Numerical model of a transferred plasma arc

JOURNAL OF APPLIED PHYSICS

VOLUME 84, NUMBER 7

1 OCTOBER 1998

Numerical model of a transferred plasma arc S. M. Aithala) and V. V. Subramaniamb) Center for Advanced Plasma Engineering, Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210

J. Paganc) and R. W. Richardsond) Department of Industrial, Systems, and Welding Engineering, The Ohio State University, Columbus, Ohio 43210

~Received 23 March 1998; accepted for publication 9 July 1998! This article presents results of two-dimensional simulations for a transferred plasma arc. Calculations are performed for the internal plasma flow within the torch body, as well as the external plasma jet impinging on a surface. The governing equations describing the plasma flow are self-consistently coupled. These equations are the compressible Navier–Stokes equations for conservation of mass and momentum ~in the radial and streamwise directions!, conservation of energy, species continuity with finite-rate ionization and recombination kinetics, and the magnetic diffusion equation for the electromagnetics. The unsteady forms of these equations are time marched to steady state using the linearized block implicit method. This model does not employ any adjustable parameters, and therefore enables direct comparison with experiments. Model predictions are compared with experimental measurements of stagnation pressure distributions recorded on a water-cooled copper plate ~workpiece!, and indicate good agreement, given the lack of adjustable parameters. © 1998 American Institute of Physics. @S0021-8979~98!07419-2#

melt.1,2 Subsequent solidification of this molten region, termed the weld pool, forms the weld or actual joint. There are several electric arc-welding techniques. Of these, plasma arc welding ~PAW! has several advantages over others such as gas-tungsten arc welding ~GTAW! and gas-metal arc welding ~GMAW!. First, PAW has a continuous forced flow of plasma gas, resulting in a high velocity plasma jet which impinges on the workpiece ~transferred arc!, thereby providing enough penetration to produce a deep weld. In contrast, GTAW and GMAW are free-burning arcs, where the flow ~if any! is induced by magnetohydrodynamic ~MHD! pumping from the arc. It is possible to operate the plasma arc in the ‘‘key-holing’’ mode. In this mode, the plasma jet penetrates the workpiece by melting through, opening up a hole in front of the traversing jet that seals upon cooling. The PAW process thus has the ability to form a deep and relatively narrow weld bead in a single pass, whereas GTAW requires multiple passes to achieve the same. Secondly, the forced flow offers the ability to limit entrainment of atmospheric oxygen from the ambient. This is particularly important when a surface is prone to oxidation. An important example of such an application is in the welding of large aluminum plates that comprise the external tank of the space transport shuttle.3 Finally, PAW can be operated in straight polarity, reverse polarity, or variable polarity mode. The discharge is initiated by applying a high frequency voltage superimposed on a dc bias between the inner electrode ~cathode! and the constricting nozzle ~anode! in order to establish a pilot arc. After the pilot arc is established, the high frequency starter is shut off. The nozzle is maintained positive ~by about 20 V! with respect to the electrode whenever the pilot arc is used. The main arc ~transferred arc! is subsequently struck between the grounded

I. INTRODUCTION

Electrical discharges with high currents on the order of hundreds of amperes occur in many applications ranging from industrial-scale arc furnaces and high power switches, to materials synthesis and processing.1 Plasma torches are an important class of electric arcs used in welding. Welding is one of the basic industrial processes used widely in manufacturing, and in fabrication of components used in a myriad of commercial applications. Despite their wide commercial use, there are few detailed models describing high-current electrical arcs from first principles. Modeling of these arcs can enable in-depth understanding of the process itself, and aid in the design of plasma torches and improvement of plasma processes. This article describes a model of such a welding arc, and compares predictions of the model with experimental measurements. Electric arc-welding processes, in general, consist of an electrode and a workpiece with opposite polarities. An arc is struck by applying an electric field between two electrodes causing current flow through the ionized gas column ~established between the electrodes!. The heat generated within the arc produces the high temperatures needed to sustain the gas in its ionized state. Thermal energy is transferred to the workpiece primarily due to particle fluxes causing it to a!

