Role of Secondary Ions on the i-v Characteristics of Oxyfuel Flame Subject to an Electric Field

This work presents a computational model that investigates the role of secondary ions in oxyfuel cutting process at preheating stage. Oxyfuel flame cutting is a century-old technique having widespread applications in heavy industries due to its low cost and unparalleled performance on metal surfaces. In this process, the cutting torches use premixed oxygen-fuel gas mixture to heat metallic work pieces, so they can be cut chemically by a high-speed jet of pure oxygen. A pictorial view of the preheating process in oxyfuel flame cutting is shown in Fig. 1. The work piece (d) is brought to a kindling temperature by means of a preheat flame (c), and no cutting oxygen (b) is applied in this stage.

However, the mechanized oxyfuel-cutting process has never benefited from the degree of autonomy due to contemporary sensing technologies' limitations at high-temperature working conditions. A potential solution to this problem is motivated by preliminary measurements demonstrating that electrical events called ‘ion currents’ associated with the flame itself can reliably indicate vital process states.

It has been well established that high-temperature reaction zones in flames conduct electricity due to the chemi-ionization process [1,2]. Prior works in the literature [3–5] indicated the current–voltage (i-v) relationship with different critical parameters (such as standoff, flowrate, fuel-oxygen ratio, etc.) of oxyfuel flame to be the salient electrical characteristic under electric field in the preheating process. Demonstrably, when a voltage is applied between the torch and work piece, the i-v curve adopts three distinct regimes [6]. Moreover, the i-v curves have been further investigated to elucidate the production rate of charged species [7,8], and the effect of electric fields on flames [9–12]. Much work was dedicated in the late twentieth century toward identifying the chemical mechanisms for the formation and recombination of these ions in flames [2,13,14], revealing that carbon-bearing fuels produce ions, H₃O⁺ as a dominant cation.

A comprehensive two-dimensional computational simulation only with the reduced combustion chemical mechanism with ion-exchange reactions has already been completed [15] to elucidate the experimental results and to investigate the electrical characteristics such as ion migrations and ion distributions. Nonetheless, the findings exhibit some magnitude of differences compared to the experimental results [6,11]. In Ref. [6], study of ion currents in the oxyfuel cutting system revealed that there are two sources of ions; primary ions that are generated in the flame itself and secondary ions that are generated at the work piece. The work unveiled that, “copper work pieces have stable and repeatable i-v characteristics regardless of the metal temperature. Meanwhile, i-v characteristics over steel work pieces were found to exhibit an unrepeatable enhanced current in positive voltage regime and drift in the floating potential.”

The secondary ions are generated chemically at the work surface; however, the mechanisms of their origin are yet to unfold. Surface chemical activity is imperative in materials other than metal catalysts. Much work has been carried out to demonstrate the sensitivity of surface kinetics [16,17]. The measurements presented here are designed to identify the dominant processes in a physical system and provide the empirical justification that is necessary for formulating a robust numerical model. Furthermore, depending on the surface material and wall temperature, chemical effects can differ [18,19]. Vlachos et al. numerically investigated the effects of radical removal by means of surface reaction on the H₂–air and CH₄–air premixed flames [20,21]. It was observed that adsorption reactions of OH, O, H, and CH₃ significantly influence extinction in methane-air combustion. Most recently, Narukawa et al. [22] and Salimath et al. [23] performed direct numerical simulations (DNS) of premixed flames impinging on the wall surface to investigate the near-wall flame behavior considering surface chemical activity. However, these investigations do not establish a sound explanation for the sporadically enhanced current in the positive regime of the i-v curve. In our latest experimental investigation [24], carbon contained in steel has been identified as an important source of secondary ions in the oxyfuel-cutting process. In addition, it was observed that the carbon content in steel burned in the oxyfuel flame is directly linked to enhanced electrical currents.

