-
1
Numerical Numerical thermofluidthermofluid modeling of
highmodeling of high--powerpowerhighhigh--pressure thermal plasma
pressure thermal plasma using reaction kineticsusing reaction
kinetics
Yasunori TanakaKanazawa University
[email protected]
http://www.ee.t.kanazawa-u.ac.jp/staffs/tanaka/
原子分子データ応用フォーラムセミナー@Toki,13 Dec 2011 IntroductionIntroduction--1
:Features of high1 :Features of high--pressure plasmaspressure
plasmasHigh-pressure plasmas for materials processing
-Te >> Th-Electron-molecule interactions are dominant to
produce ions & radicals.-Gradients of gas density and gas
temperature are low.
1. Thermally non-equilibrium plasmas(ex.Atmospheric-pressure
glow discharges APGD, DBD)
2. High power-density plasmas (Thermal plasmas /Meso-plasmas)-Te
~ Th or Te > Th-Electron-molecule, electron-atom and heavy
particle-heavy particle interactions are important to produce ions
& radicals
-High dissociation degree of gases and highly non-uniform gas
composition can be seen. -Gas density has a wide range of the
fourth order(ex. 0.0001 kg/m3 to 1.0 kg/m3) due to high gradient of
gas temperature (300 K-10000 K).
~10 MW/m3
APGD
Modulation & Continuous feeding(80%SCL-PM / CWF)
0
1
2
3
4
Rad
ial p
ositi
on [m
m]
0 2 4 6 84
3
2
1
0
Axial position [mm]
Application of thermal plasmasApplication of thermal plasmas
Nanopowder synthesis
Plasma arc cutting
Diamond film fabrication
0
5
10
40 50 60 70 80 90 100 11010
5
0
Axial position [mm]
Rad
ial p
ositi
on [m
m]
0
5
10
40 50 60 70 80 90 100 11010
5
0
Axial position [mm]
Rad
ial p
ositi
on [m
m]
0
5
10
40 50 60 70 80 90 100 11010
5
0
Axial position [mm]
Rad
ial p
ositi
on [m
m]
0
5
10
40 50 60 70 80 90 100 11010
5
0
Axial position [mm]
Rad
ial p
ositi
on [m
m]
0
5
10
40 50 60 70 80 90 100 11010
5
0
Axial position [mm]
Radi
al p
ositi
on [m
m]
Switching devices: HVCB, MCCB
Surface modification
TiO2
Introduction-2
The conventional model for thermal plasmasassumes LTE (Local
Thermodynamic Equilibrium) condition:A. Thermal equilibrium# All
temperatures including Te, Th, Tex, Trot, Tvib, etcare assumed to
be the same.
B. Chemical equilibrium# Infinite reaction rates for all
reactions
Reaction field instantaneously reaches its equilibrium. Reaction
field is determined only by T and P.
The conventional model for thermal plasmasassumes LTE (Local
Thermodynamic Equilibrium) condition:A. Thermal equilibrium# All
temperatures including Te, Th, Tex, Trot, Tvib, etcare assumed to
be the same.
B. Chemical equilibrium# Infinite reaction rates for all
reactions
Reaction field instantaneously reaches its equilibrium. Reaction
field is determined only by T and P.
Thermally & Chemically Non-EquilibriumConditions
Thermally & Thermally & Chemically Chemically
NonNon--EquilibriumEquilibriumConditionsConditions
-Very high gas flow velocity-Rapid change in state (Transient
state)-Low temp. region⇒Low reaction rate-High electric field
strength-High gradient of particle density
Modeling of Thermal PlasmasModeling of Thermal Plasmas
Actual situationsActual situations
-
2
Our work that has been done
#1. A chemically non-equilibrium model#2. A two-temperature
chemically non-equilibrium modelfor thermal plasmas was developed
using reaction kinetics. -Consideration of reaction rates for
hundreds of reactions -Thermodynamic and transport properties were
self-consistently calculated using non-CE density and Tat each
position at each calculation step.
#1. A chemically non-equilibrium model#2. A two-temperature
chemically non-equilibrium modelfor thermal plasmas was developed
using reaction kinetics. -Consideration of reaction rates for
hundreds of reactions -Thermodynamic and transport properties were
self-consistently calculated using non-CE density and Tat each
position at each calculation step.
-Results of temperature fields, particle composition -Comparison
with LTE calculation results-Results of temperature fields,
particle composition -Comparison with LTE calculation results
Results of thermal plasmas with O2 gas# Pulsed arc discharge in
dry air, # Pulsed arc discharge in dry air, # Ar+N# Ar+N22+O+O22
ICPsICPs, # Dielectric strength of hot air# Dielectric strength of
hot air(Ar+N2, Ar+N2+H2, Ar+CO2+H2, Ar+CH4+O2, SF6 plasmas have
been also developed )
-Nanopowder synthesis-Surface modification-Syngas production
Today’s presentation
Keywords:-Chemically non-equilibrium -Reaction rates-Convection
and diffusion
NonNon--equilibrium modeling of arcs/thermal plasmasequilibrium
modeling of arcs/thermal plasmas1.1. Chemically nonChemically
non--equilibrium effectsequilibrium effects2.2. Thermally
nonThermally non--equilibrium effectsequilibrium effects3.3.
