The Effect of Non-Transferred Plasma Torch Electrodes on Plasma Jet: A Computational Study

Abstract

This study explores how different electrode shapes affect plasma flow in a non-transferred plasma torch. Various cathode geometries—including conical, tapered, flat, and cylindrical—were examined alongside stepped anode designs. A 2D axisymmetric computational model was employed to assess the impact of these shapes on plasma behavior. The results reveal that different cathode designs require varying current levels to maintain a consistent power output. This paper presents the changes in electric conductivity and electric potential for different input currents across the arc formation path (from the cathode tip to the anode beginning) and relating to Ohm’s law. Significant variations in plasma jet velocity and temperature were observed, especially near the cathode tip. The study concludes by evaluating thermal efficiency across geometry configurations. Flat cathodes demonstrated the highest efficiency, while the anode shape had minimal impact.

1. Introduction

Plasma torches are thermal heat sources capable of generating plasma with high enthalpy and temperatures ranging from 2000 to 20,000 K [1,2]. Unlike other sustained heating methods, plasma torches can reach extreme temperatures and operate in a wide range of pressures, atmospheric conditions, and thermal environments [3]. They are entirely electrically powered and do not require external energy storage. These advantages make plasma torches a desirable choice for various industrial applications. However, scaling up existing designs, which typically operate between 20 kW and 8 MW, requires a deeper understanding of how electrode types and geometries influence torch performance.

DC non-transferred plasma torches typically use rod-type cathodes (RTCs) or well-type cathodes (WTCs), as shown in Figure 1 [4]. WTCs are made from high-purity, oxygen-free copper and have a simple cylindrical shape sealed at one end. These torches can operate with both oxidizing and non-oxidizing gases, making them suitable for processes like the pyrolysis of gaseous and liquid organic waste, where oxygen or steam promotes chemical reactions. In contrast, RTCs consist of a tungsten or hafnium tip attached to a copper body. Due to the volatility of tungsten oxide, which forms at lower temperatures (below 1800 K) when oxidation occurs, RTCs are limited to non-oxidizing working gases [5]. Based on the characteristics, RTC-based torches are commonly used in material processing, plasma welding, thermal spraying, and synthesis applications [6].

In an RTC plasma torch, controllable parameters, such as electric current, gas flow rate, gas composition, working gas type, and nozzle geometry, significantly affect noncontrollable outcomes such as arc voltage, arc fluctuations, arc length, torch efficiency, and electrode erosion patterns [7,8]. The effects of gas flow rate, electric current, and gas composition were analyzed in our previous studies [9,10]. In this work, we focus on how cathode and anode geometries influence arc formation, plasma jet temperature and velocity, and the overall efficiency of RTC-based non-transferred plasma torches for different geometries.

In a plasma torch, the cathode and anode are the two key electrodes that define arc behavior. The cathode acts as the electron source for sustaining the arc discharge, absorbing heat, and emitting electrons via thermionic emission. RTC torches can use various cathode tip shapes, such as conical, tapered, and flat tips, which influence arc stability and heat transfer. Numerous studies have explored these geometries separately; for example, flat cathodes were modeled by Westhoff et al. and Felipini et al. [11,12], while tapered cathodes were analyzed by Li et al., Paik et al., Selvan et al., Deng et al., Liu et al., Liang et al. [13,14,15,16,17,18,19], and conical cathodes were investigated by Huang et al. [20,21]. Although prior work shows that plasma jet velocity and temperature trends are generally similar across cathode shapes, the geometry still affects voltage at the cathode tip, arc formation, flow structure, and torch efficiency. Likewise, the geometry of the anode nozzle influences the location of the arc attachment, the velocity of the plasma jet, the energy efficiency, and air entrainment. Collares et al. found that the way electric power is delivered to the anode significantly affects efficiency [7]. Choi et al. demonstrated that stepped anodes reduce air entrainment compared to cylindrical ones [22].

The overarching goal of our research is to scale plasma burners to industrial throughput. Achieving this requires a systematic evaluation of the operating parameters that govern torch performance. In particular, electrode geometry is known to modulate arc attachment, energy transfer, and jet stability, all of which ultimately dictate thermal efficiency and longevity. In this study, we therefore perform a computational investigation of how selected cathode and anode shapes influence plasma-jet behavior in an RTC non-transferred torch. The results provide design guidance for selecting electrode profiles that are better suited to scale.