The motivation for the present work stems from two main observations. First, although numerical and experimental studies on premixed flame have been performed extensively, limited research has been conducted to better understand the underlying physics. It is therefore of a great interest to generate a better understanding of the physical behavior of the oxyfuel-cutting flames, along with a more validated current-voltage (i-v) relationship. The second observation is the fact that carbon plays a critical role in the formation of ions in the flame [2]. Thereby, based on the experimental results that have sprung recently [24], we can exploit the methodology in a computational setup to elucidate the importance of the presence of carbon at the work piece.

The objective of the present work, therefore, is to develop an efficient computational model to elucidate the role of secondary ions due to carbon in oxyfuel flame cutting and to evaluate and quantify the electrical characteristics of this approach while also highlighting the fundamental differences between the numerical and experimental results. To support this objective, we create a source term of gaseous carbon to mimic the ‘work surface’ reactions. The hypothesis is that since carbon is an important element, it will be diffusing out of the work surface (steel) and evaporating into the flame. Once in the flame, pure carbon will react very quickly to form ions.

The mole fractions of electrons and ions are infinitesimal with a maximum magnitude of ≈10⁻⁷ in the flow. Therefore, under low voltage, the velocity and temperature profiles of the flow are barely affected by the transportation of electrons and ions [25]. Once the temperature and velocity profiles are calculated along with the generation rates of electrons and ions, the electrochemical species transport equations come into play to evaluate the transportation of electrons and ions. This further provides the electron and ion number densities together with the electric potential of the 2D domain. Concurrently, the electrochemical species transportation and recombination illustrate the electron and ion current densities.

A simplified two-dimensional geometry as shown in Fig. 2 was considered to replicate the experimental setup [6]. As can be seen from the figure, we take advantage of the axial- symmetry of the system and therefore, considered only half of the physical setup and part of the work surface. Surfaces 1, 2, 7, 8, and 9 represent the torch tip on the whole, while surface 1 corresponds to an imaginary annular slot through which the premixed fuel-oxygen mixture is delivered. In this work, we present the preheating process with a stable CH₄–O₂ mixture. The torch is made from either copper or steel and has a fixed distance of 12.7 mm (≈ 0.5 in) above the work surface. This is often referred to as standoff distance.

Surface 3 represents the symmetry plane of the assembly, while surface 4 represents the work surface. Surface 4 has a width of 22 mm, with a vertical distance of 12.7 mm from surface 8. Rest surfaces are prescribed as wall boundary conditions, while surfaces 5 and 6 are treated as openings.

The initial conditions for the 2D model are summarized in Table 1. Essentially, the model is initialized through standard air composition at a pressure of 1 atm and a temperature of 300 K.

where Q is the volumetric flowrate and A denotes the area of the triangular region. The number 12 on the denominator accounts for the fact that there are 12 small triangular torch outlets.

The work surface temperature is assigned to be 600 K, which is consistent with the preheating process from the experiment [11]. The torch surface is also considered to be at the same temperature, i.e., 600 K, however with a variation in electric potential applied. Since both the torch tip and the work piece are absorbing surfaces, the number density boundary conditions will be zero, n_e (0) = n_e (L) = n_i (0) = n_i (L) = 0. The subscripts e and i correspond to electrons and ions, respectively. While some voltage, V_a is imposed at the torch, the work surface is assumed to be at ground potential. Therefore, V (0) = V_a ∈ [−10V, +10V], V (L) = 0.

Lastly, for the turbulence model, the turbulent intensity is set as 0.5 along with a turbulent length scale value of 0.001 m. Overall, detailed summary of the boundary conditions for the current study is shown in Table 2.

Borghi Diagram [27] is used to determine the flame's physical behavior. The primary parameters to locate the flame type regime are: turbulent intensity (u′), laminar flame speed (S_L), integral turbulence length (l), and laminar flame thickness (l_F).

It has been established [28,29] that for a general methane- air mixture, S_L is 0.4 ms⁻¹, whereas the laminar methane oxygen flame speed is ≈ 1.0 ms⁻¹. Cantera [30] integrated in Python was employed to calculate the laminar flame speed for this study. With the parameters defined in Table 3, S_L for the current study is found to be 4.5 ms⁻¹.

