Calculation of Electrical and Thermal Conductivities of Metallurgical Plasmas

by G. J. Dunn and T. W. Eagar

CONTENTS

Abstract ................................... . 1 Introduction ................................ 1 Theory ..................................... 1

Species Densities ............................... 1 Electrical Conductivity ........................ . 2 ......................... Thermal Conductivity . 4

Literature Review .......................... . 6 PureInertGases ............................... 6 Metal Vapor/Inert Gas Mixtures .............. .17

Conclusion ................................ .17 Acknowledgments ......................... .20 References ................................ 20

Abstract

A simplified road map of the plasma physics literature for calculation of electrical and thermal conductivities of metal- lurgical plasmas is presented. The presence of even small quantities of metal vapors in argon or helium plasmas can greatly influence these conductivities. Although very com- plex computational procedures have been developed by the- oretical physicists, in most cases, simplified formulas pro- vide estimates of plasma properties which are well within the limits of error in the physical measurements of these sys- tems.

Introduction

In recent years there has been increasing interest in modeling arc welding processes and other metallurgi- cal processes involving p1asmas.l In many cases, the published properties of pure argon or helium gases are used in calculations of transport phenomena in the arc. Since a welding arc contains significant quantities of metal vapor: and this vapor has a considerably lower ionization potential than the inert gases, the assumption of pure inert gas properties may lead to considerable error.3 Nonetheless, more sophisticated

T. W. Eagar is with the Materials Processing Center at the Massachusetts Institute of Technology Cambridge, MA.

G. 3. Dunn was with the Material Processing Center when the work was performed, hut is now with the U.S. Department of State, Washangton, D.C.

Publication of this report was sponsored by the Welding Research Coun- cil.

models of plasma transport properties in mixed gases can be a major modeling task in itself.

In the present paper, the authors review a portion of the extensive literature on calculation of plasma elec- trical and thermal conductivities. An attempt is made to simplify the analysis, to provide reasonable esti- mates of plasma properties without resorting to use of very complex models that have been developed by theoretical physicists. In most cases, the simple model is in reasonably close agreement with the more sophis- ticated models. Thus, the simplified methods de- scribed here may be more suitable for development of exploratory models in that they represent a compro- mise between the assumption of a pure plasma and the more refined, but more difficult to implement models developed by theoretical plasma physicists.

Direct measurement of the thermodynamic proper- ties of plasmas is difficult at best due to the high temperatures and severe thermal gradients which ex- ist. As a result it is more common for investigators to estimate the physical properties by calculation. The principles of such calculations have been described4 but rarely are the details of the calculation procedures presented. In the following, a simple method for calcu- lating the electrical and thermal conductivities of mul- ticomponent plasmas is presented. This is followed by a comparison of calculated and measured values for pure helium, argon and nitrogen in order to evaluate the accuracy of this simplified set of equations. Final- ly, the effect of metal vapors on the electrical and thermal conductivities of these carrier gases is de- scribed. It is hoped that these comparisons will be helpful in developing an understanding of first-order phenomena which are observed in many plasma pro- cessing operations.

Theory Species Densities

The first step in calculating plasma transport prop- erties is the determination of the plasma composition. This is performed by simultaneously solving the ideal gas law

Calculations of Conductivities of Metallurgical Plasmas 1

and a set of Saha equations that involve each atom and ion of the same atomic species, i.e.

where r denotes the stage of ionization. For single ionization, this simplifies to:

where

re,,, n; and na = particle densities (per unit volume) of electrons, ions and neutral atoms, respectively

T = temperature in Kelvin V = ionization potential of the neutral

atom me = rest mass of electron

k = Boltzmann's constant h = Planck's constant

The internal partition functions, Ze, Zi and Z a are given by5

where gi is the degeneracy or statistical weight corre- sponding t o the energy level Ei. For a monatomic gas

where S; is the total-angular-momentum quantum number of the ith electronic energy level.

Partition functions may be calculated from data given in the NBS tables of atomic energy levels com- piled by Moore.6 The degeneracy of electrons is 2.

