Simulation of Reforming Process

1
Universidade da Madeira
Theory and modelling of plasma-cathode
interaction in HID lamps a review
M. S. Benilov Departamento de Física,
Universidade da Madeira, Portugal Talk to be
delivered at the Model Inventory COST 529
workshop Funchal, April 13, 2005 Acknowledgement
s M. D. Cunha, G. V. Naidis,
N. A. Almeida, M. Carpaij...
Departamento de Física
2
Contents of the talk

• Introduction
• The model of non-linear surface heating
• - Solution on the plasma side
• - Solution inside the cathode body
• Comparison with the experimental data
• General features of the solution
• - Diffuse mode
• - Axially symmetric spot modes
• - 3D spot modes
• Stability of the diffuse mode
• Effect of metal halides
• - Variation of properties of the
  near-cathode plasma layer
• - Variation of the work function of the
  cathode surface

3
Cathodes of arc discharges diffuse and spot modes
The current is distributed over the front surface
of the cathode in a more or less uniform way the
diffuse mode.
The current is localized in a region occupying a
small fraction of the surface (cathode spot) a
spot mode.

• Cathode of an arc discharge in argon. W, R 0.75
  mm, p 4.5 bar, I 2.5 A. Courtesy of D.
  Nandelstädt, J. Luhmann, and J. Mentel.

4
History

• Multiple modes of current transfer to hot arc
  cathodes were first observed in 1951 (W. Thouret,
  W. Weizel, and P. Günther) and have been under
  the active theoretical investigation ever since.
• A self-consistent and universally recognized
  theory started to emerge 40 years later, in the
  1990s.
• What are reasons of this lag?
• - It was realized relatively late that this
  is a self-organization problem.
• - Measurements are difficult. Reliable data
  on the near-cathode voltage drop started to
  appear only in 1998. Reliable experimental data,
  referring to integral characteristics, allow more
  than one theoretical interpretation (at least,
  some people think so).
• Theoretical vs. applied physics approach the
  theory which is generally recognized by now has
  been developed on a theoretical-physics basis.

5
Theoretical vs. applied physics approach

• A model should be based on fundamental physical
  principles and not rely on empirical parameters
  and/or arbitrary theoretical suppositions such as
  a "principle" of minimum voltage.
• All simplifications must be based on estimates.
• You cannot drop a term from equations just
  because you have not enough information or
  because it complicates a solution.
• If the model neglects important effects or is
  inconsistent with fundamental physical
  principles, it is a bad model
• even if it generates beautiful pictures
  adored by the management and/or agrees with the
  experimental data.
• There has been a huge number of models violating
  these principles. Nearly all of them have been
  forgotten by now.

6
Structure of the near-cathode plasma region
7
Why are models without the sheath inconsistent?

• Some authors disregard the near-cathode
  space-charge sheath E. Fischer (1987), Flesch
  and Neiger (1998, ....)
• A solution without a sheath does not satisfy
  boundary conditions for ni ,ne on the cathode
  surface, which are governed by different
  processes
• Without a sheath, ni ne in the whole plasma
  region right up the cathode surface.
• A solution without a sheath is physically
  meaningless the total electric current is
  directed from the cathode into the plasma

Flux of plasma electrons from the quasi-neutral
plasma to the cathode Flux of ions
8
The model of non-linear surface heating
BULK PLASMA
power input
electrons
NEAR-CATHODE LAYER
ions, plasma electrons
CATHODE BODY

• There is a considerable voltage drop in a thin
  near-cathode layer
• A considerable power is deposited in the
  near-cathode layer
• Part of this power is brought to the cathode
  surface by the ions and plasma electrons while
  the rest is taken away into the bulk plasma by
  the electrons
• The energy flux to the cathode surface is
  generated in a thin near-cathode plasma layer
  which is independent of the bulk plasma
• The plasma-cathode interaction is independent of
  the bulk plasma and may be modelled independently
  The model of nonlinear surface heating

9
The model of non-linear surface heating
A complete solution can be found in two steps

• Solution on the plasma side the 1D problem
  describing the current transfer across the
  near-cathode plasma layer is solved and all
  parameters of the layer are determined as
  functions of Tw and U. In particular, functions q
  q(Tw,U) and j j(Tw,U) are found.
• Solution inside the cathode the equation of
  thermal conduction is solved with the boundary
  condition q q(Tw,U).