Working of RTC Non-Transferred Plasma Torch

This study analyzes a plasma torch equipped with a rod-type cathode and an anode. In non-transferred plasma torches, both electrodes are contained within the torch, so the electric current does not flow to an external workpiece [23]. Thermal plasma is generated when the working gas interacts with the electric arc, and then the working gas ionizes. Collisions between gas molecules and electrons produce positively and negatively charged particles, forming an electrically conductive plasma. These interactions also release heat, creating a sustained arc [24].

Figure 1a illustrates the internal layout of a DC non-transferred plasma torch with a rod-type cathode, showing the cathode, anode, and working gas flow. The current is applied at the tungsten cathode tip, establishing a voltage difference relative to the copper anode. This voltage drives the formation of an arc. A working gas, such as nitrogen, oxygen, carbon dioxide, or argon, is introduced near the cathode to elongate the arc and generate a high-voltage environment.

The current in the arc induces Joule heating, raising the local temperature and ionizing the gas. The resulting plasma expands due to heating, accelerating the gas and forming a high-velocity plasma jet. Additionally, interaction between the current and its magnetic field generates a Lorentz force, which drives the jet toward the outlet.

2. Computational Modeling

Plasma modeling is inherently complex due to the multi-species nature of the fluid, which includes electrons, ions, and neutral particles. The modeling approach depends strongly on the specific type and application of the plasma [25]. In thermal plasmas, electrons and ions are typically assumed to be in thermal equilibrium. Key parameters for thermal plasma modeling include [26,27,28,29]:

(1)

Plasma composition: Plasma properties depend on electrons and ions. The behavior of an ionized plasma is different for different materials.

(2)

Temperature: In thermal plasmas, ions, electrons, and neutrals are assumed to share the same temperature.

(3)

Plasma density: The number of particles per unit volume can determine the plasma characteristics and the interaction rate of the particles.

(4)

Pressure: Influences the density and behavior of plasma particles.

(5)

Collisional effects: In a thermal plasma, collisions are frequent and lead to equalization of the temperature of ions and electrons.

(6)

Radiation: Plays a crucial role in energy balance.

(7)

Boundary conditions: Materials, the shape of electrodes, heat flux to walls, and inflow and outflow conditions impact plasma operation and heat and mass transfer.

(8)

Chemical reactions: Ionization and recombination are to be defined accurately.

(9)

Transport properties: Properties such as thermal conductivity, viscosity, and electrical conductivity are significantly affected by temperature and plasma composition.

In this study, thermal plasma in a non-transferred plasma torch is modeled using the magnetohydrodynamic (MHD) equations within COMSOL Multiphysics v6.1. The Equilibrium Discharge Interface (EDI) model is derived from the single-fluid MHD formulation, incorporating static electric fields while disregarding induction current effects [30]. The coupled equations combine the Navier–Stokes, heat, and Maxwell equations to describe the motion of the conducting fluid in an electromagnetic field. Solving the EDI involves the following assumptions:

(1)

The modeled plasma comprises ions and electrons and satisfies charge neutrality.

(2)

Plasma is typically modeled as a locally neutral mixture that behaves as a Newtonian fluid.

(3)

A steady-state analysis is conducted under the assumption that the arc operates under steady conditions.

(4)

Laminar and quasi-incompressible flow conditions are assumed in the modeling.

(5)

The plasma is assumed to be in local thermal equilibrium (LTE), indicating that the temperature difference between electrons and heavy particles can be considered negligible under a low electric field at an atmospheric pressure.

(6)

The plasma is assumed to be optically thin, resulting in a simplified set of MHD equations.

2.1. 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

  $$\nabla ·\left(\rho \mathbf{u}\right)=0,$$

  (1)

  where the velocity (u) is defined in the axial and azimuthal directions and $\rho$ is the density.

