Trelles Finite Element Modeling of Thermal Plasma Flows...

1

FINITE ELEMENT MODELING OF THERMAL PLASMA FLOWS

Juan Pablo Trelles

Department of Mechanical Engineering, University of Minnesota

Acknowledgements: Prof. Joachim Heberlein and Prof. Emil Pfender (U of MN),

National Science Foundation, and Minnesota Supercomputing Institute

Hillsboro, OR - August 3, 2007

2

Outline

(why it is important)

(how to describe it)

(how to deal with it)

(where we are)

(what’s next)

(how to use it)

1.  Introduction & Background

2.  Mathematical Model

3.  Numerical Model

4.  Solution Approach

5.  Simulation Results

6.  Summary & Conclusions

3

Preface

“Technological applications need methods for the solution of general multiphysics-multiscale problems”

–  Methods that rely as little as possible on deep analysis of equations –  Accurate, reliable, fast solutions –  Promising: Variational Multiscale Methods (VMS)

Physical Phenomena

reactions, surface proc., phase changes, …

Mathematical Model

ODEs, PDEs, ADEs, IDEs, …

Numerical Model

FE, FD, FV, Spctrl

Solution Algorithms

BD, Newton-Krylov, AMG, Schwarz, …

knowledge

k k+1

knowledge

bottleneck for multiphysics-multiscale problems

i.e. plasmas

Technological Advancement through Modeling & Simulation:

4

Plasmas: •  Partially or fully ionized gases •  Any gas mixture with charged species, i.e. Ar + M → Ar+ + e- + M •  +99% of observable mass in the universe •  4th state of matter: solid → liquid → gas → plasma

•  Typically, span over a wide range of scales …

Definition of Plasma

cold

solid liquid gas plasma

temperature hot

5 natural technological

Plasmas:

* Plasma Science: From Fundamental Research to Technological Applications, The National Academies Press, 1995

6

•  Thermal plasmas widely used for materials processing

•  Applications show inconsistent results due to instabilities

Ø  Characteristic process time in same order as instability

Ø  Example: plasma spray coating

Motivation of this Research

Goal of this research: Develop a computational model capable of describing thermal plasma flows in industrial applications

- Need better understanding of plasma dynamics -

* www.praxair.com

torch

jet coating

powder

7

Arc Plasma Torches

plasma torch plasma jet

► arc length ∝ voltage drop

Voltage (V)

► arc dynamics → jet forcing

t3

t5 t6 t4

t1 t2

8

Dynamics of Thermal Plasma Flows

1. Imbalance electromagnetic – flow drag forces

1

1

2

voltage Δφ

time t

3. Plasma flow instabilities due to gradients

2

2. Breakdown process (reattachment) when Δφ > Δφbreakdown

3

4

voltage Δφ

time t

9 * S. Wutzke, PhD Thesis, UMN, 1967

1. attachment movement

2. new attachment appears

3. new attachment remains

anode

anode

anode

Breakdown Process experiments, simplified geometry

10

Plasma Flow Instabilities

•  Fluid: shear instability

* M. Van Dyke, An Album of Fluid Motion, 1982

arc column – cold flow interface

cathode column

anode column

•  Magnetic: kink and sausage instabilities

anode column – cold flow

•  Thermal: cold flow interaction

11

Outline







12

Plasma Models

cs

s fDtDf =

Fundamentally: kinetics or Boltzmann

],[],[

vxtvx

tDtD

∂

∂

∂

∂+

∂

∂=

rate of change in phase-space

change due to collisions

∑ −=j

jss

s Fdtxdm

2

2

Fluid (continuum) •  Boltzmann approxs. •  Multi-fluid •  Chemical & thermal non-equilibrium •  Chemical & thermal equilibrium •  Equilibrium & inviscid

Particle (discrete) •  Molecular Dynamics (MD) •  Monte Carlo (DSMC) •  Particle-in-Cell (PIC)

•  …

less accurate, simpler

more accurate

Models:

13

Plasma Models

in this research

Fundamentally: kinetics or Boltzmann

],[],[

vxtvx

tDtD

∂

∂

∂

∂+

∂

∂=

rate of change in phase-space

change due to collisions

cs

s fDtDf =∑ −=

jjs

ss Fdtxdm

2

2

14

Thermal Plasma Model

•  Local Thermal Equilibrium (LTE) or Non-Equilibrium (NLTE) •  System of Transient-Advective-Diffusive-Reactive equations (TADR):

transient + advection - diffusion - reaction = 0

1. Mass cons.

2. Species cons.

3. Momentum

4. Energy Heavies

5. Energy Electrons

6. Current cons.

7. Ampere’s law 0

000

00

2

'