For plasma pressures greater than 10 atm, the De- bye-Huckel correction for lowering of the ionization potential may become irnp~rtant .~

For a plasma mixture which may contain polyatom- ic molecules, such as nitrogen, the dissociation equilib- rium must be calculated.

where

ma = atomic mass of nitrogen E d = dissociation energy

Ouajji, et a?.? describe a method of calculating the partition functions of molecules

where Ze, Z,,, and Zr are the electronic, vibrational and rotational components, respectively.

where

hcBe n=- kT

us = symmetry factor = 2 for homonuclear molecules = 1 for heteronuclear molecules

and ue and Be are the vibrational and rotational con- stants of the molecule. These have been tabulated by Herzberg9 and JANAF.33

In the above calculations assumptions of quasi-neu- trality (ne = n), local thermodynamic equilibrium (Te = Ta = Ti), and ideal gas behavior have been made. Finkelnburg and Maeckerl0 and Olsenll have shown these assumptions to be valid for the column of atmo- spheric pressure arcs. These include welding arcs, electric furnace arcs and atmospheric pressure plasma arcs used for surfacing or consolidation of powders.

The fraction of metal contaminants is defined as

where En would include all host gas species, for exam- ple, in the case of nitrogen, N2,N2+,N,N+. This is a simplification which may be invoked for most plasma metallurgy applications because the exact metal vapor content cannot be accurately determined or con- trolled, and the purpose of the study is merely to determine the effect of a small amount of metal vapor on the inert gas plasma properties. For a closed system with known initial composition and interacting spe- cies, a more thorough calculation of plasma composi- tion may be achieved through the minimization of Gibbs free energy. Such calculations require iterative computer techniques for s ~ l u t i o n . ~

At temperatures relevant for most plasma metallur- gy applications it is possible to simplify the calcula- tions by neglecting the effects of doubly-ionized spe- cies. This is illustrated in Fig. 1, in which the composi- tion of a 1 atm Ar plasma is plotted vs. temperature. Note that secondary ionization does not become ap- preciable until T > 17,000°K Most metal vapors will become doubly ionized a t lower temperatures, but generally above the point a t which Ar is already strongly singly ionized. Therefore, for a low level of metal contamination, the electron density in the plas- ma is not significantly altered. I t is important to con- sider the first and second ionization energies of all species present in a particular application before mak- ing this simplification. Only singly-ionized species are considered in the following discussion.

Electrical Conductivity The electrical conductivity of the plasma may be

calculated using a simple formula12

2 WRC Bulletin 357

0 10000 20000 30000 40000 TEMPERATURE ( K )

Fig. 1-composition of a 1 atm argon plasma

Calculations of Conductivities of Metallurgical Plasmas

Qie is the momentum transfer cross section for elec- tron-ion interactions, and is given by13

with T in Kelvins and ne in m"3. Qae is the momentum transfer cross section for elec-

tron-neutral atom interactions and is species depen- dent. Values have been experimentally determined for a number of elements. Bederson and Kiefferl* have presented a critical review of these experimental methods, and much of the data has been compiled by Kieffer15 and Itikawa.16 The rare gases have been studied extensively and there is a fair amount of data for the alkali metals, particularly cesium, but data for other elements is rather scarce. Furthermore, the re- ported values for momentum transfer cross sections for many elements often vary by up to an order of magnitude. The reported values for common gases are less scattered, however. The cross sections of metal atoms are up to two orders of magnitude greater than those of the noble gases. This is due to the compact structure of the noble gas atoms, which have closed outer shells.l7 It is evident that any effects on the electrical conductivity due to the presence of neutral metal atoms will be directly related to their large cross sections. These effects, however, will probably be very small in atmospheric pressure plasma metallurgy ap- plications due to the low concentrations of metal va- pors present in the plasma and because of the low ionization potentials of the metal atoms: in the tem- perature range at which these atoms ionize, the cross sections of ions are approximately two orders of mag- nitude greater than those of the neutral atoms. There- fore, the ion term in the denominator of Eq. (6) soon dominates. Indeed, several authors1a20 have found that gross adjustments in the values of metal atom cross-sections yield only minor changes in the calcu- lated transport properties.

The lack of accurate cross-section data negates the value of more complex expressions for the transport properties.

