# The Effect of Non-Transferred Plasma Torch Electrodes on Plasma Jet: A Computational Study
## 2 Computational Modeling
### Governing Equations  
The fluid flow, heat transfer, and arc formation are assumed to exhibit axial symmetry. Therefore, the governing equations are expressed in a two-dimensional form. The flow inside the plasma torch has a very low density, and the Reynolds number of the flow is in the laminar flow range of 2000-2500, apart from the cathode tip region and boundary layer region of the anode. As the flow emerges from the outlet, it interacts with the atmospheric air and transitions from laminar to turbulent flow [23]. However, in the present study, the flow inside the plasma torch is studied, so a laminar flow model is used for the analysis. The governing equations are as follows:  
**Mass conservation**  
\[\
abla\cdot(\rho\mathbf{u})=0, \tag{1}\]  
where the velocity (\(\mathbf{u}\)) is defined in the axial and azimuthal directions and \(\rho\) is the density. **Momentum equation**  
\[\rho(\mathbf{u}\cdot\
abla)\mathbf{u}=\
abla\cdot\left[-p\mathbf{ I}+\mu\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}\Big{)}-\frac{2}{3}\mu( \
abla\cdot\mathbf{u})\mathbf{I}\right]+ \tag{2}\] \[+\sigma(\mathbf{E}+\mathbf{u}\times\mathbf{B})\times\mathbf{B},\]  
where \(p\) denotes the pressure, \(\mu\) is dynamic viscosity, I is the identity matrix, \(\mathbf{B}\) is the magnetic field, \(\sigma\) is electrical conductivity, \(\mathbf{E}\) is the electric field and \(\sigma(\mathbf{E}+\mathbf{u}\times\mathbf{B})\times\mathbf{B}\) represents the Lorentz force acting on the fluid. **Conservation of energy**  
The computational model adopted in this study relies on the assumption of local thermal equilibrium (LTE). Generally, the plasma within the torch can be considered to satisfy LTE conditions, except in regions close to the electrodes, specifically at the cathode tip and in the anode boundary layer [9]. The energy equation governing ions and electrons is expressed as follows:  
\[\frac{\partial}{\partial t}\left(\frac{3}{2}n_{\alpha}\kappa T_{ \alpha}\right)+\frac{\partial q_{j}^{\alpha}}{\partial x_{j}}+\frac{\partial} {\partial x_{i}}\left(\frac{5}{2}n_{\alpha}\kappa T_{\alpha}u_{i}^{\alpha}\right) \tag{3}\] \[+nq_{\alpha}E_{i}u_{i}^{\alpha}=-Q_{\alpha\beta},\]  
where  
\[Q_{\alpha\beta}=\frac{n_{\alpha}m_{\alpha}}{m_{\alpha}+m_{\beta}} \
u_{\alpha\beta}[3\kappa\big{(}T_{\alpha}-T_{\beta}\big{)}+m_{\beta} \tag{4}\] \[\big{(}u_{\beta}-u_{\alpha}\big{)}^{2}].\]  
At high temperatures, the collision frequency between species increases, making the first term in Equation (4) responsible for driving the plasma toward equilibrium. Within the plasma torch, the gas becomes ionized and reaches elevated temperatures downstream of the cathode. However, LTE conditions are not satisfied near the cathode. In this region, a fluid model becomes inadequate due to the presence of an extremely thin boundary layer composed of a collisionless plasma sheet, where charge neutrality breaks down, and a non-thermal ionization layer is formed. Furtherfrom the cathode, near the center of the plasma torch, the high temperatures cause the plasma to approach equilibrium. In contrast, near the anode, the applied water cooling lowers the temperature, preventing the establishment of LTE. The energy equation employed by COMSOL for plasma modeling is given as:  
\[\rho c_{p}\left(\frac{\partial T}{\partial t}+\mathbf{u}\cdot\
abla T \right)=\
abla\cdot\left(k\
abla T\right)+\mathbf{j}\cdot\mathbf{E}-\] \[-4\pi\varepsilon r+\frac{5}{2}\frac{k_{B}}{e}(\mathbf{j}\cdot \
abla)T-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{p}\frac{Dp}{Dt}, \tag{5}\]  
where \(\mathbf{j}\cdot\mathbf{E}\) is the Joule heating, \(4\pi\varepsilon r\) the volumetric radiation loss, \(k\) is the thermal conductivity from ordinary molecular transport, \(k_{B}\) is the Boltzmann constant and \(\frac{Dp}{Dt}\) is the work done by the pressure.