# Soft Sensor Technology for the Determination of Mechanical
## 1 Introduction  
Mechanical seals play a crucial role in the operation of centrifugal pumps. They seal the medium inside of the pump from the surrounding atmosphere in the area where the shaft passes through the pump housing. Monitoring methods have been developed recently due to their importance for the operation of the pump. [PERSON] et al. demonstrated the ability to detect damage to the mechanical seal of a centrifugal pump during operation [1]. This was achieved by analysing the vibrations of the pump. It has been recognised that damage to the mechanical seal causes higher-frequency vibration signals than those that normally occur in the centrifugal pump during operation. [PERSON] et al. analysed the power consumption of a centrifugal pump driven by an asynchronous motor under different mechanical seal conditions [2]. Their analysis was limited to the scenario of an increased leakage rate in a mechanical seal. They showed that the characteristic curves of the centrifugal pump are also affected by increasing leakage at the mechanical seal. Depending on the pump design, the mechanical seal is responsible for most of the mechanical losses inside the pump. The mechanical losses are part of the pump characteristic curves. The pump characteristic curve can be used to determine the flow rate of the pump. Because of the importance of the flow rate delivered by the pump as a process variable, there are many algorithms for determining the flow rate using the pump curve. [PERSON] et al. have studied the application of the affinity laws to the pump curve at a reduced pump speed [3]. They showed that changing the pump speed over a wide range results in significant differences between the calculated and observed pump curve. We included the mechanical powerloss of the pump in a physical model for determining the set of performance curves for radial centrifugal pumps [4]. As a result, the prediction of the pump characteristic curves at a reduced pump speed was considerably improved compared to that under the affinity laws. Frictional torque is currently measured using torque sensors that require integration onto the mating ring of the mechanical seal to determine the friction power performance. This method has been demonstrated in several papers [5; 6; 7; 8]. Soft sensing methods can be used to determine the frictional power of mechanical seals, overcoming the challenges of implementing physical sensors. In their paper, [PERSON] et al. provide an overview of the most important developments in the field of soft sensors in recent years [9]. They present how soft sensors are developed and identify current challenges. They divide soft sensor development into four phases: variable selection, preprocessing, modelling and validation. Various methods can be employed for modelling. These can be categorised into the following groups: multivariate statistical assumptions, data-driven assumptions, machine learning, filtering methods, and first-principles methods [9]. First-principles models and filtering methods are based on the physical relationships of the systems and require comprehensive knowledge. This makes their application in complex systems more challenging. Multivariate statistical, data-driven, and machine learning models are capable of accurately describing complex systems. However, they require a significant amount of data for characterisation and training. [PERSON] et al. developed a denoising diffusion probabilistic model for time series denoising (TimeDDPM) to handle the large amounts of data required for model training [10]. TimeDDPM has the ability to generate new data sets from existing samples to enhance LSTM training. [PERSON] et al. demonstrated, in two examples, that TimeDDPM can reduce prediction errors compared with other methods, such as a simple LSTM without dataset extension. In addition to the required data volumes, the measurement noise from the physical sensors is also a significant factor in the modelling. [PERSON] et al. propose a method for noise filtering [11].