Thermal Plasma Modeling · Thermal Plasma Modeling Joachim Heberlein University of Minnesota Outline: 1. Introduction to thermal plasmas 2. Generation of thermal plasmas 3. Equilibrium

Thermal Plasma ModelingJoachim Heberlein

University of Minnesota

Outline: 1. Introduction to thermal plasmas2. Generation of thermal plasmas3. Equilibrium relations4. Thermodynamic and transport functions5. Conservation equations and solution methods6. Non-equilibrium and turbulence7. Examples of recent calculations8. Electrode regions9. Conclusions

Kyoto, 24 August, 2007 Copy right remains with author

1. Introduction to thermal plasmas

• a plasma is called “thermal” if it is partially in Local Thermal Equilibrium (LTE)

• a thermal plasma is typically at pressures above 0.1 atmospheres

• collision processes dominate • high degrees of ionization (5 to 100%)• high electron densities (typically >1022 m-3)• continuum approach is used for description

Thermal plasmas are widely used for processing- high energy fluxes in controlled environment- high fluxes of reactant species

Copyright remains with the Author(s).

辻野貴志

タイプライターテキスト

Copyright remains with the author(s).

High Pressure ArcsRF DischargesShock Waves

Surface to be processed

Thermal Plasma Characteristics and Processing Paths

Example of pressure effect on equilibriumcondition

- If p�, n� and collision number�

Behavior of electron temperature (Te) and heavy-particletemperature (Th) in an arc plasma.


spacechargeregionne � ni

anodesurface

Te

Th

nene Saha

x0

arccolumn

LTEE = cont.

kinetic

non-equilibrium

Te > Th

composition

non-equilibrium

ne > ne Saha

Example of non-equilibrium in “thermal plasma”Anode region of high intensity arc

Te~12 kK

ne~1022-1023 m-3

�D~0.1-1 μm

j~ 106-108 A/m2

q~ 107-2x109 W/m2

• extent of regions and all plasma characteristics strongly dependent on macroscopic �uid �ow

2. Generation of thermal plasmas

• electric arcs - most widely used method- Joule heating of gas by passing current through it- requires electrodes- characteristics strongly depending on fluid dynamics- different configurations require different modeling

approaches

• radio frequency induction discharges- no electrodes needed- larger plasma volume generated- lower gas heating efficiency- more sensitive to process variations

• laser, shockwave, etc.Copyright remains with the Author(s).

Discharges are characterized by their voltage -current relation

Region 1 and 2:Non-self sustained discharge,electrons are generated by externalmeans

Region 3:Glow discharge: sufficient ionizingcollisions to sustain discharge

Region 4:Abnormal glow - increasing voltage

Region 5: arc: after breakdown strong increase in current density, decrease in voltage, ionization mechanism changescathode electron emission mechanism changes, cathode fall only ~10 V

Different regions of electric arcCathode region:I. Cathode fall, 10-3 mm

-ions are accelerated towards cathodeII. Cathode boundary layer, 1 to few mm

-widening of arc, lower current density

Column regionIII. Arc column, electric field, temperature

determined by interaction withsurroundings

Anode regionIV. Anode boundary layer, 1 to 2 mm

- arc constricts because of axial heatloss

V. Anode fall, 10-3 mm-electrons driven towards anode- positive for constricted attachment,negative for diffuse attachment


Different thermal plasma generator configurations

Transferred Arc Plasma Generator

• arcing between one electrode (usually cathode) and metal workpiece• high heat flux, low gas flow• high energy transfer efficiency to solid

Temperature profiles in transferred arc plasmareactor


Different thermal plasma generator configurations

Non-Transferred Plasma Generator

• plasma generation confined to torch• high bulk gas heating efficiency• wider heat flux distribution with lower peak values

Temperature distribution in a plasma jet[Boffa and Pfender, 1968]


Velocity distribution in a plasma jet[Boffa and Pfender, 1968]

Arc instabilities

Arcs are usually unstable

• MHD instabilities can lead to arc extinction• shear layer instabilities result in cold gas entrainment• anode attachment instability results in continuous

power fluctuation• electron heating instability can result in arc

constriction in anode region

Stabilization mechanisms:

• wall stabilization provides stabilizing radial gradients• convection stabilization with parallel cold flow• jet stabilization


Example of anode attachment instability in plasma torchExample of anode attachment instability in plasma torchplasma torch plasma jet

� arc length � voltage drop

� force balance: drag vs. magnetic

Voltage (V)

� arc dynamics � jet forcing

� enhanced cold flow entrainment

t3

t5 t6t4

t1 t2

3. THERMODYNAMIC EQUILIBRIUM RELATIONS

Maxwellian Velocity Distribution (most probable distribution):

