# Modelling of a Non-Transferred Plasma Torch Used for Nano-Silica Powders Production
## 2 Numerical Model
### Control Equations  
Based on the model assumptions cited above, the conservation equations of mass, momentum, and energy, as well as the electric potential and magnetic vector potential, are given by the following equations, respectively:  
\[\
abla\cdot\left(\rho\overrightarrow{u}\right)=0 \tag{1}\]  
\[\
abla\cdot\left(\rho\overrightarrow{u}\overrightarrow{u}\right)=-\
abla P +\
abla\cdot\left(\overrightarrow{\tau}\right)+\overrightarrow{l}\times \overrightarrow{B} \tag{2}\]  
\[\
abla\cdot\left(\rho h\overrightarrow{u}\right)=\
abla\left(\frac{\lambda} {c_{p}}\
abla h\right)+\overrightarrow{l}\cdot\overrightarrow{E}-4\pi e_{r} \tag{3}\]  
\[\
abla\cdot\overrightarrow{l}=\
abla\cdot\left(\sigma(-\
abla\cdot\varnothing )\right)=0 \tag{4}\]  
\[\
abla^{2}\cdot\overrightarrow{A}=-\mu_{0}\cdot\overrightarrow{l} \tag{5}\]where \(\overset{\rightarrow}{u},\
abla P,\overset{\rightarrow}{\tau},\overset{ \rightarrow}{J},\overset{\rightarrow}{B},\overset{\rightarrow}{E},h,\varnothing\) and \(\overset{\rightarrow}{A}\) are the gas velocity, pressure work, stress tensor, current density, magnetic field, electric field, plasma enthalpy, electric potential, and magnetic vector potential, respectively. The terms \(\rho,\ \lambda,\ c_{p},\ \sigma,\mu_{0}\) and \(\varepsilon_{r}\) are the gas density, thermal conductivity, specific heat, electric conductivity, permeability of free space, and the effective net emission coefficient, respectively, whereas \(\left(\overset{\rightarrow}{J}\times\overset{\rightarrow}{B}\right)\), \(\left(\overset{\rightarrow}{J}\cdot\overset{\rightarrow}{E}\right)\), and \(4\pi\varepsilon_{r}\) are the electromagnetic force (Lorentz force), the Joule heating, and the volumetric radiation loss, respectively.  
The stress tensor can be written as:  
\[\overset{=}{\tau}=\mu\left[\left(\
abla\overset{\rightarrow}{u}+\
abla \overset{\rightarrow}{u}^{T}\right)-\frac{2}{3}\
abla\cdot\overset{ \rightarrow}{u}\delta\right] \tag{6}\]  
where \(\mu\) is the dynamic viscosity, the superscript \"T\" indicates the transpose of matrix \(\overset{\rightarrow}{u}\), and \(\delta\) is the unit tensor.  
To obtain the electromagnetic field within the plasma torch, it is necessary to combine Equations (4) and (5) with the following equations:  
\[\overset{\rightarrow}{E}=-\
abla\varnothing \tag{7}\]  
\[\overset{\rightarrow}{J}=\sigma\overset{\rightarrow}{E} \tag{8}\]  
\[\overset{\rightarrow}{B}=\overset{\rightarrow}{\
abla}\times\overset{ \rightarrow}{A} \tag{9}\]  
The use of turbulence models in thermal plasma flows simulation is significantly requested due to their inherent characteristics (reactivity, large variation, and the electromagnetic effect). Hence, the standard \((K-\varepsilon)\) turbulence model developed by [PERSON] and Splalding [28] was implemented in our model, where \(K\) stands for the turbulent kinetic energy, and \(\varepsilon\) is the turbulent kinetic energy dissipation rate.  
\[\
abla\cdot\left(\rho K\overset{\rightarrow}{u}\right)=\
abla\cdot\left[ \left(\mu+\frac{\mu_{t}}{\sigma_{K}}\right)\cdot\
abla K\right]+G_{K}-\rho \varepsilon+Y_{M} \tag{10}\]  
\[\
abla\cdot\left(\rho\varepsilon\overset{\rightarrow}{u}\right)=\
abla\cdot \left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\cdot\
abla \varepsilon\right]+\frac{\varepsilon}{K}(C_{1\varepsilon}G_{K}-C_{2\varepsilon }\rho\varepsilon) \tag{11}\]  
\(Y_{M}\) represents the contribution of fluctuating dilatation. The turbulent viscosity \(\mu_{t}\), and the turbulent generation term \(G_{K}\) are defined as:  
\[\mu_{t}=\rho C_{\mu}\left(\frac{K^{2}}{\varepsilon}\right) \tag{12}\]  
\[G_{K}=\mu_{t}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}\right) \tag{13}\]  
where \(\frac{\partial u_{i}}{\partial x_{j}}\) is the production of turbulence kinetic energy. The turbulent model constants \(C_{1\varepsilon},\ C_{2\varepsilon},\ C_{\mu},\ \sigma_{K}\), and \(\sigma_{\varepsilon}\) were identified as 1.44, 1.92, 0.09, 1.0, and 1.3, respectively.