Post-doctoral Research Associate, Robinson Laboratories, The Ohio State University, 206 West 18th Avenue, Columbus, Ohio 43210. b! Associate Professor, Robinson Laboratories, The Ohio State University, 206 West 18th Avenue, Columbus, Ohio 43210; electronic mail: [email protected] c! Graduate student, Baker Systems Laboratories, The Ohio State University, Columbus, Ohio 43210. d! Associate Professor, Baker Systems Laboratories, The Ohio State University, Columbus, Ohio 43210. 0021-8979/98/84(7)/3506/12/$15.00

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© 1998 American Institute of Physics

J. Appl. Phys., Vol. 84, No. 7, 1 October 1998

FIG. 1. Schematic of plasma welding torch.

workpiece and the inner electrode by transferring the discharge. Such an arc is called a transferred arc, and a numerical model of the plasma flow in such an arc is the focus of this article. The electrode is biased negative ~by about 30 V! with respect to the workpiece in the transferred arc studied here. A typical PAW torch assembly consists of a central electrode concentric with two enclosing outer surfaces that confine the main plasma gas and shield gas flows, as shown schematically in Fig. 1. These are called the constricting nozzle and the shield gas nozzle, respectively. The constricting nozzle confines the plasma gas and acts as the anode for supporting a pilot arc ~i.e., a nontransferred arc!. The shielding gas flows through the annulus between the constricting nozzle and the shielding gas nozzle. Both the plasma and shield gases are inert. Typically, argon is the plasma gas while the shield gas is either argon itself or helium. In this article, we model the transferred arc ~in what is referred to as electrode negative operation! in the absence of the pilot arc in order to gain a fundamental understanding of the relevant physical processes. Predictions of the model are then compared with experimental measurements reported in Ref. 4. A number of groups have attempted to model plasma jets and compare their numerical results with available experimental measurements. The experimental measurements mostly comprise of temperature measurements in the plume of a free jet. The earliest attempts to model plumes of plasma torches5–7 required the specification of temperature and velocity profiles at the exit plane of the nozzle as boundary conditions. Lee and Pfender7 studied the effects of prescribing two different temperature and velocity profiles at the nozzle exit on the external flow of a free jet. The profiles chosen were such that the overall mass and energy flow rates in the two cases were identical. In spite of this, the largest difference between calculated and measured temperatures in the external plume domain was 3000 K and the largest velocity difference was 500 m/s. These are equivalent to about a 30% discrepancy in temperature and about 100% in velocity. Since these models did not include the plasma flow within the torch, the temperatures and velocities of the plasma jet at the exit plane of the torch could be obtained only by a gross specification of the mass and energy flow rate. It is clear therefore that to obtain realistic information about the flow characteristics in the plume or in an impinging external jet, it is essential to correctly model the plasma flow in the interior of the torch.

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To overcome these deficiencies in the earlier models, Scott et al.8 have modeled both the torch flow and external free jet in a single calculation. The region behind the cathode, the arc region and the plasma plume region are all included in the model. Their model is a steady, 2D model which includes the swirl component of the gas flow and a k-e turbulence model. Westhoff et al.9 have attempted a similar study, with no additional physics included in their model. Murphy and Kovitya10 have extended the model of Scott et al. to include the effects of mixing of the plasma gas with a different ambient gas. However, these models do not include any shield gas flows. Furthermore, they also do not include finite-rate ionization and recombination processes, and ionization fractions are computed from equilibrium considerations, i.e., from the Saha equation. All the aforementioned models simulate free jets ~i.e., nontransferred arcs!, and solve the governing equations at steady state. In contrast to the literature on free plasma jets, very few multidimensional numerical models exist for transferred arcs.11 Coudert et al. conducted studies on a transferred plasma arc with a flat anode at atmospheric pressure.12 While the authors compare results of their experiments with model predictions, very little information is presented about the model itself. As with other investigations, comparison of measured temperature versus predicted temperature was used to validate their model. Further, since little detail is provided about the model, no assessment can be made regarding its predictive capabilities, especially when the number of adjustable parameters is not revealed.11 A review of existing studies on transferred arcs is given in Ref. 11. In this article, we present a self-consistent model of transferred plasma arcs from first principles. The governing equations and problem formulation are described next in Sec. II and are solved to obtain the flow characteristics and species ~electron, argon ion, and neutral argon atom! concentrations. The internal flow calculations within the torch are used to obtain all the flow variables at the exit plane of the torch. Characteristics of the external plasma jet impinging on a planar surface are then calculated by using the exit plane values obtained from the internal flow calculations. The numerical method used to solve the governing equations is briefly described in Sec. III, followed by a discussion of the results in Sec. IV. A brief description of experimental measurements of stagnation pressure profiles using a watercooled copper block, as well as the procedure used to compare model predictions with measurements are given in Sec. IV. This work is summarized and concluded in Sec. V. II. PROBLEM FORMULATION AND GOVERNING EQUATIONS