'

AAutA

J

QQQDtDpqhu

th

QDtDpqhu

th

BJpuutu

Jyuty

uut

p

q

rJehe

eee

ehh

hhh

q

csss

s

∇×∇×−∇∂

∂

⋅∇−

−+−⋅∇−∇⋅∂

∂

+⋅∇−∇⋅∂

∂

×⋅∇−∇−∇⋅∂

∂

⋅∇−∇⋅∂

∂

⋅∇+∇⋅∂

∂

ηφ

ρρ

ρρ

τρρ

ρρρ

ρρρ

+ thermodynamic & transport properties + add. relations

15



1. Mass cons.

2. Species cons.

3. Momentum

4. Energy Heavies

5. Energy Electrons

6. Current cons.

7. Ampere’s law 0

000

00

2

'

'

AAutA

J

QQQDtDpqhu

th

QDtDpqhu

th

BJpuutu

Jyuty

uut

p

q

rJehe

eee

ehh

hhh

q

csss

s

∇×∇×−∇∂

∂

⋅∇−

−+−⋅∇−∇⋅∂

∂

+⋅∇−∇⋅∂

∂

×⋅∇−∇−∇⋅∂

∂

⋅∇−∇⋅∂

∂

⋅∇+∇⋅∂

∂

ηφ

ρρ

ρρ

τρρ

ρρρ

ρρρ


for thermal equil. h = hh + he


16



1. Mass cons.

2. Species cons.

3. Momentum

4. Energy Heavies

5. Energy Electrons

6. Current cons.

7. Ampere’s law 0

000

00

2

'

'

AAutA

J

QQQDtDpqhu

th

QDtDpqhu

th

BJpuutu

Jyuty

uut

p

q

rJehe

eee

ehh

hhh

q

csss

s

∇×∇×−∇∂

∂

⋅∇−

−+−⋅∇−∇⋅∂

∂

+⋅∇−∇⋅∂

∂

×⋅∇−∇−∇⋅∂

∂

⋅∇−∇⋅∂

∂

⋅∇+∇⋅∂

∂

ηφ

ρρ

ρρ

τρρ

ρρρ

ρρρ


for thermal equil. h = hh + he

no for chemical equil.


17

Plasma Composition

Ø  Chemical – non-equil. ⇒ transport eqns.

Ø  Chemical – equil. ⇒ mass action law

OK if τchem << τflow, i.e. Saha:

,...),,,,,( ehs TTpuxtn

),,( ehs TTpn

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛≈

−− eB

i

P

ee

i

ie

i

ieTkh

TmQQQ

nnn επ exp2 2

3

211

MeAMA ii ++↔+ −−1

18

Plasma Composition Equilibrium Compositions @ 1 atm

Ar-He (75-25 vol.), Th = Te

Ar, Th ≠ Te

,...),,,,,( ehs TTpuxtn

),,( ehs TTpn

he TT=θ

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛≈

−− eB

i

P

ee

i

ie

i

ieTkh

TmQQQ

nnn επ exp2 2

3

211

MeAMA ii ++↔+ −−1

Ø  Chemical – non-equil. ⇒ transport eqns.

Ø  Chemical – equil. ⇒ mass action law

OK if τchem << τflow, i.e. Saha:

19

Thermodynamic Properties

~ straightforward once composition is known, i.e.

Ar-He, Th = Te

Ar, Th ≠ Te

∑= s ssnmρ )(25

ssss sB nTnkh ερ += ∑

20

Transport Properties

Ø  Chapmank-Enskog procedure: Approx. Prob. Dist. Func. by

Apply moments + determine fluxes ⇒ equate to fluid model fluxes

i.e. 2nd order approx. viscosity *

)1(... 02210ssssss fffff ξφξξ +≈+++=

[ ] 1110

01001110

0100

2 ,000

2)2(5

21

21

ijij

ijij

j

ijij

jiijijB

qqqq

qn

qqmnqq

qTk

=−=π

µ

* See: Devoto Phys. Fluids 10(2) 1982

21

Transport Properties Ar, Th ≠ Te

Ø  Chapmank-Enskog procedure: Approx. Prob. Dist. Func. by

Apply moments + determine fluxes ⇒ equate to fluid model fluxes

i.e. 2nd order approx. viscosity *

[ ] 1110

01001110

0100

2 ,000

2)2(5

21

21

ijij

ijij

j

ijij

jiijijB

qqqq

qn

qqmnqq

qTk

=−=π

µ

* See: Devoto Phys. Fluids 10(2) 1982

)1(... 02210ssssss fffff ξφξξ +≈+++=

22

Diffusive Fluxes

Diffusive fluxes:

