# Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids
## 5 Modeling of the Discharge Development  
Numerical calculations of discharges in liquids are divided into three basic stages, allowing for the implementation of the modeling objectives described in the previous section:  
* Determination of the streamers propagation (pre-breakdown);
* Determination of discharge breakdown (arc) stage parameters (voltage, current and energy values);
* Modeling of the discharge system including the power source to select appropriate capacitor used during discharge process. The analysis presented in this paper uses a model consisting of two flat electrodes of 50 mm diameter. Active area (model of liquid juice), wherein the discharge occurs, was placed between the electrodes. Distance between the electrodes was 60 mm. The geometry of the model is shown in Figure 7a. Figure 6: Schematic diagram of the calculation algorithm. The gas bubble theory was used to determine streamers propagation in the active area. This assumption was introduced, because of the results of preliminary analysis presented in previous section of the paper. According to the obtained results, one can classify the discharge as \"slow\" [39] and theoretical description of gas bubble theory is applicable in such conditions [14]. The static models implemented in ANSYS program were used to determine electric field distributions. According to the proposed procedure, streamers development was computed using a special \"virtual\" surfaces arranged in parallel to electrodes (Figure 7b). Adopted \"virtual\" surfaces form a kind of differential division of the active area. All streamers heads can be located only on these surfaces and streamers spread are computed using a \"frame by frame\" method. So that, resolution of calculation results depends directly on the number of surfaces. The calculation procedure has been detailed further in the paper. Computations of streamers development could not be performed using only commercial FEM system (ANSYS). Some additional procedures were implemented in Authors' subprograms. Mentioned procedures were used to compute electric field gradients, voltage values at streamers heads, and to generate geometry of streamers paths and gas bubbles. According to the description of the HVED technique, a series of several tens to several hundred discharges is used [4; 9]. Therefore, the procedure for determining the streamers propagation have to be performed many times, regarding the number of analyzed impulses. The initial conditions for each impulse, in the form of variable physical parameters of the liquid environment, have been introduced based on the temperature distribution that occurs after the previous pulse was generated (Figure 6). During the modeling of the first discharge, it was assumed that the active area is characterized by homogeneous temperature, conductivity, and concentration of charges. The solution of the electric field intensity equation and the electric potential (4) for the electrostatic problem in such a system always leads to results, where the electric field intensity is homogeneous throughout the analyzed area and the electric potential distribution is characterized by iso-surfaces parallel to the electrodes (Figure 8a). \[div(\varepsilon E)=4\pi\rho \tag{4}\]  
where: **E**--electric field intensity; \(\varepsilon\)--dielectric permittivity; \(\rho\)--charge density  
Figure 8: Distribution of the electric potential in the model not including (**a**) and including the gas bubble (**c**) and the bubble model on the surface of the upper electrode (**b**). 1—electrode; 2—discharge area; 3—gas bubble. Because of homogeneous electric field distribution, discharge initiation point cannot be clearly determined. In order to initiate first discharge, the random number generator was used, implemented in a MathCAD [43] program. Node numbers (of the finite element mesh) lying on the cathode surface were exported to this software. Random node number was generated and re-entered into the ANSYS program to initiate the discharge.