Heat Transfer Mechanisms in Arcs of Various Gases at Atmospheric Pressure

Abstract

This work expands findings about the dominant heat transfer mechanisms in argon and helium arcs at atmospheric pressure for monoatomic (Ar, He, 50% Ar–50% He), diatomic (air, ${\hbox{N}}_{2}$, ${\hbox{O}}_{2}$, ${\hbox{F}}_{2}$, ${\hbox{Cl}}_{2}$), and triatomic (${\hbox{CO}}_{2}$) gases. The objective is to understand the dominant mechanisms in atmospheric plasmas through validated numerical modeling for GTAW welding process. Arcs of all gases have lengths of 10 mm and 200 A current. Five heat transfer mechanisms are considered: Joule heating, convection, radiation, conduction, and Thomson effect. Results indicate that the general structure of the arcs and dominant mechanisms are qualitatively similar for all gases; sizes change depending on the gas. The dominant energy input near the cathode is Joule heating, while that near the anode is convection. The dominant energy output always follows the same sequence: Thomson effect next to the cathode followed by convection, radiation in the arc column, and conduction near the anode. Joule heating is the most significant in Ar, while in He, it has the lowest significance. This is due to differences in electric conductivity of He (higher up to 21,000 K and lower beyond 21,000 K than other gases) and high viscosity of He, which creates a small Joule heating versus a large convection region. He transfers the most heat towards the anode by convection while ${\hbox{N}}_{2}$ has the lowest; due to the high enthalpy and viscosity of He compared to ${\hbox{N}}_{2}$. Finally, Ar has the most significant radiative emission and He the smallest due to their net emission coefficient.

Introduction

Arcs at atmospheric pressure are ubiquitous in modern technology, including arc welding, high-current switches, and electric arc furnaces. Arc welding in particular is used in applications as small as morphine pumps welded with 1 A, to aircraft carriers welded with currents approaching 1000 A, and gases involved range from pure Ar to molecular gases such as ${\hbox{CO}}_{2}$. When there are multiple options, the selection of gases is done exclusively empirically, based on past knowledge and trial and error. In some processes such as submerged arc welding, the arc gases come from the evaporation of ceramic fluxes and evaporating metals [1]. An understanding of arc behavior in general, extending beyond specific cases, will be much welcome by practitioners. In most technologies involving arcs at atmospheric pressure, heat transfer is the key consideration. In high-current switches, melting and thermal damage are to be avoided, while in welding and electric arc furnaces, the arcs are meant to melt metal (typically the anode). Heat transferred from the arc to the joint depends on arc length, input current, shielding gas, and electrode type, among other parameters [2,3,4,5].

Experimental studies of heat transfer in the arc are rarely found in the literature. Due to the emission of arc radiation, measuring thermal and electric characteristics of welding arcs is usually done by optical emission spectroscopy measurements. However, high-speed cameras with optical filters and laser illumination systems or thermocouple arrays can also be used [6,7,8]. The measurement of arc characteristics is a way to validate mathematical models of the arc, which have been the most widely used analysis tool.

The heat transfer from the arc to the workpiece is very commonly estimated by assuming that the arc is a moving heat source. The source can be either a point, a line, or a volumetric heat source. Also the heat distribution can be Gaussian [9], elliptic [10], double elliptic [11] or a combination of these [12, 13]. All these distributions have a central peak that decreases very rapidly along the workpiece domain. This approach allows estimation of characteristics such as the location of the heat affected zone and provides a way to compute residual stresses but neglects the physical phenomena inside the arc. Moreover, the heat source has to be calibrated against measurements, for example of the thermal history at locations in the workpiece.

Heat transfer inside the arc is a complex phenomenon that encompasses many mechanisms such as convection due to the plasma jet, heat conduction, and radiation from and within the arc [2, 14]. It is well established that the Joule effect and radiation are the main heat input and heat output in the arc plasma, respectively. Nonetheless, near the electrodes, the temperature difference between the electrons and other species present in the arc means that the local thermal equilibrium assumption is not valid, and mechanisms like Thomson effect, electron drift and electron absorption must be considered.