Thermal Conductivity The thermal conductivity of a mixture of gases can

be expressed as the sum. of three contributions

AT is the translational thermal conductivity and is due to the elastic collisions of particles which allow transfer of kinetic energy.21 Ap is the reactive thermal conductivity, and accounts for the diffusion of reac- tion enthalpy. For example, this heat is transferred by the diffusion of ions and electrons from the hotter areas of the plasmas to cooler areas where they recom- bine to form neutral atoms, releasing the enthalpy of the ionization reactionJ2 A similar argument holds for

dissociation reactions such as N2 = N + N. \i is the internal thermal conductivity, and accounts for the rotational and vibrational energy transported by poly- atomic molecules. For monatomic gases this term is zero. For other gases, XI, is generally a very small com- ponent of hot in the temperature range of interest8 and can be approximated as that of the pure gas.20

Translational Thermal Conductivity. The small mass of the electrons allows the decoupling of the calculations of the electron and heavy species compo- nents of the translational thermal conduct i~i ty .~~

where AH is the conductivity due to heavy species in- teractions and & is due to electron-heavy species in- teractions. Neglecting the contribution of ions, AH can be expressed as x A h , where XA is the mole fraction of atoms and \A is the thermal conductivity due to neu- tral atoms alone. D e ~ o t o * ~ has shown that for an argon plasma such as approximation yields values within a few percent of those produced by more rigorous calcu- lations.

The contribution due to neutral atoms may be cal- culated according to a solution of the Boltzmann equa- tion24

where

At = molecular weight a = a characteristic diameter in A

T* = kT/e = reduced temperature e = interaction potential

and W,^(T*) is known as the collision integral. Val- ues of Q^ for a range of reduced temperatures are given by Hirs~hfelder .~~

Jain25 has shown the above formula to be consistent with experimental data to within a few percent for argon and other noble gases.

Values of e/k and a for common gases are given by Hir~chfelder.~~ However, values for metal vapors are not available in the literature and must be estimated. The parameter e/k may be approximated asz4

where Tb is the boiling point temperature. a may be crudely estimated as the atomic diameter of the ele- ment in solid form. The accuracy of such an assump- tion is discussed by Gottlieb and Z0llweg2~ Martini27 and Robinson.28 These authors, by various methods, arrived at values of a for cesium ranging from 3.5 to 9.17A. For most plasma metallurgy applications, the effects of neutral metal atom interactions on the total thermal conductivity of the plasma are of little signifi- cance because most metal species are highly ionized in the temperature range of interest. Therefore, the above approximations will suffice.

The neutral atom translational thermal conductiv-

4 WRC Bulletin 357

ity of a mixture of gases may be calculated with an equation given by U l ~ b i n ~ ~

n

where xi is the molar fration of species i. The electron component of the thermal conductiv-

ity will dominate Eq. (11) at higher temperatures,30 when ionization becomes appreciable. It is given b91

where H = all heavy species; ions and atoms. The constant C varies from 4.1 to 5.4, depending upon the degree of ionization.

Qie is here redefined as3+

,*;. :- -i where T is in Kelvin and ne in m-3. Reactive Thermal Conductivity. An expression

for the reactive thermal conductivity of any mixture of reacting gases and inert diluents has been derived by B r o k a ~ . ~ ~

where

No = Avogadro's number i = 1 , 2 . . . N j = 1 ,2 . . .N

N = number of chemical reactions that occur

The normalized heats of reaction Mi are given by

where an, are stoichiometric coefficients for species k in the ith reaction. For the simple case of the ioniza- tion reaction, a = -1 for the ionizing atom, a = 1 for the electron and ion products, and a = 0 for other neutral and ion species.

