# Soft Sensor Technology for the Determination of Mechanical
## 1 Introduction
Ideal data, filtered for measurement noise, are used to train kernel learning models. [PERSON] et al. demonstrated, in two examples, that this approach can enhance prediction accuracy. However, it is stated that further research is needed to generalise the improvements in the estimation error, specifically regarding the influences of the incorrect modelling of the model error. [PERSON] et al. also identified hyperparameters of the models as a challenge in soft sensor development [9]. Soft sensors can serve as substitutes for variables that are challenging to measure and as backups in the case of sensor failure. [PERSON] et al. describe two data-driven methods for calculating differential pressure in a district heating network [12]. If the pressure sensor fails, the system should still be controlled by using the soft sensor as a substitute. Various convolutional neural network and long short time memory approaches, as well as transfer function approaches, were analysed. It was demonstrated that the simplest model structures yielded the best results, despite the complex interrelations within the district heating network. Recently, machine learning and data-driven models have gained popularity. This enabled the accurate determination of target values, even in complex systems. The challenges presented by low data volumes and noise behaviour were addressed with innovative methods. The review by [PERSON] et al. also identifies collecting and analysing data as a major challenge in condition monitoring [13]. [PERSON] et al. demonstrated that even in complex systems, simplified approaches can achieve decreased errors [12]. Although many models perform well in the laboratory, their implementation in industrial applications requires significant effort and a robust database, especially for data-driven models [9]. However, [PERSON] et al. 's results motivate the investigation of simple modelling approaches for complex systems. Given the critical role of mechanical seal temperature in operational performance and early failure detection, several studies have focused on the analysis of heat flows and temperature fields within mechanical seals. The studies' findings can aid in the development of a soft sensor. [PERSON] et al. demonstrated that the selection of an appropriate material combination for mechanical seal rings can effectively reduce the sliding face temperature during dry running compared with alternative material combinations [14]. They calculated the temperature fields in the mechanical seal analytically. Therefore, the half-infinite model was used. Their conclusion was that the lower sliding surface temperatures were due to the lower coefficient of friction (COF). In the initial phase, the analytical model reproduces the temperature-time behaviour of the sliding surface with deviations. However, the calculated temperature at the sliding surface in a steady-state shows good agreement. [PERSON] et al. investigated the heat flows at a mechanical seal [8]. The frictional torque of the mechanical seal was also measured using a torque sensor. The seal was installed in a centrifugal pump. The local Nusselt number was determined at several positions. The results were compared with the literature data. They found that surrogate models, such as those that use cylinder geometries to calculate the Nusselt number, do not adequately represent the actual Nusselt numbers observed in mechanical seals. The work of [PERSON] has shown that the predominant consideration in determining heat flows within mechanical seals is the heat transfer to the surrounding fluid [15]. In his analysis, [PERSON] used a one-dimensional model of the heat flows transferred from the sliding surface to the surrounding fluid. The concept of the model is based on the fin theory described in [16]. Transmission efficiencies are used to account for the effects of the multi-dimensional temperature field. [PERSON] and [PERSON] (2007) also used this approach [17]. [PERSON] and [PERSON] have analysed a mechanical seal using numerical simulation [18].