NonNon--MaxwellianMaxwellian EEDFEEDF
Assumption of local thermodynamic equilibrium (LTE)
conditionParticle composition can be determined only by local
temperature and pressure through the mass action law.-Solving Saha
equations & Guldberg-Waage equations-Minimization of Gibb’s
free energy of a system
How to consider chemically nonHow to consider chemically
non--equilibriumequilibrium
A conventional model for thermal plasmas
∑
∏−∏−=+⋅∇+
∂
∂
= ==
L N
ii
rN
ii
ffj
rjjjj
j rif
i nnmYtY
1 11)()(
)(
lllll
ll ββ ααββρρ
Ju
-Reaction rates are finite Time scale of a system~Reaction
time-Convection and diffusion effects are not negligible for
particle composition determination
Convection Diffusion Production by reactions
Mass conservation of each of species j:
Actual situations
This equation should be solved to determine particle
compositionThis equation should be solved to determine particle
compositionconsidering chemically nonconsidering chemically
non--equilibrium effects.equilibrium effects.
Te~Th
Consideration of chemically nonConsideration of chemically
non--equilibriumequilibrium
∑
∏−∏−=+⋅∇+
∂
∂
= ==
L N
ii
rN
ii
ffj
rjjjj
j rif
i nnmYtY
1 11)()(
)(
lllll
ll ββ ααββρρ
Ju
Convection Diffusion Production by reactions
Mass conservation of species j:
#Disadvantages in treatment of chemically non-equilibrium(i) A
large amount of data concerning reaction rates
with a wide temperature range are necessary.(ii) Thermodynamic
and transport properties
cannot be calculated in advance as functions of T & P. They
should be calculated at each position with local T & P and
particle composition calculated.
(iii) Numerical instability often occurs because of a wide range
of reaction rates.
An implicit method is usually adopted to solve the mass
conservation equations.
#Disadvantages in treatment of chemically non-equilibrium(i) A
large amount of data concerning reaction rates
with a wide temperature range are necessary.(ii) Thermodynamic
and transport properties
cannot be calculated in advance as functions of T & P. They
should be calculated at each position with local T & P and
particle composition calculated.
(iii) Numerical instability often occurs because of a wide range
of reaction rates.
An implicit method is usually adopted to solve the mass
conservation equations.
Examples of chemically non-equilibrium modeling will be
presented.Examples of chemically non-equilibrium modeling will be
presented.
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3
ExEx--1. 1. ModellingModelling of pulsed arc discharges in airof
pulsed arc discharges in air
Assumptions(1) Pressure surrounding the arc is the atmospheric
pressure,
i.e. 101325 Pa.(2) The one-temperature model is adopted, i.e.
T=Te=Th.(3) Axisymmetric structure.(4) Optically thin.(5)
Turbulence is neglected.(6) The electric field has only an axial
component,
while magnetic field has only an azimuthal one.(7) The dry-air
plasma has 13 species:
N2, N2+, N, N+, O2, O2+, O, O+, NO, NO+, and e.(8) Reaction
rates depend on Arrehius’ law.(9) Chemically equilibrium(CE) is not
dictated.
Ez
r
Hθ
Example 1: A pulsed arc discharge in air with Ipeak=72A, tf/th=
1.0/12 µs.
Y.Tanaka, PSST, 14, 134, 2005
Simple model-Mass conservation:
-Momentum conservation:
-Energy conservation:
Governing equationsGoverning equations
( ) ( ) 2211 02 θσµυηυηυρυρυ HErr
rrrr
pr
rrt z
−−
∂∂
∂∂
+∂∂
−=∂
∂+
∂∂
radz
N
j
jjj PEr
Yh
CDr
rrrh
Cr
rr
hrrr
ht
−+
∂
∂⋅
−
∂∂
+
∂∂
∂∂
=
+
∂∂
+
+
∂∂
∑=
2
1 pp
22
11
)2
(1)2
(
σκρκ
υυρυρ
ｍｍ
( ) 01 =∂
∂+
∂∂
rr
rtυρρ
-Mass conservation of species j:( ) ( ) ( ) 11
1 11∑ ∏∏
= ==
−⋅−=
∂∂
∂∂
−∂
∂+
∂∂ L
l
N
ii
rl
N
ii
fl
fjl
rjlj
jj
jj rilf
il nnmrY
Drtrr
Yrrt
Y ββ ααββρυρρ
nnnnnn NOOONNe 22 +++++− ++++=-Charge neutrality: -Equation of
state: kTnp
jj∑=
∫∞=
02 rdr
IEzπσ
-Ohm’s law:
-Ampere’s law:ξξσθ dEr
H zr
∫= 01
Solving these eqs. enables one to drive nj without CE
assumption.