Hk is the heat content of species k a t temperature T and is obtained from

where H¡ - Gg8 is the species standard enthalpy relative to that at T = 298.15 K and is the heat

of formation a t 298.15 K. Values of qgg may be obtained from the JANAF Thermochemical Tables,33 which also list valus of the standard enthalpies to 6000 K of electrons and most elements and ions. Values for PT - G8 to 20,000 K may be calculated from the NBS Tables of Atomic Energy Levels6 according toS3

x nigi ex~(-uilkT) joules 1 g, exp(-ui/kT) mole0K

(20)

where gi and ui are as defined previously. The Rij elements in Eq. (15) were rewritten30 as

where xi and a i k are as defined previously and B k i is the reduced mass and is given by

This factor renders any terms in Eq. (21) due to electrons negligible. In many applications the calcula- tions may be further simplified by assuming that cer- tain reactions, for example the inert gas and metal vapor ionization reactions, occur within mutually ex- clusive temperature ranges.

The cross sections for interactions between neutral atoms are calculated according to8

The collision integral Q L ~ , ? ~ ) * (T*) is tabulated in Hir~chfelder.~~

The characteristic diameters a and interaction po- tentials efor collisions between neutral atoms of unlike species are calculated according to the relationships30

uii + 0]! 0.. ZÑÑÑÑÑÃ

11 (24)

The cross sections for ion-neutral collisions are giv- en by30

QCTR is the cross section for charge transfer. Values for the noble gases and alkali metals have been report- ed by Rapp and Francis34 and for nitrogen by Capitelli and D e ~ o t o . ~ ~ Cross sections for other elements can be interpolated from their ionization potentials.

QP is the polarization cross section and is calculated from the static dipole polarizability of the neutral atomz0

with the polarizability term, a, given in A3 and Boltz- mann's constant in eV/K. Values of a have been tabu- lated by Teachout and Pack.36

Calculations of Conductivities of Metallurgical Plasmas 5

For nonresonant charge transfer Qia(tota1) can be approximated as Qp. The cross section for collisions between charged particles is given by Eq. (16). Cross sections for collisions involving molecular species such as Ny are discussed by Capitelli and D e ~ o t o . ~ ~

The calculated thermal conductivities of argon and helium37 at atmospheric pressure are plotted in Figs. 2 and 3. In each case the contributions due to each com- ponent are also plotted. For argon, the electron com- ponent dominates, except at low temperatures where the degree of ionization is very low. The reactive ther- mal conductivity, due to the ionization reaction, peaks a t about 14,000°K causing a maximum in the total thermal conductivity. For helium the very large heavy species translational component dominates up to about 15,000°K where the degree of ionization be- comes appreciable.

Literature Review Pure Inert Gases

The calculation of plasma transport properties has been the subject of extensive study. The noble gases and nitrogen have enjoyed particular attention. In- deed, the pursuit of more precise theory has far out- paced the development of experimental techniques which can settle the theoretical disputes. In this sec- tion we will present a brief review of calculated and experimentally measured electrical and thermal con- ductivities of argon and helium at p = 1 atm. This review is intended as a vehicle for analysis and evalua- tion of the calculation methods we have presented. In the latter part of this section we will review some of the publications which have addressed the practical sce- nario of a noble gas or nitrogen plasma containing metal vapor contaminants.

The electrical conductivity of argon as calculated by several researchers is plotted in Fig. 4. It can be seen that these curves differ by as much as a factor of two at some temperatures. The calculations of the agree closely with those of Cambe15 and with those of D e v o t ~ ~ ~ at high temperatures. Olsen's valuesll are significantly larger a t high temperatures and signifi- cantly lower at low temperatures. Glickstein's values39 are very much lower at all temperatures. In Fig. 5 our calculations are compared with experimental data giv- en by E r n r n o n ~ . ~ ~ Further experimental data can be found in refs. 41-59. Further theoretical calculations can be found in refs. 60-63. The calculations by vari- ous workers of the electrical conductivity of helium are compared in Fig. 6. Close agreement of the authors37 with Devoto and LiG4 is apparent. Again, Glickstein's values^ are lower. The calculations of Lick and Em- r n ~ n s ~ ~ are significantly higher in the temperature range 12,000-17,000°K Calculations have also been performed by Kannappan and B o ~ e , ~ ~ but they present only their results for p = 0.1 atm.

Experimental data for helium are far scarcer than for argon. The only data found in the literature were those reported by Hwang and Emmon@ for p = 10

atm. They compared these data to the calculations of Lick and E m m o n ~ ~ ~ (for p = 10 atm) and noted that the experimental values were lower by a s much as a factor of two.