• Momentum equation

$$\begin{array}{c}\hfill \rho \left(\mathbf{u}·\nabla \right)\mathbf{u}=\nabla ·\left[-p\mathbf{I}+\mu \left(\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T\right)-{\displaystyle \frac{2}{3}}\mu \left(\nabla ·\mathbf{u}\right)\mathbf{I}\right]+\\ \hfill +\sigma \left(\mathbf{E}+\mathbf{u}\times \mathbf{B}\right)\times \mathbf{B},\end{array}$$

(2)

• where p denotes the pressure, $\mu$ is dynamic viscosity, I is the identity matrix, B is the magnetic field, $\sigma$ is electrical conductivity, E is the electric field and $\sigma \left(\mathbf{E}+\mathbf{u}\times \mathbf{B}\right)\times \mathbf{B}$ 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:

  $$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{3}{2}}{n}_{\alpha }\kappa T_{\alpha }\right)+{\displaystyle \frac{\partial q_j^{\alpha }}{\partial x_j}}+{\displaystyle \frac{\partial }{\partial x_i}}\left({\displaystyle \frac{5}{2}}{n}_{\alpha }\kappa T_{\alpha }{u}_i^{\alpha }\right)\\ \hfill +n{q}_{\alpha }{E}_i{u}_i^{\alpha }=-Q_{\alpha \beta },\end{array}$$

  (3)

  where

  $$\begin{array}{c}\hfill Q_{\alpha \beta }={\displaystyle \frac{n_{\alpha }{m}_{\alpha }}{m_{\alpha }+m_{\beta }}}{v}_{\alpha \beta }[3\kappa \left(T_{\alpha }-T_{\beta }\right)+m_{\beta }\\ \hfill {\left(u_{\beta }-u_{\alpha }\right)}^2].\end{array}$$

  (4)

• 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. Further from 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:

  $$\begin{array}{c}\hfill \rho c_p\left({\displaystyle \frac{\partial T}{\partial t}}+\mathbf{u}·\nabla T\right)=\nabla ·\left(k\nabla T\right)+\mathbf{j}·\mathbf{E}-\\ \hfill -4\pi \epsilon r+{\displaystyle \frac{5}{2}}{\displaystyle \frac{k_B}{e}}\left(\mathbf{j}·\nabla \right)T-{\left({\displaystyle \frac{\partial ln\rho }{\partial lnT}}\right)}_P{\displaystyle \frac{Dp}{Dt}},\end{array}$$

  (5)

  where $\mathbf{j}·\mathbf{E}$ is the Joule heating, 4$\pi \epsilon r$ the volumetric radiation loss, k is the thermal conductivity from ordinary molecular transport, $k_B$ is the Boltzmann constant and ${\displaystyle \frac{Dp}{Dt}}$ is the work done by the pressure.

• Maxwell equations

$$\nabla ·\mathbf{j}=Q_{j,v}$$

(6a)

$$\mathbf{j}=\sigma \mathbf{E}+\sigma \mathbf{u}\times \mathbf{B}$$

(6b)

$$\mathbf{E}=-\nabla V.$$

(6c)

• where j is the current density (A/m²), $\sigma$ is electric conductivity (S/m) and V is electric potential. Here $Q_{j,v}=0$, which is a volumetric current source term. As in the current equations, there is no source within the plasma. Further, the self-induced magnetic field is calculated from Ampere’s law:

$$\nabla \times \mathbf{H}=\mathbf{j}$$

(7a)

$$\mathbf{B}=\nabla \times \mathbf{A}$$

(7b)

In the above equation, B is the magnetic flux density, A the magnetic vector potential, and H is the magnetic field. B and H are related using relative the permeability $\left({\mu }_r\right)$ defined by the material property.

2.2. Geometry and Mesh

In our previous work, computational results from the EDI model were validated against temperature measurements obtained via emission spectroscopy. These comparisons confirmed that the temperature profile at the plasma torch outlet is symmetric [9]. Based on this, a 2D axisymmetric geometry was used for the current simulations, as shown in Figure 2. The overall length of the plasma torch is 70 mm, the inlet radius is 2.5 mm, the outlet diameter for a cylindrical anode is 6 mm, and the stepped anode is 8 mm. Furthermore, the overall length of the cathode is 9.1 mm, the bottom diameter is 9mm, and the top diameter is a maximum of 6 mm. However, for some cathode shapes, the diameter is smaller than 6 mm because of the curved surface. Although the overall dimensions of the torch remained constant in all case studies, the cathode and anode shapes were varied. Figure 3 presents the different cathode designs used in the analysis (with a cylindrical anode), while Figure 2 (bottom) illustrates the stepped anode paired with a flat cathode for the anode geometry study.