Successively, the molecular diffusion rate, D is then computed to be 1.768056 × 10⁻⁵ m for a Schmidt number of 1.0 [27] by Eq. (3). Then, with the know values for D and S_L, the flame thickness, l_F is found to be 3.929 × 10⁻⁶ m. Finally, the turbulent intensity (u′) is chosen to be 0.5%.

Now that all the variables have been enumerated, the ratios $l/lF$and $u′/SL$are then assessed. The ‘green square’ in Fig. 3 illustrates our present study which has a value of ≈ 255 along the x-axis (⁠$l/lF$⁠) and ≈ 12.5 along y-axis (⁠$u′/SL$⁠). It is observed that the final value of our current study falls within the thin reaction zone in the Borghi diagram. Peters [27] demonstrated that when the flame is in the thin reaction zone, the preheat region of the flame will be bordered by turbulence. As such, to model the oxyfuel flame preheat phase, turbulence models are employed in the computational setup rather treating the flame regime as laminar.

where $Sct$ is the turbulent Schmidt number (=$μtρDt$⁠). Here, $μt$ is the turbulent viscosity and $Dt$ is the turbulent diffusivity.

In Eqs. (16)–(17), $Gk$ represents the generation of turbulence kinetic energy due to the mean velocity gradients. $Gb$ is the generation of turbulence kinetic energy due to buoyancy. $YM$ represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. $C2$ and $C1ε$ (= 1.44) are constants. $σk$ (= 1.0) and $σε$ (= 1.2) are the turbulent Prandtl numbers for k and ε, respectively. $Sk$ and $Sε$ are user-defined source terms. In Eq. (23), S is the modulus of mean strain rate tensor, S, which is defined by Eq. (24).

$τω$ is known as the wall shear stress.

A chemical kinetic mechanism is required for the computational model that includes both neutral species and ionic species reactions. Through chemi-ionization, the mechanism should concede the production and depletion of ions. Additionally, ion-molecule reactions, and dissociative recombination should also be exerted by this mechanism.

In the current study, the reactions of the methane-oxygen (CH₄–O₂) flame are combined with GRI 3.0 [40] and additional ionization reactions from Belhi et al. [41] to describe the chemistry of ions in flames. The thermodynamic properties of the charged species are taken from Burcat database [42]. Table 4 presents the coefficients magnitudes [10] of the ionization reactions. The transport properties are extracted from a methane-air reaction presented by Chen et al. [43].

In Eqs. (32) and (33), $Na$ is the Avogadro constant, (= 6.02214086 × 10²³∕mol), $cion$ is the molar concentration of ions, and $ce$ is the molar concentration of electrons. As can be seen, the electric current density is related to the electron species' mobility, diffusivity, space, and bulk velocity, whereas the ion charge density is determined based on the molar concentration of ion species.

A 2D CFD model for the domain as shown in Fig. 2 was developed using ANSYS FLUENT 2020R2. Two different metallic surfaces, steel, and copper were considered as the work surface (surface 4) in the 2D model to imitate the experimental setup [6]. The material properties of both steel and copper are listed in Table 5. Furthermore, while considering the steel surface, a source term of gaseous carbon is assigned at surface 4 to mimic the ‘work surface’ reactions. In this approach, the combustible carbon element is assigned a definite mass flowrate of 1.78 × 10⁻⁵ kg/s based on the analytical expression employed in the experimental investigation, $m˙=π4D2ρχF$ [24]. The reaction rates were then tracked at different radial flame lines using the modified Arrhenius' equation as shown in Ref. [10]. Carbon comprises for around 0.20% in low-carbon steel [46]. The hypothesis is that since carbon is an important element, it will be diffusing out of the work surface (steel) and evaporating into the flame. Once in the flame, pure carbon will react very quickly to form ions.