)( 32 δµτ

uuu t ⋅∇−∇+∇−=

∑ ≠−∇−= es sshhh JhTq

κ'

eeeee JhTq

−∇−= κ'due to reactions

23

Diffusive Fluxes

Diffusive fluxes:

)( 32 δµτ

uuu t ⋅∇−∇+∇−=


κ'

eeeee JhTq

−∇−= κ'

Mass diffusion: Self-Consistent Effective Binary Diffusion (SCEBD)

due to reactions

∑≠

+−=sj

jjj

jss

ss

ss G

TRD

yGTRD

J ''

inter-species transport

24

Diffusive Fluxes

Diffusive fluxes:

)( 32 δµτ

uuu t ⋅∇−∇+∇−=


κ'

eeeee JhTq

−∇−= κ'

ee

q JmeJ

−≈

∑≠

+−=sj

jjj

jss

ss

ss G

TRD

yGTRD

J ''

pe

e

e

q

e

eq E

enpE

enBJ

enpBuEJ

σσσ =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇+≈

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

×−

∇+×+= ...

Mass diffusion: Self-Consistent Effective Binary Diffusion (SCEBD)

Generalized Ohm’s law: (consistent with SCEBD)

due to reactions

inter-species transport

effective electric field

25

Electromagnetic Equations

•  Maxwell’s equations: 1) Ampere’s law: 2) Faraday’s law:

3) Ohm’s law: 4) Gauss’ law: 5) No magnetic monopoles: 0

0

)(

0

=⋅∇

=⋅∇

×+=

∂∂−=×∇

=×∇

B

J

BuEJ

tBE

JB

q

pq

p

q

σ

µ

26



3) Ohm’s law: 4) Gauss’ law: 5) No magnetic monopoles:

•  3) in 1) → in 2) ⇒ Magnetic Induction eqn.

0

0)()(

=⋅∇

=×∇×∇+××∇−∂

∂

B

BButB

η

0

0

)(

0

=⋅∇

=⋅∇

×+=

∂∂−=×∇

=×∇

B

J

BuEJ

tBE

JB

q

pq

p

q

σ

µ

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3) Ohm’s law: 4) Gauss’ law: 5) No magnetic monopoles: 0

0

)(

0

=⋅∇

=⋅∇

×+=

∂∂−=×∇

=×∇

B

J

BuEJ

tBE

JB

q

pq

p

q

σ

µ

•  3) in 1) → in 2) ⇒ Magnetic Induction eqn.

0

0)()(

=⋅∇

=×∇×∇+××∇−∂

∂

B

BButB

η

)0 from (suggested

priori) a 0(

=⋅∇∂∂−−∇=

=⋅∇×∇=

qpp JtAE

BAB

φ 0)()(

0

=×∇×−∂

∂⋅∇+∇⋅∇

=×∇×∇+×∇×−∇+∂

∂

AutA

AAutA

p

p

σφσ

ηφ

•  Or, in term of potentials

28

Source Terms

Joule heating: Due to net current flow; main term driving “electric arcs”

)( BuEJQ qJ ×+⋅=

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Source Terms


)( BuEJQ qJ ×+⋅=

tA

enpEE pe

ep ∂

∂−−∇=

∇+≈

φ* effective field and potential:

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Source Terms


∑ ≠ −= es esheeses

eBeh TTnmm

kQ δν )(3

)( BuEJQ qJ ×+⋅=

Energy exchange:

Due to collisions between electrons and all other species

tA

enpEE pe

ep ∂

∂−−∇=

∇+≈

φ

Radiation losses: Net volumetric emission coefficient

* effective field and potential:

31

Outline







32

Numerical Approximation

§  The “Holy Grail” of numerical methods (?):