We consider the plasma flow to be laminar and to consist of neutral Ar atoms, singly ionized atoms, and electrons. The presence of multiply ionized atoms (Ar11, Ar111, etc.) is excluded since for the gas temperatures and densities of interest here, their number densities are expected to be much smaller than the number density of singly ionized argon atoms. This assumption will be verified a posteriori. The assumption of a laminar flow is justified by the fact that the maximum Reynold’s number for the transferred arc studied

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in this article is less than 100. Such a low value of the Reynold’s number is due to the low flow rates of interest here ~6 l per minute versus the 60 l per minute used in the only other work on a transferred arc, see Ref. 12!. The electronimpact ionization and three-body recombination processes are described by the following overall reaction:

] ] rv2 1 ] ~ ru !1 ~ rru2!1 ~ r uw ! 2 ]t r ]r ]x r 52 1

Ar1e 2 Ar11e 2 1e 2 . In the present work, emphasis is placed on the plasma jet impinging on the workpiece. Phase change at the welding surface is not taken into consideration. This work does not couple modeling of the weld pool with the arc, but rather focuses on the gas-phase processes in detail. The workpiece is considered a boundary of the fluid domain. The plasma is assumed to be quasineutral (i.e., n i 'n e ) since the representative Debye length is much smaller than the characteristic dimensions that are involved. Sheath regions adjacent to surfaces or plasma-cold gas boundaries are not considered here, and the standard MHD approximations are assumed to hold.13 The geometry is two-dimensional and axisymmetric, and it is assumed that only the azimuthal component of the magnetic induction, B u , is significant. This is correct for a transferred arc with no pilot arc present, where the current mainly flows from the electrode ~cathode! to the workpiece ~anode!. Finally, it is assumed that a single temperature can represent the plasma. This is justified by examining the maximum deviation between electron and heavy particle temperatures, which is given by ~see p. 166 of Ref. 1! T e 2T p m H l 2e e 2 E 2 5 , Te 24m e k 2B T 2e

]r 1 ] ] 1 ~ rru !1 ~ r w ! 50, ]t r ]r ]x

~1!

G

FS

] ]u ]w 1 h ]x ]x ]r

DG S 1

D

2h ]u u 2 , r ]r r

~2!

] ] 1 ] ~ rw !1 ~ r r uw ! 1 ~rw2! ]t r ]r ]x 52

F

G DG

]p ] ]w 2 1 2 h “.u 2h ]x ]x ]x 3

1 j rB u1

F S

]u ]w 1 ] 1 hr r ]r ]x ]r

~3!

,

]ne 1 ] ] 1 ~ un e r ! 1 ~n w! ]t r ]r ]x e

S

D

1

1 ] rD e ] 2rn e m e E r8 2 ~ n kT ! r ]r kT ] r e

1

] De ] 2n e m e E 8e 2 ~ n kT ! 5n˙ e , ]x kT ] x e

S

D

~4!

]eT 1 ] ] 1 ~@ e T 1 p # ur ! 1 ~@ e T 1 p # w ! ]t r ]r ]x 5

S D S D

]T ] ]T 1 ] kr 1 k 1 h F1 j r wB u 2 j x uB u r ]r ]r ]x ]x 1

where T e is the electron temperature, T is the heavy particle temperature, m H is the atomic mass of the heavy particles, l e is the electron mean free path, e is the charge on the electron, E is the magnitude of the electric field, m e is the mass of an electron, and k B is Boltzmann’s constant. For an atmospheric pressure, high-intensity argon arc with E51300 V/m, l e 5331026 m, m Ar /m e 573104 , and T e 530 000 K, all of which are characteristics of the present problem, the deviation between T e and T computed from the above formula is about 1%. This justifies the assumption of local thermodynamic equilibrium ~LTE! or thermal equilibrium. This last assumption of a single-temperature plasma could be invalid at the boundaries of the discharge, despite the fact that the pressure is atmospheric. Although our formulation allows thermal nonequilibrium effects to be considered, we nevertheless ascribe a single temperature to the heavy particles ~atoms and ions! and electrons in the interest of simplicity. Given these assumptions, the governing equations for conservation of mass, momentum ~neglecting the swirl component!, species continuity, and energy can be written as14