Based on the above, it cannot be denied that heat transfer phenomena inside the arc play a fundamental role in welding technology. The topic has been researched, but not to the extent that accords with its importance. For example, Lu et al. [15] identified the best-operating conditions in a GMAW process to maximize heat transfer between the arc and aluminum plates using a three-dimensional numerical model that includes the filler wire, arc plasma and a meltable workpiece. They found that 99% of the energy generated in the arc is due to Joule heating. Furthermore, the arc consumes 14–19% of the total energy, so the arc efficiency is 66–77% in this system.

Schnick et al. [16] conceptualized the arc columns as two concentric cylinders. The inner cylinder had argon and metal vapor present from the wire evaporation; meanwhile, the outer cylinder was considered to be composed only of argon. In the inner cylinder, they found that even a small amount of metal vapor increases the radiation losses significantly and thus decreases the arc temperature. They also used a 2D axisymmetric CFD model to simulate GMAW welding and found that local radiation and resistive heating values have the same order of magnitude as convection and heat conduction. Thus, these components cannot be neglected in the global energy balance for MIG arcs.

Recently, maps of the dominant heat transfer mechanism were developed by Velazquez et al. [17], creating a simple way to present the main features of heat transfer in TIG welding using helium and argon as shielding gases. The maps were obtained by processing the results of numerical simulations and they provided the isotherms and isopleths of radiative emission of the arc, and the distribution of the dominant heat transfer mechanisms inside the arc and their relative importance in the form of color maps. They analyzed the changes in these maps that occurred when current input and arc length were varied. While the arc length modified the region the workpiece in which convection dominated, the arc current impacts the region and degree of dominance of all mechanisms, in particular the Joule effect. Also, they predicted that shielding gas would modify the region and degree of each mechanism’s dominance. The details on the construction of such maps may be seen in [17].

The purpose of the work is not the development of model, which has been already developed and validated extensively in the literature. Instead, the numerical model is applied to assess quantitatively the relative importance of each of the heat transfer mechanisms in the arc column for nine different plasma gases, including monoatomic, diatomic, triatomic and a mixture. This constitutes the originality of the work.

Methodology

Mathematical Model

This work builds on the model of the arc at atmospheric pressure developed originally in [18], and used recently in [17, 19, 20]. The model considers the arc to be axisymmetric (Fig. 1), which is true for electric arc furnaces, high current switches, and a large number of welding applications such as GTAW. Because this model has been validated extensively in the literature, its key features will be described only briefly.

Fig. 1

Computational domain of the DC electric arc regions

The main assumptions made in the model are listed in Table 1, which result in the governing equations listed in Table 2, with the boundary conditions listed in Table 3. The notation is given in “Appendix 2”. It is worth mentioning that due to the LTE assumption, energy fluxes associated with the electron and ion drift and diffusion fluxes (apart from that connected with the electric current density) are not considered. Also, the inclusion of sheath voltages in the model could modify the properties of the arcs formed from the different gases.

Table 1 List of simplifying assumptions used in the development of the mathematical model

Table 2 Governing equations

Table 3 Boundary conditions (see Fig. 1 to locate every boundary)