t : time, ρ: mass density, r : radial position, υ : radial
velocity, p: pressure, σ: electrical conductivity, µ0:
permeability, Ez: electric field strength, Hθ: magnetic field
strength, h: enthalpy, κ: thermal conductivity, Dj: effective
diffusion coefficient, Cpm: effective specific heat, Yj: mass
fraction, Prad: radiation loss, βjl: stoichiometric coefficient, αl
: reaction rate
t : time, ρ: mass density, r : radial position, υ : radial
velocity, p: pressure, σ: electrical conductivity, µ0:
permeability, Ez: electric field strength, Hθ: magnetic field
strength, h: enthalpy, κ: thermal conductivity, Dj: effective
diffusion coefficient, Cpm: effective specific heat, Yj: mass
fraction, Prad: radiation loss, βjl: stoichiometric coefficient, αl
: reaction rate
)2,2(ijΩπ
1 : 2O2→ 2O + O22 : O2 + NO → 2O + NO3 : O2 + N2→ 2O + N24 : O2
+ O → 3O 5 : O2 + N → 2O + N 6 : NO + O2→ N + O + O27 : 2NO → N + O
+NO 8 : NO + N2→ N + O + N29 : NO + O → N + 2O 10: NO + N → 2N + O
11: N2 + O2→ 2N + O212: N2 + NO → 2N + NO 13: 2N2→ 2N + N214: N2 +
O → 2N + O 15: N2 + N → 3N16: N2 + e-→ 2N + e-17: N2 + O → NO +
N18: NO + O → N + O219: N + O → NO+ + e-20: 2N → N2+ + e-21: 2O →
O2+ + e-
22: O + e- O+ + 2e-23: N + e- N+ + 2e-24: NO+ + O N+ + O225: O2+
+ N N+ + O226: NO + O+ N+ + O227: O2+ + N2 N2+ + O228: O2+ + O O+ +
O229: NO+ + N O+ + N230: NO+ + O2 O2+ + NO31: NO+ + O O2+ + N32: O+
+ N2 N2+ + O33: NO+ + N N2+ + O34: N2+ + N N2 + N+35: N2+ + N2 + e-
2N236: N2+ + N + e- N2 + N37: N+ + N2 + e- N + N238: N+ + N + e-
2N39: O2 + e 2O + e40: O2 + e O2+ + 2e41: NO + e NO+ + 2e42: N2 + e
N2+ +2e
Reactions in dryReactions in dry--air plasmasair plasmas
42 forward reactions &their backward reactions
Totally 84 reactionswere taken into
account
Rate coefficients -Forward reactions: by Arrhenius lawusing data
of C.Park (NASA)-Backward reactions:by the principle of detailed
balancing
∑≠
−=
jiji
i
jj
Dx
YD
1
)1,1()1(
12
)(831
ijji
ji
ij mmmmkT
Ω
+=
∆ ππ
∑≠
∆=
ejejj
e
nn
kTe
)1(
2
σ
)1(
1
ijij p
kTD∆
=
-Thermodynamic and transport properties depends on not only
T& p but also particle composition nj at each time step at each
position.
Rapid transient phenomena in pulsed arc discharges
↓Chemical non-equilibrium
k :Boltzmann const., T: temperature, e: electronic charge, nj:
density of species j, mj: mass of species j, πΩij: collision
integral between i-j, xi:mole fraction of species j, Yj: mass
fraction of species j, p : pressure (Pa)
k :Boltzmann const., T: temperature, e: electronic charge, nj:
density of species j, mj: mass of species j, πΩij: collision
integral between i-j, xi:mole fraction of species j, Yj: mass
fraction of species j, p : pressure (Pa)
Transport properties of dryTransport properties of dry--air arc
plasmasair arc plasmas--11First order approximation of
Chapman-Enskog method
-Electrical conductivity
-Effective diffusion coefficient
Transport properties
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4
( )
∆+
∂∂
+= fjjj
j HZTkTkT
mh ln
251 2
j
N
jjhYh ∑
=
=1
∑=
∂∂
=N
j
jj T
hYC
1pm( ) ( )
( )2/1/54.245.0/1
1ji
jijiij mm
mmmm+
−⋅−+=ξ
∑∑=
=
∆=
N
jN
iiji
jj
n
nm
1
1
)2(η
)2,2()2(
12
)(1651
ijji
ji
ij mmmmkT
Ω
+=
∆ ππ
∑∑=
=
∆=
N
iN
jijjij
i
n
nk1
1
)2(415
ξκ
Zj: internal partition function of species j, ∆Hfj: standard
enthalpy of formation
Zj: internal partition function of species j, ∆Hfj: standard
enthalpy of formation
Transport properties of dryTransport properties of dry--air arc
plasmasair arc plasmas--22
-Thermal conductivity
-Viscosity
Thermodynamic properties Transport properties
Electric current waveformElectric current waveform
The similar waveform to that in the experiment was adopted for
comparison.
-Peak value of current=72A
-Time duration to peak= 1.0 µs
-Time duration to half current=12.0µs0 1 2 3 4 5 6 7 8
01020304050607080
Ipeak=72 A, tf/th=1.0/12.0 µs
Cur
rent
(A)
Time (µs)
Time evolution in temperatureTime evolution in temperature
(x, y)=(0,0) : center axis of the arc
An increase in temperature on center axisarises from increasing
joule heating there.
Electric current
-Atmospheric air-11 species(N2, N2+, N, N+,O2, O2+, O, O+,NO,
NO+,e)
-84 kinds of reactions
-Atmospheric air-11 species(N2, N2+, N, N+,O2, O2+, O, O+,NO,
NO+,e)
-84 kinds of reactions
Time evolution in pressureTime evolution in pressure
A rapid decrease in pressure around the axis can be seen.A
discontinuous surface is generated Shock-wave
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5
Time evolution in mass densityTime evolution in mass density
A rapid decrease in ρ around the axis and an increase in ρ
surrounding the arc are due to the generated shock-wave.