The governing equations are solved by employing a finite volume approach. Second-order spatial discretization scheme was adopted to discretize the governing equations in the whole domain. Moreover, coupled solution method and the coupled pressure–velocity coupling algorithm were used to solve the governing equations. The PRESTO! scheme designed for flows involving steep pressure gradients was chosen for the pressure equation. The realizable k–$ε$ turbulence model was considered the viscous model. Stiff chemistry solver along with the eddy dissipation concept (EDC) was introduced to the CFD code to solve for the turbulence-chemistry interaction.

where $Cτ$ is a time scale constant.

where ∗ denotes fine-scale quantities and, $Cξ$is the volume fraction constant, and v is the kinematic viscosity.

In Situ Adaptive Algorithm (ISAT) [37,49] was selected as the integration method for the chemistry solver since ISAT can accelerate the chemistry calculations by two to three orders of magnitude, thus offering substantial reductions in run-times. Finally, the convergence criterion requirement is set to be 10 × 10⁻⁶ as the residual term for the equations. The parameters for the current CFD setup are summarized in Table 6.

Simulations for the current computational model were carried out in one of the high-performance computing systems, ‘TinkerCliffs’, from the Advanced Research Computing (ARC) division at Virginia Tech. TinkerCliffs has an AMD EPYC 7702, Intel Xeon Platinum 9242 CPU model @ 2.0–2.3 GHz with 41,984 cores.

For the purpose of lateral comparison, we create somewhat virtual flame inspection lines in the 2D domain as shown in Fig. 4. Another motivation to do so is the fact that the color schemes in such 2D model are difficult to analyze since the variation of magnitude of the number densities will be substantial.

As shown, we have three segmented vertical lines, and a horizontal line over surface 4. The vertical lines, in our case, the flame lines have a total length of ≈ 14.25 mm. Flame lines 1 and 3 have increased resolution along a distance of ≈ 2 mm from surfaces 1 and 4, respectively, meaning they are divided into much finer nodal points. This is done because the process parameters and electrical characteristics change rapidly depending on where you are in the flow, and these changes are very swift near surfaces 1 (torch/inlet area) and 4 (work surface). As such, another vertical line, a line probe, is also considered which is ≈ 5.5 mm above the work surface.

All things considered, the results presented herein are extracted along these flame lines, if not indicated otherwise. Furthermore, they also act as the axial and radial distribution lines for the current 2D analysis. Furthermore, the results are discussed through combustion part (Temperature, velocity, etc.) and subsequent eletrochemical part, to impart a sound understanding.

The temperature contour plot near the torch tip for the two-dimensional domain is shown in Fig. 5. The geometry of the flame is characterized with having ‘inner’ and ‘outer’ cones that describe the luminous zone formation at the front.

As shown, the premixed methane-oxygen flame generated in the two-dimensional domain has a V-shaped flame. The profile also illustrates that the temperature is high beneath the V-shape slit. According to Nernst–Einstein equation, the electrons and ions will have a high diffusion rate. Additionally, the profile of temperature at the reaction zone changes [31] for accounting turbulence.

Furthermore, the axially distributed temperature profile as shown in Fig. 6 illustrates that at flame front (≈ 5.7 mm), where the mixture of fuel (CH₄) and air (O₂) are well premixed, the temperature is maximum.

The radial velocity is much more comprehensible than the axial velocity in the flow for a two-dimensional flame, and the phenomena are shown in Fig. 7. The velocity declines over a 1 mm length scale close to the work surface. At a sufficiently high voltage, the flow field can be shifted toward the electric bias areas.

As indicated in chemical kinetic mechanism, Hydronium (H₃O⁺) and formyl (HCO⁺) cations are the two ions in the ionization reactions with H₃O⁺ ion being the abundant one [50]. The generation rate of H₃O⁺ is as good as that of the electrons (e⁻), whereas HCO⁺ has a substantially lower generation rate.