Exact solution independently of the size of the discretization … in a fast & reliable manner … for arbitrary problems (i.e. TADR)

y exact solution

discrete approximation (point-wise exact)

x or t

Ø  Need to consider solution approach (i.e. solve Ax = b) Ø  No such method exists yet

33

Finite Differences Finite Volumes Finite Elements

Discretization Methods

0Y =)(R ∫Ω =Ω⋅ 0YW d)( R∫Ω =Ω 0Y d)( R

approximate equation approximate solution

•  Most common, weighted residual methods with local support:

Ø  If implemented correctly, all methods perform ~ same Ø  Challenge for all methods: multi-scale problems

stencil control volume finite element

e1 e2

e3 e4 out in (i,j)

(i+1,j)

(i-1,j)

(i,j+1)

(i,j-1)

34

Multiscale Phenomena

Typical example:

x or t

boundary layers, sheaths

chemical reactions, nucleation

shocks, chemical fronts

turbulence, wave scattering

y

“different characteristic sizes needed to describe different parts of the

process”

* Ed. Wiley, 2000

35

A Simple Multiscale Problem

xy =

0)0( ,1' == yy

1st order ODE: Solution:

36

A Simple Multiscale Problem

1st order ODE: Solution:

0)1( ,0)0( ,1''' ===+− yyyyε

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−=

)1exp(1)exp(1

εεxxy

boundary layer

xy =

0)0( ,1' == yy

Perturbed ODE (ε → 0): Solution:

37

Multiscale Problems

Why are multiscale problems difficult? Ø  Every numerical method (FD, FV, FE, spectral, etc.) fails

unless Δx < O(ε)

Solution: Ø  Design methods that take into account the effect of the

smallest (unsolvable) scales into the large (solvable) scales

Why? Ø  Smallest scale needs to be resolved:

Δx < smallest spatial scale & Δt < smallest temporal scale Ø  Unmanageable in general TADR equations in 3D

38

•  System of TADR equations:

The TADR System

0YSYSYKYAYA 010 ==+−∇⋅∇−∇⋅+∂∂ )()()()(reactivediffusiveadvectivetransient

R t

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

A

TTup

A

Tup

p

e

h

φφ

or Y

•  Many problems can be treated as TADR, i.e. −  Incompressible & compressible flows −  Multi-fluid models −  Boltzmann approximations

•  Inherently Multi-Scale (whenever a term dominates over the others)

•  This is ~ arbitrary. In a code, we really need:

functions of 010 SSKAA , , , , ... ,, , , , YYYX ∇t

here:

39

•  Variational form TADR system:

•  Scale decomposition:

Variational Multiscale Methods (I)

' and ' WWWYYY +=+=

0YWYW ==Ω⋅∫Ω ))(,()( RR d

total = large + small

40

•  Then is equivalent to:

•  Variational form TADR system:

•  Scale decomposition:

Variational Multiscale Methods (I)

' and ' WWWYYY +=+=

0-)( :from defined SYY LR L* =

0YWYW ==Ω⋅∫Ω ))(,()( RR d

0YW =))(,( R

scale) (large equation scale smallscale) (small equation scale large

)','()) (,'( and )',())(,(ff ==

=+=+ 0YWYW0YWYW LRLR

total = large + small

41

•  Large Scales: using duality

Variational Multiscale Methods (II) )',()',( YWYW ∗= LL

0YWYW =+ ∗ )',())(,( LR (large scales equation depends on Y’)

42




•  Small Scales: equivalent weak form

solve using Green’s function or … 'd) ('' )'(' )',( Ω−= ∫Ω XXX YY Rg

) (' )) (,'()','( YYYWYW RLRL −=→−=

) (' YτY R−= (τ approx. of integral operator)

43




•  Small Scales: equivalent weak form

solve using Green’s function or … 'd) ('' )'(' )',( Ω−= ∫Ω XXX YY Rg

) (' )) (,'()','( YYYWYW RLRL −=→−=

) (' YτY R−= (τ approx. of integral operator)

•  Finally: equation for large scales only

0YτWYW =− ∗ ))(,())(,( RLR

44

•  Generalization:

P = Ladv SUPG P = L GLS P = -L* VMS

Stabilized & Multiscale Finite Element Methods

n4 n3

n2 element

n1 (e)

node Ω

Ω’