Calculations of the thermal conductivity of argon are compared in Fig. 7. Tabaczynski6$ has shown that his values agree very closely with those of D e v o t ~ . ~ ~ The calculations of the authors37 agree with Tabac- zynski's fairly well except a t lower temperatures, where they are significantly higher. This discrepancy is due to a higher calculated electron component of the conductivity on our part. The equation used69 was different in form from the equation presented in this paper. We believe that the latter will provide a better agreement with Tabaczynski and Devoto. The curve attributed to Cambe15 is the result of oversimplified theory. This exception aside, it can be seen that calcu- lations by the various researchers have all yielded curves of the same basic shape, with about as much variance as was evident in the electrical conductivity comparisons.

The calculations of the authors3I are compared in Fig. 8 with experimental data of Emmons,4O Bues, et a1.F and Becker and B~z.~O It can be seen that above 10,OOO°K%h data continues to increase rapidly with temperature, while the theoretical curve levels off. This discrepancy is due to the effect of radiative heat transport within the gas.71 Most measurements of the thermal conductivity are in fact the sum of the true thermal conductivity and the effective radiative ther- mal conductivity. This latter quantity is dependent upon the radial dimensions of the arc studied. Thus, Emmons' data40 from an arc of 5 mm radius are greater in value than those reported by Bues, et aLY4l for an arc of 2.05 mm radius. Asinovskii and K i r i l i ~ ~ ~ ~ analyzed this dependence on arc radius and extrapolated their data back to R = 0 to obtain the true thermal conduc- tivity. Devoto71 notes that the values thus obtained lie within 30% of theory, which he estimates to be within experimental error. Knopp, et ~ 1 . : ~ have shown that similar data for nitrogen agree more closely with the- ory when radiative loss is considered (see Fig. 9).

Further experimental data for argon can be found in refs. 48-50,57 and 74.

Calculations of the thermal conductivity of helium are compard in Fig. 10. All have the same general shape. The greatest discrepancy is between the calcu- lations of Devoto and Li6* and those of the authors,37 which are up to 25% smaller.

The calculations of the authors37 are compared with low temperature experimental data75776 in Fig. 11. Agreement is satisfactory. No high temperature data was found in the literature.

In light of current experimental accuracy, it is con- cluded that the simplified theory presented in this work is adequate for analysis of the relative effects of metal vapor contaminants in inert gases. The calcula- tions of the authors37 for pure inert gases are in satis- factory agreement with those of other researchers. Un- til more precise experimental study is achieved, higher

6 WRC Bulletin 357

TEMPERATURE (K ) Fig. 2-Thermal conductivity of a pure argon gas at one atmosphere Pressure. AT= total thermal conductivity; \A = heavy species component, A. = electron component, and AÃ = reactive component

Calculations of Conductivities of Metallurgical Plasmas 7

TEMPERATURE (K) Fig. 3-Thermal conductivity of a pure helium gas at one atmosphere pressure. AT = total thermal conductivity; \A = heavy species component, XÃ = electron component, and AR = reactive component

8 WRC Bulletin 357

Dunn

Cam

5000 10000 15000

TEM PERATURE ( K )

Fig. 4-Argon electrical conductivity as calculated by Glick~tein?~ CambeL5 Olsenl' and the authors3' at one atmosphere pressure

Calculations of Conductivities of Metallurgical Plasmas 9

~ m m o n s da ta

-- Dunn a n d Eagar

0 5000 10000 I5000 20000

TEMPERATURE ( K )

Fig. 5-Comparison of calculated argon electrical c~nductivity~~ with experimental data of Emmons40

10 WRC Bulletin 357

TEMPERATURE (K) Fig. 6-Helium electrical conductivity as calculated by lick stein?^ Lick and Emrn0ns,6~ Devoto and LP4 and the authors3'

Calculations of Conductivities of Metallurgical Plasmas

Kan

- Dunn -

nappan and Bose

\ Glickstein

TEMPERATURE ( K ) Fig. 7-Argon thermal conductivity as calculated by ~ i c k s t e i n , ~ ~ ~arnbe l ,~ Kannapan and B ~ s e , ~ ~ Tabac~ynski~~ and the authors37