Average Velocity:

Average Kinetic Energy:

Definition of Temperature:

dN(v)

N=

4�v2dv

(2�kT/m)3

2

exp(-mv 2 /2kT)

v = (8kT/�m)1

2

v2= 3kT/m

3

2kT =

1

2mv2


Maxwellian velocity distribution for twodifferent temperatures

THERMODYNAMIC EQUILIBRIUM RELATIONS

Boltzmann Distribution of Excited States

Saha Equation (Mass Action Law For Ionization)

ns

n=

gs

Zexp(-Es/kT)

gs = statistical weight of state S

Z = partition function = �grexp(-E r/kT)

neni

n=

2gi

Z

(2�mekT)32

h3 exp(-E i/kT)


THERMODYNAMIC EQUILIBRIUM RELATIONS

• Additional requirement: blackbody radiationaccording to T– seldom obtained in terrestrial plasmas– use concept of Local Thermal Equilibrium (LTE)

all relations applicable except radiation field

• “Thermal Plasmas” are approximating LTE conditionsexcept in boundaries

• Pressure reduction leads to non-equilibriumconditions

Dalton' s Law : p = pn + pe + pi

Perfect Gas Law : pV = NkT

DEVIATIONS FROM THERMAL EQUILIBRIUM

• High density, temperature gradients– Diffusion faster than equilibration– Important in arc fringes, at surfaces

• High electric fields– Charge carriers acquire energy faster than they can equilibrate– Important in low pressure discharges

• Fast flow velocities– Macroscopic motion faster than equilibration– “Frozen flow conditions”– Important in high velocity jets

Most important deviations• Te � Th because of slow electron-heavy particle equilibration• Frozen recombination reactions• Ground state - excited states non-equilibrium


4. THERMODYNAMIC AND TRANSPORTPROPERTIES

(Boulos, Fauchais and Pfender, 1994)

Difference to normal gases:• Dissociation, ionization increase energy density

• Electrons lead to higher electrical, thermal conductivities

• Dissociation, ionization increases energy transport– Peaks in thermal conductivity–

• Viscosity increases because of larger momentum transfercollision cross-section

• Higher radiation transport because of high population ofexcited states

� =� trh

+ �tre

+ �in + �react

Thermodynamic and transport properties

Need to determine:(1) Composition - minimization of Gibb’s free energy

G = H - TSSaha equation for ionization reaction of noble gas

(2) Thermodynamic functions using partition function and mixture rules- density, enthalpy, specific heat

(3) Transport coefficients using Chapman-Enskog approach for solving Boltzmann equation- thermal conductivity, viscosity, electrical

conductivity, diffusion coefficient- need collision cross sections, interaction

potentials or collision integrals

(4) Radiation properties - emission coefficient


Temperature dependence of the composition (species numberdensities) of an argon plasma at atmospheric pressure (starting from

one mole of Ar at room temperature)

Temperature dependence of thecomposition (species numberdensities) of a nitrogen plasma atatmospheric pressure (startingfrom one mole of N2 at roomtemperature)

Temperature dependence of thecomposition (species numberdensities) of an Ar - H2 (20 vol%)plasma at atmospheric pressure


Zt = Ztr � Zrot � Zvib � Zel � Zchem

Partition Function(sum over all energy states)

=V

h3 2�mkT( )3

2 � �el � �chem

Zel = gs

s

� exp �Es /kT( ) Zchem = exp �Echem /kT( ) Zion = exp �Ei ��Ei

kT

�

� �

�

�

is reduction of ionization energy due to overlapping energy levels.�Ei

U int = RT� lnZt

�T

�

� �

�

� �

V

h = u + pv

p = RTln Zt

�V

�

� �

�

� �

T

h = RT� lnZt

�T

�

�

� �

V

+ V� lnZt

�V

�

�

� �

T

�

�

�

��

Specific Internal Energy: Specific enthalpy

with

Neglecting rotational and vibrational energy states

Enthalpies for various plasmas


Variation of specific heats with temperature [K.S.Drellishak, 1963]

cp =�h

�T

�

� �

�

� �

p

Thermal Conductivity

Heat flux � = thermal conductivity

from simplified kinetic theory

q = ��dT

dz

� =13

nv cvl =23

cv

�

kT

�m

n = number density of particles= average thermal velocity

cv = speci�c heat at constant volume= mean free path= collision cross section for momentum transfer

v l�


Thermal Conductivity

For reacting gases, e.g. with dissociation, ionization,transport of reaction energy must be considered

= translational thermal conductivities of heavy particlesand electrons, respectively