The cathode tip is made of tungsten, while the remainder of the cathode and the entire anode are composed of copper. Nitrogen was selected as the working gas for all simulations. Its properties, such as transport and thermodynamic characteristics under equilibrium discharge conditions, were defined in COMSOL, as described in detail in the thermal plasma book and our earlier publication [9,29].

A triangular mesh was used for all geometries. The number of mesh elements varied based on the specific cathode and anode configurations, as shown in

Table 1. Mesh statistics.

Geometry	Cathode Shape	No. of Elements	Skewness
Cathode a	conical	177,458	0.95
Cathode b	tapered	31,119	0.9
Cathode c	tapered	176,934	0.95
Cathode d	tapered	179,032	0.94
Cathode e	flat	223,145	0.94
Cathode f	flat	227,094	0.94
Stepped anode	flat	248,541	0.96

. Boundary layers were not included, since the EDI model does not accurately capture near-wall physics. The associated near-wall property variations are discussed in our prior work [9].

2.2.1. Cathode Shapes

The shape of the cathode tip is critical, as it significantly affects arc stability and overall torch performance [31]. This study examines various cathode tip geometries, sourced from prior literature (cathodes a–d) and experimental designs at RISE (cathodes e and f) [11,15,20,32]. Figure 3 illustrates the different cathode shapes used in this work. In the present study, all the cathodes are 9.1 mm in height, and the bottom radius is 4.5 mm. The top distance from the symmetry axis is adjusted based on the shape of the cathode, but the maximum distance is 3 mm.

(1)

Conical cathode (Figure 3a): Features a pointed apex directed toward the anode. This shape focuses the arc, enhancing energy density and making it effective for cutting and melting.

(2)

Tapered cathodes (Figure 3b–d): Similar to conical tips but with a wider base and narrower end. These shapes provide a stable and focused arc, suitable for high-power operations.

(3)

Flat cathode (Figure 3e,f): Consists of flat, disc-shaped ends aligned perpendicular to the arc. These offer stable arc behavior and are typically used in lower-power (around 20 kW) torches.

2.2.2. Anode Shapes

Literature indicates that anode nozzle geometry significantly affects air entrainment in non-transferred plasma torches. Choi et al. and Samareh et al. found that stepped nozzles reduce the amount of ambient air mixed into the plasma jet compared to cylindrical nozzles [22,33]. This reduction leads to higher plasma enthalpy and temperature. Additionally, the stepped design promotes a smoother axial velocity profile by minimizing flow disturbances caused by air intrusion. To investigate these effects, the present study includes case studies comparing cylindrical and stepped anode geometries, as shown in Figure 2 (bottom). These simulations aim to evaluate how anode shape influences plasma flow characteristics and identify which configuration offers greater thermal efficiency.

2.3. Boundary Conditions and Solvers

Table 2. Models and boundary conditions.

Model	Solved Regions	Boundary Conditions
Electric currents	Fluid and anode	$j_n$ at cathode tip
		Ground = anode
Magnetic fields	Fluid and anode	Field components =
		in-plane
		vector potential
		${\Psi }_o$ = 1 A/m
Laminar flow	Fluid	$u_{phi}=5.53$ m/s
		$u_z=5.53$ m/s
Heat transfer	cathode, anode	$T_{cathode}=3500$ K
in solids and fluids	and fluid	$T_{ustr}=300$ K
		$h_{anode}=$
		${10}^4$ W/$\left({\mathrm{m}}^2\mathrm{K}\right)$
		$T_{ext}=500$ K

summarizes the models and boundary conditions used in this study. These components were coupled using COMSOL’s multiphysics framework. The EDI model integrates electric current and heat transfer to account for enthalpy transport, Joule heating, and volumetric radiation losses. Electromagnetic surface losses were mapped as boundary heat sources and applied separately to the cathode and anode regions.

The Lorentz force, which accelerates the plasma jet, was implemented through coupling between the magnetic field and laminar flow modules, following the approach described by Westhoff et al. [11]. The boundary conditions are the same for all the case studies, but the current density ($j_n$) is varied between $1\times {10}^4$ and $1\times {10}^6$, to perform simulations for different powers. Further, the area of the cathode tip on which the current density is defined varies for different cathodes and is shown in Table 1. It should be noted that even though the current density is defined until a certain distance from the symmetry axis, the voltage starts to develop from the tip of the cathode near the symmetry axis. Based on the solver comparisons of our previous work, the PARDISO solver was chosen with a fully coupled method, offering better convergence and computational efficiency [9].

