# Soft Sensor Technology for the Determination of Mechanical
## 1 Introduction
### Derivation of Mechanical Seal Heat Flow Model  
Figure 1a shows the design of a mechanical seal made up of several components used in a centrifugal pump. The mating ring is the stationary element of the mechanical seal and is sealed to the pump housing by means of an O-ring. The primary ring rotates with the pump shaft and is pressed against the mating ring by a spring via the pump impeller disc. An elastomer prevents leakage between the pump shaft and the primary ring. When the primary ring is rotating and the pumped mediums are present, the gap between the primary ring and the mating ring increases. Only a small amount of leakage escapes from the pump due to the small size of the gap, in the range of one micron [20]. [PERSON] and [PERSON] and [PERSON] (2007) use the analytical description of heat flows in fins to calculate the temperature of the sliding surface (\(\theta_{0}\)). They define transmission efficiencies (\(E\)) for different mechanical seal geometries in accordance with Equation (1) [15; 17]. Equation (1) establishes a relationship between the heat flow (\(\dot{Q}\)) through the mechanical seal and the heat flow expected at a temperature difference between a reference temperature and the temperature of the medium inside the pump. This is calculated from the heat transfer coefficient between the mechanical seal and the surrounding medium (\(\alpha\)), the area of the  
mechanical seal in relation to the surrounding medium (\(A\)), and the difference between the reference temperature (\(\theta_{\text{ref}}\)) and the temperature of the medium inside the pump (\(\theta_{\text{M}}\)). \[E:=\frac{\dot{Q}}{\alpha A(\theta_{\text{ref}}-\theta_{\text{M}})} \tag{1}\]  
The model used to derive the heat flows in the mechanical seal is shown in Figure 1b. The model was developed based on the form coefficient method. This method has been described, for example, by [PERSON] [21]. The form coefficient method balances heat flows at specific points. Heat flows or temperatures can therefore be calculated for stationary temperature fields in bodies. The sliding surface temperature (\(\theta_{0}\)) increases due to the friction power (\(P_{\text{R}}\)). It is assumed that the heat flows generated in the rings can only be dissipated to the medium inside the pump with the temperature, \(\theta_{\text{M}}\). All other surfaces should be considered to be insulating. In the mating ring, heat flows from surface 0 to surface 2 (\(\dot{Q}_{20}\)) where a temperature sensor can be mounted. There is also a heat flow from the sliding surface  
Figure 1: (a) Design of a mechanical seal with its components in the application in a centrifugal pump; (b) heat flow model of a mechanical seal with a specification of the surface temperatures (\(\theta_{i}\)), the heat flows in the rings of the mechanical seal (\(\dot{Q}_{ij}\)), the convective heat flows between the mechanical seal and the medium inside the pump (\(\dot{Q}_{Cj}\)), the frictional power of the mechanical seal (\(P_{\text{R}}\)), and the characteristic width (\(W_{i}\)) and length (\(L_{i}\)) of the rings of the mechanical seal. to surface 1 (\(\dot{Q}_{10}\)) where heat is transferred to the medium. Heat flows from surface 2 to surface 1 due to the temperature gradient (\(\dot{Q}_{12}\)). Heat flows in the primary ring from the seal face to face 3 (\(\dot{Q}_{30}\)) where the heat transferred to the seal ring is dissipated. It is assumed that the sliding surface temperature in the mating and primary rings is the same, as suggested by [PERSON] [15]. In addition to the form coefficient method being used, transmission efficiencies are also calculated for stationary temperature fields [15; 17; 21].