= reactive contribution to thermal conductivity

= internal energy transport of atoms (i.e. excited states)

� =� trh +� tr

e +�react +� int

� trh , � tr

e

�react

� int

Contributions to the thermal conductivity of anitrogen plasma


Thermal conductivities of H2-Ar mixtures

Electrical conductivity of various gases


Viscosity

Simplified kinetic theory

particle momentum x number of particles crossing determinemomentum transfer and viscosity

only valid for low degrees of ionization

vz

z

n = particle densitym = particle mass

= average thermal velocity= mean free path= total momentum transfer cross section

v l�

μ =13

n � m �v �l =13

8�

mkT�

Viscosity

For neutral atoms (low temperatures)

For appreciable ionization (xi > 0.03)

i.e. T > 10,000 K for Ar, H2, N2 and T > 17,000 K for He

long range Coulomb forces become important, thenμ decreases with increasing T

�H�H <�He�He <� N�N <� Ar�Ar


Variation of viscosity with temperature[C. Gorce, 1975; IUPAC Report, 1982]

5. Conservation equations and solution methods

• conservation of mass, momentum and energy

• typically used with boundary layer assumptions- axial gradients << radial gradients

• derived for arc column region

• description of electrode regions require modification of approach


Formulation for LTEFormulation for LTE

Fluid (conservation eqns.) +Electromagnetic (Maxwell’s eqns.) +Thermodynamic & Transport Properties

1. Total mass:

2. Mass averaged momentum:

3. Total thermal energy:

4. Current conservation:

5. Ampere’s law:

��

�t+� � �u = 0

��u

�t+ u � �u

�

�

� � = ��p �� + j � B

�Cp

�T

�t+ u � �T

�

�

� � =� � ��T( ) + j � E '�Ur + 5

2

kB

ej � �T

( ) 0=��

�2A = �μ0 j

Conservation equations for steady state, 2-dimensional arc, with boundary layer assumptions

MASS CONSERVATION

MOMENTUM CONSERVATION

ENERGY CONSERVATION

OHM’S LAW

PERFECT GAS LAW

fully developed: Elenbaas-Heller

�

�z�u( ) +

1

r

�

�rr�v( ) = 0

� u�u

�z+ v

�u

�r

�

� �

= ��p

�z+

1

r

�

�rrμ

�u

�r

�

� � + jr��

� u�h

�z+ v

�h

�r

�

� �

=1

r

�

�rr�

Cp

�h

�r

�

�

� + �Ez

2�Pr

I = 2�E z �rdr0

R

�

p = nrkTr

�

1

r

�

�rr�

Cp

�h

�r

�

� �

�

� + �E z

2� Pr = 0


n

ne,ni

Es

Ei

h

me

pn, pe, pi

p

N

T,TeTu

v

Number density of atoms in energy state sTotal number density of atomsNumber densities of electrons, ions, respectivelyEnergy of the state sIonization energyBoltzmann constantPlanck’s constantElectron massPartial processes of neutrals, electrons, ions, respectively

Total pressureSystem volumeTotal number of atoms, ions, electrons in systemEquilibrium temperature, electron and heavy particle temperature, respectivelyVelocity of atom, ion or electron

u

v

μ

�j

Bcp

�

E1

Ur,P

k,kB

e

μe

�

A

DensityVelocity, axial velocityRadial velocity componentviscositystress tensorcurrent densitymagnetic inductionspecific heat at constant pressuretotal thermal conductivityelectric fieldvolumetric radiation loss

Electron chargeElectrical conductivityPermeability of free spaceElectric potentialMagnetic vector potentialNumber densities of species r

�

�

nr

ns

V

Nomenclature

Finite Differences Finite Volumes Finite Elements

0Y =)( hR �� =� 0YW dhh )(R�� =� 0Y dh )(R

approximate equation approximate solution

• System of equations (cons. mass, mom., energy, etc.) written as:

• Formulate problem for Yh, the discrete counterpart of Y (vector of unknowns)

• Most common, weighted residual methods with local support:

stencil control volume finite element

� If implemented correctly, all methods perform ~ same

� Challenge for all methods: multi-scale and multi-physics problems

DiscretizationDiscretization MethodsMethods

0Y =)(R


Example: Temperature distribution from Elenbaas-Heller equation

GAS = Argon; RADIUS = 2.0 mm

Current (A) 100 200 300

Power (KW/m) 150 370 760

Wall heat flux (W/m2) 9.50E + 6 2.50E + 7 5.20E + 7

Example: Temperature distribution from Elenbaas-Heller equation

GAS = Hydrogen; RADIUS = 2.0 mm

Current (A) 100 200 300

Power (KW/m) 410 750 1030

Wall heat flux (W/m2) 3.00E + 7 5.70E + 7 7.50E + 7


Importance of magnetic effectsinteraction of self magnetic field with radial current density results in

pressure gradient, flow acceleration (e.g. cathode jet)