A rapid decrease in ρ around the axis and an increase in ρ
surrounding the arc are due to the generated shock-wave.
Time evolution in Time evolution in particle composition
distributionparticle composition distribution
N2N2+NN+O2O2+OO+NONO+e-
A rapid change in composition arising from dissociation and
ionization reactions can be seen until about t=1.0 µs. From
t>1.0 µs, the composition change is due mainly to the radial
transport.
A rapid change in composition arising from dissociation and
ionization reactions can be seen until about t=1.0 µs. From
t>1.0 µs, the composition change is due mainly to the radial
transport.
Dominant reactions for Dominant reactions for
generation/disappearance of speciesgeneration/disappearance of
species
0.0 0.2 0.4 0.6 0.8 1.0-20
-10
0
10
20
N2 + O NO + N
N + O NO+ + e-
[x104]
2N N2+ + e-
N2 + e- 2N + e-
N + e- N+ + 2e-
Radial position (mm)
Mas
s pro
duct
ion
rate
(kg
m-3 s-
1 )
NO + N N2 + O
NO+ + e- N + O
N + O2 NO + O
N2+ + e- 2N
N+ + 2e- N + e-
Mass production rate of N atomby each reactionat t=0.4 µs after
ignition
Mass production rate of N atomby each reactionat t=0.4 µs after
ignition
Gen
.D
isap
p. Gen.: N2+e 2N + eDisapp.:N+e N+ +e
Dominant reactions for generation/disappearance Dominant
reactions for generation/disappearance of species at 0.4 of species
at 0.4 µµs after arc ignitions after arc ignition
NO+ + e-→ N + ON + O → NO+ + e-NO+
O+ + 2e-→ O + e-N+ + 2e-→ N + e-
O + e-→ O+ + 2e-N + e-→ N+ + 2e-e
-
NO + N → N2 + ON2 + O → NO + NNOO+ + 2e-→ O + e-O + e-→ O+ +
2e-O+O + e-→ O+ + 2e-O+ + 2e-→ O + e-OO2+ + e-→ 2O2O → O2+ + e-O2+N
+ O2→ NO + O2O + N2→ O2 + N2O2
N+ + 2e-→ N + e-N + e-→ N+ + 2e-N+N + e-→ N+ + 2e-N2 + e-→ 2N +
e-NN2+ + e-→ 2N2N → N2+ + e-N2+N2 + e-→ 2N + e-NO + N → N2 +
ON2
DisappearanceGeneration
NO N2O2
O NO+ N
N2+
N+O+
O2+
Heavy particle chemistryat t=0.4 µs
-
6
Comparison with experimental resultsComparison with experimental
results--1:1:Neutral particle density, electron densityNeutral
particle density, electron density
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00
2
4
6
8
10
12[x1025]
t=2.0 µs
t=0.8 µs
Neu
tral
par
ticle
den
sity
(m-3)
Radial position (mm)
t=0.4 µs
Experimental results were obtained with a two-wavelength
laser-interferometer by Akazaki et al.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.00.00.10.20.30.40.50.60.70.80.9
t=0.4 µs@LTE
[x1024]
t=2.0 µs
t=0.8 µs
t=0.4 µs
Ele
ctro
n de
nsity
(m-3)
Radial position (mm)
Good agreements between calculation results and experimental
ones.Good agreementsGood agreements between calculation results and
experimental ones.
Red:Present cal. results Red:Present cal. results
Comparison with experimental resultsComparison with experimental
results--22Arc radius, expanding velocity of shock waveArc radius,
expanding velocity of shock wave
rs: Radial position of shock wave surfacerc: Arc channel radius
defined as the radius in which 90% electrons is included.
0 1 2 3 4 5 6 70.00.51.01.52.02.53.03.54.0
Measured
Calculated
rs
rc
Rad
ius
r c, r
s (m
m)
Time (µs)
10-1 100 1010.00.51.01.52.02.53.03.5
Vel
ocity
(km
s -1)
Time (µs)
d rs/dt
Good agreementsGood agreementsbetween calculation results and
experimental ones.
Red:Calculated result
SummarySummaryDevelopment of 1-D Chemical non-equilibrium model
of pulsed arc dischargesin dry air at atmospheric pressure (84
reactions considered)
Development of 1-D Chemical non-equilibrium model of pulsed arc
dischargesin dry air at atmospheric pressure (84 reactions
considered)
Time evolutions in temperature, pressure, mass density, particle
composition in a pulsed arc discharge was obtained.
Shock-wave generationDominant reactions for each particle
Comparison with experimental results:-Neutral particle density,
-Electron density, -Arc conducting radius and its expanding
velocity, -Radial position of shock-wave surface and its expanding
velocity
Good agreements were obtained.
Keywords:-Chemically non-equilibrium -Reaction rates-Convection
and diffusion
NonNon--equilibrium modeling of arcs/thermal plasmasequilibrium
modeling of arcs/thermal plasmas1.1. Chemically nonChemically
non--equilibrium effectsequilibrium effects2.2. Thermally
nonThermally non--equilibrium effectsequilibrium effects3.3.
