# Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids
## 6 Modeling of the Breakdown (Arc) Stage  
Further analyses were carried out for the breakdown stage, where the plasma channel occurs between the electrodes (channel \"connects\" the electrodes). It was assumed, during this stage, that all \"secondary\" streamers (streamers that not reach the electrode surface) were neglected (Figure 14). To compute the current and voltage values in the time domain, it was necessary to determine the electrical conductivity of the plasma, where charge carriers are in the form of electrons and ionized particles. It was required to establish the influence of thermodynamic parameters (mainly temperature and pressure) on the conductivity value. For highly ionized plasma, the conductivity value can be computed using formulas (11), and (8) for weakly ionized plasma [39, 46]. \[\sigma_{D}=0.591\frac{\left(4\pi\varepsilon\right)^{2}(kT)^{3/2}}{\varepsilon^ {2}m^{1/2}}\frac{1}{ln\theta}, \tag{11}\]  
where: \(T\)--temperature, \(k\)--Boltzmann constant, \(m\)--mass of the particle, \(\theta\)--electron concentration, \(e\)--elementary energy  
It was assumed that in the presented system there is a plasma characterized by medium ionization degree. Electrical conductivity of the plasma was estimated as the average value from formulas (8) and (11) [39, 47]. Figure 15 presents the conductivity values as a function of the plasma temperature. Because of the mentioned relation, arc stage analysis requires coupled electromagnetic and thermal calculations. Figure 14: The geometry of the discharge obtained on the basis of streamers propagation analysis (**a**) and adopted for the analysis of the breakdown stage (**b**). 1—electrodes; 2—discharge. Figure 13: Densities of currents in streamers at various times (**a–d**) of the discharge development. Dimensions of the plasma channel were determined automatically. In the case of a circular arc, the equation for radius of the plasma channel can be specified as [47]:  
\[\tau_{l}=\left(\frac{2}{\pi^{2}}\right)^{1/4}\cdot\left(\frac{1}{\sigma\rho h} \right)^{1/4}\cdot\left(2g\frac{\rho_{a}}{\rho}\right)^{1/8}\cdot z^{1/8}\cdot l ^{1/2}, \tag{12}\]  
where: \(I\)--current, \(z\)--arc length, \(g\)--gravity, \(h\)--specific enthalpy of the arc plasma, \(\rho\)--arc plasma density  
According to formula (12), to determine the plasma channel radius, it is necessary to input discharge current value and iterative calculations must be performed to use it. The values of discharge currents were introduced on the basis of the results obtained in previous iteration steps of the conducted analyzes. In the first step, the discharge diameter was assumed as 10 \(\upmu\)m (the value was the same as in the pre-breakdown modeling). The arc plasma of the given conductivity was introduced to the numerical model in the form of a solid body (Figure 14b). The physical phenomena occurring in the plasma have been omitted. Nevertheless, this simplification is acceptable in relation to determination of active power of plasma channel. Static calculations of the electromagnetic problem were carried out to compute the discharge conduction current (Figure 16a,b) and the heat sources distribution (Figure 16c). These computations were performed iteratively, with the iteration frequency depending on the thermal analysis results discussed further in the paper. Based on the conduction currents density (\(J\)) and the distribution of heat sources (\(p_{V}\)), it was possible to compute the resistance of the arc (13). The proposed method uses the total active power (\(P\)) and current (\(I\)) values, obtained by adding its elementary values in individual finite elements in the  
Figure 16: Current density vectors (**a**), current density distribution (**b**), and heat sources distribution (**c**) in the breakdown (arc) stage of discharge analysis. Figure 15: Plasma conductivity as a function of temperature. discharge model (\(i=1\,\dots\,N\)).