F

]p ] ]u 2 1 2 h ¹.u 2 j x B u 2h ]r ]r ]r 3

j 2r 1 j 2x

s

,

~5!

where r is the mass density, p is the pressure, u is the radial component of velocity, w is the axial or streamwise component of velocity, t is time, r is the radial coordinate, x is the axial or streamwise coordinate, B u is the azimuthal or tangential component of the magnetic induction vector, j r is the radial component of the current density vector, j x is the axial or streamwise component of the current density vector, F is the viscous dissipation, E r8 5E r 2wB u , E 8x 5E x 1uB u , k is the thermal conductivity, s is the electrical conductivity, and h is the coefficient of viscosity. Note that the energy equation is written in terms of the total internal energy per unit volume, defined as e T 5e1 r (u 2 1 v 2 1w 2 )/2, and the internal energy per unit volume, e, is given by e53/2n T k B T1n˙ e e i , where n T is the total number density ~i.e., n T 5n A 12n e ), k B is Boltzmann’s constant, T is the temperature, and e i is the ionization potential of the argon atom. The detailed expression for F is given in Ref. 15. A twotemperature description of an atomic plasma can be easily accomplished in the above formulation by defining an electron temperature T e that is different from the heavy particle temperature T(ÞT e ). An additional conservation equation describing the evolution of T e in space and time would then have to be provided. The above equations are the compress-

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ible Navier–Stokes equations but including the Lorentz body force terms ( j x B u andj r B u ) in the momentum equations and additional terms in the energy equation representing ohmic heating and work done by the Lorentz body forces. In the above equations, the pressure is p5n T k B T, and n˙ e 5k f n e n A 2k r n 3e represents the net production of electrons due to electron impact ionization ~with a rate constant k f ) and three-body recombination ~with a rate constant k r ) processes. The rate constants k f and k r are obtained from Ref. 16. The mass density r is related to the neutral atom number density n A and electron ~or ion! number density n e through the following algebraic relation:

r 5m A n A 1m i n i 1m e n e 'm A ~ n A 1n e ! 5m A n 0 .

~6!

In order to ensure that the above algebraic constraint is satisfied at each point at all times, the species continuity equation for the neutral atom number density is discarded. n A is then evaluated using Eq. ~6!. The above Eqs. ~1! through ~6! represent six equations for the seven unknown quantities r , u, w, T, n e , n A , and B u . The seventh equation is obtained from Maxwell’s equations, which can be combined to yield a single wave equation for the magnetic induction. However, since only quasi-steady or steadystate solutions are of interest here, this wave equation can be reduced to a parabolic or elliptic form.13 Inclusion of the time derivative in the parabolic form via the m 0 s ( ] B/ ] t) term would force us to resolve extremely small time scales involving the inverse of the speed of light. This would be very computationally intensive, and unnecessary when seeking a steady-state solution or when time scales of interest are much longer than the time required for an electromagnetic wave to traverse the domain. Using Ampere’s law @ “3B 5 m 0 s (E1v3B) # , expanding the resulting cross products, and using the vector identity: “3 ~ “3B! 52¹ 2 B1“ ~ “–B! 52¹ 2 B, Maxwell’s equations can be written in strong conservation form for zero space-charge density for the present problem, as follows: 2¹ 2 B5 m 0 ~ “3j! . By definition, plasmas exhibit the property of overall electrical neutrality on length scales much larger than the Debye length. Therefore on these scales, the electron and positive ion number densities can be taken to be approximately equal or that the net space-charge density is zero. Using Ohm’s law ~a constitutive equation of the form j5 s (E1v3B), the above equation can be written as follows for a single component of the magnetic induction, B u :

S D S D

SD

1 ] r ]Bu Bu ] 1 ] 1 ]Bu 1 1 r ]r s ]r ]x s ]x r ]r s 2m0

] ] ~ wB u ! 2 m 0 ~ uB u ! 50. ]x ]r

~7!