• Cathode arc attachment hassmaller diameter than arc

• Current density gradientgenerates pressure gradient

�p(r, z) = jr

R� (r,z).B(r, z)dr

= jr

R� (r, z)

μo

rj

o

r� ( � r ,z) � r d � r

�

� �

�

�

for j(r, z) =I

�R(z)2

�p(r, z) =μ0Ij(z)

4�1

r2

R2

�

� � �

� � �

maximum velocities of 10 2to 103 m/s

Energy transport by radiation

• radiation is important transport mechanism at plasma temperatures- emission and absorption- line radiation and continuum

• correct treatment requires determination of absorption in every volume element of irradiation from entire plasma- solution of radiation transfer equation- for large number of wavelength intervals

• different modeling approaches with simplifying assumptions exist- assume optically thin, only emission is counted at calculated T

integrated over all wavelengths- use net emission coefficient based on simplified plasma

geometry, integrated over all wavelengths


Argon emission coefficients at 1 atm.

Spectral emission coefficient Total net emission coefficient,L=optical path length

Menart, 1996

6. Non-equilibrium and turbulenceModeling non-equilibrium conditions requires

• two energy conservation equations, one each for electronsand for heavy species- assuming electrons have Te, all heavy species Th- need momentum transfer cross section Qeh

heavy particle energy

electron energy

• species conservation equations including diffusion fluxes- rate equations determine composition- need diffusion coefficients for fluxes Js

• properties for different ratios Te/Th, for different non-equilibriumcompositions- significantly increases computational effort

ehhhh Qqhut

h&

rr+��=��+

�

� '��

rqeheee QEJQqhut

h&

rr&

rr��+��=��+

�

� '��

csss

s nJnut

n&

rr+��=��+

�

�)(


Approach for property calculations in 2-temperature plasma

Effect of kinetic non-equilibrium

Number densities in Ar plasma Thermal conductivity of oxygen plasma

Pfender and Heberlein, 2007Copyright remains with the Author(s).

Turbulence ModelingTurbulence Modeling

Challenging because turbulence is characterized by

large span of scales (i.e. flow features from l1 … ln)

Three common approaches:

1. Direct Numerical Simulation (DNS)• Solves all the scales of the flow (very expensive!)

• Unfeasible for industrial-type problems

• Requires no “modeling” of turbulence (i.e. no extra equations, assumptions, etc.)

• Large Eddy Simulation (LES)• Solves for the large scales of the flow and models the small scales

• Turbulence model needed to approximate the small scales

1. Reynolds-Averaged Navier-Stokes (RANS)• Most common approach for industrial-type problems

• Models all scales of the flow

• Many models developed, usually a model is adequate for a specific problem

• Common models: 0 eqns: mixing length; 1 eqn: Spalart-Allmaras;

2 eqns: k-�, k-� RNG, k-�; 7 eqns: Reynolds Stresses

l1ln

TheThe k-k-�� modelmodel• As most models, relies on Boussinesq hypothesis: turbulence is mostly dissipative �

model it as an “extra” diffusion mechanism (a.k.a. turbulent viscosity μt)

• Models transport of turbulent kinetic energy k and its dissipation �:

• Need to solve additional transport equations for k and �:�

�μ2k

t =

��μ�

μ�

��+�

�

�

�=+

�

�ijijt

k

t EEkgraddivkdivtk

.2)()( U

Rate of increase

Convectivetransport

Rate ofproduction

Diffusivetransport

Rate ofdestruction

kCEE

kCgraddivdiv

t ijijtt

2

21.2)(

)( ��μ

��

�

μ��

��

�

�+��

�

�=+

�

� U

Rate of increase

Convectivetransport

Rate ofproduction

Diffusivetransport

Rate ofdestruction

� This are equations of the “standard” k-� model: fully turbulent steady-state flow, no

body forces, constant properties, etc.