When a sufficiently strong electric bias voltage is applied, the charge carriers can be driven away to the vicinity of the absorbing surfaces. Boucher [51] experimentally confirmed that the metal surfaces absorb free electrons due to their rapid transport in the gas phase and produce a positively charged ‘sheath’.

The electrical species transport model results in the number density profiles of electrons and ions. The axial distribution of the number densities for both H₃O⁺ and e⁻ are shown in Fig. 8. This is done since the two-dimensional color scheme is difficult to analyze as the magnitude of the densities changes extensively. The magnitudes change very rapidly near the surfaces (inlet and work surfaces). As can be seen from Fig. 8, the maximum magnitude of species number density is 10¹⁸∕m3, and this tones with the physical measurement [26]. The sheaths' saturation can be observed by constraining the torch and work surface number densities boundary condition to zero. The electrons' (e⁻) and ions' (H₃O⁺) number densities peak at the flame front which is ≈ 5.7 mm. Figures 8(a)–8(c) provides pictorial illustration of the number densities at different electric bias voltages and are discussed below:

• When a voltage of −10V is applied at the torch, we can observe a swift increase in the number of ions (H₃O⁺) at the torch surface which vanishes more rapidly at the work surface than the electrons (e⁻). Even though the applied voltage is driving them (e⁻) in that direction, e⁻ vanishes almost entirely at the work surface. This is the formation of a ‘sheath’.

• Things become a bit interesting when a torch voltage of 0V is applied as shown in Fig. 8(b). In this case, there is a slow expansion of H₃O⁺ at the beginning of the torch tip surface, while the increment of the number of e⁻ is swift. The radial velocity leads electrons away. However, e⁻ disappear quickly at the work surface than H₃O⁺. The charge carrier pair of H₃O⁺ and e⁻ differs in mass by some 35 thousand times [6]. In addition, the heavy positive ions are in the power of bulk fluid velocity, whereas the tiny electrons respond to the electric field and are at liberty to diffuse. Thence, the electrons move more comfortably than the ions in CH₄–O₂ flame. Sheaths are formed in both surfaces (torch and work surface) because of the number of differences between ions and electrons, suggesting that the grid was able to resolve the Debye length.

• Figure 8(c) corresponds to the number densities of H₃O⁺ and e⁻ when a torch electric bias voltage of +10V is applied. In contrast to the −10V case, when a positive electric bias is applied, +10V in this case, there is a swift increase in number densities of e⁻ at the torch surface, whereas they vanish quickly at the work surface. Simultaneously, the ions (H₃O⁺) start to dwindle at the torch surface with an abrupt drop at around 1.8 mm. This is where the torch ends, and a radial velocity appears. Nonetheless, H₃O⁺ ions remain significant at the work surface.

where $qi$ and $ni$ are the charge number and the number density of species i, respectively. $ϵ0$ is the permittivity of free space and is equal to 8.8542 × 10¹² farad/m.

The electric potential for three different voltages, −10 V, 0 V, and +10 V are shown in Fig. 9. These are plotted axially along the flame line once the number density of electrons and ions has been calculated. It can be observed that even though an electric bias voltage of 0 V is applied, due to sheaths' charge accumulation, the electrical potential rises above 0 V. When a positive voltage (+10 V) or relatively strong electric potential is applied, the concentration of negative ions becomes insignificant. This leads to one of the sheaths dominating as electrons evacuate quickly leaving a positive charge in their wake. However, negative ions are dominant when a negative voltage of −10 V is applied. Due to ‘ohmic losses’ along the flame's length, a significant voltage drop can be observed in Fig. 9.