The model is self-consistent What is specified
is not a distribution of the energy flux from the
plasma over the surface but rather a dependence
of the energy flux density on the local surface
temperature, this temperature being unknown
apriori! The model should have multiple
solutions, describing different modes of current
transfer from the plasma to the cathode.
10
The model of non-linear surface heating

• The model was apparently first suggested more
  than 40 years ago (W. L. Bade and J. M. Yos,
  Technical Report of Avco Corporation, USA, 1963)
  and was apparently re-discovered more than once.
  However, in general this model was half-forgotten
  and advances in theoretical description of the
  plasma-cathode interaction for many years have
  been limited.
• A self-consistent and universally recognized
  theory based on this model started to emerge only
  in the 1990s.
• Developments on the theoretical side
• - A modern theoretical description of the
  near-cathode plasma layer was developed (M. S.
  Benilov and A. Marotta 1995, R. Rethfeld et al
  1996).
• - It was shown that the model gives multiple
  solutions describing different modes of current
  transfer from the plasma to the cathode (M. S.
  Benilov 1998). Switching of mode mechanisms by
  hand became unnecessary.
• Developments on the experimental side
• - Detailed electrical and thermal
  measurements have been performed (J. Mentel et al
  1998, ). Near-cathode voltages of up to 50V
  were reported!
• - Theoretical results predicted by the model
  of non-linear surface heating have been confirmed.

11
The model of non-linear surface heating
By now, this model has been successfully used by
many authors e.g.,

• - J. Wendelstorf 1999
• - S. Coulombe 2000
• - R. Bötticher and W. Bötticher 2000, 2001a,
  2001b
• T. Krücken 2001
• W. Graser 2001
• M. S. Benilov and M. D. Cunha 2002, 2003a, 2003b
  
• R. Bötticher, W. Graser, and A. Kloss 2004 (a
  reproducible diffuse-spot transition was achieved
  in the experiment and was quantitatively
  correctly described by 3D non-stationary
  simulations)
• L. Dabringhausen 2004 (simulations of
  steady-state 3D spots)
• S. Lichtenberg, L. Dabringhausen, J. Mentel, and
  P. Awakowicz 2004 (simulations of steady-state 3D
  spots)
• M. S. Benilov, M. D. Cunha, and G. V. Naidis 2004
  (a multiple-species MH plasma)
• M. Galvez 2004 (collisional sheath simulations
  of the whole lamp)
• K. C. Paul et al 2004 (collisional sheath
  simulations of the whole lamp)
• - G. M. J. F. Luijks, S. Nijdam, and H. A. v.
  Esveld 2004.

12
Solution on the plasma side

• The ionization layer a hydrodynamics description
• Equations
• Equation of motion of the ion fluid accounting
  for ion inertia, ion pressure gradient, electric
  field force, friction force due to elastic
  collisions of ions with neutral particles,
  friction force due to ionization of neutral
  particles.
• Equation of balance of the electron energy in the
  ionization layer.
• Boundary conditions
• On the plasma side of the ionization layer
  ionization equilibrium.
• On the edge of the space-charge sheath the Bohm
  criterion.
• A complete solution is still missing. Frank
  Scharfs talk deals with this problem.
• The space-charge sheath a kinetic description
• Equations
• A kinetic equation describing the motion of ions,
• The Poisson equation.
• Boundary condition at the sheath edge the Bohm
  criterion

13
Solution on the plasma side
Density of the energy flux from the plasma to the
cathode surface vs. the surface temperature.
Tungsten cathode in the argon plasma, p 1 atm.
From M. S. Benilov and M. D. Cunha 2002.
14
Solution on the plasma side

• In the range Ult 20V, the dependence of qp on Tw
  for fixed U includes a growing and falling
  sections separated by a maximum

Reason of the non-monotony is overcoming of
the increase of combined ion and plasma electron
heating by an increase of thermionic cooling
which occurs when the plasma approaches full
ionization

• At higher U, the dependence changes a plateau
  appears, which subsequently gives way to two
  maxima and a minimum

Reasons of the maxima are, respectively,
non-monotony of the dependence of the ion current
on the electron temperature which is caused by a
deviation of the ion current from the diffusion
value and rapid increase of the plasma electron
heating which is subsequently overcome by
thermionic cooling.
15
Solution inside the cathode body

• Inside the cathode body the thermal conduction
  equation
• Ñ (k ÑT) 0
• At the cathode surface density of the energy
  flux from the plasma (or to the cold gas) is a
  given function of the local surface temperature
  and of the near-electrode voltage drop
• q q(Tw,U)