Solution and Generation of the Maps

The model was cast in the CFD software PHOENICS version 2017, and numerically solved using an entirely orthogonal grid of $60\times 60$ cells that was the output of a grid sensitivity analysis previously presented [18]. For the work, nine different simulations were performed keeping the same arc length of 10 mm and the arc current of 200 A; the type of gas is varied. The physical properties of the nine gases (air, helium, argon, 50–50% He–Ar mixture, ${\hbox{CO}}_{2}$, ${\hbox{O}}_{2}$, ${\hbox{N}}_{2}$, ${\hbox{F}}_{2}$ and ${\hbox{Cl}}_{2}$) were computed using the methods presented by Murphy et al. [23,24,25,26]. “Appendix 1” provides details of the calculations made to obtain the physical properties of ${\hbox{F}}_{2}$ and ${\hbox{Cl}}_{2}$. Some plasma gases (air, ${\hbox{O}}_{2}$, ${\hbox{F}}_{2}$, ${\hbox{Cl}}_{2}$) are impractical for welding, particularly with a tungsten cathode, but are helpful to gain a theoretical understanding of the arc physics and particularly the heat transfer mechanisms as a function of the plasma properties. FORTRAN subroutines were written to compute the electromagnetic problem, the source terms of the momentum and energy conservation equations, as well as some boundary conditions. Several subroutines were created to calculate the five heat transfer mechanisms in Watts per cubic meter inside the arc (positive or negative terms). Then, as stated in our previous work [17], a subroutine in Matlab® was used to construct the map of the dominant heat transfer mechanisms, showing the intensity of the largest heat transfer mechanism in each cell for both inputs and outputs of energy. Also, the shape of the arc, defined by the isotherm at which the conductivity approaches zero (the external isotherm) is drawn, and the relative importance of the dominant mechanisms to the rest of the mechanisms is shown through isolines. Every run took around 50,000 iterations and approximately 2 h of computation time to provide a converged solution (when the conservation equation’s imbalances reached a level below 0.001%) on an Intel® Xeon® CPU E31245 3.30 GHz computer. The model was extensively validated by comparing the predicted results against other simulations and experimental data of most of the arc characteristics and arc-weld pool interactions, as can be seen in previous published works [18, 27].

Results and Discussion

Dominance maps were produced for the nine gases analyzed, keeping the arc current and length constant at 200 A and 10 mm, respectively. In those maps, the colors indicate the major or dominant mechanism (red is radiation, light blue is conduction, green is convection, purple is the Thomson effect, and orange is Joule heating). The color bar of each mechanism corresponds to the value of power density in each cell normalized by the maximum value of that mechanism in the whole domain. The shape of the arc (visible arc) is depicted by the outer isotherm (dotted line) and the isotherm for which the conductivity is 2850 S m$^{-1}$ is also plotted (dashed line) since in previous works, it was used to define the shape of the arc [19, 20]. Because electrical conductivity behaves differently for each gas, the isotherms need to be established accordingly for each map. Table 4 contains the value for both isotherms of all the gases considered in this work.

Table 4 Temperature (K) corresponding to conduction and visible isotherms of all the gases simulated

Finally, the relative importance of the dominant heat transfer mechanism is presented through isolines. For example, the isoline with a label of 0.8, means that 80% of the total heat input or output is due to the dominant heat transfer mechanism [17], and since these lines cross through several regions, the comparison can be made. We grouped the maps by the nature of the gas at standard conditions, Fig. 2 show monoatomic gases such as Ar, He, and Ar–He mixtures, Fig. 3 shows diatomic gases such as ${\hbox{N}}_{2}$, ${\hbox{O}}_{2}$, ${\hbox{F}}_{2}$, ${\hbox{Cl}}_{2}$, and air; and finally, the only triatomic gas, ${\hbox{CO}}_{2}$, is displayed in Fig. 4. The maps come in two parts, bisected by the axis of symmetry: the left-hand side corresponds to energy inputs and the right-hand side shows the outputs. Regardless of the type of gas, the mechanisms for the input and output of energy are similar. The inputs, from the cathode to the anode, are Joule heating and convection, while the outputs, from the cathode to the anode, are the Thomson effect, convection, radiation and conduction. What changes in the maps is the area over which each major mechanism dominates and its intensity.

