# Modelling of a Non-Transferred Plasma Torch Used for Nano-Silica Powders Production
## 2 Numerical Model
### Calculation Domain and Boundary Conditions  
The geometry of the DC plasma torch consists typically of a tungsten tip rod-type cathode and a copper anode. The nitrogen gas was input through a nozzle with a diameter of 8.2 mm on either side of the cathode. The length of the anode nozzle channel and the exit diameter were 225 mm and 17.2 mm, respectively (Figure 1). The nozzle at the end of the torch was used to spread out the heated gas in the form of a plasma jet. The axisymmetric boundary conditions performed in this study are mentioned in Table 1. Inflow boundary conditions are typically specified by imposing values of known properties of the gas injection, typically velocity and temperature. At the torch inlet, the gas was injected at 300 K, with a uniform axial \((u_{zin})\) and zero radial velocity component \((u_{rin})\). A swirl flow was required to enhance gas mixing, and an inflow tangential velocity component \((u_{gin})\) was introduced. In addition, the turbulent inlet parameter of the kinetic turbulent energy, \(\left(K=0.005\times u_{in}^{2}\right)\), and its rate of dissipation \(\left(\epsilon=0.1\times K^{2}\right)\) were considered. The inflow mass flow rate of the plasma gas can be obtained by integrating the axial velocity at the surface of the torch inlet, namely:  
\[Q_{in}=\int_{S_{inlet}}u_{zin}\ ds=u_{zin}S_{inlet}\overset{\rightharpoonup}{z} \tag{14}\]  
where \(S_{inlet}\) is the inlet surface and \(\overset{\rightharpoonup}{z}\) is the unit vector along the torch axis. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Boundary** & \(\mathbf{P}\) & \(\overset{\rightarrow}{\mathbf{u}}\) & \(\mathbf{T}\) & \(\emptyset\) & \(\overset{\rightarrow}{\mathbf{A}}\) \\ \hline Inlet & \(P=P_{in}\) & \(u_{in}\) & \(T_{in}=300\) K & \(\partial_{n}\varnothing=0\) & \(\partial_{n}\overset{\rightarrow}{A}=0\) \\ Cathode & \(\partial_{n}P=0\) & \(\overset{\rightarrow}{u}=0\) & \(T=T_{cells}(r)\) & \(-\rho\partial_{n}\varnothing=J_{cells}(r)\) & \(\partial_{n}\overset{\rightarrow}{A}=0\) \\ Anode & \(\partial_{n}P=0\) & \(\overset{\rightarrow}{u}=0\) & \(-k\partial_{n}T=h_{\infty}(T-T_{w})\) & \(\varnothing=0\) & \(\partial_{n}\overset{\rightarrow}{A}=0\) \\ Torch wall & \(\partial_{n}P=0\) & \(\overset{\rightarrow}{u}=0\) & \(\partial_{n}T=0\) & \(\partial_{n}\varnothing=0\) & \(\partial_{n}\overset{\rightarrow}{A}=0\) \\ Outlet & \(P=P_{out}\) & \(\partial_{n}\overset{\rightarrow}{u}=0\) & \(\partial_{n}T=0\) & \(\partial_{n}\varnothing=0\) & \(\overset{\rightarrow}{A}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions of the numerical model. Figure 1: (**a**) The geometry of the plasma torch. (**b**) Revolution of the 3D geometry of the torch. All dimensions are in mm. The temperature and the current density implemented on the cathode surface of the torch can be estimated by Gaussian profiles:  
\[T_{cath}(r)=500+3000\exp\left[-\left(\frac{r}{2R_{cath}}\right)^{n_{cath}}\right] \tag{15}\]  
\[J_{cath}(r)=J_{max}\exp\left[-\left(\frac{r}{R_{cath}}\right)^{n_{cath}}\right] \tag{16}\]  
where \(r\) is the radial distance from the torch axis, and \(n_{cath}\), \(J_{max}\), and \(R_{cath}\) are parameters that satisfy the distribution of the working current and temperature profiles over the cathode surface. The values of shape parameters for three cases are given in Table 2.  
In non-transferred arc torches, an overall convective heat transfer is used for the modelling of heat transfer of the regions of the anode far from arc attachment, namely:  
\[-k\partial_{n}T=h_{w}(T-T_{w}) \tag{17}\]  
The term (\(\partial_{n}\equiv n\cdot\
abla\)), denotes the derivative in the direction of the outer normal to the surface, where \(n\) represents the outer normal to the surface boundary. \(h_{w}=1\ \times\ 10^{4}\ \mathrm{W}/(\mathrm{m}^{2}\,\mathrm{K})\) is the convective heat transfer coefficient at the anode surface, and \(T_{w}=500\ \mathrm{K}\) is the temperature of cooling water. At the torch exit, \(P_{out}\) was set equal to the atmospheric pressure.