• Methods of solution
• Home-made codes
• ANSYS
• Femlab

16
Comparison with the experimental data
Current-voltage characteristics of the diffuse
mode on a tungsten cathode in the argon plasma. p
2.6 atm. Lines modelling. Points experiment.
1, ? h 24 mm, R 0.75 mm. 2, ? h 25
mm, R 0.5 mm. 3, ? h 15 mm, R 0.3 mm.
17
Comparison with the experimental data
Temperature in the center of a tungsten cathode
operating in the diffuse mode in the argon
plasma. R 0.75 mm. p 2.6 atm. Lines
modelling. Points experiment. ? h 24
mm. ? h 29 mm. ? h 19 mm.
18
Comparison with the experimental data
Power removed by heat conduction and irradiated
from the cathode surface in the diffuse mode on a
tungsten cathode in the argon plasma. R 0.5 mm.
p 2.6 atm. Lines modelling. Points
experiment. 1, 2 Power removed by heat
conduction. 3, 4 Power irradiated from the
cathode surface. 1, 3, ?, h 30 mm. 2,
4, ?, ? h 20 mm.
19
Comparison with the experimental data
D. Nandelsätdt, M. Redwitz, L. Dabringhausen, J.
Luhmann, S. Lichtenberg, and J. Mentel 2002,
Fig. 12 Comparison of the total power losses
obtained by pyrometric measurements (points) with
the model by Benilov and Cunha (lines). p 2.6
atm argon pure tungsten cathodes, h 20 mm. 1,
? R 0.3 mm. 2, ? R 0.5 mm. 3, ? R
0.75 mm (the cathode operated in a spot mode!)
20
Comparison with the experimental data
D. Nandelsätdt, M. Redwitz, L. Dabringhausen,
J. Luhmann, S. Lichtenberg, and J. Mentel 2002,
Fig. 13 Comparison of the measured cathode fall
with the models by several authors. p 10 atm
xenon pure tungsten cathode, R 0.6 mm, h 14
mm.
21
General pattern of various modes
Current-voltage characteristics of different
modes of current transfer. W, R 2 mm, h 10
mm, Ar, 1 bar
22
General features of the solution the diffuse mode
a
b
c
d
e
W, R 2 mm, h 10 mm, Ar, 1 bar
23
General features of the solution 1st 2D spot mode
a
b
c
d
W, R 2 mm, h 10 mm, Ar, 1 bar
24
General features of the solution 3D spot modes

a
b
c
d
W, R 2 mm, h 10 mm, Ar, 1 bar
25
General features of the solution 3D spot modes

a
b
c
d
W, R 2 mm, h 10 mm, Ar, 1 bar
26
Stability limit of the diffuse mode

• The first bifurcation point represents the limit
  of stability of the diffuse mode the diffuse
  mode is stable at IgtIbif and unstable at IgtIbif .
• This point can be calculated in a relatively
  simple way.

W, R 2 mm, h 10 mm Ar, 1 bar
Limit of stability
27
Stability limit of the diffuse mode

• The stability limit decreases rapidly with a
  decrease of the cathode radius. The same trend is
  observed in the experiment (Neumann 1987).

Current-voltage characteristics of the diffuse
mode and limits of its stability. Tungsten
cathode. Argon plasma, p 1bar.
28
Stability limit of the diffuse mode

• The stability limit decreases rapidly with an
  increase of the cathode height. The same trend is
  observed in the experiment (Neumann 1987).

Current-voltage characteristics of the diffuse
mode and limits of its stability. Tungsten
cathode. Argon plasma, p 1bar.
29
Stability limit of the diffuse mode

• A small decrease of the work function originates
  a strong decrease of the stability limit. The
  same trend is observed in the experiment (Neumann
  1987).

R 1 mm, h 14 mm
Current-voltage characteristics of the diffuse
mode and limits of its stability. Tungsten
cathode. Argon plasma, p 1bar.
30
Stability limit of the diffuse mode

• An increase of the argon pressure result in an
  increase of the stability limit. The same trend
  is observed in the experiment (Lichtenberg et al
  2002).
• A change from argon to xenon and from xenon to
  mercury results in an increase of the stability
  limit. The latter is observed in the experiment
  (Thouret 1951).

Current-voltage character- istics of the diffuse
mode and limits of its stability. Tungsten
cathode. R 1 mm, h 14 mm.
31
Metal halide plasmas

• The presence of metal halides in the plasma
• affects the current transfer in two ways
• Through variation of properties of the
  near-cathode plasma layer due to the presence of
  metal halides in the gas phase
• Through variation of the work function of the
  cathode surface due to formation on the surface
  of an alkali metal monolayer

32
Near-cathode layer in multi-species plasmas

• The equations are reformulated in terms of
  effective coefficients, the most important of
  these coefficients being the effective rate
  constant of ionization of a neutral particle by
  electron impact and the effective mean cross
  section for momentum transfer in elastic
  collisions between a positive ion and a neutral
  particle.
• The effective coefficients are calculated by
  means of averaging the corresponding individual
  coefficients over plasma partial composition
  evaluated at the plasma edge of the ionization
  layer.
• Partial composition of a plasma at the edge of
  the ionization layer is determined from equations
  of local balance of production and loss of every
  plasma species in volume reactions for given
  elementary composition of the mixture and given
  values of the heavy-particle temperature Th,
  electron temperature Te, and gas pressure p.
• There is no detailed balancing between direct and
  reverse reactions at the edge of the ionization
  layer, therefore the calculation requires, in
  addition to thermodynamic data, also kinetic data
  (in contrast to the case of an LTE plasma,
  ThTe).