Fig. 2

Heat transfer predominance mechanism for a Ar, b He, c 50–50% Ar–He. Color identification: radiation = red, convection = green, conduction = light blue, Thomson = purple, Joule heating = orange. In each case, the edge of the arc (arc shape) is presented along with the relative importance of the dominant heat transfer mechanisms through isolines (Color figure online)

Fig. 3

Heat transfer predominance mechanism for a air, b ${\hbox{N}}_{2}$, c ${\hbox{O}}_{2}$, d ${\hbox{Cl}}_{2}$, e ${\hbox{F}}_{2}$. Color identification: radiation = red, convection = green, conduction = light blue, Thomson = purple, Joule heating = orange. In each case, the edge of the arc (arc shape) is presented along with the relative importance of the dominant heat transfer mechanisms through isolines (Color figure online)

Fig. 4

Heat transfer predominance mechanism for ${\hbox{CO}}_{2}$. Color identification: radiation = red, convection = green, conduction = light blue, Thomson = purple, Joule heating = orange. In each case, the edge of the arc (arc shape) is presented along with the relative importance of the dominant heat transfer mechanisms through isolines (Color figure online)

Analyzing the differences in the mechanisms from the largest to the minor areas for each color (mechanism) has to consider the physical properties of every gas and the arc characteristics that determine the magnitude of the heat transfer mechanism. The most relevant mechanisms are discussed below.

Summary of Results

Table 5 presents a summary of some basic characteristics of each gas. The maximum temperature and velocity in the arc are provided. Most of the arcs have maximum temperatures of 20,000 to 23,000 K. Air, ${\hbox{O}}_{2}$ and ${\hbox{N}}_{2}$ have the highest velocities, followed by He and ${\hbox{CO}}_{2}$. Other parameters such as the arc column voltage, and those related to the interactions of the arc and the weld pool, the total heat flow to the weld pool, maximum pressure and maximum shear stress, are also presented. He, ${\hbox{N}}_{2}$, air and ${\hbox{CO}}_{2}$ exhibit the highest arc voltages and since arc current is constant, it is concluded that these gases have the highest average Ohmic resistances of those studied. A consequence of high Ohmic resistance is that these gases deliver more heat to the anode than the other gases, indicating the Joule heat’s importance in the heat transferred from the arc to the weld pool. It is difficult to conclusively establish the best gas for heating since we kept the arc length and arc current unchanged but the results presented give guidance for a better shielding gas selection.

Table 5 Characteristic values of the arc and its interaction with the weld-pool for the shielding gases studied in this work

Finally, the gases that apply most pressure and shear stress to the weld pool are ${\hbox{N}}_{2}$, air, and ${\hbox{CO}}_{2}$, while the gases causing lowest shear are ${\hbox{Cl}}_{2}$, He, ${\hbox{F}}_{2}$ and Ar, i.e., the former gases are more likely to perturb the weld pool than the latter gases. This magnitude of the pressure and shear stress is correlated to the vertical extent of the region of Joule heating dominance. When Joule heating dominance extends to close to the workpiece, this indicates that the arc remains constricted, increasing pressure and shear stress.

Joule Effect

The Joule effect is the main heating mechanism near the cathode. He and Ar are two extreme cases; Ar has the largest Joule zone and intensity while He has the least extensive and intense Joule contribution. Helium has two important properties that distinguish it from the other gases. Both arise from its high ionization energy. First, the electrical conductivity of He is lower than the other gases up to about 20,000 K as can be seen in Fig. 5a and the viscosity is higher above 10,000 K as can be seen in Fig. 5b. The low electrical conductivity of helium means that the region conducting electricity is relatively narrow near the cathode compared to the argon arc, which means that the Joule heating region is smaller in He than in Ar. The high viscosity implies that the flow spreads radially at a relatively low axial position (i.e. closer to the cathode). The two effects combine to give a relatively small region of Joule heating dominance and a correspondingly large region of convective heat transfer dominance as it will be seen in the next section.

Fig. 5

Thermophysical properties of Ar and He as a function of temperature, a electrical conductivity, b dynamic viscosity

Near the cathode, the arc’s most elevated temperatures and the highest current densities are present. Since the arc operates at the same current, it is clear that the Joule effect comes from the electric resistance (inversely proportional to the electric conductivity) and consequently, Ar presents more electric resistance than He in this region.