NonNon--MaxwellianMaxwellian EEDFEEDF
-
7
Cross-section of plasma torch and reactor
Plasma torch Reactor chamber
8-turn coil
Quartz tubeCooling water
155mm
910mm
Gasinjection
Water-cooled SUS wall
Calculation space
330mm 150mm
35mm
Frequency: 450 kHz, RF.Power: 13.5, 27.0 kW Pressure: 1, 0.5,
0.3 atmGas:Sheath gas Ar/additional gas=98/2 slpm( e.g.)Additional
gas N2=2, N2+H2=1+1, N2+O2=1.56+0.44 slpm(air)
Frequency: 450 kHz, RF.Power: 13.5, 27.0 kW Pressure: 1, 0.5,
0.3 atmGas:Sheath gas Ar/additional gas=98/2 slpm( e.g.)Additional
gas N2=2, N2+H2=1+1, N2+O2=1.56+0.44 slpm(air)
Calculation conditions
(1) Steady state(2) Axisymmetric structure(3) Laminar flow,
therefore the turbulent effect is negligible.(4) Optically thin(5)
Chemically non-equilibrium condition is allowed.(6) Te can be
different from Th
2T-NCE Modeling of Ar plasmas with molecular gases
For Ar+N2+O2 plasma:13 species: N2, N2+, N, N+, O2, O2+, O, O+,
NO, NO+, Ar, Ar+, e
For Ar+N2+O2 plasma:13 species: N2, N2+, N, N+, O2, O2+, O, O+,
NO, NO+, Ar, Ar+, e
Assumptions
Particles considered
Governing equations0=)(∇+
∂∂ uρρ ・
t-Mass conservation:
( ) ( ) [ ]∗ℜ+
∂∂
•∇+∇•∇+∂∂
−=•∇+∂
)(∂rHEz
uzpu
tu &&
θσµηηρρ
0uu
( ) ( ) [ ]*022)( zHErvw
rrrp
t&&
θσµρυηηυηυρρυ ℜ++−
∂∂
•∇+∇•∇+∂∂
−=•∇+∂
∂ uu
( ) ( )r
rrw
rvwww
tw
∂∂
−+∇•∇=•∇+∂
∂ )()( ηρηρρ u
-Momentum conservation:
Axial:
Radial:
Swirl:
( ) ( ) ( )[ ] eh0
htrh
ee
EQYhDThth LN
jjjj
rf
+∆−∇′•∇+∇•∇=′•∇+∂
)′(∂ ∑∑)=⋅( lll
l
ββ
ρλρρ u
-Energy conservation for heavy particles:
-Energy conservation for electrons:( ) eh
0radee
ee
treeeee
ee251
25 EQPEET
mTTnTn
t
LN
j rf−∆−−+
•∇−∇•∇=
•∇+
∂∂ ∑∑
)≠⋅(
∗
lll
l
ββθθσκλκκ Γ2
5u
-Mass conservation for each particle:
( )∑ ∏∏
−⋅−+∇•∇=•∇+
∂
∂ L
l
N
ii
rl
N
ii
fl
fjl
rjljjjj
j rilf
il nnmYDYtY ββ ααββρρ
ρ)()(
)(u
etc…-Maxwell eq. for vector potential:
-Equation of state-Charge neutrality
θθ ωσµ AjA && 02 =∇
Reaction heat
Energy transfer between h & e
1: O2 + O2→ O + O +O22: O2 + NO → O + O + NO3: O2 + N2 → O + O +
N2 4: O2 + O → O + O + O5: O2 + N → O + O + N6: O2 + Ar → O + O +
Ar7: NO + O2 → N+ O + O2 8: NO + NO → N + O +NO9: NO + N2 → N + O +
N2
10: NO + O → N + O + O11: NO + N → N + O + N12: NO + Ar→ N + O +
Ar13: N2+O2→N+N+O214: N2+NO →N+N+O15: N2+N2→N+N+N216: N2+O
→N+N+O17: N2+N →N+N+N18: N2+Ar →N+N+Ar19: N2+e →N+N+e20: N2+O
→NO+N
Ex.: Reactions in Ar-N2+O2 plasma21: NO + O → N +O222: N + O →
NO+ + e23: N + N → N2++e24: O + O → O2+ + e25: O + e → O+ + e +
e26: N + e → N+ + e + e27: NO+ + O → N+ + O2 28: O2+ + N → N+ +
O229: NO + O+ → N+ + O230: O2+ + N2→ N2+ + O231: O2+ + O → O+ +
O232: NO+ + N → O+ + N233: NO+ + O2→O2+ +NO34: NO+ + O → O2+ +N35:
O+ + N2→N2++O36: NO+ + O →O2+ +N37: N2+ + N →N2 + N+38: N2++ N2 + e
→N2+N239: N2++ N + e →N2+N40: N+ +N2 + e →N2+N
41: N++ N +e → N +N42: O2 + e → O+O + e43: O2 + e → O2++e+e44:
NO + e → NO++e+e45: N2 + e → N2++e +e46: Ar++e+e→ Ar + e 47: Ar+Ar→
Ar+ +e+Ar
94 reactions were considered.94 reactions were considered.
