transient heat conduction equation [22]. The equation assumes a uniform temperature distributed around the mechanical seal, i.e., that the thermal conductivity, \(\lambda\), of the mechanical seal is constant. The heat conduction equation was solved using Matlab's PDE Modeler. \[\frac{1}{r}\frac{\partial}{\partial r}\left(r\lambda\frac{\partial\theta}{ \partial r}\right)+\frac{\partial}{\partial x}\left(\lambda\frac{\partial \theta}{\partial x}\right)=0 \tag{21}\]  
Figure 4 shows the calculated transmission efficiencies from Equations (7) and (8) for the three mechanical seals. To calculate the transmission efficiency, the averaged surface temperatures were used. A constant friction power of \(P_{\text{R}}=100\) W was applied to each of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **PO001** & **PO002** & **PO003** \\ \hline inner diameter (\(d_{i}\)) & 38 mm & 38 mm & 50 mm \\ \hline outer diameter mating ring (\(D_{\text{GR}}\)) & 54 mm & 54 mm & 69 mm \\ outer diameter primary ring (\(D_{\text{GL}}\)) & 46 mm & 46 mm & 58 mm \\ characteristic length mating ring (\(I_{\text{CR}}\)) & 11.3 mm & 11.3 mm & 10.5 mm \\ characteristic length primary ring (\(I_{\text{GL}}\)) & 5 mm & 5 mm & 5.4 mm \\ characteristic width mating ring (\(W_{\text{GR}}\)) & 5 mm & 5 mm & 4.5 mm \\ characteristic width primary ring (\(W_{\text{GL}}\)) & 3.5 mm & 3.5 mm & 3.9 mm \\ \hline material mating ring & SiC1 & SiC2 & SiC1 \\ material primary ring & SiC1 & SiC1 & SiC1 \\ heat conductivity mating ring (\(\lambda_{\text{GR}}\)) & 120\(\frac{\text{W}}{\text{mK}}\) & 100\(\frac{\text{W}}{\text{mK}}\) & 120\(\frac{\text{W}}{\text{mK}}\) \\ heat conductivity primary ring (\(\lambda_{\text{GL}}\)) & 120\(\frac{\text{W}}{\text{mK}}\) & 120\(\frac{\text{W}}{\text{mK}}\) & 120\(\frac{\text{W}}{\text{mK}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the three mechanical seals under analysis: PO001, PO002 and PO003. Figure 3: Surface images of the sliding surface of the PO002 seal made of the material SiC2 (**left**-hand side) and PO001 made of the material SiC1 (**right**-hand side). the rings as a boundary condition at the sliding surface. A constant medium temperature of \(\vartheta_{\text{M}}=20\)\({}^{\circ}\text{C}\) was also assumed. As suggested by [PERSON] and [PERSON] (2007) [17], the transmission efficiencies are plotted against the parameter \(\dfrac{L}{\text{W}}\sqrt{\dfrac{\alpha\,\text{W}}{\lambda}}\). In addition to the heat transfer coefficient and thermal conductivity, the characteristic lengths (\(L\)) and widths (\(W\)) from Table 1 also contribute to the parameter. At the surfaces to the medium (surfaces 1 and 3), a constant heat transfer coefficient was considered. Depending on the parameter \(\dfrac{L}{\text{W}}\sqrt{\dfrac{\alpha\,\text{W}}{\lambda}}\), \(\alpha\) was varied. The convective heat flows (\(\dot{Q}_{\text{C},i}\)) at the surfaces were considered due to the assumed constant medium temperature and heat transfer coefficient, neglecting the time dependence of the temperature field, as shown in Equation (21). The transmission efficiency, \(E_{20}\) (red graph), converges to zero as the heat transfer coefficient increases. It differs from the other heat transmission coefficients shown as it does not exhibit values greater than one for small heat transfer coefficients. The limiting case of Equation (8) shows that the following:  
\[\underset{\alpha\to 0}{\lim}E_{20}=\infty. \tag{22}\]  
Since the only source of heat dissipation in the model is heat transfer between the medium and the mechanical seal, this seems logical. Moreover, all the mechanical seals show that \(E_{10}\approx E_{30}\).