## 2 Materials and Methods
### Estimation of Heat Fluxes  
After evaluating all the terms on the right-hand side of (3), the value of \(Q_{k}\) is determined. To assess the degree of thermal impact on the material and construct temperature profiles, it is advisable to calculate the heat flux density. The average heating power of the cathode was then determined as  
\[P_{k}=\frac{Q_{k}}{t}, \tag{8}\]  
where \(P_{k}\)--average heating power of the cathode over time; \(t\)--duration of hardening (10 s). The heat flux density, which corresponds to the average heating power, is then calculated as follows:  
\[q=\frac{P_{k}}{S}, \tag{9}\]  
where \(S\)--area of the contact surface of the sample with the electrolyte.  
At the same location where the heat flux \(q\) is applied, intense heat exchange with the surrounding electrolyte also takes place. The electrolyte washes the lower surface of the sample, removing heat through convection. Consequently, the temperature distribution within the material was formed by the interaction of the heating source and the cooling effect of convective heat exchange.  
To correctly take this effect into account in numerical modeling, it is necessary to determine the heat transfer coefficient \(\alpha\). This coefficient relates the heat flow to the temperature difference between the sample and the electrolyte, as described by [PERSON]'s law:  
\[q_{c}=\alpha\cdot\left(T_{surf}-T_{el}\right), \tag{10}\]  
where \(q_{c}\)--density of heat flow going into the electrolyte; \(T_{surf}\)--surface temperature of the sample; \(T_{el}\)--electrolyte temperature.  
To estimate the heat transfer coefficient, the method of analyzing dimensionless similarity numbers is employed, using Reynolds, Prandtl, and Nusselt numbers. In the scenario where the electrolyte flows out through the upper base of the anode cone and washes the surface of the part from below, followed by radial spreading, the characteristic size used for calculating the similarity numbers will be the average hydraulic size \(Z\) (since the surface is a rectangle measuring 20 mm by 25 mm). This size is calculated using the formula  
\[Z=\frac{2\cdot a\cdot b}{(a+b)}. \tag{11}\]  
Since the electrolyte flows out of the upper base of the cone anode with a diameter \(D\), with a flow rate of \(G=10\) L/min, the flow rate of the electrolyte at the surface of the sample is determined as  
\[v=\frac{4\cdot G\cdot 10^{-3}}{60\cdot\pi\cdot D^{2}}\;(\text{m/s}). \tag{12}\]  
Then the Reynolds number is defined as  
\[Re=\frac{v\cdot Z}{v}, \tag{13}\]where \(v\)--flow rate of the electrolyte at the surface of the sample; \(Z\)--characteristic size of the streamlined body; \(v\)--kinematic viscosity of the electrolyte solution. The Prandtl number for an electrolyte solution is calculated using the formula  
\[Pr=\frac{c\cdot\mu}{\lambda}, \tag{14}\]  
where \(c\)--specific heat capacity of the electrolyte solution; \(\mu\)--dynamic viscosity of the electrolyte solution; \(\lambda\)--thermal conductivity of the electrolyte solution.  
The flow characteristics observed in the experiment and subsequent calculations definitively indicate laminar flow within the system under investigation. Based on this, according to the literature [23], the formula for calculating the Nusselt number is as follows:  
\[Nu=0.664\cdot Re^{0.5}\cdot Pr^{1/3}. \tag{15}\]  
This formula is valid for flow around a plate when the Reynolds number is less than 500,000. Following this, the heat transfer coefficient \(\alpha\) is determined using the formula  
\[\alpha=\frac{Nu\cdot\lambda}{Z}. \tag{16}\]  
The required calculation parameters (\(L\), \(c\), \(\lambda\), \(\rho\), \(\mu\), \(v\)) for the sodium carbonate solution were taken from [24].