The contour plots for the electric potential over the domain are shown in Figs. 10(a) and 10(b) for +10 V and −10 V, respectively. These plots also illustrate the differences between positive and negative torch surface potential. Belhi et al. [45] described that charged species are driven by the local gradient of the electric potential when an electric field is applied. In this way, the positive charges are transported from the reaction zone and negative charges move in the opposite direction. It is evident from Fig. 10 that in both figures the flame is nearly uniform in potential. The flame front is much more electrically conductive because of the high density of ions there. Considering Fig. 10(a) as an instance, when +10 V is applied at the torch tip, it can be observed that the electric potential does not decrease proportionately due to the presence of high density of the electrical species. However, for a negative voltage of −10 V, the gradients are perceivable near the work surface as shown in Fig. 10(b). This is known as ‘saturation’. The more mobile of the two charge carriers (e⁻) vacates the region near the negative surface, growing the sheath, and the only means to further increase current is to increase the transport of the less mobile (H₃O⁺) through the sheath. However, that happens very slowly compared to e⁻ so the current only increases very slowly with voltage.

The ion and electron current densities are extracted for the upstream boundary based on different torch voltages and are calculated by Eqs. (30) and (31). In simulating the response of flames subject to an electric field, an accurate prediction of current densities is imperative. The cross-sectional average current densities do not change throughout the domain by virtue of Kirchoff's law, however, the cross-sectional average flux of individual species can vary. In the inner cone where ion densities are very high, both electrons and ions will have their peak current densities. As can be seen in Fig. 11, there is a hint of the onset of saturation in the current density plots. However, the ion current does not go to zero at positive voltage, meaning that a recirculation is delivering positive ions to the torch tip.

To further investigate this prospect, axial variation of equivalence ratio is considered. Since the F/O ratio is at 0.833 (volumetric), the tentative equivalence ratio is 1.6, denoting a rich flame. Therefore, there is more fuel than there is oxygen to burn it. This is done because the maximum flame temperature occurs at rich conditions, and that effect is severely exaggerated in atmospheric flames without nitrogen diluent. Entrained oxygen from the surrounding air helps complete combustion at the plate surface. It can be noted from Fig. 12 that once the reaction starts, CH₄ isn't the only fuel and O₂ isn't the only oxidizer. The combustion will result in different species and radicals as well. The oxygen will be consumed first, and there will be leftover fuel. However, there should not be much of the original reactants left once in the outer cone. The bulk of the flow will be intermediate species.

Additional H₃O⁺ along with the oxygenated cation, C₂H₃O⁺ account for the recirculation of positive ions to the torch tip resulting in an ion current density lower than zero for the positive voltage regime as shown in Fig. 11.

The ion current density and the electron current density are integrated over the work surface to calculate the total electric current. Figure 14 presents the current-voltage (i-v) characteristics for the premixed oxyfuel flame subject to an electric field. The i-v curve adopts three distinct regimes for different electric bias voltages applied.

• Regime1: Negative Saturation. The region at the torch tip is starved of electrons, so currents have to be carried by heavy positive ions being driven upstream against the flow. Regime 1 appears when there is a sufficient growth in negative currents. It is evident from Fig. 14 the current for both copper and steel surfaces changes linearly with an almost perfect agreement. However, this segment characteristic is strongly governed by the F/O ratio [11].

• Regime2: Ohmic. In this slim regime, no saturation occurs, thus the name ‘Ohmic’, and both current densities follow Ohm's law. The availability of both positive and negative charge carriers is abundant in this regime. This narrow region acts as a zone where the current rapidly transitions between the two saturation regimes (regimes 1 and 3). While Ohmic behavior is present in all regimes, regime 2 has the most influence. The slope of the curve of regime 2 (or, Ohmic regime resistance) is sensitive to the F/O ratio, with a constant sensitivity with respect to the standoff distance.

• Regime3: Positive Saturation. Regime 3 appears when the positive current sufficiently grows, and when voltages are sufficiently positive to saturate the downstream sheath, meaning the electrical current there is due to the delivery of positive charge. Figure 14 shows that copper surfaces experience a stable and repeatable i-v characteristics in regime 3. This is because copper oxide is mechanically stable, and its melting temperature is higher than copper. It forms a stable protective layer over the metal and that halts chemical activity at the surface. Meanwhile, steel surface exhibits unrepeatable i-v characteristics with sporadically enhanced current in the same regime and drift in the floating potential.