� But yet, very often used for more complex flows, i.e. thermal plasmas


Plasma jet consists of hot and cold fluid parcels exchanging energy and momentum

Two fluid approach to simulate large scale turbulence

(P.C. Huang et al., 1995)

Two-fluid turbulence simulation results

Comparison of time averaged results with experimental data


Two-fluid simulation results of temperature and velocitydistributions and particle heating

Plasma and Particle Temperatures

Plasma and Particle Velocities

Particle Trajectories

• all particles injected with same velocity• particles see widely varying plasma temperature and velocities• strong effect on particle properties and trajectories

7. Examples of recent calculations

Two example calculations

(1) Highly constricted arc in plasma cutting torch

• two-dimensional geometry

• assuming non-equilibrium, Te�Th, ne affected by diffusion

• oxygen as plasma gas

(2) Time dependent three dimensional plasma spray torch

• describes anode attachment instability

• assuming kinetic non-equilibrium

• argon as plasma gasCopyright remains with the Author(s).

highly constricted arcnozzle diameter ~2 mm

200 A, 150 V arc

transferred to work piece

oxygen is plasma gas

(1)Plasma cutting torch

Ghorui et al., 2007

Courtesy of Hypertherm Inc.

Modeling domain that of a plasma cutting torch

• models including a downstream region have little influence on results


Boundary condition at cathode surface

• assumed current density profile derived fromexperimental temperature measurements


2T-Non-equilibrium Model

1. Electrons have Te , all heavy particles have Th

2. Two separate energy equations are solved: one for electrons and the other for ions, in addition to momentum, mass, species and charge conservation

3. Electrons receive energy through Ohmic heating

4. Heavy particles receive energy from electrons through collisions

5. Viscous dissipation term appears only in heavy particle equation

6. Radiation loss appears only in electron energy equation

Chemical non-equilibrium modeling approach


Chemical non-equilibrium

• Net rate of accumulation or

depletion of species inside a

plasma volume will influence

2-T chem. Equil. rate equations:

•Inside plasma, it is assumed for any species k: .5 <Qk<2

•Properties are tabulated as a function of Te for every:

p, (Te/Th), ZA, ZI, ZD in discrete steps.

Equilib.

rate SA

LTE model results


Non-equilibrium model results

temperature distributions: Te upper half, Th lower half

• distinct difference between Te and Th in entrance regionand close to wall


• comparison with experiment shows acceptable agreement- modeling results at nozzle exit, experimental 2 mm downstream


Collisional coupling between Te and Th

• maximum in collision frequency coincides with temperatureregion where Te and Th are closest

• collision frequencies for different Te/Th ratios and different non-equilibrium factors

Axial temperature distributions


Non-equilibrium model resultsRadial ne distribution

• comparison with experiment shows acceptable agreement

Radial ne distributions with and withoutcomposition non-equilibrium

• noticeable effect at intermediate radii



Axial velocity

• strongest acceleration near nozzle exit


Axial current density distribution

• strong variations in nozzle entrance region


(2) Time dependent model of plasma spray torch(2) Time dependent model of plasma spray torchTrellesTrelles and Heberlein, 2006and Heberlein, 2006

Numerical Approach: Stabilized FEMNumerical Approach: Stabilized FEM

0YSYSYKYAYA 010 ==+��+�� )()()()(

reactivediffusiveadvectivetransient

R43421434214342143421

t

total = large + small

� ��

=�� 0YôWYW ')()()(

'

dd RPR +

• Stabilized and Multi-scale Methods: 'YYY +=

• Solution: �-method, Globalized Inexact-Newton, Pre-Cond. GMRES

• System of transient – advective – diffusive – reactive equations:

Computational Domain Computational Domain �

- capture arc + jet

- hexahedral elements

(8 nodes / element)

- unknowns per node:

9 for LTE model

10 for NLTE model

cathode anode

arcjet

cathode

anode

torchinside


Boundary Conditions Boundary Conditions

p ur T � Ar

Side 1: inlet 0pp = inuu rv= inTT = 0, =n� 0=iA

Side 2: cathode 0, =np 0=iu cTT = 0, =n� 0, =niA

Side 3: cathode tip 0, =np 0=iu cTT = cn j=� ,�� 0, =niA

Side 4: outlet 0, =np 0, =niu 0, =nT 0, =n� 0=iASide 5: anode 0, =np 0=iu ( )wwn TThT �=�� 0=� 0, =niA

Gas Current [A] Flow Rate [slpm] Injection Torch 1 Ar-H2 600 60 Straight

Torch 2 Ar-H2 600 60 Straight

Torch 3a Ar-H2 600 60 Straight

Torch 3b Ar-H2 600 60 Swirl

Cases StudiedCases Studied

side 1: inlet

side 2: cathode

side 3: cathode tip

side 5: anode

side 4: outlet

Arc and Jet DynamicsArc and Jet Dynamics

undulating andfluctuating nature of jetcaptured by simulation

movement of arc � jet forcing

Schlieren imageplasma jet turbulence


Arc Dynamics: Approach 1 (LTE)Arc Dynamics: Approach 1 (LTE)old attachment

new attachment forms new attachment remains

� Too large voltage drop !!!