12 WRC Bulletin 357

4 Emrnons

Bues

1. A Becker and Bez -

/ --- / Theory

TEMPERATURE ( K ) Fig. 8-Comparison of calculated argon thermal conductivit~~' with experimental data of E m r n ~ n s , ~ ~ Sues4' and Becker and Bez70

Calculations of Conductivities of Metallurgical Plasmas

TEMPERATURE (K) Fig, S-Thermal conductivity of nitr~gen.'~ Solid line: theory. Dashed line: experimental data. Symbols: data with radiative loss correction

WRC Bulletin 357

Devoto and Li \ /G

\ L i c k and Ernrnons

Dunn and Eagar

0 5000 10000 5000 20000

TEMPERATURE ( K ) Fig. 10-Helium thermal conductivity as calculated by Gli~kstein?~ Lick and E t n m o n ~ , ~ ~ Devoto and Li64 and the authors

Calculations of Conductivities of Metallurgical Plasmas 15

Coll ins

-- Dunn and Eagar

0 5000 I0000

TEMPERATURE ( K )

Fig. 1 1-Comparison of calculated helium thermal conductivitP7 with experimental data of C01Iins'~ and Blais and Mann76

16 WRC Bulletin 357

order calculations cannot be evaluated. Furthermore, until more accurate cross-section data become avail- able, calculations based upon more intricate theory will continue to be of purely academic consequence.

Metal VaporIInert Gas Mixtures The properties of an inert gas plasma can be strong-

ly affected by the presence of an easily ionized con- taminant. This is illustrated in Fig. 12, in which the electrical conductivities of helium plasmas containing various concentrations of aluminum vapor are plotted. The conductivity can be increased by several orders of magnitude by even very small amounts of aluminum, which ionizes at a much lower temperature than heli- um.

The seeding of common gases with easily ionized metal vapors has been studied extensively for many years in the field of magnetohydrodynamics. HarrisT7 experimentally measured the electrical conductivity of cesium-seeded nitrogen, helium, neon and argon plasmas, and reported satisfactory agreement with theoretical calculations by Frost.78

The widespread use of copper as an electrical con- tact material, for example in circuit breakers, has prompted a number of studies of cop~er-a i r? ,~~ cop- p e r - n i t r ~ g e n ~ ~ ? ~ ~ , ~ ~ and copper-argon p l a ~ m a s . ~ ~ ~ ~ ~ Ouajji, et U Z . ? ~ have applied their calculated conduc- tivities for air-copper mixtures to an Elenbaas-Heller mode1 of the electric arc column, and have shown that the axial temperature distribution and electric field of a plasma in air are strongly affected by the presence of a small amount of copper vapor. Haddad, et al.:4 have studied an argon plasma contaminated with cerium. G l i ~ k s t e i n ~ ~ has investigated argon and helium arcs containing aluminum. Johnson and Pfendera5 have studied a nitrogen plasma contaminated with sodium. Abbaoui, et aLS6 have calculated the electrical and thermal conductivities of argon plasmas contaminated with a variety of metals, and have shown that the ionization potential of the metal contaminant is the predominant criterion in determining the magnitude of the increase in conductivity of the plasma.

However, it must be stressed that the effect of a single contaminant on a pure inert gas plasma is not necessarily a valid indication of the effect of the con- taminant in an actual plasma, in which other contami- nants may already exist. For example, attempts have been made to explain the effect of certain trace metals on gas tungsten arc welds (GTAW) in terms of their effect in the arc pla~ma.~~*M Plasma calculations were performed to demonstrate the dramatic effect of these easily-ionized elements in pure argon plasmas. How- ever, in practice, the welding arc is not purely argon, but is already contaminated with a variety of metal vapors emanating from the electrode^.^^ Most metals have ionization potentials significantly lower than that of argon, and therefore enhanced conductivities will be a pre-existing condition. It has been shown that the effect of trace elements is significantly diminished when an initial vaporlinert gas mixture is consid-

ered.37 This is illustrated in Fig. 13, in which the effect of an addition of 5% aluminum to a 10% Fe/He plasma is shown to be negligible, in contrast to the very strong effect illustrated in Fig. 12, in which a pure helium plasma results in a conductivity which is already en- hanced at low temperatures.