At this juncture, the food for thought is, whether the enhanced currents in regime 3 for steel surface could somehow be a by-product of the oxidation reaction itself? Oxidation is ruled out as the cause of the increased currents because the direction of electrical current is backwards, and a mechanism for the formation of free ions was not perceived. The additional positive ions generated due to the diffusion of carbon (from steel surface) in the flame are delivered to the torch tip through a recirculation. What if that carbon is also generating additional ions over the work surface?

When a lean-stoichiometric flame is considered, meaning abundant O, Reaction (R7) is dominant [54,61], whereas Reaction (R8) presides when C₂H₂ is ample [60,62].

The additional cations generated through carbon diffusing out of the steel workpiece provide a recirculation of positive ions to the torch tip and increase net electric current in regime 3 of the i-v curve due to the presence of those ions.

An experimental investigation was undertaken by Martin et al. [24]. This investigation was conducted using two experiments. In the first, ferrous wires with varying carbon content were burned while holding all other conditions constant. In the second, carbon was added directly by blowing fluidized graphite powder into the flame. The outcomes are shown in Fig. 16.

Figure 16(a) shows the variation of net current when carbon content is varied using different wires. Vertical error bars represent the maximum and minimum of the windowed mean value throughout the test. Horizontal error bars represent the most extreme variation in carbon content permissible by the material specification. It was concluded that the addition of even trace amounts of carbon in the outer cone (regardless of the source) can be expected to roughly double the positive saturation current.

Current signals for various mass flow rates of graphite are shown in Fig. 16(b). The points represent the mean signal, vertical error bars are constructed from rms noise, and horizontal error bars represent a 95% confidence interval on the mass flowrate measurement. There is good confidence that the trend is linear. Despite the rms signal amplitude and the mass measurement uncertainty, the mean values rise linearly with little scatter. The extreme current was found to be repeatable, though a clean signal with stainless steel wire was never achieved at this feed rate.

The resulting i-v curves for the experimental investigation are compared with the computational results and are shown in Figs. 17(a)–17(b). The experimental results were taken for volumetric flow rates of 20 scfh and 25 scfh, respectively, whereas the computational result corresponds to an inlet velocity of 12 ms⁻¹. The reason there is so much scatter for the experimental curves is because the standoff is high (= 12.7 mm), and the signals are so small. This is only about 5µA of current in regime 3, which is very small. At lower standoffs, the work will be in a region with more ions, so the signals are much stronger. When the flame is moved off the edge of a plate/work surface, the current is not significantly changed until more than half of the flame is off the work. This suggests that the electrical resistance of the outer cone is more or less uniform, and the current can enter virtually any part of the cone.

It can be readily observed from the figures that, while regime 1 matches linearly for both experimental and computational model, a degree of variation can be seen in regime 3. However, at the same time, the variation is compromised by increasing the volumetric flowrate from 20 scfh to 25 scfh. The probable grounds for such variation in the i-v plots could be due to:

• The initial velocity prescribed at the inlet might not be approximating the physical flow rates perfectly. Regime 3 is quite sensitive to the velocity. Decreasing the velocity will tend to decrease regime 3 current. It may also slightly increase regime 1 current; however, those data seem to be relatively insensitive to flow rate; they are much more sensitive to F/O ratio.

• The ions' mobility might be more substantial than the values assigned in the computational model. Because the mobility is quite small, the recombination rate has a bigger part to play in regime 3. In this regime, all of the electrons are driven away from the work, so the current is determined by the number of ions that survive to reach the plate. It is still weakly sensitive to voltage because the ‘sheath’ where electrons are absent grows deeper into the flame, where ions are more abundant, so more survive to reach the work.

• A lower velocity in the hot zone gives ions more time to recombine before they reach the work surface. This results in lower ion concentrations and lower saturation currents.