The current densities j r and j x appearing in Eqs. ~2!, ~3!, and ~5! are computed from Ampere’s law (“3B5 m 0 j):

j r 52

1 ]Bu , m0 ]x

j x 52

1 ]Bu . m0 ]r

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and

It is also possible to reformulate Maxwell’s equations in terms of the scalar potential governed by a Poisson equation. The resulting equation for the electric potential f is analogous to Eq. ~7!. The governing Eqs. ~1!–~7!, are nondimensionalized for both internal flow and external flow using the same scales. These are a length scale of 3.175 mm, a temperature scale of 10 000 K, a pressure scale of 1.08773105 Pa for the 100 A case, a pressure scale of 1.16253105 Pa for the 150 A case, and a velocity scale of 1861 m/s, corresponding to the isentropic speed of sound in argon at 10 000 K. The nondimensionalized equations are then solved numerically in conjunction with appropriate boundary and initial conditions to obtain the distributions of velocity, temperature, magnetic induction, and species concentrations both within the torch, as well as in the external jet flow impinging on a workpiece. The solution strategy adopted here is to split the domain into two parts—one representing the internal flow up to the exit plane of the torch, and the other representing the external plasma jet issuing from the exit plane. Calculated values for all the dependent variables from the internal flow are then used as boundary conditions for the external flow. In the interest of attaining high computational speeds, the magnetic transport equation is not solved over the entire domain in both internal and external flows. In the internal flow, Eq. ~7! is solved only within the region denoted as BCDEFG in Fig. 2~a!. It has been observed experimentally that the spread of the jet in the radial direction is negligible.4 Hence, the extent of the computational domain for the electromagnetics calculation in the external flow is restricted to twice the extent of the nozzle radius ~ACGH in Fig. 2~b! denotes the region over which the magnetic transport equation is solved!. The boundary conditions necessary to complete the formulation of this problem are given in Tables I and II, corresponding to the internal flow and external flow domains shown in Figs. 2~a! and 2~b!, respectively. The heat flux into the workpiece is modeled as a convective boundary condition, and is given explicitly in footnote 10 of Table II. The radial distribution of gas temperature at the surface is then determined from this boundary condition. The unsteady form of Eqs. ~1! through ~5!, in conjunction with Eqs. ~6! and ~7! are solved using a time-marching method which is described in Sec. III. Equation ~6! being algebraic is imposed at each grid point at each time step. Equation ~7! being elliptic is solved iteratively at each time step. Consequently, initial conditions also need to be supplied in order to complete the formulation of the problem. For the internal flow calculations, the radial velocity is set to zero everywhere initially, while the axial velocity is given a linear variation from a small value at the inlet ~10 m/s! to a higher subsonic value ~1675 m/s! at the exit. A subsonic value at the exit is prescribed for the initial velocity

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FIG. 2. ~a! Physical domain used for the internal flow simulations. The plane DE denotes the exit plane of the torch. ~b! Physical domain used for the external flow simulations. The torch exit plane is denoted as AB, the shield gas exit plane is denoted as BD, DE is the torch body, EF is the outflow boundary, FGH represents the surface of the water-cooled copper plate, and AH denotes the centerline or axis of the torch.

regardless of whether the final solution is supersonic or subsonic. An initial linear temperature distribution is prescribed over the whole domain ~ranging from 1000 K at the inlet to 14 500 K at the exit, along the centerline!. This temperature distribution is then given a parabolic variation in the radial