3. Results

This section outlines the findings from case studies aimed at assessing how the shapes and sizes of cathodes and anodes affect plasma torch operation. Previously, we validated the reliability of our computational model by comparing the flow property variations with experimental results. In the comparison, it was observed that there is a good match between the experimental and computational results at the center of the plasma torch, and the details are published in our previous paper [9]. Consequently, this paper aims to understand the impacts of the geometry changes; therefore, only computational results from various case studies and some of the results are explained by referring to the literature works.

3.1. Case 1: Study of Cathode Shapes

This section presents the variations in velocity, temperature, pressure, and current density observed in a non-transferred plasma torch for different cathode shapes. The simulations cover conical, tapered, and flat-tip cathodes, as illustrated in Figure 3.

3.1.1. Current and Voltage

Current and voltage are key operational parameters in arc plasma torches. Figure 4a presents simulated current–voltage characteristics for different cathode shapes. For input currents below 30 A, the relationship between current and voltage is linear. However, above 30 A, an inverse trend emerges; the voltage decreases as the current increases, consistent with the findings by Ramasamy et al., who attributed this to increased arc conductance [34].

While the overall trend remains consistent across cathode designs, voltage values vary slightly depending on shape. In the low-current region (<30 A), cathodes a, d, and f show higher peak voltages than b, c, and e. These differences are related to how each geometry influences arc resistance. At higher currents, flat cathodes exhibit the highest voltages, followed by tapered and conical shapes.

In the model, current density is prescribed at the cathode tip, and voltage is computed via Ohm’s law. To analyze variation in arc behavior, a reference line is defined from the cathode tip to the start of the anode (Figure A1). Along this line, temperature, electrical conductivity, and electric potential are extracted for different input currents that are shown in Figure 4c legends.

Figure 4c show that both conductivity (dash and dotted line) and temperature (solid line) peak near the cathode tip and decline toward the anode. These values also increase with higher current input. In contrast, Figure 4b shows that the electric potential (dashed line)increases steadily from the cathode to the anode, reflecting Ohm’s law; as conductivity drops, potential must rise to maintain current.

3.1.2. Arc Formation

This section examines how cathode shape affects arc formation and the resulting current and voltage behavior at the cathode tip. The electric arc forms between the cathode and anode when the applied current concentrates at the cathode tip. As noted by Westhoff et al. and Untaroiu et al., the arc naturally aligns along the shortest path between the electrodes [11,35]. Their studies also showed that the current density on the inner wall of the anode peaks between the cathode tip and the nozzle, allowing the arc length to be estimated from the center of this distribution.

Figure 5 shows the current density variations plotted on the inner walls of the anode (which is 14 mm from the inlet to the outlet). The plot shows that the arc attaches to the anode at approximately 14 mm from the cathode tip. This attachment location is consistent across all cathode shapes, indicating that arc length does not vary significantly with geometry. However, the arc’s intensity differs; cathodes b, e, and f exhibit lower current densities compared to a, c, and d, suggesting shape-dependent arc stability and energy concentration. Further, the plot shows three peaks; the reason might be that once the arc attaches to the anode, it detaches, and further, after a certain distance, it tries to attach again, and that point might indicate the increase in the current density [21].

3.1.3. Temperature and Velocity Changes

The result of ionization close to the cathode tip causes a sharp rise in both temperature and velocity in that region. Further, as the plasma moves downstream, these values gradually decline toward the outlet, mainly due to cooling by the anode. This pattern of temperature and velocity variation has been widely reported in previous studies using various analytical approaches [11,15,32].

Figure 6a shows temperature profiles for different cathode shapes at a fixed power input of 14 kW using a cylindrical anode. It is worth noting that the current density required at the cathode tip to achieve 14 kW power is not uniform across all cathodes. To elucidate, the current density required at the cathode tip for the attainment of 14 kW power is as follows for the respective cathodes: cathode a (${10}^6$ A/m²), cathode b ($8·{10}^6$ A/m²), cathode c ($8·{10}^5$ A/m²), cathode d ($1.5·{10}^6$ A/m²), cathode e ($3·{10}^6$ A/m²), and cathode f ($2·{10}^6$ A/m²), respectively. The figure highlights a strong dependence of cathode shape on temperature near the tip, while outlet temperatures remain relatively consistent. Cathodes b, e, and f produce higher tip temperatures than d, a, and c.