Reaction rate coefficients:#Temperature-dependent
approximation1. Arrhenius type: α=aTbexp(-c/T) 2. Lennard-Jones
type:
α=aexp(b+cT+dT2)exp(-c/T)#. Energy Distribution
Function:
Reaction rate coefficients:#Temperature-dependent
approximation1. Arrhenius type: α=aTbexp(-c/T) 2. Lennard-Jones
type:
α=aexp(b+cT+dT2)exp(-c/T)#. Energy Distribution
Function: ∫∞
=
0
2/1
)()(2 εεεεσα dfmef
ll
Maxwellian for heavy particles
-
8
Ex. Rate coefficients of electron impact reactionsEx. Rate
coefficients of electron impact reactions
∂∂
−=∑ εεεσδε
ε nnmm
NENeJ
sss 2)()/2(3
)/(22/1
22f
∑
∂∂
−
−
−=s
gg
esnkT
kTn
Mm
mNJ
εεεεσδε
2)(22
2
2/1el
[ ])(2)( 2/12/1 εσεε sjsj mR
=
2/10 )()( εεε fnn e=
ee,
0elf )()()(
−
+
∑ ∫
∂∂
+∂
∂+
∂∂
−∂∂
−=∂
∂
js
sjs
ndnR
NJJ
tn sj
εε
εεε
εεε εε
ε
: Energy gain by electron by E
: Energy loss in elastic collisions
Rate of inelastic collisions such as excitation, ionization
EEDF: Electron energy distribution function
Electron-electron collision term
∫∞
=
0
2/1
)()(2 εεεεσα dfmef
ll
Rate coefficient of electron impact reactions
Two-term expansion of 0-D Boltzmann eq.10-2 10-1 100 101 102
10-2310-22
1x10-2110-2010-1910-18
Electr.ex
Ioni.Momen.
Vib.ex
Cro
ss se
ctio
n (m
2 )
Electron energy (eV)
e-N2e-N2
50 100 150 200 25010-12
10-11
1x10-10
1x10-9
Electron impactionization98%Ar-2%air Eq.Tg=3000 K
N2
NO
ArO2
NO
Rat
e co
effic
ient
[m3 /s
]E/N [x10-17 V m2]
50 100 150 200 25010-12
10-11
1x10-10
1x10-9 Eq.Comp.@Tg=4000 K
Eq.Comp.@Tg=3500 K
Eq.Comp.@Tg=300-3000 K
Electron impactionization of N98%Ar-2%air Eq.
Rat
e co
effic
ient
[m3 /s
]
E/N [x10-17 V m2]
Transport properties of thermal plasma
∑∑=
=
∆
=N
jN
jijj
jj
n
nm
1
1
)2(η
∑=
∆= N
jejjej
e
n
nk
1
)2(
tre 4
15
ξλ
The first order approximation of Chapman-Enskog method
The first order approximation of Chapman-Enskog method
∑
−=
≠
N
ji ij
j
jj
DxY
D1
Transport and thermodynamic properties for heavy particles and
electrons are calculated at each position using non-equilibrium
composition and collision integrals.
Transport and thermodynamic properties for heavy particles and
electrons are calculated at each position using non-equilibrium
composition and collision integrals.
∑∑≠
=
∆=
eiN
jijjij
i
n
nk
1
)2(
trh 4
15
ξλ
∑≠
∆= N
ejejj
e
ee
n
nkTe
)1(
2
σ
∑≠
+=
ejh
j
je
e
e kTmY
kTmY
pρ-Mass density
-Effective diffusion coefficient
-Translational thermal conductivity for heavy particles
-Translational thermal conductivity for electrons
-Electrical conductivity
-Viscosity
Influenced markedly by temperature and composition
2D-Temp. distribution by 1T-CE and 2T-NCE methods
0 200 400 600 8006040200
9000
Axial position (mm)
Rad
ial p
ositi
on (m
m)
0204060
7000 K
80009000
2T-NCE2T-NCE
1T-CE:LTE1T-CE:LTE
Ar-air plasmaP=0.3atm
0 200 400 600 800604020
0
9000 Th
Axial position (mm)
Rad
ial p
ositi
on (m
m)
0204060 Te
7000 K
80009000
0 200 400 600 800604020
0
9000 Th
Axial position (mm)R
adia
l pos
ition
(mm
)
0204060 Te
7000 K
8000900098%Ar+2%[email protected] [email protected]
98%Ar+2%[email protected] [email protected]
0 200 400 600 8006040200
Th
Axial position [mm]
0204060
Te6000 K
7000 K8000 K9000 K
Rad
ial p
ositi
on [m
m]
98%Ar+2%[email protected] [email protected]
98%Ar+2%[email protected] [email protected]
Temperature distribution of Ar-N2+O2 & Ar-N2 plasmas
-
9
Radial T distribution for Ar-N2+O2 & Ar-N2 plasma
98%Ar+2%N298%Ar+2%N2
0 5 10 15 20 25 30 3502468
10
Th
Te
Tem
pera
ture
[kK
]
Radial position [mm]
Te>Th: thermal non-eq.