33
MH plasma composition

• The plasma-producing gas in metal halide (MH)
  lamps represents a mixture of molecular and
  atomic species composed of Hg and (some of) the
  following elements Na, Tl, Dy, Sc, Cs, I, and
  Ar.
• The main neutral gas-phase species in MH lamps
  are Hg, Na, Tl, Cs, Dy, Sc, I, NaI, TlI, CsI,
  DyI, DyI2, DyI3, ScI, ScI2, ScI3.
• Ion species produced in ionization of atoms are
  Hg, Na, Tl, Dy, Sc, Cs, I. As the
  ionization potentials of metal iodides are
  substantially higher than those of corresponding
  metal atoms (e.g., 5.14 eV for Na and 7.54 eV for
  NaI, 6.11 eV for Tl and 8.47 eV for TlI, 3.89 eV
  for Cs and 7.25 eV for CsI), the presence of
  molecular ions is neglected.
• Attachment of electrons to atoms I results in
  appearance of ions I-.
• Thus, the present model takes into account the
  following species Hg, Na, Tl, Cs, Dy, Sc, I,
  NaI, TlI, CsI, DyI, DyI2, DyI3, ScI, ScI2, ScI3,
  Hg, Na, Tl, Dy, Sc, Cs, I, I-, e.

34
MH plasma diffuse discharge

• An addition of metal halides to the mercury
  plasma results in a small decrease of the cathode
  surface temperature.

Maximal temperature of the cathode surface.
h14,5 mm, R1mm, p 5 bar. MH plasma
HgNaTlI0.890.0050.050.055.
35
MH plasma diffuse discharge

• The near-cathode voltage drop in the MH plasma,
  UMH, is slightly lower than UHg, the latter being
  in turn slightly lower than UAr.
• The stability limit in Hg is higher than that in
  Ar, in accord with a general tendency (lower
  near-cathode voltages correspond to a higher
  stability limit).
• An addition of metal halides results in a
  considerable expansion of the range of stability
  of the diffuse mode. The reason decrease of the
  derivative ?q/?Tw.

Current-voltage characteristics and the stability
limit. h14,5 mm, R1mm, p 5 bar. MH plasma
HgNaTlI0.890.0050.050.055.
36
Variation of the work function

• Even smalls amounts of the alkali vapor in the
  plasma, of the order of 1 and less, can produce
  a significant effect on the work function.
• The decrease of the work function produced by
  adding Cs to the mercury plasma is stronger than
  that produced by adding Na and comes into play at
  higher values of the surface temperature.

Work function of tungsten covered by a monolayer
of Na or Cs in Na-Hg or Cs-Hg plasmas. p5bar.
Calculation with the use of the data from J.
Almanstötter, B. Eberhard, K. Günther, and T.
Hartmann 2002 and Welton 2002
37
Na-Hg plasma effect of variation of work function

• Formation of the sodium monolayer affects the
  diffuse mode of current transfer in the same
  direction that the presence of metal atoms in the
  gas phase does Tw and U decrease, range of
  stability of the diffuse mode expands.
• Formation of the sodium monolayer produces only a
  moderate effect.

Lines maximal temperature of the cathode surface
and current- voltage characteristics. Diffuse
mode on a W cathode, p5bar. Points stability
limit of the diffuse mode with ZNa0.08 on a
cathode covered by a monolayer of Na (the full
circle) in the Na-Hg plasma with ZNa0.08 on a
clean cathode (the open circle) in the Hg plasma
(the square).
38
Cs-Hg plasma effect of variation of work function

• Formation of the cesium monolayer changes the
  plasma-cathode interaction dramatically the
  temperature of the cathode surface decreases very
  strongly, the diffuse-mode current-voltage
  characteristic becomes N-S-shaped.
• The question of stability of the diffuse mode in
  the Cs-Hg plasma requires an additional
  mathematical treatment.

Maximal temperature of the cathode surface and
current-voltage characteristics. Diffuse mode of
discharge on a tungsten cathode.
39
Conclusions

• Considerable advances in understanding the
  interaction of high-pressure plasmas with
  thermionic cathodes have been achieved during the
  last decade.
• A self-consistent and universally accepted theory
  and simulation methods of plasma-cathode
  interaction have developed and validated by a
  comparison with the experiment.
• The most important features of the modern
  simulation approach are

• The approach is based on first physical
  principles and does not involve adjustable or
  empiric parameters
• Different modes of current transfer and
  transitions between them can be simulated in a
  completely self-consistent way
• Possibility of modelling of complex 3D
  geometries, also with account of non-stationary
  effects
• Possibility of modelling of complex mixtures,
  such as MH plasmas.