Improved Approach: Non-Equilibrium ModelImproved Approach: Non-Equilibrium Model

• Thermal non-equilibrium (Te � Th) (NLTE):

� If Te = Th� = LTE model


Arc Dynamics: Approach 3 (NLTE)Arc Dynamics: Approach 3 (NLTE)

attachment

time

NLTE LTE

Comparison with ExperimentsComparison with Experiments

• Voltage frequencies NLTE & LTE can match

• BUT … more realistic voltage drops with NLTE model

• Wide spectra in exp. data due to pure Ar & new anode

0 100 200 300 400 50023

24

25

26

time [ μs]

voltage dr

0 100 200 300 400 500

27

30

33

36

time [ μs]

��

p [V]

0 100 200 300 400 50040

50

60

70

time [ μs]

�� [V]

0 10 20 300

0.5

1

frequency [kHz]

Power [a.u.]

0 10 20 300

0.5

1

frequency [kHz]

Power [a.u.]

0 10 20 300

0.5

1

frequency [kHz]

Power [a.u.]

EXP.

EXP.

NLTE

NLTE

LTE

LTEfp ~ 5.3 f p ~ 5.7


Electric Potentials and FieldsElectric Potentials and Fields

• Non-LTE model produces more realistic voltage drops

Er max

Pressure and VelocityPressure and Velocity

• Formation of cathode jet

• Cold flow avoids entering hot plasma• Inflection point in velocity profiles � K-H instability (?)


8. Electrode Regions

Anode region

• drop of temperature and electrical conductivity near surface posesproblem for describing current transfer

• modeling must consider all diffusion effects, charge flux influenceson electric fields

• column fluid flow (mass and energy transport) affect anode region• column models usually assume region with artificially high electrical

conductivity between column and surface

Cathode region

• cathode electron emission model required• always strong space charges, strong cathode fall• numerous detailed models exist dividing cathode region into space charge sheath, ionization and thermalization zone• column models usually assume current density distribution at

cathode boundary

Relations Describing Arc Characteristics inAnode Region

Conservation equations including separate electron energy equations

Maxwell’s equations

Generalized Ohm’s law (without B-field, thermodiffusion)

= const.

Heat loss from arc to anode requires heat into anode region- Increased dissipation: �(�)�, E�, R�� constriction

- Energy transport into anode region by convection R� or |grad ne| �, E� � diffuse attachment

Decrease of E in anode region can mean negative anode fall- Predicted theoretically, confirmed by some experiments

j =� E +1

ene

dpe

dx

�

� �

�

� � I = 2� jrdr = 2� � E +

1

ene

dpe

dx

�

�

�

�

o

R

�o

R

� rdr


Contributions to the total current density in anargon arc (I = 200 A)

(Jenista et al., 1997)

• electron density gradient becomes principal current driver

• electric �eld reverses to reduce electron �ux

1. �E2. (1/ene)dpe/dx3. � dTe/dx

Predicted Te and Th pro�les (left) and measured Te pro�le (right)

• Te remains high

200 A, argon, 1 atm 100 A, argon, 1 atm

Temperature pro�les in anode boundary layer

Jenista et al., 1997, Yang et al., 2006.Copyright remains with the Author(s).

Different Contributions to Anode Heat Flux

q = j�w +5

2

k

e+�

�

�

� �

�

jTe �ka

dT

dx� ke

dTe

dx+ ji (� i ��w)

Comparison of calculated and experimental anode heat �uxdistributions


Constricted Mode:

• anode jet and

cathode flow form

stagnation layer

away from anode

Diffuse mode:

• anode surface

serves as stagnation

plane

Arc Anode Attachment: Constricted vs. Diffuse Mode

Streamlines in Anode Boundary Layer for TwoAnode Attachment Modes

Argon, 1 atm, 200 A

• arc constriction for increased gas heating

• anode jet forces cold gas entrainment(thermal pinch)

• high �ow compresses thermalboundary layer

• increases arc diameter in stagnation region


Temperature Distributions of Ions and Neutrals forTwo Anode Attachment Modes

Argon, 1 atm, 200 A

• increased energy dissipation can leadto maximum for Th, Te, ne

• monotonic drop in Th, Te, ne

Electric potential distributions for a constricted and adiffuse attachment of an argon arc (I = 200 A)

• potential difference between column and anode positive for constricted mode, about zero for diffuse mode

• potential gradient shows increase for constricted mode, monotonic drop for diffuse more

• both modes show negative gradients immediately in front of anode

Van

Vcol

Vse


Anode heat �ux distributions in an argon arc (I = 200 A)

