# Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids
## 8 Conclusions  
The article deals with modelling and simulation issues of pulse discharges occurring in liquid environments, used in many techniques. As it was mentioned in the introduction, there is a lack of universal and accurate models to simulate all stages of discharges (initiation, pre-breakdown, and arc). Most popular models of discharges in liquids (especially dielectric) were shortly described. Because of the lack of information about the parameters of discharges in juices, preliminary tests were performed. Based on the authors' laboratory tests, most important electrical parameters of discharges were performed. Additionally, specific transient characteristic of discharge propagations were determined. Considering the obtained results, discharge was classified as \"slow.\" In such type of discharge, gas bubble theory model was used to describe the phenomena of streamers propagation paths. In the paper, the numerical method and algorithm for pre-breakdown were characterized. Subsequently, analysis of electric potential gradients in vicinity of the streamer's heads were used to determine the streamer position. The proposed method, based on gradients determination and probability distribution, was used to establish total discharge path and time characteristic of discharge growth. The mentioned phase of discharge is described in Section 5 of the paper. It is worth to highlight that the proposed method enables to determine energy values based on momentary values of current and voltage.  
In the Section 6, authors' solutions in the scope of modelling of breakdown (arc) stage of discharge were detailed. For the given geometry of discharge path (determined in previous steps of the analysis), the total energy generated in the arc plasma was established using iterative calculations. Plasma conductivity and the cross section of plasma channel were computed using analytical formulas described in Section 6. Coupled analysis of electromagnetic and thermal fields (in FEM system) were performed to determine the total power of the arc and transient temperature characteristics. Circuit models were used to simulate the power source in the form of RC system. Different capacity values of series capacitor enable to form different arc current characteristics. The CFD models were used in the last stage, where temperature distribution in the time between next discharges was computed. Based on the results, it was possible to consider nonlinear material parameters for the next discharge distribution modelling.  
Results presented in the paper concern discharges in juices and other liquids. All algorithms can be classified as original Authors' solutions. It has to be highlighted, that similar solutions are rarely presented in scientific papers. Obviously, discussed models and algorithms, can be extended to obtain results characterized by higher accuracy. However, obtained results are promising and are characterized by high versatility.  
Conceptualization, methodology, software, writing--original draft preparation, visualization: M.W.; methodology, validation, P.K.; formal analysis, resources: S.K., writing--review and editing, supervision: S.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.