# Modelling of a Non-Transferred Plasma Torch Used for Nano-Silica Powders Production
## 4 Results and Discussion
### Flow of Plasma in the Torch
An increase in the current led to an increase in the voltage of the plasma arc, which reinforced the electric power of the plasma. This linear variation in plasma voltage with plasma current shows that the plasma closely obeys [PERSON]'s law. Moreover, it can be clearly seen that the voltage increased with the increasing nitrogen flow rate, and the plasma was stable when the current and the nitrogen flow rate varied. Figure 5: Distribution of plasma velocity for different current values ((a) 100 A, (b) 150 A, (c) 200 A) in the torch. To study the efficiency of the plasma torch, we must consider that a portion of the input electric power was transferred to the plasma arc, and the rest of energy was extracted by the cooling water. The energy balance for the plasma torch is written as:  
\[E_{exp}=U_{exp}I-Q_{lost} \tag{20}\]  
where \(E_{exp}\) is the effective power transferred to the plasma arc, \(U_{exp}I\) is the electric power supplied by the DC generator, and \(Q_{lost}\) represents the energy lost by the cooling water. \[Q_{lost}=4.18\times C_{pw}\times\dot{m}_{w}(T_{out}-T_{in}) \tag{21}\]  
where \(C_{pw}\) is the specific heat capacity of water (1 cal/cm\({}^{3}\)), 4.18 is the conversion factor for converting cal/s into watts, \(\dot{m}_{w}\) is the amount of cooling water flow rate (sccm), and \(T_{out}\) and \(T_{in}\) are the outlet and inlet temperatures, respectively. However, for numerical simulation, the energy of the plasma leaving the torch can be calculated by integrating the plasma enthalpy at the surface of the torch output:  
\[E_{cal}=\iint\dot{m}_{g}hds \tag{22}\]  
The electrothermal efficiency can be obtained by dividing the energy transferred to the plasma gas by the electric power, mainly:  
\[\eta=\frac{E_{exp}}{U_{exp}I}=\frac{E_{cal}}{U_{cal}}I \tag{23}\]  
Figure 7 illustrates the numerical results and experimental measurements of electrothermal efficiency of the plasma torch for different working currents and at different mass flow rates. At lower mass flow rates, the increase in current led to an increase in the degree of ionization, which increased the plasma pressure inside the torch; then, the plasma velocity increased, and more plasma came out from the nozzle exit of the torch. Thus, the efficiency increased with current. However, at a higher gas flow rate, the amount of plasma coming out from the torch did not increase significantly because the increase in the plasma pressure inside the torch due to the increase in the current caused a higher collision of the plasma particles with the anode wall (limitation of the velocity) and led to more heat loss from the plasma to the walls, decreasing the total efficiency of the torch [36]. Figure 6: Plasma voltage as a function of current. Furthermore, in Figure 8, experimental measurements and numerical predictions of electrothermal efficiency display that the plasma efficiency decreased linearly when the enthalpy increased. The comparison between the value of numerical results and the measured one showed the same variation tendency of the efficiency, but the values of the numerical model were less than those of experimental measurements. Indeed, the measured value of total energy included both the energy transferred to the plasma arc and the heat dissipated by cooled water. Experimentally, the energy extracted by the cooling water was under-evaluated because a part of heat dissipated into the environment. In this case, the effective thermal efficiency was overestimated. In addition, Figure 9 shows that the specific enthalpy of nitrogen gas increased rapidly as temperature increased, which led to greater electrical energy needed. Figure 4:0 Numerical and experimental variation in thermal efficiency as a function of enthalpy with a gas flow rate Q = 233.3 sccm. Figure 4:1 Numerical and experimental variation of thermal efficiency with working current (100 A, 150 A, 200 A) for different gas flow rates (Q = 233.3 sccm and Q = 100 sccm).