# Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids
## 7 Modeling of Temperature Distribution in the Time between Discharges  
The temperature distribution after the discharge extinction, can be used to re-analyze the generation of the next discharge, carried out by using the principles given in Section 5 of this paper. For example, time required for the proposed solutions, despite the significant time consumption, enables a full analysis of discharges in liquid environments. It is worth to point out that for simplifications of numerical analysis, two different heat transfer models were used during discharge phase and break between the next discharges analysis:  
* The \"static\" approach without fluid movements modeling (15) was used to compute heat transfer process during breakdown (in the pre-breakdown (streamer) stage, thermal analysis was not performed). The special criterion value of thermal conductivity (16) was used to include thermal convection heat transfer in the fluid [49];
* The CFD model was used for heat and mass transfer analysis in the time between next discharges. Coupled equations for continuity of flow (18), momentum (19) (here presented in one dimension of Cartesians system), and energy (20) balances were computed in Ansys program. A typical turbulent k-\(\varepsilon\) model was in use. Temperature field distribution in the last step of discharge analysis was used as initial condition for CFD modeling. \[\frac{\partial t}{\partial\tau}=\frac{\mathrm{p}_{V}}{c\rho}+\frac{1}{c\rho}[ \
abla(\lambda\
abla\mathrm{t})] \tag{15}\]  
\[\lambda_{e}=\left(1+\frac{C_{1}Ra_{s}^{n}}{Ra_{s}+C_{2}}\right)\lambda_{s} \tag{16}\]  
where: \(t\)--temperature, \(\tau\)--time, \(c\)--specific heat, \(\rho\)--mass density, \(\lambda\)--thermal conductivity, \(\lambda_{e}\)--equivalent thermal conductivity (including fluid velocity), \(Ra_{s}\)--the Rayleigh number, \(C_{1}\), \(C_{2}\), \(n\)--factors depended on fluid flow type (laminar, turbulent). \[\frac{\partial\rho}{\partial\tau}+\
abla\cdot\left(\rho\overrightarrow{v} \right)=0, \tag{17}\]  
\[\rho\frac{dw_{x}}{d\tau}=\rho g_{x}-\frac{\partial p}{\partial x}+\mu\left( \frac{\partial^{2}w_{x}}{\partial x^{2}}+\frac{\partial^{2}w_{x}}{\partial y^ {2}}+\frac{\partial^{2}w_{x}}{\partial z^{2}}\right)+\frac{1}{3}\mu\frac{ \partial}{\partial x}\left(\frac{\partial w_{x}}{\partial x}+\frac{\partial w _{y}}{\partial y}+\frac{\partial w_{z}}{\partial z}\right) \tag{18}\]  
\[\frac{\partial}{\partial\tau}(\rho h)+\
abla\cdot\left(\rho h\overrightarrow{v }\right)=\
abla[(\lambda+\lambda_{t})\
abla T]+p_{V}, \tag{19}\]  
where: \(\rho\)--fluid mass density; \(p\)--hydrostatic pressure; \(\rho g\)--mass forces; \(\mu\)--dynamic viscosity; \(h\)--enthalpy; \(\lambda\)--thermal conductivity; \(\lambda t\)--thermal conductivity for turbulent flow; \(T\)--temperature; \(p_{V}\)--volumetric density of heat sources and loses. Because of the fact that the electrical permittivity and resistivity values of analyzed liquid depend on the temperature value (both parameters decrease in temperature function), calculations of thermal energy propagation in the time between the discharges seems to be a very important factor [50]. If the break time between initiation of the next discharge is too short, a large heterogeneity of temperature distribution is observed in the liquid sample. In such conditions (Equations (4) and (5)), streamers will probably propagate in the channel of the previous discharge or its vicinity. Transient thermal analysis between discharges enables to determine minimal time interval, after which, next discharge will propagate randomly, independently from the pattern of previous breakdown process. Exemplarytemperature and fluid velocity distributions computed after first discharge (presented in Figures 16 and 17) are presented in Figure 19.  
Performed CFD analysis were used to determine the electric field intensity distributions in the analyzed workload (sample of liquid juice). Temperature distributions were introduced as initial conditions of the electrostatic problem.