Inclusion of O2 causes shrinkage of a plasma
z=155mm, mid coil
P=0.3atm P=0.3atm
0 5 10 15 20 25 30 3502468
10
Th
Te
Tem
pera
ture
[kK
]
Radial position [mm]
98%Ar+2%air(1.56%N2+0.44%O2)98%Ar+2%air(1.56%N2+0.44%O2)
P=0.3atm13.5kW
0 5 10 15 20 25 30 3502468
10
ThTe
Tem
pera
ture
[kK
]Radial position [mm]
98%Ar+1%N2+1%H298%Ar+1%N2+1%H2
Power balance in Ar-N2 plasma
0 10 20 3002468
10
Th
Te
Tem
pera
ture
[kK
]
Radial position [mm]
98%Ar+2%N2 plasma@[email protected]
z=155mm, mid coil
0 5 10 15 20 25 30 35-60
-40
-20
0
20
40
60
R-cond. -div(heΓe)
-Σ∆Ql -Prad -Eeh
σ|Eθ|2
Pow
er [M
W/m
3 ]
Radial position [mm]0 5 10 15 20 25 30 35
-60
-40
-20
0
20
40
60
Diff.R-cond.Z-conv. -Σ∆Ql
EehR-conv.
Pow
er [M
W/m
3 ]
Radial position [mm]
Electrons
Heavy particles
0 200 4006040200
Axial positi
0204060 8000 K9000 K
Rad
ial p
ositi
on [m
m]
N atom mass fraction by 1T-CE and 2T-NCE methods
Ar-air plasmaP=0.3atm
0 5 10 15 20 25 30 351017
10181019102010211022
1x10231024
1x1025
O+
O N+
NO
Ar+, eN O2
N2Ar
Num
ber
dens
ity [m
-3]
Radial position [mm]0 5 10 15 20 25 30 3510
17
10181019102010211022
1x10231024
1x1025
NO+O+
O N+
NO
Ar+, eNO2
N2
Ar
Num
ber
dens
ity [m
-3]
Radial position [mm]
0 200 400 600 800604020
0
0.002
0.004 1T-CE
Axial position (mm)
Rad
ial p
ositi
on (m
m)
0204060 2T-CE
0.004
0.0060.008 2T-NCE2T-NCE
1T-CE:LTE1T-CE:LTE
2T-NCE2T-NCE 1T-CE(LTE)1T-CE(LTE)
98%Ar+2%air ICP
N atom
N atom
Conclusions
-A 2D-2T-NCE model: > it was developed for high power
argon induction thermal plasmas with molecular gases (Ar+N2,
Ar+N2+H2, Ar+N2+O2, Ar+CO2+H2, Ar+CH4+O2 ).
-Non-equilibrium effects:>Thermally and chemically
non-equilibrium condition
can be seen near the wall in the plasma torch region.
>Non-equilibrium affects prediction
of distribution of particle composition.
-Full coupling with Boltzmann equation-Transport of excited
particles-Detailed model of radiation transport -Separate treatment
for electron and ion
-Full coupling with Boltzmann equation-Transport of excited
particles-Detailed model of radiation transport -Separate treatment
for electron and ion
Future works
-
10
Keywords:-Non-Maxwellian electron energy distribution function
(EEDF)
-Boltzmann equation- Low electron density situation
with high electric field strength
NonNon--equilibrium modeling of arcs/thermal plasmasequilibrium
modeling of arcs/thermal plasmas11 Chemically nonChemically
non--equilibrium effectsequilibrium effects22 Thermally
nonThermally non--equilibrium effects in ICPequilibrium effects in
ICP33 NonNon--MaxwellianMaxwellian EEDFEEDF
Introduction Introduction NonNon--MaxwellianMaxwellian
EEDFEEDF
Critical electric field strength for hot air, air+Cu, CO2, SF6
at 300-3500K
Equilibrium composition calc.+Boltzmann eq.
Critical electric field strength for hot air, air+Cu, CO2, SF6
at 300-3500K
Equilibrium composition Equilibrium composition
calc.+calc.+BoltzmannBoltzmann eqeq..
Dielectric properties of hot gasDielectric properties of hot
gas
For further reliability and compactness of a CB, prediction of
dielectric property of hot gas is important for various gases.
GWP of SF6=23900 Candidates (SF6 mixed gas, N2,CF3I, CO2,
Air)
The present work
-There remains a hot gas with temperatures around 3000K after
arc interruption.
-Transient recovery voltage(TRV) is applied to the hot gas.
Y.Tanaka, JPD, 37, 2004,IEEE DEI, 12, 2005.
Circuit breakers
Electrical breakdown can occur through the hot gas.
Non-Maxwellian EEDF
Procedure on prediction of dielectric strength of hot
gasProcedure on prediction of dielectric strength of hot gas
Target: Gas kind SF6, Air, CO2, Air+CuTemperature
300-3500KElectric field Uniform
Estimation of critical electric field, which gives α=α−η=0 ,at
different gas temperatures.
Calculation of effective ionization coefficient α=α−ηof hot
gases by solving Boltzmann equationusing electron impact
cross-section & composition data.
Particle compositions (ex. equilibrium composition) of hot gases
are calculated.