• Constricted mode brings 4 fold increase in peak heat �uxfor approximately the same total heat transfer

• Electrons from stationary hotspot (> 3500K)

• Spot is heated through ionbombardment

• High melting point materialse.g. W, C, Mo, Zr

• Typical current densities ~

Thermionic Cathode Emission Mechanism

104� /cm2

Current density given by Richardson-Dushman

for most metals work function = energy requirement for release of one electron

= 4-5 V for electronegative metals (Cu, Ag, W)1.5 - 4 V for electropositive metals (Th, Ca, Ba)

j = ��2exp(-e �eff /kT) �eff = �w - ( eE4�� 0

)1

2

� � 60 �/cm2k2

�w =


Thermionic current density as function of temperature

• at 104 A/cm2, a reduction of � by 2 V results in a cathode temperature reduction of about 1700 K

Arc cathode model(Zhou and Heberlein, 1994)

Cathode

Heat conduction equation

B1 B2

qJem Te°Ti

Cathode spot

Space charge zone

Ionization zone Plasma

ne=niJi

Jed

qev

Energy balance equation at the cathode surface

Energy balance equation at the ionization zone

Generalized Saha equation

Electron emission equation at the cathode surface

Steenbeck Minimum principle

Relation between current & current densities in the space charge zone

W


Cathode temperature distributionComparison of modeling and experimental results

• Fast evaporation of thoria from the cathode spot may increase the work function of 2% ThO2-W cathodes.

(6.4 mm, truncated, 200 A, Ar)

1000

1500

2000

2500

3000

3500

4000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Experiment (ThO2-W)

Modeling (ThO2-W)

Modeling (W)Te

mpe

ratu

re (K

)

Distance from the Cathode Tip (mm)

Effect of Cathode Diameter on Temperatureand Heat Transfer

• Cathode tip temperature is primarily a function of workfunction and of arc parameters

• Cathode tip cooling at 100 A is primarily through electronemission


Cathode tip temperatures

• Comparison between experimental & theoretical results of cathode tip temperature.

3000

3200

3400

3600

3800

4000

0 100 200 300 400 500 600

Modeling result6.4 mm, cone shape cathode3.2 mm, cone shape cathode6.4 mm, truncated cathode3.2 mm, truncated cathode

Tem

per

atu

re (

K)

Arc Current (A)

9. Conclusions• Thermal plasma models rely on simplifications for obtaining results

- some very good predictions have been obtained

• major issues are non-equilibrium regions, instabilities- 2-temperature properties, diffusion fluxes- time dependent calculations

• limited availability of transport properties of gas mixtures- very cumbersome to calculate

• present state of computer technology limits advances- run times of more than a month for realistic 3-D, non-

equilibrium model

• electrode models usually describe effects in limited parameter range

• solution for contradictions with some experimental results will require novel modeling approaches- non-continuum formulation

• models for high currents/high current densities still neededCopyright remains with the Author(s).

Acknowledgments:

E. PfenderJ. MenartJ. JenistaT. AmakawaP.C. HuangS. GhoruiJ.P. TrellesX. Zhou

Selected Bibliography

1. General ReviewsBoulos, M.I., Fauchais, P., and Pfender, E. (1994). Thermal Plasmas: Fundamentals and Applications, Vol. I, Plenum

Publishing, New York and London.

Pfender, E. and Heberlein, J., (2007), “Heat Transfer Processes and Modeling of Arc Discharges”, Advances in Heat Transfer,Academic Press, New York, NY.

2. PropertiesChapman, S. and Cowling, T.G. (1970). “The Mathematical Theory of Non-uniform Gases,” 3rd edn. Cambridge Universisty

Press, Cambridge.

Ramshaw, J.D. and Chang, C.H. (1991). Plasma Chem. Plasma Process 11, 395.

Murphy, A.B. (1997). IEEE Tans. Plasma Sci. 25, 809.

Murphy, A.B. and Arundell, C.J. (1994). Plasma Chem. Plasma Process 14, 451.

Murphy, A.B. (1995). Plasma Chem. Plasma Process 15, 279.

Aubreton, J., Bonnefoi, C., and Mexmain, J.M. (1986). Rev. Phys. Appl. 21, 365.

Devoto, R.S. (1973). Phys. Fluids 16, 616.

Murphy, A.B. (2000). Plasma Chem. Plasma Process 20, 279.

Aubreton, J., Elchinger, M.F., and Fauchais, P. (1998). Plasma Chem. Plasma Process 18(1), 1.

