# Soft Sensor Technology for the Determination of Mechanical
## 1 Introduction
### Derivation of Mechanical Seal Heat Flow Model
Generally, heat flows from surface \(j\) to surface \(i\) (\(\dot{Q}_{ij}\)) can be represented below based on the form coefficient method [21]:  
\[\dot{Q}_{ij}:=k_{ij}\big{(}\theta_{i}-\theta_{j}\big{)}. \tag{2}\]  
The form factor (\(k_{ij}\)) includes factors for conduction (e.g., thermal conductivity, \(\lambda\)) and convective heat transfer (e.g., the heat transfer coefficient, \(\alpha\)). When convective heat transfer occurs, \(k_{ij}:\mathbb{R}^{2}\rightarrow\mathbb{R}\)  applies, as the heat transfer coefficient depends on the medium temperature and the flow state of the convective flow. The heat transfer coefficients for mechanical seals have already been studied, as described in the introduction. These can be represented by the Nusselt number (\(Nu\)), as follows [22]:  
\[Nu:=\frac{\alpha L}{\lambda}. \tag{3}\]  
A characteristic length (\(L\)) and the thermal conductivity of the medium are used to standardise the heat transfer coefficient. As shown by [PERSON] and [PERSON], there are correlations that can be used to describe the averaged Nusselt number (\(\overline{Nu}\)) for mechanical seals [18]. Contrary to Equation (3), the averaged Nusselt numbers apply to the whole area considered. In general, Equation (4) [22] can be used to describe the functional relationships of the averaged Nusselt numbers:  
\[\overline{Nu}:=cRe^{m}Pr^{h}\bigg{(}\frac{Pr}{Pr_{W}}\bigg{)}^{b}. \tag{4}\]  
Depending on the geometry and considering the heat transfer conditions, the coefficients \(c,\ m,\ h\), and \(b\) vary. \(Re\) is the Reynolds number, \(Pr\) is the Prandl number of the medium at its temperature, and \(Pr_{W}\) is the Prandl number of the medium at the temperature of the solid wall. Instead of describing the heat flows, \(\dot{Q}_{ij}\), with the form coefficients in accordance with Equation (2), Equations (5) and (6) are introduced:  
\[\dot{Q}_{ij}:=a_{ij}\big{(}\theta_{i}-\theta_{j}\big{)}. \tag{5}\]  
\[a_{ij}:=\alpha A_{i}E_{ij} \tag{6}\]  
The form coefficient is now \(k_{ij}\dot{=}a_{ij}\). The advantage of this representation is that it explicitly includes the heat transfer coefficient, so that the Nusselt number approaches can be applied. The transmission efficiency from surface \(j\) to surface \(i\) (\(E_{ij}\)) now describes the influence of geometry on the specific heat flows. This is calculated as follows:  
\[E_{ij}:=\frac{\theta_{i}-\theta_{\text{M}}}{\theta_{j}-\theta_{\text{M}}}\ \forall\ i\
eq 2 \tag{7}\]  
It can be seen from Equation (7) that the transmission efficiency, \(E_{20}\), is excluded from the definition. This is because there is no heat transferred from the mechanical seal to the medium at face 2. Figure 1b shows that \(\dot{Q}_{20}=\dot{Q}_{12}\). From this context, \(E_{20}\) can be derived:  
\[E_{20}:=\frac{\theta_{1}-\theta_{\text{M}}}{\theta_{0}-\theta_{2}}. \tag{8}\]Furthermore, the temperature at the sliding surface can be calculated from \(\dot{Q}_{20}=\dot{Q}_{12}\) in accordance with Equation (9), taking into account Equation (10):  
\[\vartheta_{0}:=(a_{0}+1)\vartheta_{2}-a_{0}\vartheta_{\text{M}} \tag{9}\]  
Figure 1:0