Convection Near the Anode

The other major source of positive energy in the arc is the convection, which dominates near the anode as an input. The convective heat transfer is computed as the product $\rho V h$. Therefore, it can be stated that the arc produces heat by the Joule effect near the cathode and delivers heat to the workpiece by convection. In this case, the extreme cases are He (highest delivery by convection) and ${\hbox{N}}_{2}$ (lowest convection dominance zone). Figure 6 shows the velocity profiles and enthalpy contours of the two gases since the convective heat transfer depends directly on the velocity and enthalpy. Although the ${\hbox{N}}_{2}$ plasma jet velocity is greater than the helium arc velocity near the anode, the region inside the He enthalpy contour of value $2.9\times 10^{7}$ J is significantly larger than that inside the corresponding ${\hbox{N}}_{2}$ enthalpy contour, so the convection dominance zone is larger in the He map. Besides, as stated in the previous section, the high viscosity of helium at temperatures greater than 10,000 K means that the region over which the convective flow is important is larger in He than in other gases. The helium viscosity can be seen in Fig. 5b.

Fig. 6

Enthalpy contours for He (a) and ${\hbox{N}}_{2}$ (b). Velocity profile for He (c) and ${\hbox{N}}_{2}$ (d)

Radiation

Radiation is the primary energy output mechanism in the arc column. Figure 7 shows the net emission coefficient for the two extreme cases of Ar (highest radiation intensity and extent) and He (lowest intensity and extent). Ar emits radiation several orders of magnitude more strongly than He at a given temperature, which explains the difference.

Fig. 7

Net emission coefficient for Ar and He as a function of temperature

Conduction

The output by heat conduction near the anode counterbalances the convection delivered in the boundary layer zone, which is produced by the impingement of the plasma jet and subsequent radial deflection. Helium and fluorine are the extreme cases, with He having the lowest and ${\hbox{F}}_{2}$ the highest heat losses by conduction. This mechanism is explained by conductive heat flux to the workpiece, which has a much lower temperature than the plasma. The temperature gradient near the anode is much greater (one order of magnitude) in ${\hbox{F}}_{2}$ than in He, as shown in Fig. 8a and b. Another important factor is the thermal conductivity at low temperatures, which, for molecular gases such as ${\hbox{F}}_{2}$, has peaks in the temperature range close to the workpiece. The peaks arise from molecular dissociation reactions, as can be seen in Fig. 8c.

Fig. 8

Temperature gradient vector plot for He (a) and ${\hbox{F}}_{2}$ (b), thermal conductivity for He and ${\hbox{F}}_{2}$ (c)

Conclusions

The electric arcs formed at atmospheric pressure from nine gases have been explored through numerical simulation. The gases are monoatomic (Ar, He, 50% Ar–50% He), diatomic (air, ${\hbox{N}}_{2}$, ${\hbox{O}}_{2}$, ${\hbox{F}}_{2}$, ${\hbox{Cl}}_{2}$), and triatomic (${\hbox{CO}}_{2}$). The parameters used for all simulations were 200 A and 10 mm arc length.

The simulations indicate that for all gases, the dominant heat transfer mechanisms are the same in comparable regions of the arc; however, the relative size of the regions of dominance varies with the properties of the gases in a predictable way.

The Joule effect is the dominant heat input mechanism near the cathode, and has the largest area of dominance in Ar, and smallest in He. Convection is the dominant heat input mechanism near the anode, with the largest area of dominance in He, and smallest in ${\hbox{N}}_{2}$. Radiation is the primary energy output in the arc column, and has the largest area of dominance in Ar, and smallest in He. The magnitude of radiated power in Ar is several orders of magnitude greater than He.

Conduction is the dominant energy output from the arc near the anode, consistent with the heat transfer through a boundary layer to the surface of the workpiece. The highest conduction rate is observed for ${\hbox{F}}_{2}$, consistently with its highest thermal gradient and the lowest for He. Molecular gases tend to have higher conduction rates.