Similarly, Figure 6b presents plasma jet velocity for the same conditions. Again, cathodes b, e, and f show higher velocities, suggesting that flat cathodes enhance energy transfer and jet momentum. In contrast, conical or tapered cathodes with smaller tip areas tend to produce lower plasma temperatures and velocities.

3.2. Case 1: Study of Anode Shapes

This section compares flow structure, velocity, temperature, and current density for cylindrical and stepped anode nozzles. The corresponding geometries are shown in Figure 2 and a flat-tip cathode, Figure 3e. Simulations are conducted at two input power levels: 6.2 kW and 18.4 kW.

3.2.1. Flow Structure

Figure 7 shows streamlines and velocity vectors for both cylindrical and stepped anode nozzles. In Figure 7a, the cylindrical nozzle produces smooth, undisturbed streamlines. In contrast, Figure 7b reveals a small region of flow separation near the step in the stepped nozzle, consistent with earlier findings by Choi et al. [22].

Figure 7c displays velocity vectors within the stepped nozzle. The plot indicates that velocity gradients near the torch wall are lower in the stepped configuration than in the cylindrical one. These differences might affect the behaviors and characteristics of the plasma jet.

3.2.2. Velocity Variations

Figure 8a shows axial velocity along the symmetry axis from the cathode tip to the outlet for both anode types, under input powers of 6.2 kW and 18.4 kW. For a given cathode shape, the peak velocity near the cathode tip remains similar across the two anode geometries. However, as the jet moves toward the outlet, differences emerge; cylindrical nozzles show an 18% velocity drop, while stepped nozzles exhibit a sharper 42% reduction. This accelerated decay is attributed to flow separation in the stepped region, which disrupts the velocity profile.

Figure 8b compares radial velocity at the outlet and it could be observed that the stepped nozzle yields a radial velocity approximately 38% lower than the cylindrical one. This difference is primarily influenced by inlet velocity and thermal expansion, which shape the outlet jet profile. To further characterize the flow, Reynolds numbers at the nozzle exit were calculated: 488 and 406 for cylindrical nozzles, and 364 and 300 for stepped nozzles. According to Fincke et al., a turbulent shear layer typically forms when the Reynolds number exceeds 400 [36]. These results suggest that cylindrical nozzles produce more turbulent and faster jets than stepped nozzles.

3.3. Temperature and Current Density Variations

In Figure 9a, we observe the temperature variation within both the stepped and cylindrical nozzle anodes. Notably, the temperature variation remains consistent across both nozzle types, characterized by an increase at the cathode tip and a subsequent decrease towards the outlet. This thermal behavior in our computational model is regulated by factors such as the surface areas of the cathode tip, inlet velocity, and inlet power. Given that these conditions are uniform for both stepped and cylindrical nozzle configurations, no discernible differences in temperature profiles emerge with respect to anode shapes.

Figure 9b presents the variations in current density along the inner walls of the anode, extending from the cathode tip to the outlet. The plot distinctly highlights the initial peak occurring at the cathode tip, positioned at 9.1 mm. The second peak, found at 14 mm, designates the location where the arc attaches to the anode surface. Notably, the stepped nozzle anode exhibits a distinctive pattern of increase and subsequent decrease in current density at 19 mm, coinciding with the stepped region. Beyond this point, the current density in the stepped nozzle decreases significantly. In contrast, the cylindrical anode portrays a more stable and uniform current density variation, resulting in a more stable arc.

