# The Effect of Non-Transferred Plasma Torch Electrodes on Plasma Jet: A Computational Study
## 3 Results
### Thermal Efficiency  
Efficiency in the context of a plasma torch refers to its ability to effectively heat the plasma gas. The thermal efficiency of a plasma torch holds substantial importance in assessing its processing performance, serving as a measure of the energy effectively utilized in the process [37]. Several factors influence the efficiency of a plasma torch, including the variation of plasma jet temperature, velocity, and kinetic energy with distance from the torch exit. One approach to enhance the efficiency of a plasma torch involves constriction, which leads to an augmented current density and electric fields [2]. The calculation of thermal  
Figure 9: (**a**) Temperature and (**b**) current density variations in cylindrical and stepped nozzle anodes. efficiency for a plasma torch takes into account the input power transferred to a thermal plasma jet, the flow rate, and the temperature rise of cooling water passing through the torch [11,38]. The thermal efficiency, calculated based on heat loss, is determined by applying the Gauss divergence theorem to the energy equation as presented in Section 2.1 (Equation (5)). Each term in Equation (5) corresponds to the following:  
\[\oint_{s}\mathbf{\hat{n}}\cdot\left(\rho c_{p}\mathbf{u}T\right)ds=\int_{s_{out }}\mathbf{\hat{n}}\cdot\rho c_{p}\mathbf{u}Tds+\int_{s_{in}}\mathbf{\hat{n}} \cdot\rho c_{p}\mathbf{u}\cdot Tds=Q_{net} \tag{8}\]  
\[\oint_{s}\mathbf{E}\times\mathbf{H}\cdot\mathbf{\hat{n}}\;ds=\int\mathbf{j} \cdot\mathbf{E}\;dv=Q_{input}. \tag{9}\]  
The left-hand term in the above equation is electromagnetic flux into the whole torch through the cathode surface. The right-hand term is Ohmic heating. The ordinary heat transfer term corresponds to  
\[\oint_{s}\mathbf{\hat{n}}\cdot k\
abla Tds-\text{Ordinary transport}. \tag{10}\]  
The enthalpy transport due to the current term is represented by  
\[\int_{v}\frac{5}{2}\frac{k_{B}}{e}\
abla\cdot(\mathbf{j}\cdot T)dv=\oint_{s} \frac{5}{2}\frac{k_{B}}{e}\mathbf{\hat{n}}\cdot(\mathbf{j}T)ds. \tag{11}\]  
The pressure-volume work term is as follows:  
\[\oint\hat{n}\cdot\frac{\partial(ln\rho)}{\partial(lnT)}\mathbf{u}P\,ds \tag{12}\]  
\[\int_{v}4\pi c_{r}\;dv. \tag{13}\]  
The above Equations (10)-(13) represent the heat and radiation losses. Thus, the efficiency is calculated as follows:  
\[Q_{net}=Q_{input}-Q_{loss} \tag{14}\]  
\[\eta=\frac{Q_{net}}{Q_{input}}=1-\frac{Q_{loss}}{Q_{input}} \tag{15}\]  
From the computational analysis, the efficiency is calculated from the heat losses, input current, and voltage, using the following formulation:  
\[\eta(\%)=1-\frac{Q_{loss}}{I(A)\cdot V(V)}\cdot 100. \tag{16}\]  
The heat loss is mainly due to the cooling of the anode, and it is calculated from the mass flow rate, heat capacity, and difference in temperature at the inlet and outlet. Table 3 delineates the efficiencies of plasma torches featuring distinct cathode and anode geometries. Notably, the data reveal that the utilization of a flat cathode geometry yields higher efficiency compared to plasma torches employing other cathode shapes. It is interesting to note that just by changing the shape of the cathode tip, there is a 10% variation in the thermal efficiencies of the plasma torch. Furthermore, the experimental studies from our previous works on plasma torches with cathode shape e showed an efficiency variation between 70% and 80% for different input powers, which is close to the computational result presented in this paper. In contrast, the efficiencies of plasma torches utilizing a stepped nozzle and cylindrical nozzle anodes exhibit negligible differences because there is no change in the temperature variations (shown in Figure 9a), suggesting that the anode shape  
may not exert a substantial impact on efficiency. Instead, the input power emerges as a significant influencing factor. Various literature studies establish a general efficiency range for non-transferred plasma torches between 40 and 60% [39].