∫∫ ΩΩ=Ω⋅+Ω⋅

'')()()( 0YτWYW dd RPR

•  Requires minor modification to a standard FEM •  Solves many “difficult” problems: i.e. incompatible discretizations •  Applicable to other methods: Finite Volumes, Spectral

“A Framework for the Solution of General Multiphysics-Multiscale Problems”

Ø  BUT … still left: define good τ

45

Intrinsic Time Scales Matrix, τ

•  Formally: •  Empirically:

λs characteristic scales of each operator; i.e. in 1D:

and some adequate norm | ⋅ | needs to be defined

11 )/( −− −∇⋅∇−∇⋅+∂∂=≈ 1SKAτ tL

21

)( 2222 −+++= reactdiffadvtrans λλλλ 10 SKAAτ

tttrans Δ≈

∂

∂≈

1λ

xxadv Δ≈

∂

∂≈

1λ 22

2 1xxdiff

Δ≈

∂

∂≈λ 1≈reactλ

Ø  Better & more accurate formulations possible – but more expensive Ø  Above τ has proven efficient and robust … see Simulation Results

46

Outline







47

Discrete System

Ø  Newton Method: Ø  Residual vector:

Ø  Jacobian matrix: (approx., frozen coeff.)

∫∫

∫∫∫

ΩΩ

ΓΩΩ

∇⋅∇+⋅

+⋅+∇⋅∇+−−∇⋅+∂∂⋅=

=

ee

eee te

ee

)( )()(

)()())((

2

YKNYτN

YqqNYKNSYSYAYANRes

ResRes

DC

10010

RP

A

∫∫

∫∫∫

ΩΩ

ΓΩΩ

∇⋅∇+⋅

⋅+∇⋅∇+−∇⋅+⋅=

=

ee

eeee

ee

)( )()(

)()())((

2

NKNNτN

NqNNKNNSAANJac

JacJac

DC

110

RP

ζ

A

spy(Jace) for Hexahedra

node

YResJacResYJac0YRes ∂∂≈−=Δ⇒→ , )(

* N = N(X) basis function

48

Solver Layout

Need to solve:

Loop: Time stepping - Second order implicit predictor-multicorrector

Loop: Solution non-linear system - Globalized Newton-Krylov method

Loop: Solution linear system - Preconditioned Generalized Minimal Residual (GMRES)

end end

end

0YRes0YYXRes =→= )( ),,,( t

11

1

++

+

Δ+=

≤Δ+

kkkk

kk

YYY

ResYJacRes

λ

γ

)()(

11 bPxAP

bAx−− =

=

0YYXRes →),,,( t

49

Time Stepping

•  Solution of by 2nd order, implicit, predictor multi-corrector, with control of high frequency amplification

0YYRes

YYY

YYY

YYYY

=

+−=

+−=

+−=Δ

−

++

++

++

++

),(

)1(

)1(

)1(

1

1

11

mf

m

f

nn

nmnmn

nfnfn

nfnfnn

t

αα

α

α

αα

αα

αα

0YYRes =),(

•  higher order → BD methods (i.e. Sundials’ IDA solver)

•  α-method or BD only need: Res and Jac for a given

Ø  Need solution of non-linear system at each time step t

const

n

nnn Δ

=∂

∂=YYYY

ς , ,

t tn tn+1 tn+αf

yn+αf

t tn tn+1 tn+αm

dy/dtn+αf

Δt αfΔt Δt αmΔt

y(t) dy/dt(t)

dy/dtn

dy/dtn+1 yn+1

yn

50

Solution of Non-Linear System

•  Needed: minimum function evaluations (expensive for complex physics → matrix–free, pseudo-trans. not very attractive)

•  Globalized Inexact Newton:

>> Forcing term γ: Eisenstat-Walker >> Backtracking: Armijo condition >> Line search λ: Parabolic - three point interpolation

11

1

++

+

Δ+=

≤Δ+

kkkk

kk

YYY

ResYJacRes

λ

γ

* Backtracking essential when Δt still large (too large change in solution)

51

Solution of Linear System

•  Scaling: (unavoidable, unless dimensionless variables are used)

)( ,~~ 11 ADbxAbDAxD diag==→= −−

)~(_ ,~~~~ ~~ 11 APbxAbPxAP 000 diagblock==→= −−

... , ~~~~ 11 ==→= −− PbxAbPxAP

•  Pre-preconditioning: (something to do to A and b before linear solve)