Equilibrium composition of 99%Air+1%Cu at 1.0 Equilibrium
composition of 99%Air+1%Cu at 1.0 atmatm
1000 1000010-6
10-510-410-310-210-1100
Cu2 N2+
N3
NO+
O-
3000300 30000
O2+
Cu2+Cu+
e-
O+N+
Cu(g)
NO2
N
N2O
O
NO
O2Cu(cr,l)
N2
Mol
e fr
actio
n
Temperature (K)
Dominant particles: N2, O2, NO, N2O, N, O, Cu(cr,l), Cu(g), Cu+,
e
Drastic increase in Cu(g)
-
11
Calculation of effective ionization coefficientCalculation of
effective ionization coefficient
∂∂
−=∑ εεεσδε
ε nnmm
NENeJ
sss 2)()/2(3
)/(22/1
22f
∑
∂∂
−
−
−=s
gg
esnkT
kTn
Mm
mNJ
εεεεσδε
2)(22
2
2/1el
[ ])(2)( 2/12/1 εσεε sjsj mR
=
2/10 )()( εεε fnn e=
ee,
0elf )()()(
−
+
∑ ∫
∂∂
+∂
∂+
∂∂
−∂∂
−=∂
∂
js
sjs
ndnR
NJJ
tn sj
εε
εεε
εεε εε
ε
: Energy gain by electron by E
: Energy loss in elastic collisions
Rate of inelastic collisions such as excitation, ionization
EEDF: Electron Energy Distribution Function
Electron-electron collision term
∑ ∫∑ ∫∞∞
−
=
−=
sattachs
sionis dfm
edfme
NN 0
2/1
d0
2/1
d
)()(21)()(21 εεεσδυ
εεεσδυ
ηαα
Effective ionization coefficient
Two-term expansion of 0-D Boltzmann eq.
ElectronElectron--ion collisions and electronion collisions and
electron--electron collisionselectron collisions
Λ=
+−→
ln4
)()()1()(
2
4ei
iiii
επσ
εσδεσδεσ
eei
ssss
(Coulomb cross section)
Electron-ion collisions
Obtained by solving the Fokker-Planck equation using the
Rosenbluth potentials
Obtained by solving the Fokker-Planck equation using the
Rosenbluth potentials
Electron-electron collisions
−
∂∂
+
−
∂∂
∂∂
∂∂
+=
∂∂
− εεεϕ
εεεεϕε
εα
2223 2/1
2/322/1
ee
nnnnntn
Electron impact cross sections for Air+CuElectron impact cross
sections for Air+Cu
10-2 10-1 100 101 10210-22
10-21
10-20
10-19
10-18
Electr.exVib.ex.
Diss.ex.
Ioni.
Momem.
Cro
ss se
ctio
n (m
2 )
Electron energy (eV)
10-2 10-1 100 101 10210-2310-22
1x10-2110-2010-1910-18
Electr.ex
Ioni.Momen.
Vib.ex
Cro
ss se
ctio
n (m
2 )
Electron energy (eV)
e-O2e-O2
e-N2e-N2
10-2 10-1 100 101 10210-2310-2210-2110-2010-1910-18
Electr.ex
Vib.ex
Ioni.
Momem.
Cro
ss se
ctio
n (m
2 )
Electron energy (eV)
e-NOe-NO
10-2 10-1 100 101 10210-22
10-21
10-20
10-19
10-18
Electr.ex. Ioni.
Momem.
Cro
ss se
ctio
n (m
2 )
Electron energy (eV)
e-Cue-Cu
Effective ionization coefficient Effective ionization
coefficient vsvs E/NE/N and and TTgg
100%Air100%Air 99% Air + 1% Cu99% Air + 1% Cu
50 100 150 200 250-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Tg=2000 K
( α -
η ) /
N (
m2 )
[x10-21]
100%Air
99%Air+1.0%Cu
E/N (Td)50 100 150 200 250
-0.10.00.10.20.30.40.50.60.70.80.91.0
4000K3500K
3000K
2500K2000K
300-1000K
[x10-21]
(α-η
) / N
(m
2 )
E/N (Td)
-
12
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.55060708090
100110120130
[x103]
100% Air
99%Air+1%Cu
Red
uced
cri
tical
ele
ctri
c fie
ld
stre
ngth
(E/
N) cr
(T
d)
Gas temperature (K)
Critical electric field strength of air Critical electric field
strength of air vsvs TTgg-Ecr is proportionto density N.
Ionization potential:-N2 15.58 eV-O2 12.07 eV-NO 9.25 eV-Cu 7.73
eV
-Decline in (E/N)crfrom 1500K arises from NO ionization and O2
dissociation.
-Remarkable decayin (E/N)cr results fromCu(g) production.
1000100
101
102
99%Air-1%Cu
Mesured by Rothhardt
SF6
the present work for 0.1 MPa
Mesured by Rothhardt
Measured by Shade
The present work for 0.1 MPa-SF6
Calculated by Cliteur for 0.7MPa-SF6
CO2
Air
500300 2000 5000
Cri
tical
red
uced
ele
ctri
c fie
ld
stre
ngth
E
/p
(kV
/cm
/atm
)
Gas temperature (K)
Comparison of critical electric field strength Comparison of
critical electric field strength vsvs TTgg
SummarySummary
-Chemically non-equilibrium effect：
It can affect the prediction of particle composition and also
temperature, in particular, around the fringe of plasmas.
-Thermally non-equilibrium effect : Te>ThTe>Th can be seen
around the fringe of plasmas.It also affects particle composition
there for thermal plasmas.
-Non-Maxwell EEDF: Hot gas to which electric field applied
Dielectric strength of hot gas can be predicted by
Boltzmannequation solution with high-temperature composition.
NonNon--equilibrium models should be adopted equilibrium models
should be adopted for advanced understanding of thermal plasmas/arc
plasmas.for advanced understanding of thermal plasmas/arc
plasmas.