Although further considerations are needed (e.g. deterioration or melting and evaporation of the electrode) before applying the results to welding applications, the present work helps to understand the physical mechanisms of heat transfer present in the arc, and giving a foundation to analyze the effect of novel arc gas compositions.

Appendices

Appendix 1: Calculation of Thermophysical Properties

The thermophysical properties of the plasmas were obtained using standard methods under the assumption of LTE. The composition of the plasma was calculated as a function of temperature by the method of Gibbs free energy minimization. The thermodynamic properties (density, specific heat, enthalpy) were then obtained from the composition and the thermodynamic properties of each species present. The transport coefficients (viscosity, thermal conductivity, electrical conductivity) were calculated using the Chapman–Enskog method  [28]. More detailed descriptions of the methods used have been given previously  [24,25,26, 29, 30].

The species considered and sources of the collision integrals have been given for argon  [24, 30], helium and argon–helium  [26], nitrogen  [25, 29], and oxygen and carbon dioxide  [25] plasmas. Note that the C–C collision integrals used in the calculation of carbon dioxide plasma properties have been updated to those given by Stallcop et al. [31]. The equivalent data have not been published for fluorine and chlorine plasmas and are presented here.

The chlorine species considered were ${\hbox{Cl}}_{2}$, Cl, Cl$^{-}$, ${\hbox{Cl}}^{+}$, Cl$^{2+}$, Cl$^{3+}$ and Cl$^{4+}$. The thermodynamic properties of the atoms and positive ions were calculated from the electronic energy levels in the NIST database  [32]. The thermodynamic properties for ${\hbox{Cl}}_{2}$ and Cl$^{-}$ were taken from the JANAF tables  [33].

The fluorine species considered were ${\hbox{F}}_{2}$, F, F$^{-}$, ${\hbox{F}}^{+}$, F$^{2+}$ and F$^{3+}$. The thermodynamic properties of the atoms and positive ions were calculated from the electronic energy levels given by Moore  [34]. The thermodynamic properties for ${\hbox{F}}_{2}$ and F$^{-}$ were taken from the JANAF tables  [33].

The ${\hbox{Cl}}_{2}$–${\hbox{Cl}}_{2}$, ${\hbox{F}}_{2}$–${\hbox{F}}_{2}$, Cl–Cl and F–F collision integrals, and the Cl–Cl$^{+}$ and F–F$^{+}$ elastic interaction cross-section, were calculated using the phenomenological potential  [35]. The data required to calculate the potential are given in Table 6; the effective electron number was calculated using the expression recommended by Cambi et al. [36]. The charge-exchange cross sections for the Cl–Cl$^{+}$ and F–F$^{+}$ interactions were obtained from the data of Sakabe and Izawa  [37]. The Cl–Cl$^{+}$ and F–F$^{+}$ collision interactions were obtained from the charge-exchange and elastic cross-sections using standard methods  [24].

Table 6 Data required for calculation of phenomenological potential

Calculation of electron-neutral collision integrals requires momentum transfer cross-section data [24]. For e-${\hbox{Cl}}_{2}$, these were taken from Rescigno [40], while the data for e-Cl were taken from Saha  [41], following the recommendation of Christophorou and Oltoff [42]. For e-F, the momentum transfer cross-section data of Robinson and Geltman  [43] were used, while for e-${\hbox{F}}_{2}$, the total elastic cross-section data of Schneider and Hay  [44] were adopted.