3.4. 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 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:

$${\oint }_s\widehat{\mathbf{n}}·\left(\rho c_p\mathbf{u}T\right)ds={\int }_{s_{out}}\widehat{\mathbf{n}}·\rho c_p\mathbf{u}Tds+{\int }_{s_{in}}\widehat{\mathbf{n}}·\rho c_p\mathbf{u}·Tds=Q_{net}$$

(8)

$${\oint }_s\mathbf{E}\times \mathbf{H}·\widehat{\mathbf{n}}ds=\int \mathbf{j}·\mathbf{E}dv=Q_{input}.$$

(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

$${\oint }_s\widehat{\mathbf{n}}·k\nabla Tds-\mathrm{Ordinary}\mathrm{transport}.$$

(10)

The enthalpy transport due to the current term is represented by

$$\begin{array}{c}\hfill {\int }_v{\displaystyle \frac{5}{2}}{\displaystyle \frac{k_B}{e}}\nabla ·(\mathbf{j}·T)dv={\oint }_s{\displaystyle \frac{5}{2}}{\displaystyle \frac{k_B}{e}}\widehat{\mathbf{n}}·\left(\mathbf{j}T\right)ds.\end{array}$$

(11)

The pressure–volume work term is as follows:

$$\oint \widehat{n}·{\displaystyle \frac{\partial \left(ln\rho \right)}{\partial \left(lnT\right)}}\mathbf{u}Pds$$

(12)

$${\int }_{v}4\pi {\epsilon }_{r}dv.$$

(13)

The above Equations (10)–(13) represent the heat and radiation losses. Thus, the efficiency is calculated as follows:

$$Q_{net}=Q_{input}-Q_{loss}$$

(14)

$$\eta ={\displaystyle \frac{Q_{net}}{Q_{input}}}=1-{\displaystyle \frac{Q_{loss}}{Q_{input}}}$$

(15)

From the computational analysis, the efficiency is calculated from the heat losses, input current, and voltage, using the following formulation:

$$\eta \left(\%\right)=1-{\displaystyle \frac{Q_{loss}}{I\left(A\right)·V\left(V\right)}}·100.$$

(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. Thermal efficiencies of non-transferred plasma torch with different cathode shapes and anodes.

Geometry	Input Power (kW)	Inlet Velocity (m/s)	Efficiency (%)
	Cathode a	14	11.06	76
Cathode b	14	11.06	75.9
Cathode c	14	11.06	75.4
Cathode d	14	11.06	73
Cathode e	14	11.06	81
Cathode f	14	11.06	72
Stepped nozzle	6.2	11.06	54.5
Stepped nozzle	18.4	11.06	80.8
Cylindrical nozzle	6.2	11.06	54.5
Cylindrical nozzle	18.4	11.06	81

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]. However, it is acknowledged that arc power, gas flow rate, and anode cooling intricacies play pivotal roles in determining plasma torch efficiency. Shicong et al. conducted experimental studies, concluding that the thermal efficiency of plasma torches typically falls within the range of 65–78% [40]. Further, from the computational study presented in this paper, the efficiency range of the plasma torch spans between 50 and 80%, with the power input and cathode shape identified as influential factors. It is noteworthy that this efficiency range appears slightly higher than those reported in prior works. This discrepancy can be attributed to the present study’s focus on calculating efficiency based on the internal flow within the plasma torch while excluding the effects of plasma jet interaction with the atmosphere upon exiting.

4. Conclusions

This study presented a computational investigation of cathode and anode geometries in non-transferred plasma torches using steady-state MHD modeling on a 2D axisymmetric domain in COMSOL Multiphysics (v6.1). The impact of electrode shape on flow structure, temperature, velocity, and current density in the plasma jet was systematically analyzed.

Three cathode configurations were evaluated: conical, tapered, and flat. The results showed that while the arc length remained largely unchanged across cathode types, the intensity of the arc varied depending on the shape. Cathode geometry significantly influenced local plasma temperature at the cathode tip, though outlet conditions (temperature and velocity) were similar among all cathodes. Further, flat-shaped cathodes exhibited flow separation near the tip, and an in-depth investigation of the phenomenon is explained in our previous work [41].

The analysis of cylindrical and stepped anode nozzles demonstrated that stepped anodes introduce flow separation, resulting in reduced plasma velocity and arc current. In contrast, cylindrical nozzles produced more stable plasma jets and higher axial velocities.

One interesting finding of this work is that the cathode shape can cause a difference in the thermal efficiencies by 10%. Efficiency analysis revealed that flat cathodes offered slightly better thermal efficiency compared to other geometries, highlighting that cathode e can be used for further studies related to the upscaling. However, anode geometry had a minimal effect on efficiency. Overall, input power and gas flow rate were found to be the dominant factors influencing the thermal efficiency of non-transferred plasma torches, with electrode shapes playing a secondary role.