Aubreton, J., Elchinger, M.F., Rat, V., Fauchais, P., and Murphy, A.B. (2003). Two-temperature combined diffusion coefficientsin argon-helium thermal plasmas. High Temp. Mater Process 7, 107-113.

Van de Sanden, M.C.M., Schram, P.P.J.M., Peeters, A.G., Van der Mullen, J.A.M., and Kroesen, G.M.W. (1989).Thermodynamic generalization of the Saha equation for a two-temperature plasma. Phys. Rev. A 40, 5273-5276.


3. RadiationMenart, J.A. (1996). Theoretical and Experimental Investigations of Radiative and Total Heat Transfer in Thermal Plasmas,PhD Thesis, Department Mechanical Engineering, University of Minnesota.

Gleizes, A., Gonzalez, J.J., Linani, B., and Raynal, G. (1993). Calculation of net emission coefficient of thermal plasmas inmixtures of gas with metallic vapour. J. Phys. D: Appl. Phys. 26, 1921-1927.

Lowke, J.J. and Capriotti, E.R. (1969). Calculation of temperature profiles of high pressure electric arcs using the diffusionapproximation for radiation transfer. J. Quant. Spectrosc. Radiat. Trans. 9, 207-236.

Lowke, J.J. (1974). J. Quant. Spectosc. Radiat. Trans. 16, 111-122.

4. Modeling ResultsHsu, K.C. and Pfender, E. (1981). Calculation of thermodynamic and transport properties of a two-temperature argon plasma.In 5th International Symposium on Plasma Chemistry, Heriot-Watt University, Edinburgh, Scotland.

Gonzalez, J.J., Freton, P., and Gleizes, A. (2002). Comparisons between two-and three-dimensional models: gas injection andarc attachment. J. Phys. D.: Appl. Phys. 35, 3181-3191.

Li, H.P. and Chen, X. (2001). Three-dimensional modeling of a dc non-transferred arc plasma torch. J Phys. D: Appl. Phys. 34,L99-L102.

Baudry, C., Vardelle, A. and Mariaux, G. (2005). Numerical Modeling of a dc non-transferred plasma torch: movement of the arcanode attachment and resulting anode erosion. High Temp. Mater. Process 9(1), 1-18.

Trelles, J.P., Pfender, E., and Heberlein, J. (2006). Multiscale finite element modeling of arc dynamics in a DC plasma torch. Inprint.

Trelles, J.P. and Heberlein, J. (2006). Simulation results of arc behavior in different plasma spray torches, 15(4).

Chen, D.M., Hsu, K.C., and Pfender, E. (1981). Two-temperature modeling of an arc plasma reactor. Plasma Chem. PlasmaProcess 1(3), 295-314.

Lowke, J.J., Morrow, R., and Haidar, J. (1997). A simplified unified theory of arcs and their electrodes. J Phys. D: Appl. Phys.30, 2033-2042.

5. Electrode ModelsJenista, J., Heberlein, J.V.R., and Pfender, E. (1997). Numerical model of the anode region of high-current electric arcs.IEEE Trans. Plasma Sci. 25(5), 883-890.

Amakawa, T., Jenista, J. Heberlein, J., and Pfender, E. (1998). Anode-boundary-layer behaviour in a transferred, high-intensity arc. J. Phys. D: Appl. Phys. 31(20), 2826-2834.

Jenista, J., Heberlein, J., and Pfender, E. (1997). Model for anode heat transfer from an electric arc. In Proceedings of theFourth International Thermal Plasma Processes Conference, p. 805.

Morrow, R. and Lowke, J.J. (1993). A one-dimensional theory for the electrode sheaths of transfer in gas metal arc welding.Aust. Weld. J. 42(1), 32-35.

Benilov, M.S. (1997). Analysis of thermal non-equilibrium in the near cathode region of atmospheric-pressure arcs. J. Phys.D: Appl. Phys. (28), 286-294.

Lichtenberg, S., Dabringhausen, L., Langencheidt, O., and Mentel, J. (2005). The plasma boundary layer of HID-cathodes:modeling and numerical results. J Phys. D: Appl. Phys. 38, 3112-3127.

Benilov, M., Carpaij, M., and Cunha, M. (2006). 3D modeling of heating of thermionic cathodes by high -pressure arcplasmas. J Phys. D: Appl. Phys. 39, 2121-2134.

Zhou, X. and Heberlein, J. (1994). Analysis of the arc-cathode interaction of free-burning arcs. Plasma Sources Sci. Technol.3(4), 564-574.


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Thermal Plasma Modeling · Thermal Plasma Modeling Joachim Heberlein University of Minnesota Outline: 1. Introduction to thermal plasmas 2. Generation of thermal plasmas 3. Equilibrium