The net emission coefficients of Ar, ${\hbox{O}}_{2}$, C, ${\hbox{Cl}}_{2}$ and ${\hbox{F}}_{2}$ were calculated using the method of Cram [45]. Those for He were taken from Cressault et al.[46] and for ${\hbox{N}}_{2}$ from Ernst et al. [47]. For ${\hbox{CO}}_{2}$, air and Ar–He, a mole-fraction-weighted average of the constituent atoms was used. In all cases, an absorption length $R_{\text {p}}$ of 1 mm was selected. Note that Ernst et al. did not provide data for $R_{\text {p}}=1$ mm, so their $R_{\text {p}}=3$ mm data was multiplied by a factor of 1.5 to account for the decreased absorption. The choice of $R_{\text {p}}$ is in accordance with the advice of Cressault et al. [48], who recommended a value corresponding to the radius at which the arc temperature has fallen to about 80% of its value on axis, but noted that the choice was not critical. For the Ar and He arcs, whose temperature profiles are shown in Refs. [17] and [20] respectively, the radii at which the temperature falls to 80% of its maximum value are about 0.8 and 1.3 mm, respectively. Given the uncertainties noted by Cressault et al. [48], $R_{\text {p}}=1$ mm is a reasonable approximation.

Appendix 2: List of Symbols

Magnitude	Description	Units
$B_{\theta }$	Azimuthal magnetic flux density	(Tesla)
b	Subscript indicating property at the edge of the momentum boundary layer
$C_{\text {p}}$	Specific heat	(J kg$^{-1}$ K$^{-1}$)
$dV_{\text {j}}$	Volume element in the arc region	(m$^{3}$)
e	Electron charge $1.602176634 \,10^{-19}$	(C)
h	Enthalpy	(J kg$^{-1}$)
I	Current	(A)
$J_{\text {r}}$	Radial current density component	(A m$^{-2}$)
$J_{\text {z}}$	Axial current density component	(A m$^{-2}$)
$J_{\text {c}}$	Cathode current flux density	(A m$^{-2}$)
k	Thermal conductivity	(W m$^{-1}$ K$^{-1}$)
$k_{\text {B}}$	Boltzmann’s constant $1.380649 \times \,10^{-23}$	(m$^{2}$ kg s$^{-2}$ K$^{-1}$)
P	Pressure	(N m$^{-2}$)
$\text {Pr}_{\text {w}}$	Prandtl number	(–)
$Q_{\text {a}}$	Heat flux by anode fall	(W m$^{-2}$)
$Q_{\text {c}}$	Heat flux by cathode fall	(W m$^{-2}$)
$Q_{\text {cond}}$	Heat flux by condensation of electrons	(W m$^{-2}$)
$Q_{\text {conv}}$	Convective heat flux	(W m$^{-2}$)
$Q_{\text {e}}$	Heat flux by Thomson effect	(W m$^{-2}$)
$Q_{\text {rad,i}}$	Heat flux by radiation	(W m$^{-2}$)
r	Radial coordinate	(m)
$r_{\text {i,j}}$	Vector linking each surface element at the anode to each volume element in the arc	(m)
$R_{\text {c}}$	Cathode spot radius	(m)
$\dot{S}_{\text {h}}$	Source term in the energy conservation equation	(W m$^{-3}$)
$S_{\text {r}}$	Net radiative emission coefficient	(W m$^{-3}$)
T	Temperature	(K)
$V_{\text {c}}$	Cathode fall	(V)
$v_{\text {r}}$	Radial velocity	(m s$^{-1}$)
$v_{\text {z}}$	Axial velocity	(m s$^{-1}$)
w	Subscript indicating property at the anode surface
z	Axial coordinate	(m)
$V_{\text {LTE}}$	Voltage fall along the centerline in the LTE part of the column	(V)
Greek
$\alpha$	Ratio of electron temperature and plasma temperature at anode fall	(–)
$\mu$	viscosity	(kg m$^{-1}$ s$^{-1}$)
$\mu _{\text {0}}$	Vacuum magnetic permeability $1.256 637 062 12 \times \,10^{-6}$	(H m$^{-1}$)
$\rho$	Plasma density	(kg m$^{-3}$)
$\sigma$	Electric conductivity	(S m$^{-1}$)
$\phi$	Electric potential	(V)
$\phi _{\text {w}}$	Work function of the workpiece	(V)
$\Psi$	Solid angle between $r_{\text {i,j}}$ and the vector normal to the anode surface	(sr)