## 2 Materials and Methods
### Energy Balance  
In order to study the effect of electrolytic plasma hardening for 10 s on the temperature distribution in 20 GL steel samples in three modes of 150 V, 200 V and 250 V, the values of the amount of heat released during the current flow during EPT were calculated for each of the three modes. The amount of heat, in turn, was calculated based on the dynamics of electric power during EPT by integrating the function P(t) over time. Since the nature of the function P(t) was complex with many local minima and maxima, the integration was carried out using the trapezium formula  
\[Q_{e}\approx\sum_{i=0}^{N-1}P_{i}\cdot\Delta t_{i}\approx\sum_{i=0}^{N-1} \frac{P_{i}+P_{i+1}}{2}\cdot\Delta t_{i}, \tag{1}\]  
where \(Q_{e}\)--amount of heat released during the EPH process; \(P_{i}\) and \(P_{i+1}\)--adjacent values of electric power on the \(P(t)\) graph; and \(\Delta t_{i}\)--time difference equal to \(\Delta t_{i}=t_{i+1}-t_{i}\) and determined separately for each value \(P_{i}\) and \(P_{i+1}\). The graphs of the function \(P(t)\) were derived by multiplying the corresponding values of the functions \(I(t)\) and \(U(t)\) using the formula  
\[P(t)=U(t)\cdot I(t). \tag{2}\]  
To estimate the heat flux density during electrolytic plasma hardening of 20 GL steel, it is necessary to consider the law of conservation of energy. Research [14; 15; 16] has established that the conversion of electrical energy into heat occurs in the plasma layer on the processed surface of the cathode sample. Based on these assumptions, the energy distribution during EPH of the cathode sample is presented as follows:  
\[Q_{k}=\ Q_{e}-Q_{p}-Q_{r}-Q_{h}, \tag{3}\]  
Figure 2: Electrical schematic diagram of measuring EPH parameters. where \(Q_{k}\)--energy spent on heating the plasma layer and the cathode; \(Q_{r}\)--amount of heat calculated using Formula (1); \(Q_{r}\)--energy spent on evaporating the electrolyte; \(Q_{r}\)--radiation energy of the plasma layer; \(Q_{h}\)--energy spent on heating the electrolyte. It is now necessary to evaluate the terms included in Equation (3). The values for \(Q_{c}\) are obtained from Formula (1). Given that the thickness of the vapor-plasma shell is small compared to its perimeter, the radiation of the plasma layer can be neglected [17], thus \(Q_{r}=0\). The term \(Q_{h}\) can be calculated using the formula  
\[Q_{h}=c\cdot m\cdot\Delta T, \tag{4}\]  
where \(c\)--specific heat capacity of the electrolyte solution; \(m\)--mass of the entire solution in the reservoir; \(\Delta T\)--temperature increase during a single quenching of the sample over a time of \(t=10\) s.  
To estimate the heat used for electrolyte evaporation in the cathode area, it is necessary to determine the nature and volume of the vapor-gas envelope formed during electrolytic plasma treatment. According to a number of experimental studies, the vapor-gas envelope at cathodic polarity has a pulsating nature [18; 19; 20]. Initially, intensive electrolyte evaporation occurs near the cathode, forming a vapor-gas shell. Upon reaching a certain threshold, the shell breaks down and a discharge occurs, destroying the shell. The cycle then repeats, accumulating a new shell. This pulsation was confirmed by oscillographic recordings of current strength and visual observations of the process. On the I graph, each cycle corresponds to the duration between two successive local minima and maxima. This discharge dynamic supports the idea that numerous local phase transitions of liquid into steam occur during the EPT process (for example, lasting 10 s), resulting in shell formation. Therefore, the heat \(Q_{v}\) should be estimated by considering multiple cycles rather than a single shell volume. The average discharge frequency extracted from the analysis of the \(I(t)\) graph can be used to estimate the number of cycles. Consequently, during the processing time, numerous complete cycles of shell accumulation and destruction can be observed, each accompanied by partial evaporation of the electrolyte volume around the cathode.