A two-dimensional model has been developed using the ANSYS FLUENT CFD code and is capable of analyzing the transportation of electrons' and ions' in a premixed CH₄–O₂ flame subject to electric bias voltages. This essentially imitates the preheating stage of oxyfuel flame-cutting process. The CFD solver on FLUENT was incorporated with a chemical kinetic mechanism and transport property evaluations. The simulations provided reasonable outcomes of the combustion and electrochemical properties. The transport results illustrate the current-voltage (i-v) relationship. Primarily, the role of secondary ions on the i-v relationship has been investigated and have been compared to the physical experiment. The aforementioned results permit following remarks:

• A source term of gaseous carbon has been assigned at the work surface (surface 4) for steel that ‘mimics’ the surface reaction phenomena.

• Secondary ions in the oxyfuel-cutting flame are produced in part (if not entirely) by carbon.

• Ion current density experiences a recirculation that delivers positive ions to the torch tip.

• Sporadically enhanced electric currents observed in the positive saturation are due to the additional ions generated by carbon reaction. Once in the flame, pure carbon reacts quickly to form ions.

• Some degree of difference has been observed in the i-v relationship, however, this can be compromised by playing around with the process parameters such as inlet velocity, F/O ratio, standoff distance, etc.

• National Science Foundation (NSF) (Grant Nos. 1900540 and 1900698; Funder ID: 10.13039/100000001).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A =

area (m²)

$ci$ =

molar concentration of species i (kmol m⁻³)

$Di$ =

diffusivity of species i (m² s⁻¹)

$Ea$ =

activation energy (J mol⁻¹)

K =

mobility (m² V⁻¹ s⁻¹)

k =

rate constant

$ni$ =

number density of species i (m⁻³)

$qe$ =

fundamental charge (1.6E-19) (C)

$Q$ =

volumetric flow rate (m³/s)

$Ru$ =

gas constant (J mol⁻¹ K⁻¹)

$Sct$ =

turbulent Schmidt number

t =

time (S)

T =

temperature (K)

U =

velocity (ms⁻¹)

$Yi$ =

mass fraction of species i

$y+$ =

dimensionless wall distance

$ε0$ =

vacuum permittivity (8.9E-12) (C V⁻¹ m⁻¹)

$λD$ =

Debye length (μm)

$σi$ =

space charge density of species i (C m⁻³)

$B$ =

Boltzmann constant (1.38E-23) (J K⁻¹)

β =

temperature exponent

μ =

dynamic viscosity (Pa · s)

ρ =

density (kg/m³)

τ =

stress tensor (N/m²)

A grid convergence study was performed to compute the numerical uncertainty of the computational model. The grid convergence index (GCI) method [65], based on Richardson extrapolation (RE), was chosen to estimate the discretization error within each grid. GCI is a well-known approach to compute grid uncertainty and the articles from [66–70] illustrate the applications of the same.

In this study, three different structured grids were examined, formed by quadrilateral grid elements. Table 7 presents the grid converge index results for the three grids examined, with pictorial illustrations in Fig. 18. It is evident from both the table and figures that the results do not vary significantly as the mesh density goes from ≈ 539 thousands to ≈ 808 thousands. Thereby, for the analysis of the current computational model, a mesh density with 539,076 nodes and 537,500 elements was considered.

where $λD$ is the Debye length, $ε0$ is the permittivity of free space, $kB$ is Boltzmann's constant, which is 1.380649 × 10²³ J/K, $qe$ is the electrical charge of a particle, which is 1.602 × 10⁻¹⁹ coulombs, $ne$ is the density of electrons, $Te$ is the temperature of the electrons, $zi$ is the rate of ionic charge compared to the charge of electrons, $ni$ is the density of atomic species i, and $Ti$ is the temperature of ions. The temperature of electrons and ions are assumed equal to the flame temperature (⁠$Te$ ≈ T) from Clements and Smy's approximation [71,72]. With the current mesh density, the value of $λD$ obtained was well within the experimental observations, 3.8 − 38 μm [26].