direction. An initial pressure distribution is also prescribed, with the pressure held constant (at 1.087310 5 Pa) until a specified downstream distance (x52.7 mm), followed by a linear decrease to a specified value of 1.0133105 Pa at the exit. The total number density, n T at each axial section is then calculated from these initial pressure and temperature distributions. The initial electron number density is prescribed at a constant value of 7.831018 m23. The initial number density profiles of the atoms and mass density are then calculated from the algebraic relationship, Eq. ~6!. The specific values used for these initial guesses do not impact the final converged results, though the initial profiles and magnitudes do need to be physically reasonable and selfconsistent in order for the time-marching procedure to be numerically stable. No initial distributions are required for the magnetic induction as it is governed by an elliptic equation @Eq. ~7!#. For the external flow, values are initially prescribed for u ~20 m/s! and w ~20 m/s! throughout the domain. The static pressure within the domain is given the ambient value of 1 atmosphere initially. The initial temperature is given the value of 10 000 K throughout the domain. The electron number density is given a small constant value (7.831018 m23) throughout the domain. The initial value of density is then evaluated from the temperature and pressure using the equation of state ( P5n T k B T). We evaluate the coefficient of viscosity, thermal conductivity, electrical conductivity, and the species diffusivities using mean-free-path theory.17 The electrical resistivity is calculated as the sum of an electron-neutral resistivity and a Spitzer–Harm resistivity.17 In colder regions of a plasma flow, the discretized form of Eq. ~7! can be confronted with the loss of diagonal dominance. Hence, stable numerical solutions cannot be obtained below a minimum value of the electrical conductivity. This minimum value is called the

TABLE I. Boundary conditions for the internal flow problem. Please refer to Fig. 2~a!. ABCD ~Orifice or nozzle wall!

r u w T ne

]r / ] r50 0 0 c

] 2 n e / ] r 2 50

DE ~Exit plane!

EF ~Centerline!

FGH ~Electrode or cathode!

a

]r / ] r50 0 ] w/ ] r50 ] T/ ] r50 ] n e / ] r50

] 2 r / ] r 2 50 0 0 ] 2 T/ ] r 2 50 ] 2 n e / ] r 2 50

] 2 u/ ] x 2 50 ] 2 w/ ] x 2 50 ] 2 T/ ] x 2 50 ] 2 n e / ] x 2 50

HA ~Inlet! b

0 ] 2 w/ ] x 2 50 d

] 2 n e / ] x 2 50

Boundary conditions for B u , with electromagnetics calculation over GBCDEFG

Bu

GBCD

DE ~Exit plane!

EF ~Centerline!

FG ~along cathode surface!

B u 5 m 0 I/2p r

] B u / ] x50

0

2 B u 5r m 0 I/2p r cathode

n T is obtained from P exit51 atmosphere, and extrapolated value of T exit . n e is obtained by implicit extrapolation, and r is calculated from r 5m A (n e 1n A ). b n T is obtained from the equation of state, P5n T k B T. P is obtained from the prescribed total pressure P 0 and total temperature T 0 , assuming isentropic flow conditions exist between the stagnation state and the inlet, and T is obtained from the prescribed total temperature T 0 and the extrapolated value of w at the inlet. n 0 is obtained by implicit extrapolation, and r is calculated from r 5m A (n e 1n A ). c T is computed from the boundary condition ] T/ ] r u (r5r 0 )52(h/k) @ T(r5r 0 )2T coolant# , where h/k53.47 31022 m, and T coolant5300 K. d Obtained from prescribed total temperature T 0 at the inlet, and the extrapolated value of w.

a

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TABLE II. Boundary conditions for the external flow problem. Please refer to Fig. 2~b!. AB ~Exit plane of torch!

BD ~Exit plane of shield gas flow!

a

b

b

b

a

c

0 0 500 K ] 2 n e / ] r 2 50

] 2 u/ ] r 2 50 ] 2 w/ ] r 2 50 ] T/ ] r50 ] 2 n e / ] r 2 50

r u w T ne

Bu

a

c

a

500 K

a

a

AB

BC ~Torch body!

a

B u 5 m 0 I/2p r

DE ~Torch outerbody!

EF ~outflow!

CG

B u 5 m 0 I/4p r cathode

FH ~Plane of workpiece!

]r / ] x50 0 0 d

] n e / ] x50

HA ~Axis or centerline of torch!

]r / ] r50 0 ] w/ ] r50 ] T/ ] r50 ] n e / ] r50

GH ~along workpiece!

AH ~Centerline!