Shayler and Fanglg and Abbaoui, et U Z . , ~ ~ attempted to simplify the thermal conductivity calculations by assuming that the effect of a small amount of metal vapor on the thermal conductivity of a nitrogenlg or argonS6 plasma would be manifested primarily in an augmentation of the electron component of the trans- lational thermal conductivity. This assumption allows the considerable simplification of foregoing the calcu- lations of the reactive and heavy species translational components of the thermal conductivity; these are as- sumed to be approximately equal to those of the pure gases, values for which are readily available in the literature. As discussed by Abdelhakim, e t aZ.FO such a simplification may lead to unacceptable error, as a decrease in the latter components is not considered. This is particularly true for diatomic gases such as nitrogen, for which the dissociation reaction leads to a very large reactive thermal conductivity at low tem- peratures, and for gases such as helium, for which the heavy species translational thermal conductivity is the dominant component. Abdelhakim20 and the au- t h o r ~ ~ ~ have shown, for nitrogen and helium, respec- tively, that, due to the decreased number of base gas species and resulting decrease in AR or AH, the effect of a metal vapor contaminant is to decrease the total thermal conductivity. The simplifications of Shayler and Fang and Abbaoui can only predict an increase. Nevertheless, in many applications, the fraction of metal vapors in the plasma may be very small, in which case such approximations may suffice for gases such as argon, which has neither a large heavy species transla- tional thermal conductivity nor a large reactive ther- mal conductivity at low temperatures. For gases which do, such as helium and nitrogen, a better assumption for very small (<I%) concentrations of metal vapor may be that the thermal conductivity of the mixture is approximately equal to that of the pure gas, as the decrease in reactive or heavy species translational con- ductivity will tend to be compensated by the increase in the electron translational c o n d u c t i ~ i t y . ~ , ~ ~ Such simplifications must be approached with caution.

Conclusion

A simplified set of equations for the calculation of plasma thermal and electrical conductivities has been presented. Knowledge of collision cross sections and basic thermochemical data is required. Our goal in delineating these equations is to simplify the process of calculating metallurgical plasma properties. Such calculations will contribute towards the understand- ing of developing plasma processing applications. In particular, the consideration of the effects of small

CaZcu1ation.s of Conductivities o f Metallurgical Plasmas 17

0 5000 I0000 I5000 20000

TEMPERATURE (K)

Fig. 12-Electrical conductivities of helium/aluminum gas mixtures

18 WRC Bulletin 357

amounts of metal vapors on the properties of the inert gas plasma is fundamental to the realistic modeling of any metallurgical plasma process.

A comparison of thermal and electrical conductiv- ities of argon and helium calculations by the methods outlined in the paper with theoretical calculations of other authors and with experimental data shows satis- factory agreement. Although more rigorous theory is available, the accuracy of current collisian cross-sec- tion data does not justify its use.

Acknowledgment

The authors wish to thank the Office of Naval Re- search for support of this work under Contract NOOOl4-80-C-0384.

References

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2. Glickstein, S. S., in Trends in Welding Re- search in the United States, S. A, David, Ed., 1982, ASM, Metals Park, 0hi0,p. 3.

3. Block-Bolten, A. and Eagar, T. W., ibid., p. 53. 4. Fauchis, P., Boulos, M. and Pfender, E., in Plas-

ma Technology in Metallurgical Processing, J. Fein- man, Ed., 1987, AIME, Warrendale, PA, p. 11.

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18. Dunn, G. Metal Vapors in Gas Tungsten Weld- ing Arcs. Sept. 1984, SM/SB Thesis, MIT, Cambridge, MA.

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22. Gross, B,, Grycz, B. and Miklossy, K., Plasma Technology, American Elsevier Publishing Co., New York, 1969.

23. Devoto, R. S., Physics of Fluids, lO(lO), 2105- 2122 (1967).

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Calculations of Conductivities of Metallurgical Plasmas