•  Preconditioning: (as usual)

ILU(0) ILU(tol) Add. Schwz. EBE-GS BlkDiag

Scaling X X X X X

Pre-PreCond X X

PreCond X X X X

best (least expensive)

(EBE)

•  Generalized Minimal Residual (GMRES) with restarts + …

52

Outline







53

Computational Domain Ω

-  hexahedral trilinear elements -  torch or torch + jet -  d.o.f. per node: 9 for LTE

10 for NLTE cathode

anode

cathode anode

arc jet

torch inside

54

Solution Parameters/Statistics

Time Advancement: Δt ~ 0.1 – 1.0 µs nt ~ 500 – 1000 (each reattachment period ~ 100 µs)

Non-linear Solver: ~ 5-7 Inexact Newton ites. /Δt

η0 = 1.0-2, γmax = 1.0-3 Backtracking needed @ beginning of reattachment

Stop: |Res|/|Res0| ≤ 0.1 & |ΔY|/|Y| ≤ 1.0-3

Linear Solver: Krylov space 30 x 5 restarts = 150 ites. /In. Newton

Scaling + Blck Diag. Pre-Preconditioner + No Preconditioner

nt: 474, t: 2.27e-04

> Reattachment process begins

ite: 1, out/in: 2/ 4, fevals: 4, red: 2, flag: 0, lam: 0.25, eta: 1.00e-02, errY: 3.62e-02, errR: 1.00e+00

ite: 2, out/in: 1/14, fevals: 1, red: 0, flag: 0, lam: 1.00, eta: 5.62e-03, errY: 1.08e-02, errR: 7.85e-01




errSS: 2.06e-02, dtnew = 3.06e-07, SFAC = 1.07

55

Temperature Distribution: LTE Model

Instantaneous temperature distribution inside the torch Conditions: Ar-He (75-25), 800 A, 60 slpm, straight injection

anode attachment

56

Arc Dynamics: LTE Model old attachment

new attachment forms new attachment remains

Ø  Too large voltage drop !!!

57

ü  reattach. cond.

Arc Dynamics: LTE + Reattachment Model

attachment growth new attachment

Ø  More realistic voltage drop

58

Arc Dynamics

Ø  Electric potential over 14000 K isosurface

t = 360 µs t = 362 µs

t = 363 µs t = 368 µs

original attachment

formation of new attachment

new attachment

59

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Exp.

Exp.

Num. A

Num. A

Num. B

Num. Bfp ~ 3.30 fp ~ 3.23 fp ~ 8.98

Comparison with Experiments

Ø  LTE + Reattachment Model can match Voltage Drop Frequency OR Magnitude BUT NO BOTH

Eb = 5⋅104 V/m Eb = 2⋅104 V/m

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Temperature Distributions: LTE & NLTE

NLTE

LTE

Ar, 400 A, 60 slpm, 60 slpm

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Pressure and Velocity

•  Formation of cathode jet ( ) •  Cold flow avoids entering hot plasma

•  Inflection points in velocity profiles

→ Kelvin-Helmholtz instability (?) plasma

cold flow

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Electric Potentials and Fields

•  NLTE model produces more realistic voltage drops

Er max

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Arc Dynamics: LTE vs. NLTE

attachment

time

NLTE LTE

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Comparison 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

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EXP.

EXP.

NLTE

NLTE

LTE

LTEfp ~ 5.3 fp ~ 5.7

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Arc and Jet Dynamics

Conditions: Ar, 400 A, 60 slpm

Simulations reveal complex structure of

fluctuating jet

Schlieren image plasma jet turbulence

66 simulation (Th) experiment

Arc Movement as Jet Forcing time

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Outline







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1)  Developed: n-dimensional, transient, fully coupled, Stabilized/

Multiscale-FEM solver for TADR eqns.

2)  Solver applied to Thermal Equilibrium (LTE) and Thermal Non-

Equilibrium (NLTE) simulations of thermal plasma flows.

First time NLTE model applied to arc dynamics

3)  Results of modeling of thermal plasma flows: §  Main aspects of arc dynamics revealed §  Reasonable agreement with experiments

4)  Results NLTE model match experiments better than LTE

5)  Non-equilibrium description essential for realistic arc modeling

Summary & Conclusions

Trelles Finite Element Modeling of Thermal Plasma Flows...

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