] B/ ] x50

0

a

Obtained from the solution of the internal flow problem. n T is obtained from the equation of state ( P5n T k B T), after setting the p51 atmosphere and T is obtained from its respective boundary condition. n e is obtained by implicit extrapolation, and r is calculated from r 5m A (n e 1n A ). c Quadratic distribution is prescribed for w at the exit of the shield gas channel ~surrounding the outer surface of r orifice! such that * r 2 2w p rdr yields a volume flow rate of 25 ft3/h ~CFH!. The radial velocity component is 1 then prescribed as u52w tan@a(r)#, where a (r) changes depending on the angles that the respective nozzle walls make with the streamwise ~x! direction. d Boundary condition for temperature is ] T/ ] x u (x5L)52(h/k) @ T(x5L)2T plate# , where h/k 575e 25r/r 0 m21, r 0 512.7 mm, L is the standoff distance, and T plate5500 K. b

conductivity ‘‘floor.’’ The value of this ‘‘floor’’ is not an adjustable parameter for a given set of operating conditions ~i.e., flow rate, total current, and electrical power input!. A specific value can be assigned to the conductivity floor based on the input electrical power, consistent with Poynting’s theorem. This will be addressed in more detail in Sec. IV. III. NUMERICAL METHOD

The linearized block implicit ~LBI! method developed by Briley & McDonald is used to solve Eqs. ~1!–~6! numerically.18 This method was originally developed for the solution of the compressible Navier–Stokes equations for nonreacting flows, with some limited extension to reacting flows.19 We have previously extended the LBI method to include state-specific kinetics20,21 and plasma flows.22,23 The governing equations are written in strong conservation form and nondimensionalized. They are then discretized in time using the Crank–Nicolson scheme, and then linearized about the previous time step using a Taylor series first-order accurate in time. These linearized equations ~and boundary conditions! are then discretized in space to second-order accuracy by central differencing, after transforming from a uniform grid in the physical domain to a uniform grid in the computational domain.14 The two-dimensional operator is then split consistently into two one-dimensional operators using the Douglas–Gunn alternating direction implicit ~ADI! scheme.24 This procedure results in a series of block tridiagonal matrices ~with block sizes equal to the number of unknowns at each point!, which can be solved efficiently using standard LU decomposition methods.25 The magnetic transport equation is solved separately at each time step of the time-marching process, using the odd-even Jacobi iterative scheme.

The LBI method is an implicit method that correctly preserves the coupling between the different governing equations, but suffers from the need to add artificial or numerical dissipation.18 Such addition of artificial dissipation is necessary for overall stability of the numerical scheme, and when inaccuracies arise from the way boundary conditions are prescribed.18 This difficulty is due to the central differencing used by the method to discretize the convective terms in the governing equations. It has been observed that numerical solutions exhibit nonphysical behavior if the cell Reynold’s number is greater than 2.18 This numerical instability can be avoided by using upwind differencing for the convective terms, or by adding a certain amount of second-order or fourth-order dissipation explicitly.26 We have found that, in practice, the amount of artificial dissipation that is added based on the cell Reynold’s number criterion, is overly restrictive. Consequently, we employ the second-order dissipation term used by Briley and McDonald18 but multiplied by a constant less than 1, and a second-order dissipation term resembling that of Rusanov ~see p. 235 of Ref. 27!. Several methods do exist ~e.g., flux-corrected transport or FCT, total variational diminishing or TVD schemes! that add and subsequently remove numerical dissipation at various stages of the calculation.28 However, the LBI method allows the governing equations to be efficiently solved on vector and parallel computers and for the dissipation level to be controlled during the time-marching process. The exact form of artificial dissipation is often undisclosed in the literature, and therefore we describe it in detail here. It must be made clear at the outset that the addition of such a term to the discretized form of the governing equations is important for stability of the particular timemarching scheme used. The fully converged solution is independent of the magnitude of artificial dissipation beyond a

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certain minimum amount. In this work, terms of the form m r ] 2 f / ] r 2 1 m x ] 2 f / ] x 2 are added to the discretized and nondimensionalized forms of the governing Eqs. ~1!–~5!, resulting in additional implicit and explicit terms in the final linearized difference equations. The specific expressions for the non-dimensional coefficients, m r and m x for the internal flow within the torch are

H

0.00025r ~ Dr ! for T